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1,600 | 2,454 | Probability Estimates for Multi-class
Classification by Pairwise Coupling
Ting-Fan Wu
Chih-Jen Lin
Department of Computer Science
National Taiwan University
Taipei 106, Taiwan
Ruby C. Weng
Department of Statistics
National Chenechi University
Taipei 116, Taiwan
Abstract
Pairwise coupling is a popular multi-class classification method that
combines together all pairwise comparisons for each pair of classes. This
paper presents two approaches for obtaining class probabilities. Both
methods can be reduced to linear systems and are easy to implement. We
show conceptually and experimentally that the proposed approaches are
more stable than two existing popular methods: voting and [3].
1
Introduction
The multi-class classification problem refers to assigning each of the observations into one
of k classes. As two-class problems are much easier to solve, many authors propose to use
two-class classifiers for multi-class classification. In this paper we focus on techniques that
provide a multi-class classification solution by combining all pairwise comparisons.
A common way to combine pairwise comparisons is by voting [6, 2]. It constructs a rule
for discriminating between every pair of classes and then selecting the class with the most
winning two-class decisions. Though the voting procedure requires just pairwise decisions,
it only predicts a class label. In many scenarios, however, probability estimates are desired.
As numerous (pairwise) classifiers do provide class probabilities, several authors [12, 11, 3]
have proposed probability estimates by combining the pairwise class probabilities.
Given the observation x and the class label y, we assume that the estimated pairwise class
probabilities rij of ?ij = p(y = i | y = i or j, x) are available. Here rij are obtained
by some binary classifiers. Then, the goal is to estimate {pi }ki=1 , where pi = p(y =
i | x), i = 1, . . . , k. We propose to obtain an approximate solution to an identity, and
then select the label with the highest estimated class probability. The existence of the
solution is guaranteed by theory in finite Markov Chains. Motivated by the optimization
formulation of this method, we propose a second approach. Interestingly, it can also be
regarded as an improved version of the coupling approach given by [12]. Both of the
proposed methods can be reduced to solving linear systems and are simple in practical
implementation. Furthermore, from conceptual and experimental points of view, we show
that the two proposed methods are more stable than voting and the method in [3].
We organize the paper as follows. In Section 2, we review two existing methods. Sections
3 and 4 detail the two proposed approaches. Section 5 presents the relationship among the
four methods through their corresponding optimization formulas. In Section 6, we compare
these methods using simulated and real data. The classifiers considered are support vector
machines. Section 7 concludes the paper. Due to space limit, we omit all detailed proofs.
A complete version of this work is available at http://www.csie.ntu.edu.tw/
?cjlin/papers/svmprob/svmprob.pdf.
2
Review of Two Methods
Let rij be the estimates of ?ij = pi /(pi + pj ). The voting rule [6, 2] is
X
?V = argmaxi [
I{rij >rji } ].
(1)
j:j6=i
P
A simple estimate of probabilities can be derived as pvi = 2 j:j6=i I{rij >rji } /(k(k ? 1)).
The authors of [3] suggest another method to estimate class probabilities, and they claim
that the resulting classification rule can outperform ?V in some situations. Their approach
is based on the minimization of the Kullback-Leibler (KL) distance between r ij and ?ij :
X
l(p) =
nij rij log(rij /?ij ),
(2)
i6=j
Pk
where i=1 pi = 1, pi > 0, i = 1, . . . , k, and nij is the number of instances in class i or
j. By letting ?l(p) = 0, a nonlinear system has to be solved. [3] proposes an iterative
procedure to find the minimum of (2). If rij > 0, ?i 6= j, the existence of a unique global
minimal solution to (2) has been proved in [5] and references therein. Let p? denote this
point. Then the resulting classification rule is
?HT (x) = argmaxi [p?i ].
It is shown in Theorem 1 of [3] that
p?i
>
p?j
if and only if p?i > p?j , where p?j =
2
P
s:s6=j
rjs
k(k ? 1)
;
(3)
? are sufficient if one only requires
that is, the p?i are in the same order as the p?i . Therefore, p
? can be derived as an approximation
the classification rule. In fact, as pointed out by [3], p
to the identity by replacing pi + pj with 2/k, and ?ij with rij .
X pi + p j
X pi + p j
pi
pi =
(
)(
)?ij
(4)
)=
(
k ? 1 pi + p j
k?1
j:j6=i
3
j:j6=i
Our First Approach
? is an approximate solution to (4). Instead
Note that ?HT is essentially argmaxi [?
pi ], and p
of replacing pi + pj by 2/k, in this section we propose to solve the system:
pi =
X pi + p j
(
)rij , ?i,
k?1
subject to
j:j6=i
k
X
pi = 1, pi ? 0, ?i.
(5)
i=1
? denote the solution to (5). Then the resulting decision rule is
Let p
?1 = argmaxi [?
pi ].
As ?HT relies on pi + pj ? k/2, in Section 6.1 we use two examples to illustrate possible
problems with this rule.
To solve (5), we rewrite it as
k
X
Qp = p,
i=1
Pk
(
rij /(k ? 1)
pi = 1, pi ? 0, ?i, where Qij = P
s:s6=i ris /(k ? 1)
if i 6= j,
(6)
if i = j.
Observe that j=1 Qij = 1 for i = 1, . . . , k and 0 ? Qij ? 1 for i, j = 1, . . . , k, so there
exists a finite Markov Chain whose transition matrix is Q. Moreover, if rij > 0 for all
i 6= j, then Qij > 0, which implies this Markov Chain is irreducible and aperiodic. These
conditions guarantee the existence of a unique stationary probability and all states being
positive recurrent. Hence, we have the following theorem:
Theorem 1 If rij > 0, i 6= j, then (6) has a unique solution p with 0 < pi < 1, ?i.
With Theorem 1 and some further analyses, if we remove the constraint pi ? 0, ?i, the
linear system with k + 1 equations still has the same unique solution. Furthermore, if any
one of the k equalities Qp = p is removed, we have a system with k variables and k
equalities, which, again, has the same single solution. Thus, (6) can be solved by Gaussian
elimination. On the other hand, as the stationary solution of a Markov Chain can be derived
by the limit of the n-step transition probability matrix Qn , we can solve p by repeatedly
multiplying QT with any initial vector.
Now we reexamine this method to gain more insight. The following arguments show that
the solution to (5) P
is a global minimum
P of a meaningful optimization problem. To begin,
we express (5) as j:j6=i rji pi ? j:j6=i rij pj = 0, i = 1, . . . , k, using the property that
rij + rji = 1, ?i 6= j. Then the solution to (5) is in fact the global minimum of the
following problem:
min
p
k
X
X
X
(
rji pi ?
rij pj )2
i=1 j:j6=i
subject to
k
X
pi = 1, pi ? 0, ?i.
(7)
i=1
j:j6=i
Since the object function is always nonnegative, and it attains zero under (5) and (6).
4
Our Second Approach
Note that both approaches in Sections 2P
and 3 involve solving
optimization problems using
P
the relations like pi /(pi + pj ) ? rij or j:j6=i rji pi ? j:j6=i rij pj . Motivated by (7), we
suggest another optimization formulation as follows:
min
p
k
1X X
(rji pi ? rij pj )2
2 i=1
subject to
j:j6=i
k
X
pi = 1, pi ? 0, ?i.
(8)
i=1
Pk
In related work, [12] proposes to solve a linear system consisting of i=1 pi = 1 and
any k ? 1 equations of the form rji pi = rij pj . However, pointed out in [11], the results
of [12] strongly depends on the selection of k ? 1 equations. In fact, as (8) considers all
rij pj ? rji pi , not just k ? 1 of them, it can be viewed as an improved version of [12].
Let p? denote the corresponding solution. We then define the classification rule as
?2 = argmaxi [p?i ].
Since (7) has a unique solution, which can be obtained by solving a simple linear system, it
is desired to see whether the minimization problem (8) has these nice properties. In the rest
of the section, we show that this is true. The following theorem shows that the nonnegative
constraints in (8) are redundant.
Theorem 2 Problem (8) is equivalent to a simplification without conditions p i ? 0, ?i.
Note that we can rewrite the objective function of (8) as
(P
2
1 T
s:s6=i rsi
min = p Qp,
where Qij =
p
2
rji rij
if i = j,
if i 6= j.
(9)
From here we can show that Q is positive semi-definite. Therefore, without constraints
pi ? 0, ?i, (9) is a linear-constrained convex quadratic programming problem. Consequently, a point p is a global minimum if and only if it satisfies the KKT optimality condition: There is a scalar b such that
Q e p
0
=
.
(10)
1
eT 0 b
Here e is the vector of all ones and b is the Lagrangian multiplier of the equality constraint
Pk
i=1 pi = 1. Thus, the solution of (8) can be obtained by solving the simple linear system
(10). The existence of a unique solution is guaranteed by the invertibility of the matrix
of (10). Moreover, if Q is positive definite(PD), this matrix is invertible. The following
theorem shows that Q is PD under quite general conditions.
Theorem 3 If for any i = 1, . . . , k, there are s 6= i and j 6= i such that
then Q is positive definite.
rsi rsj
ris
6=
rji rjs
rij ,
In addition to direct methods, next we propose a simple iterative method for solving (10):
Algorithm 1
1. Start with some initial pi ? 0, ?i and
2. Repeat (t = 1, . . . , k, 1, . . .)
pt ?
Pk
i=1
pi = 1.
X
1
[?
Qtj pj + pT Qp]
Qtt
(11)
j:j6=t
normalize p
(12)
until (10) is satisfied.
Theorem 4 If rsj > 0, ?s 6= j, and {pi }?
i=1 is the sequence generated by Algorithm 1,
any convergent sub-sequence goes to a global minimum of (8).
As Theorem 3 indicates that in general Q is positive definite, the sequence {p i }?
i=1 from
Algorithm 1 usually globally converges to the unique minimum of (8).
5
Relations Among Four Methods
The four decision rules ?HT , ?1 , ?2 , and ?V can be written as argmaxi [pi ], where p is
Pk
derived by the following four optimization formulations under the constants i=1 pi = 1
and pi ? 0, ?i:
?HT : min
p
?1 :
min
k X
k
X
[
(rij pj ? rji pi )]2 ,
(14)
i=1 j:j6=i
min
p
k
k X
X
(rij pj ? rji pi )2 ,
(15)
(I{rij >rji } pj ? I{rji >rij } pi )2 .
(16)
i=1 j:j6=i
?V : min
p
(13)
j:j6=i
p
?2 :
k X
k
X
1 1
[
(rij ? pi )]2 ,
k 2
i=1
k X
k
X
i=1 j:j6=i
Note that (13) can be easily verified, and that (14) and (15) have been explained in Sections
3 and 4. For (16), its solution is
c
,
pi = P
j:j6=i I{rji >rij }
where c is the normalizing constant;? and therefore, argmaxi [pi ] is the same as (1). Clearly,
(13) can be obtained from (14) by letting pj ? 1/k, ?j and rji ? 1/2, ?i, j. Such approximations ignore the differences between pi . Similarly, (16) is from (15) by taking the
extreme values of rij : 0 or 1. As a result, (16) may enlarge the differences between pi .
Next, compared with (15), (14) may tend to underestimate the differences between the p i ?s.
The reason is that (14) allows the difference between rij pj and rji pi to get canceled first.
Thus, conceptually, (13) and (16) are more extreme ? the former tends to underestimate
the differences between pi ?s, while the latter overestimate them. These arguments will be
supported by simulated and real data in the next section.
6
6.1
Experiments
Simple Simulated Examples
[3] designs a simple experiment in which all pi ?s are fairly close and their method ?HT
outperforms the voting strategy ?V . We conduct this experiment first to assess the performance of our proposed methods. As in [3], we define class probabilities p1 = 1.5/k,
pj = (1 ? p1 )/(k ? 1), j = 2, . . . , k, and then set
pi
+ 0.1zij if i > j,
pi + p j
rji = 1 ? rij
if j > i,
rij =
(17)
(18)
where zij are standard normal variates. Since rij are required to be within (0,1), we truncate
rij at below and 1 ? above, with = 0.00001. In this example, class 1 has the highest
probability and hence is the correct class.
Figure 1 shows accuracy rates for each of the four methods when k = 3, 5, 8, 10, 12, 15, 20.
The accuracy rates are averaged over 1,000 replicates. Note that in this experiment all
classes are quite competitive, so, when using ?V , sometimes the highest vote occurs at two
?
For I to beP
well defined, we consider rij 6= rji , which is generally true. In addition, if there is
an i for which j:j6=i I{rji >rij } = 0, an optimal solution of (16) is pi = 1, and pj = 0, ?j 6= i.
The resulting decision is the same as that of (1).
1.05
1
1
1
0.95
0.95
0.95
0.85
0.8
0.75
1.05
0.9
Accuracy Rates
0.9
Accuracy Rates
Accuracy Rates
1.05
0.85
0.8
0.75
0.9
0.85
0.8
0.75
0.7
0.7
0.7
0.65
0.65
0.65
0.6
2
3
4
log k
5
6
2
(a) balanced pi
7
0.6
2
3
4
log k
5
6
2
(b) unbalanced pi
7
0.6
2
3
4
log k
5
6
7
2
(c) highly unbalanced pi
Figure 1: Accuracy of predicting the true class by the methods ?HT (solid line, cross
marked), ?V (dash line, square marked), ?1 (dotted line, circle marked), and ?2 (dashed
line, asterisk marked) from simulated class probability pi , i = 1, 2 ? ? ? k.
or more different classes. We handle this problem by randomly selecting one class from
the ties. This partly explains why ?V performs poor. Another explanation is that the rij
here are all close to 1/2, but (16) uses 1 or 0 instead; therefore, the solution may be severely
biased. Besides ?V , the other three rules have done very well in this example.
Since ?HT relies on the approximation pi + pj ? k/2, this rule may suffer some losses
if the class probabilities are not highly balanced. To examine this point, we consider the
following two sets of class probabilities:
(1) We let k1 = k/2 if k is even, and (k + 1)/2 if k is odd; then we define p1 =
0.95?1.5/k1 , pi = (0.95?p1 )/(k1 ?1) for i = 2, . . . , k1 , and pi = 0.05/(k?k1 )
for i = k1 + 1, . . . , k.
(2) If k = 3, we define p1 = 0.95 ? 1.5/2, p2 = 0.95 ? p1 , and p3 = 0.05. If
k > 3, we define p1 = 0.475, p2 = p3 = 0.475/2, and pi = 0.05/(k ? 3) for
i = 4, . . . , k.
After setting pi , we define the pairwise comparisons rij as in (17)-(18). Both experiments
are repeated for 1,000 times. The accuracy rates are shown in Figures 1(b) and 1(c). In
both scenarios, pi are not balanced. As expected, ?HT is quite sensitive to the imbalance of
pi . The situation is much worse in Figure 1(c) because the approximation pi + pj ? k/2 is
more seriously violated, especially when k is large.
In summary, ?1 and ?2 are less sensitive to pi , and their overall performance are fairly
stable. All features observed here agree with our analysis in Section 5.
6.2
Real Data
In this section we present experimental results on several multi-class problems: segment,
satimage, and letter from the Statlog collection [9], USPS [4], and MNIST [7]. All data
sets are available at http://www.csie.ntu.edu.tw/?cjlin/libsvmtools/
t. Their numbers of classes are 7, 6, 26, 10, and 10, respectively. From thousands of
instances in each data, we select 300 and 500 as our training and testing sets.
2
We consider support vector machines (SVM) with RBF kernel e??kxi ?xj k as the binary
classifier. The regularization parameter C and the kernel parameter ? are selected by crossvalidation. To begin, for each training set, a five-fold cross-validation is conducted on
the following points of (C, ?): [2?5 , 2?3 , . . . , 215 ] ? [2?5 , 2?3 , . . . , 215 ]. This is done by
modifying LIBSVM [1], a library for SVM. At each (C, ?), sequentially four folds are
Table 1: Testing errors (in percentage) by four methods: Each row reports the testing errors
based on a pair of the training and testing sets. The mean and std (standard deviation) are
from five 5-fold cross-validation procedures to select the best (C, ?).
Dataset
k
satimage
6
segment
7
USPS
10
MNIST
10
letter
26
?HT
mean
std
14.080 1.306
12.960 0.320
14.520 0.968
12.400 0.000
16.160 0.294
9.960 0.480
6.040 0.528
6.600 0.000
5.520 0.466
7.440 0.625
14.840 0.388
12.080 0.560
10.640 0.933
12.320 0.845
13.400 0.310
17.400 0.000
15.200 0.400
17.320 1.608
14.720 0.449
12.560 0.294
39.880 1.412
41.640 0.463
41.320 1.700
35.240 1.439
43.240 0.637
?1
mean
14.600
13.400
14.760
12.200
16.400
9.480
6.280
6.680
5.200
8.160
13.520
11.440
10.000
11.960
12.640
16.560
14.600
14.280
14.160
12.600
37.160
39.400
38.920
32.920
40.360
std
0.938
0.400
1.637
0.000
0.379
0.240
0.299
0.349
0.420
0.637
0.560
0.625
0.657
1.031
0.080
0.080
0.000
0.560
0.196
0.000
1.106
0.769
0.854
1.121
1.472
?2
mean
14.760
13.400
13.880
12.640
16.120
9.000
6.200
6.920
5.400
8.040
12.760
11.600
9.920
11.560
12.920
15.760
13.720
13.400
13.360
13.080
34.560
35.920
35.800
29.240
36.960
std
0.784
0.400
0.392
0.294
0.299
0.400
0.456
0.271
0.580
0.408
0.233
1.081
0.483
0.784
0.299
0.196
0.588
0.657
0.686
0.560
2.144
1.389
1.453
1.335
1.741
?V
mean
15.400
13.360
14.080
12.680
16.160
8.880
6.760
7.160
5.480
7.840
12.520
11.440
10.320
11.840
12.520
15.960
12.360
13.760
13.520
12.440
33.480
33.440
35.000
27.400
34.520
std
0.219
0.080
0.240
1.114
0.344
0.271
0.445
0.196
0.588
0.344
0.160
0.991
0.744
1.248
0.917
0.463
0.196
0.794
0.325
0.233
0.325
1.061
1.066
1.117
1.001
used as the training set while one fold as the validation set. The training of the four folds
consists of k(k ? 1)/2 binary SVMs. For the binary SVM of the ith and the jth classes,
using decision values f? of training data, we employ an improved implementation [8] of
Platt?s posterior probabilities [10] to estimate rij :
rij = P (i | i or j, x) =
1
1 + eAf?+B
,
(19)
where A and B are estimated by minimizing the negative log-likelihood function. ?
Then, for each validation instance , we apply the four methods to obtain classification
decisions. The error of the five validation sets is thus the cross-validation error at (C, ?).
After the cross-validation is done, each rule obtains its best (C, ?).? Using these parameters, we train the whole training set to obtain the final model. Next, the same as (19),
the decision values from the training data are employed to find rij . Then, testing data are
tested using each of the four rules.
Due to the randomness of separating training data into five folds for finding the best (C, ?),
we repeat the five-fold cross-validation five times and obtain the mean and standard deviation of the testing error. Moreover, as the selection of 300 and 500 training and testing
instances from a larger dataset is also random, we generate five of such pairs. In Table 1,
each row reports the testing error based on a pair of the training and testing sets. The results show that when the number of classes k is small, the four methods perform similarly;
however, for problems with larger k, ?HT is less competitive. In particular, for problem
letter which has 26 classes, ?2 or ?V outperforms ?HT by at least 5%. It seems that for
[10] suggests to use f? from the validation instead of the training. However, this requires a further
cross-validation on the four-fold data. For simplicity, we directly use f? from the training.
?
If more than one parameter sets return the smallest cross-validation error, we simply choose one
with the smallest C.
?
problems here, their characteristics are closer to the setting of Figure 1(c), rather than that
of Figure 1(a). All these results agree with the previous findings in Sections 5 and 6.1. Note
that in Table 1, some standard deviations are zero. That means the best (C, ?) by different
cross-validations are all the same. Overall, the variation on parameter selection due to the
randomness of cross-validation is not large.
7
Discussions and Conclusions
As the minimization of the KL distance is a well known criterion, some may wonder why
the performance of ?HT is not quite satisfactory in some of the examples. One possible explanation is that here KL distance is derived under the assumptions that n ij rij ?
Bin(nij , ?ij ) and rij are independent; however, as pointed out in [3], neither of the assumptions holds in the classification problem.
In conclusion, we have provided two methods which are shown to be more stable than both
?HT and ?V . In addition, the two proposed approaches require only solutions of linear
systems instead of a nonlinear one in [3].
The authors thank S. Sathiya Keerthi for helpful comments.
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1,601 | 2,455 | Autonomous helicopter flight
via Reinforcement Learning
Andrew Y. Ng
Stanford University
Stanford, CA 94305
H. Jin Kim, Michael I. Jordan, and Shankar Sastry
University of California
Berkeley, CA 94720
Abstract
Autonomous helicopter flight represents a challenging control problem,
with complex, noisy, dynamics. In this paper, we describe a successful
application of reinforcement learning to autonomous helicopter flight.
We first fit a stochastic, nonlinear model of the helicopter dynamics. We
then use the model to learn to hover in place, and to fly a number of
maneuvers taken from an RC helicopter competition.
1 Introduction
Helicopters represent a challenging control problem with high-dimensional, complex,
asymmetric, noisy, non-linear, dynamics, and are widely regarded as significantly more
difficult to control than fixed-wing aircraft. [7] Consider, for instance, the problem of designing a helicopter that hovers in place. We begin with a single, horizontally-oriented main
rotor attached to the helicopter via the rotor shaft. Suppose the main rotor rotates clockwise (viewed from above), blowing air downwards and hence generating upward thrust.
By applying clockwise torque to the main rotor to make it rotate, our helicopter experiences an anti-torque that tends to cause the main chassis to spin anti-clockwise. Thus,
in the invention of the helicopter, it was necessary to add a tail rotor, which blows air
sideways/rightwards to generate an appropriate moment to counteract the spin. But, this
sideways force now causes the helicopter to drift leftwards. So, for a helicopter to hover in
place, it must actually be tilted slightly to the right, so that the main rotor?s thrust is directed
downwards and slightly to the left, to counteract this tendency to drift sideways.
The history of helicopters is rife with such tales of ingenious solutions to problems caused
by solutions to other problems, and of complex, nonintuitive dynamics that make helicopters challenging to control. In this paper, we describe the successful application of
reinforcement learning to designing a controller for autonomous helicopter flight. Due to
space constraints, our description of this work is necessarily brief; a detailed treatment is
provided in [9]. For a discussion of related work on autonomous flight, also see [9, 12].
2 Autonomous Helicopter
The helicopter used in this work was a Yamaha R-50 helicopter, which is approximately
3.6m long, carries up to a 20kg payload, and is shown in Figure 1a. A detailed description
of the design and construction of its instrumentation is in [12]. The helicopter carries an
Inertial Navigation System (INS) consisting of 3 accelerometers and 3 rate gyroscopes
installed in exactly orthogonal x,y,z directions, and a differential GPS system, which with
the assistance of a ground station, gives position estimates with a resolution of 2cm. An
onboard navigation computer runs a Kalman filter which integrates the sensor information
from the GPS, INS, and a digital compass, and reports (at 50Hz) 12 numbers corresponding
to the estimates of the helicopter?s position ( ), orientation (roll , pitch , yaw
),
velocity ( ) and angular velocities (
).
(a)
(b)
Figure 1: (a) Autonomous helicopter. (b) Helicopter hovering under control of learned policy.
Most Helicopters are controlled via a 4-dimensional action space:
: The longtitudinal (front-back) and latitudinal (left-right) cyclic pitch controls. The rotor plane is the plane in which the helicopter?s rotors rotate. By
tilting this plane either forwards/backwards or sideways, these controls cause the
helicopter to accelerate forward/backwards or sideways.
: The (main rotor) collective pitch control. As the helicopter main-rotor?s blades
sweep through the air, they generate an amount of upward thrust that (generally)
increases with the angle at which the rotor blades are tilted. By varying the tilt
angle of the rotor blades, the collective pitch control affects the main rotor?s thrust.
: The tail rotor collective pitch control. Using a mechanism similar to the main
rotor collective pitch control, this controls the tail rotor?s thrust.
Using the position estimates given by the Kalman filter, our task is to pick good control
actions every 50th of a second.
3 Model identification
To fit a model of the helicopter?s dynamics, we began by asking a human pilot to fly the
helicopter for several minutes, and recorded the 12-dimensional helicopter state and 4dimensional helicopter control inputs as it was flown. In what follows, we used 339 seconds
of flight data for model fitting, and another 140 seconds of data for hold-out testing.
There are many natural symmetries in helicopter flight. For instance, a helicopter at (0,0,0)
facing east behaves in a way related only by a translation and rotation to one at (10,10,50)
facing north, if we command each to accelerate forwards. We would like to encode these
symmetries directly into the model rather force an algorithm to learn them from scratch.
Thus, model identification is typically done not in the spatial (world) coordinates
, but instead in the helicopter body coordinates, in which
the , , and axes are forwards, sideways, and down relative to the current position
of the helicopter. Where there is risk of confusion, we will use superscript and to
distinguish between spatial and body coordinates; thus, is forward velocity, regardless of
.
orientation. Our model is identified in the body coordinates
which has four fewer variables than . Note that once this model is built, it is easily
converted back using simple geometry to one in terms of spatial coordinates.
Our main tool for model fitting was locally weighted linear regression (e.g., [11, 3]). Given
a dataset
where the ?s are vector-valued inputs and the ?s are the realvalued outputs to be predicted, we let be the design matrix whose -th row is , and let
be the vector of ?s. In response to a query at , locally weighted linear regression makes
, and is a diagonal matrix with
the prediction
, where
(say)
, so that the regression gives datapoints near
a larger weight. Here,
determines how weights fall off with distance from , and
was picked in our experiments via leave-one-out cross validation. 1 Using the estimator for
. By
noise
given in [3], this gives a model
, where
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(b)
(c)
Figure 2: (a) Examples of plots comparing a model fit using the parameterization described in the text
(solid lines) to some other models (dash-dot lines). Each point plotted shows the mean-squared error
between the predicted value of a state variable?when a model is used to the simulate the helicopter?s
dynamics for a certain duration indicated on the -axis?and the true value
of that state variable (as
measured on test data) after the same duration.
Top
left:
Comparison
of
-error to model not using
, etc. terms. Top right: Comparison of -error to model omitting intercept (bias) term. Bottom:
Comparison of and to linear deterministic model identified by [12]. (b) The solid line is the true
helicopter state on 10s of test data. The dash-dot line is the helicopter state predicted by our model,
given the initial state at time 0 and all the intermediate control inputs. The dotted lines show two
standard deviations in the estimated state. Every two seconds, the estimated state is ?reset? to the
true state, and the track restarts with zero error.
Note that the estimated state is of the full, highdimensional state of the helicopter, but only is shown here. (c) Policy class. The picture inside
the circles indicate whether a node outputs the sum of their inputs, or the
of the sum of their
inputs. Each edge with an arrow in the picture denotes a tunable parameter. The solid lines show
the hovering policy class (Section 5). The dashed lines show the extra weights added for trajectory
following (Section 6).
<
applying locally-weighted regression with the state and action as inputs, and the onestep differences (e.g.,
) of each of the state variables in turn as the target output,
this gives us a non-linear, stochastic, model of the dynamics, allowing us to predict as
a function of and plus noise.
We actually used several refinements to this model. Similar to the use of body coordinates
to exploit symmetries, there is other prior knowledge that can be incorporated. Since both
and are state variables, and we know that (at 50Hz) , there
is no need to carry out a regression for . Similarly, we know that the roll angle of
the helicopter should have no direct effect on forward velocity . So, when performing
regression to estimate , the coefficient in corresponding to can be set to 0. This allows
us to reduce the number of parameters that have to be fit. Similar reasoning allows us to
conclude (cf. [12]) that certain other parameters should be , or (gravity), and these
were also hard-coded into the model. Finally, we added three extra (unobserved) variables
to model latencies in the responses to the controls. (See [9] for details.)
,
Some of the (other) choices that we considered in selecting a model include whether to use
the
,
and/or
terms; whether to include an intercept term; at what frequency to
identify the model; whether to hardwire certain coefficients as described; and whether to
use weighted or unweighted linear regression. Our main tool for choosing among the models was plots such as those shown in Figure 2a. (See figure caption.) We were particularly
interested in checking how accurate a model is not just for predicting from , but
how accurate it is at longer time scales. Each of the panels in Figure 2a shows, for a model,
the mean-squared error (as measured on test data) between the helicopter?s true position
and the estimated position at a certain time in the future (indicated on the -axis).
The helicopter?s blade-tip moves at an appreciable fraction of the speed of sound. Given the
(
R
B
QR
R
and the presence of temporally close-by samples?which will be spatially
close-by as well?may
make data seem more abundant than in reality (leading to bigger !#" than might be optimal for test
data). Thus, when leaving out a sample in cross validation, we actually left out a large window (16
seconds) of data around that sample, to diminish this bias.
danger and expense (about $70,000) of autonomous helicopters, we wanted to verify the
fitted model carefully, so as to be reasonably confident that a controller tested successfully
in simulation will also be safe in real life. Space precludes a full discussion, but one of
our concerns was the possibility that unmodeled correlations in might mean the noise
variance of the actual dynamics is much larger than predicted by the model. (See [9] for
details.) To check against this, we examined many plots such as shown in Figure 2, to
check that the helicopter state ?rarely? goes outside the errorbars predicted by our model
at various time scales (see caption).
D
4 Reinforcement learning: The P EGASUS algorithm
We used the P EGASUS reinforcement learning algorithm of [10], which we briefly review
here. Consider an MDP with state space , initial state , action space , state
, reward function
, and discount . Also let some
transition probabilities
family of policies
be given, and suppose our goal is to find a policy in with
high utility, where the policy of is defined to be
S +B
S 8 ;B
S B
!
#"
where the expectation is over the random sequence of states !$!$ $ visited over time
when is executed in the MDP starting from state .
These utilities are in general intractable to calculate exactly, but suppose we have a computer simulator of the MDP?s dynamics?that is, a program that inputs and outputs
#% drawn from
. Then a standard way to define an estimate & of is via
!
$
!
$
$
Monte Carlo: We can use the simulator to sample a trajectory
, and by taking the
!
on this sequence, we obtain
empirical sum of discounted rewards
one ?sample? with which to estimate . More generally, we could generate ' such sequences, and average to obtain a better estimator. We can then try to optimize the estimated
utilities and search for ? )(
+* & , .?
NQK L N 7
S B
B
Unfortunately, this is a difficult stochastic optimization problem: Evaluating & , involves
a Monte Carlo sampling process, and two different evaluations of & , will typically give
slightly different answers. Moreover, even if the number of samples ' that we average over
is
arbitrarily large, & will fail with probability 1 to be a (?uniformly?) good estimate of
, . In our experiments, this fails to learn any reasonable controller for our helicopter.
The P EGASUS method uses the observation that almost all computer simulations of the
form described sample %
by first calling a random number generator to get one
(or more) random numbers - , and then calculating % as some deterministic function of the
input and the random - . If we demand that the simulator expose its interface to the
random number generator, then by pre-sampling all the random numbers - in advance and
fixing them, we can then use these same, fixed, random numbers to evaluate any policy.
Since all the random numbers are fixed, &
./ is just an ordinary deterministic func
tion, and standard search heuristics can be used to search for 0(
* & , . Importantly,
this also allows us to show that, so long as we average over a number of samples ' that
is at most polynomial in all quantities of interest, then with high probability, & will be a
"2143 ). This also allows us to give guarantees
uniformly good estimate of ( "#&
on the performance of the solutions found. For further discussion of P EGASUS and other
work such as the variance reduction and gradient estimation methods (cf. [6, 5]), see [9].
M
F
NK LON 7
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5 Learning to Hover
One previous attempt had been made to use a learning algorithm to fly this helicopter, using
5 -synthesis [2]. This succeeded in flying the helicopter in simulation, but not on the actual
helicopter (Shim, pers. comm.). Similarly, preliminary experiments using 6
and 687
controllers to fly a similar helicopter were also unsuccessful. These comments should not
be taken as conclusive of the viability of any of these methods; rather, we take them to be
indicative of the difficulty and subtlety involved in learning a helicopter controller.
x?velocity (m/s)
y?velocity (m/s)
1.5
z?velocity (m/s)
0.4
1
1
0.2
0.5
0.5
0
0
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20
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5
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15
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66
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5
10
15
20
25
30
20
25
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z?position (m)
y?position (m)
x?position (m)
7
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15
Figure 3: Comparison of hovering performance of learned controller (solid line) vs. Yamaha licensed/specially trained human pilot (dotted line). Top: velocities. Bottom: positions.
We began by learning a policy for hovering in place. We want a controller that, given the
current helicopter state and a desired hovering position and orientation
,
computes controls
to make it hover stably there. For our policy class ,
we chose the simple neural network depicted in Figure 2c (solid edges only). Each of
the edges in the figure represents a weight, and the connections were chosen via simple
reasoning about which control channel should be used to control which state variables. For
instance, consider the the longitudinal (forward/backward) cyclic pitch control , which
causes the rotor plane to tilt forward/backward, thus causing the helicopter to pitch (and/or
accelerate) forward or backward. From Figure 2c, we can read off the control control as
<
B 5 KMK
+B MN 5 KMK
;B B N ;B $
Here, the ?s are the tunable parameters (weights) of the network, and 5 KMK
!#"%$& '"%! is defined to be the error in the -position (forward direction, in body coordinates)<
between where the helicopter currently is and where we wish it to hover.
We chose a quadratic cost function on the (spatial representation of the) state, where 2
(*) +-,.0/1)32546) /
,%89:25;) / ,%89:25<=) / ,%89:2?>4 8=9:2:>; 89:2?>< 89:2A@5)CB*/B ,%8D, E
(1)
7
7
7
This encourages the helicopter to hover near F
, while also keeping the veloc
ity small and not making abrupt movements. The weights G GIH , etc. (distinct from the
weights parameterizing our policy class) were chosen to scale each of the terms to be
7
: < B
S M S B :
B
roughly the same order of magnitude. To encourage small actions and smooth control of
JG -K
G ML
the helicopter, we also used a quadratic penalty for actions:
G N
G DO
, and the overall reward was
.
Using the model identified in Section 3, we can now apply P EGASUS to define approx
imations & , to the utilities of policies. Since policies are smoothly parameterized in
the weights, and the dynamics are themselves continuous in the actions, the estimates of
utilities are also continuous in the weights.3 We may thus apply standard hillclimbing al
gorithms to maximize & in terms of the policy?s weights. We tried both a gradient
B
B:/1B
B
.SRE RT
The
if RE 7R
7 error term is computed with
BU. appropriate
/VRE RT wrapping about PMQ rad, so that /V
rad, and the helicopter is currently facing
rad, the error is 0.02, not PMQ
P-Q
P rad.
3
Actually, this is not true. One last component of the reward that we did not mention earlier was
that, if in performing the locally weighted regression, the matrix WYX5Z[W is singular to numerical
precision, then we declare the helicopter to have ?crashed,? terminate the simulation, and give it
a huge negative (-50000) reward. Because the test checking if W X Z[W is singular to numerical
]?) ,
precision returns either 1 or 0, \ Q has a discontinuity between ?crash? and ?not-crash.?
2
10
9.5
9.5
9
9
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8
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8
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66
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80
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64
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68
67.5
?80
?81
Figure 4: Top row: Maneuver diagrams from RC helicopter competition.
[Source:
www.modelaircraft.org]. Bottom row: Actual trajectories flown using learned controller.
ascent algorithm, in which we numerically evaluate the derivative of & with respect to
the weights and then take a step in the indicated direction, and a random-walk algorithm
in which we propose a random perturbation to the weights, and move there if it increases
& , . Both of these algorithms worked well, though with gradient ascent, it was important
to scale the derivatives appropriately, since the estimates of the derivatives were sometimes
numerically unstable.4 It was also important to apply some standard heuristics to prevent
its solutions from diverging (such as verifying after each step that we did indeed take a step
uphill on the objective & , and undoing/redoing the step using a smaller stepsize if this was
not the case).
The most expensive step in policy search was the repeated Monte Carlo evaluation to obtain
& , . To speed this up, we parallelized our implementation, and Monte Carlo evaluations
using different samples were run on different computers, and the results were then aggre
gated to obtain & . We ran P EGASUS using 30 Monte Carlo evaluations of 35 seconds
$ . Figure 1b shows the result of implementing and
of flying time each, and
running the resulting policy on the helicopter. On its maiden flight, our learned policy was
successful in keeping the helicopter stabilized in the air. (We note that [1] was also successful at using our P EGASUS algorithm to control a subset, the cyclic pitch controls, of a
helicopter?s dynamics.)
We also compare the performance of our learned policy against that of our human pilot
trained and licensed by Yamaha to fly the R-50 helicopter. Figure 5 shows the velocities and
positions of the helicopter under our learned policy and under the human pilot?s control. As
we see, our controller was able to keep the helicopter flying more stably than was a human
pilot. Videos of the helicopter flying are available at
http://www.cs.stanford.edu/?ang/nips03/
R
6 Flying competition maneuvers
We were next interested in making the helicopter learn to fly several challenging maneuvers. The Academy of Model Aeronautics (AMA) (to our knowledge the largest RC helicopter organization) holds an annual RC helicopter competition, in which helicopters have
to be accurately flown through a number of maneuvers. This competition is organized into
Class I (for beginners, with the easiest maneuvers) through Class III (with the most difficult
maneuvers, for the most advanced pilots). We took the first three maneuvers from the most
challenging, Class III, segment of their competition.
Figure 4 shows maneuver diagrams from the AMA web site. In the first of these maneuvers
4
A problem exacerbated by the discontinuities described in the previous footnote.
R
(III.1), the helicopter starts from the middle of the base of a triangle, flies backwards to the
lower-right corner, performs a pirouette (turning in place), flies backwards up an edge
of the triangle, backwards down the other edge, performs another pirouette, and flies
backwards to its starting position. Flying backwards is a significantly less stable maneuver
than flying forwards, which makes this maneuver interesting and challenging. In the second
maneuver (III.2), the helicopter has to perform a nose-in turn, in which it flies backwards
out to the edge of a circle, pauses, and then flies in a circle but always keeping the nose of
the helicopter pointed at center of rotation. After it finishes circling, it returns to the starting
point. Many human pilots seem to find this second maneuver particularly challenging.
Lastly, maneuver III.3 involves flying the helicopter in a vertical rectangle, with two
pirouettes in opposite directions halfway along the rectangle?s vertical segments.
How does one design a controller for flying trajectories? Given a controller for keeping
a system?s state at a point 5
, one standard way to make the system move
through a particular trajectory is to slowly vary F
along a sequence of set
points on that trajectory. (E.g., see [4].) For instance, if we ask our helicopter to hover
at , then a fraction of a second later ask it to hover at $ , then at
$
and so on, our helicopter will slowly fly in the -direction. By taking this
procedure and ?wrapping? it around our old policy class from Figure 2c, we thus obtain a
computer program?that is, a new policy class?not just for hovering, but also for flying
arbitrary trajectories. I.e., we now have a family of policies that take as input a trajectory,
and that attempt to make the helicopter fly that trajectory. Moreover, we can now also
retrain the policy?s parameters for accurate trajectory following, not just hovering.
Since we are now flying trajectories and not only hovering, we also augmented the policy
class to take into account more of the coupling between the helicopter?s different subdynamics. For instance, the simplest way to turn is to change the tail rotor collective
pitch/thrust, so that it yaws either left or right. This works well for small turns, but for large
turns, the thrust from the tail rotor also tends to cause the helicopter to drift sideways. Thus,
we enriched the policy class to allow it to correct for this drift by applying the appropriate
cyclic pitch controls. Also, having a helicopter climb or descend changes the amount of
work done by the main rotor, and hence the amount of torque/anti-torque generated, which
can cause the helicopter to turn. So, we also added a link between the collective pitch control and the tail rotor control. These modifications are shown in Figure 2c (dashed lines).
We also needed to specify a reward function for trajectory following. One simple
choice for would have been to use Equation (1) with the newly-defined (time-varying)
5 F
. But we did not consider this to be a good choice. Specifically, consider
making the helicopter fly in the increasing -direction, so that F
starts off as
(say), and has its first coordinate slowly increased over time. Then, while
will indeed increase, it will also almost certainly lag conthe actual helicopter position
sistently behind 5 . This is because the hovering controller is always trying to ?catch up?
A may remain large, and the helicopter will
to the moving 5 F
. Thus,
5
cost, even if it is in fact flying a very accurate trajectory in the
continuously incur a
increasing -direction exactly as desired. It would be undesirable to have the helicopter
risk trying to fly more aggressively to reduce this fake ?error,? particularly if it is at the cost
of increased error in the other coordinates. So, we changed the reward function to penalize deviation not from 5
, but instead deviation from
, where
is the ?projection? of the helicopter?s position onto the path of the idealized,
desired trajectory. (In our example of flying in a straight line, for a helicopter at
,
.) Thus, we imagine an ?external observer? that
we easily see
looks at the actual helicopter state and estimates which part of the idealized trajectory the
helicopter is trying to fly through (taking care not to be confused if a trajectory loops back
on itself), and the learning algorithm pays a penalty that is quadratic between the actual
position and the ?tracked? position on the idealized trajectory.
We also needed to make sure the helicopter is rewarded for making progress along the
R
R
SR R R R
.R R R R R
.R R R R R
.R R R R
<
<
8
- R R R
trajectory. To do this, we used the potential-based shaping rewards of [8]. Since, we are
already tracking where along the desired trajectory the helicopter is, we chose a potential
function that increases along the trajectory. Thus, whenever the helicopter?s
makes forward progress along this trajectory, it receives positive reward. (See [8].)
Finally, our modifications have decoupled our definition of the reward function from
5 F
and the evolution of F
in time. So, we are now also free
to consider allowing
to evolve in a way that is different from the path of
the desired trajectory, but nonetheless in way that allows the helicopter to follow the actual,
desired trajectory more accurately. (In control theory, there is a related practice of using the
inverse dynamics to obtain better tracking behavior.) We considered several alternatives,
but the main one used ended up being a modification for flying trajectories that have both
a vertical and a horizontal component (such as along the two upper edges of the triangle in
III.1). Specifically, it turns out that the (vertical)-response of the helicopter is very fast:
To climb, we need only increase the collective pitch control, which almost immediately
causes the helicopter to start accelerating upwards. In contrast, the and responses are
much slower. Thus, if
moves at upwards as in maneuver III.1, the helicopter will tend to track the -component of the trajectory much more quickly, so that it
accelerates into a climb steeper than , resulting in a ?bowed-out? trajectory. Similarly,
an angled descent results in a ?bowed-in? trajectory. To correct for this, we artificially
slowed down the -response, so that when
is moving into an angled climb
or descent, the
portion will
evolve normally with time, but the changes to
will be delayed by seconds, where here is another parameter in our policy class, to be
automatically learned by our algorithm.
Using this setup and retraining our policy class? parameters for accurate trajectory following, we were able to learn a policy that flies all three of the competition maneuvers fairly
accurately. Figure 4 (bottom) shows actual trajectories taken by the helicopter while flying
these maneuvers. Videos of the helicopter flying these maneuvers are also available at the
URL given at the end of Section 5.
8
8
8
References
[1] J. Bagnell and J. Schneider. Autonomous helicopter control using reinforcement
learning policy search methods. In Int?l Conf. Robotics and Automation. IEEE, 2001.
[2] G. Balas, J. Doyle, K. Glover, A. Packard, and R. Smith. 5 -analysis and synthesis
toolbox user?s guide, 1995.
[3] W. Cleveland. Robust locally weighted regression and smoothing scatterplots. J.
Amer. Stat. Assoc, 74, 1979.
[4] Gene F. Franklin, J. David Powell, and Abbas Emani-Naeini. Feedback Control of
Dynamic Systems. Addison-Wesley, 1995.
[5] Y. Ho and X. Cao. Pertubation analysis of discrete event dynamic systems. Kluwer,
1991.
[6] J. Kiefer and J. Wolfowitz. Stochastic estimation of the maximum of a regression
function. Annals of Mathematical Statistics, 23:462?466, 1952.
[7] J. Leishman. Principles of Helicopter Aerodynamics. Cambridge Univ. Press, 2000.
[8] A. Y. Ng, D. Harada, and S. Russell. Policy invariance under reward transformations:
Theory and application to reward shaping. In Proc. 16th ICML, pages 278?287, 1999.
[9] Andrew Y. Ng. Shaping and policy search in reinforcement learning. PhD thesis,
EECS, University of California, Berkeley, 2003.
[10] Andrew Y. Ng and Michael I. Jordan. P EGASUS: A policy search method for large
MDPs and POMDPs. In Proc. 16th Conf. Uncertainty in Artificial Intelligence, 2000.
[11] C. Atkeson S. Schaal and A. Moore. Locally weighted learning. AI Review, 11, 1997.
[12] Hyunchul Shim. Hierarchical flight control system synthesis for rotorcraft-based unmanned aerial vehicles. PhD thesis, Mech. Engr., U.C. Berkeley, 2000.
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1,602 | 2,456 | Approximate Policy Iteration
with a Policy Language Bias
Alan Fern and SungWook Yoon and Robert Givan
Electrical and Computer Engineering, Purdue University, W. Lafayette, IN 47907
Abstract
We explore approximate policy iteration, replacing the usual costfunction learning step with a learning step in policy space. We give
policy-language biases that enable solution of very large relational
Markov decision processes (MDPs) that no previous technique can solve.
In particular, we induce high-quality domain-specific planners for classical planning domains (both deterministic and stochastic variants) by
solving such domains as extremely large MDPs.
1
Introduction
Dynamic-programming approaches to finding optimal control policies in Markov decision
processes (MDPs) [4, 14] using explicit (flat) state space representations break down when
the state space becomes extremely large. More recent work extends these algorithms to
use propositional [6, 11, 7, 12] as well as relational [8] state-space representations. These
extensions have not yet shown the capacity to solve large classical planning problems such
as the benchmark problems used in planning competitions [2]. These methods typically
calculate a sequence of cost functions. For familiar STRIPS planning domains (among
others), useful cost functions can be difficult or impossible to represent compactly.
The above techniques guarantee a certain accuracy at each stage. Here, we focus on inductive techniques that make no such guarantees. Existing inductive forms of approximate
policy iteration (API) select compactly represented, approximate cost functions at each iteration of dynamic programming [5], again suffering when such representation is difficult.
We know of no previous work that applies any form of API to benchmark problems from
classical planning.1 Perhaps one reason is the complexity of typical cost functions for
these problems, for which it is often more natural to specify a policy space. Recent work
on inductive policy selection in relational planning domains [17, 19, 28], has shown that
useful policies can be learned using a policy-space bias, described by a generic knowledge
representation language. Here, we incorporate that work into a practical approach to API
for STRIPS planning domains.
We replace the use of cost-function approximations as policy representations in API2 with
direct, compact state-action mappings, and use a standard relational learner to learn these
mappings. We inherit from familiar API methods a (sampled) policy-evaluation phase
using simulation of the current policy, or rollout [25], and an inductive policy-selection
phase inducing an approximate next policy from sampled current policy values.
1
Recent work in relational reinforcement learning has been applied to STRIPS problems with
much simpler goals than typical benchmark planning domains, and is discussed below in Section 5.
2
In concurrent work, [18] pursued a similar approach to API in attribute-value domains.
We evaluate our API approach in several STRIPS planning domains, showing iterative
policy improvement. Our technique solves entire planning domains, finding a policy that
can be applied to any problem in the domain, rather than solving just a single problem
instance from the domain. We view each planning domain as a single large MDP where
each ?state? specifies both the current world and the goal. The API method thus learns
control knowledge (a ?policy?) for the given planning domain.
Our API technique naturally leverages heuristic functions (cost function estimates), if
available?this allows us to benefit from recent advances in domain-independent heuristics for classical planning, as discussed below. Even when greedy heuristic search solves
essentially none of the domain instances, our API technique successfully bootstraps from
the heuristic guidance. We also demonstrate that our technique is able to iteratively improve policies that correspond to previously published hand-coded control knowledge (for
TL-plan [3]) and policies learned by Yoon et al. [28]. Our technique gives a new way of
using heuristics in planning domains, complementing traditional heuristic search strategies.
2
Approximate Policy Iteration
We first review API for a general, action-simulator?based MDP representation, and later,
in Section 3, detail a particular representation of planning domains as relational MDPs and
the corresponding policy-space learning bias.
Problem Setup. We follow and adapt [16] and [5]. We represent an MDP using a generative model !S, A, T, C, I", where S is a finite set of states, A is a finite set of actions, and
T is a randomized ?action-simulation? algorithm that, given state s and action a, returns a
next state t. The component C is an action-cost function that maps S ? A to real-numbers,
and I is a randomized ?initial-state algorithm? with no inputs that returns a state in S. We
sometimes treat I and T (s, a) as random variables.
For MDP M = !S, A, T, C, I", a policy ? is a (possibly stochastic) mapping from S to A.
?
(s) and the Q-cost function Q?M (s, a) are the unique solutions to
The cost function JM
?
?
Q?M (s, a) = C(s, a) + ?E[JM
(T (s, a))], where JM
(s) = E[Q?M (s,? (s))],
representing the expected, cumulative, discounted cost of following policy ? in M starting
from state s, and where 0 ? ? < 1 is the discount factor. In this work, we seek to
?
(I)], due to the complexity of the problems we consider.
heuristically minimize E[JM
Given a current policy ?, we can define a new improved policy PI[?](s) by
argmina?A Q?M (s, a). The cost function of PI[?] is guaranteed to be no worse than that
of ? at each state and to improve at some state for non-optimal ?. Exact policy iteration
iterates policy improvement (PI) from any initial policy to reach an optimal fixed point.
?
(policy evaluation) and then
Policy improvement is divided into two steps: computing JM
computing Q?M and selecting the minimizing action (policy selection).
Approximate Policy Iteration. API, as described in [5], heuristically approximates policy iteration in large state spaces by using an approximate policy-improvement operator
trained with Monte-Carlo simulation. The approximate operator performs policy evaluation by simulation?evaluating a policy ? at a state s by drawing some number of sample
trajectories of ? starting at s?and performs policy selection by constructing a training set
of samples of either the J or Q cost functions from a ?small? but ?representative? set of
states and then using this training set to induce a new ?approximately improved? policy.
The use of API assumes that states and perhaps actions are represented in factored form
(typically, a feature vector) that facilitates generalizing properties of the training data to the
entire state and action spaces. Due to API?s inductive nature, there are typically no guarantees for policy improvement?nevertheless, API often ?converges? usefully, e.g. [24, 26].
We start API by providing it with an initial policy ?0 and a real-valued heuristic function
H, where H(s) is interpreted as an estimate of the cost of state s (presumably with respect
to the optimal policy). We note that H or ?0 may be trivial, i.e. always returning a constant
or random action respectively. For API to be effective, however, it is important that ?0 and
H combine to provide guidance toward improvement. For example, in goal-based planning
domains either ?0 should occasionally reach a goal or H should provide non-trivial goaldistance information. In our experiments we consider scenarios that use different types of
initial policies and heuristics to bootstrap API.
Given ?0 , H, and an MDP M = !S, {a1 , . . . , am }, T, C, I", API produces a policy sequence by iterating steps of approximate policy improvement?note that ?0 is used in only
the initial iteration but the heuristic is always used. Approximate policy improvement
computes an (approximate) improvement ? " of a policy ? by attempting to approximate
the output of exact policy improvement, i.e. ? " (s) = argmina?A Q?M (s, a). There are two
steps: estimating Q-costs for all actions at a representative set of states, and using resulting
data set to learn an approximation of ? " . Figure 1 gives pseudo-code for our variant of API.
Step 1: Q-Cost Estimation via Rollout. (see [25]) Given ?, we construct a training set D,
? a1 ), . . . , Q(s,
? am )".
describing an improved policy ? " , consisting of tuples !s,? (s), Q(s,
?
?
For each sampled state s and action a, the term Q(s, a) refers to QM (s, a) as estimated by
drawing ?sampling width? trajectories of length ?horizon? from s and computing the average discounted trajectory cost over the sampled trajectories, where the cost of a trajectory
includes the value of the heuristic function at the horizon state. To get a ?representative
? estimates) within
set? of states, we include each state s visited by ? " (as indicated by the Q
?horizon? steps from one of ?training set size? states drawn from the initial distribution.3
?
Step 2: Learn Policy. Select ? " with the goal of minimizing the cumulative Q-cost
for ? "
over D (approximating the same minimization over S in exact policy iteration). Traditional
API uses a cost-function space learning bias in this selection?in Section 3 we detail the
policy-space learning bias used by our technique. By labeling each training state with
the associated Q-costs for each action, rather than simply with the best action, we enable
the learner to make more informed trade-offs. We note that the inclusion of ?(s) in each
training example enables the learner to normalize the data, if desired?e.g. our learner (see
Section 3) uses a bias that focuses on states where large improvement appears possible.
3
API for Relational Planning
In order to use our API framework, we represent classical planning domains (not just single
instances) as relationally factored MDPs. We then describe our compact relational policy
language and the associated learner for use in step 2 of our API framework.
Planning Domains as MDPs. We say that an MDP !S, A, T, C, I" is relational when S
and A are defined by giving the finite sets of objects O, predicates P , and action types Y .
A fact is a predicate applied to the appropriate number of objects. A state in S is a set of
facts (taken to be ?true? in the state), and S is all such states. An action is an action type
applied to the appropriate number of objects, and the action space A is the set of all actions.
A classical planning domain is specified by providing a set of world predicates, action
types, and an action simulator. We simultaneously solve all problem instances of such a
planning domain4 by constructing a relational MDP as described below.
Let O be a fixed set of objects and Y be the set of action types from the planning domain.
Together, O and Y define the MDP action space. Each MDP state is a single problem
3
It is important that states are sampled from ? ! rather than ? to match the training distribution to
the implied ?test set? distribution.
4
As an example, the blocks world is a classical planning domain, where a problem instance is an
initial block configuration and a set of goal conditions. Classical planners attempt to find solutions to
specific problem instances of a domain.
API (n, w, h, H,? 0 )
// training set size n, sampling width w,
// horizon h, initial policy ?0 ,
// cost estimator (heuristic function) H.
? ? ?0 ;
loop
D ? Draw-Training-Set(n, w, h, H,? );
? ? Learn-Decision-List(D);
until satisfied with ?;
//e.g. until change is small
Return ?;
Draw-Training-Set(n, w, h, H,? )
// training set size n, sampling width w,
// horizon h, cost estimator H, current policy ?
D ? ?; E ? set of n states sampled from I;
for each state s0 ? E // Draw trajectory of
// sample states from s0
s?s ;
0
for i = 1 to h
Q? (s) ? Policy-Rollout(?, s, w, h, H);
a ? action maximizing Q? (s, a);
D ? $s,? (s), Q? (s)% ? D;
s ? state sampled from T (s, a);
Return D;
Policy-Rollout (?, s, w, h, H) // Computes estimate of Q? (s)
// policy ?, state s, sampling width w, horizon h, cost estimator H
Initialize Q? (s), a vector indexed by the actions in A, to zeroes;
for1 each action a in A
for2 sample = 1 to w
s! ? s;
for3 step = 1 to h
Q? (s, a) ? Q? (s, a) + C(s! , ?(s! ));
s! ? a state sampled from T (s! , ?(s! )) // end for3
Q? (s, a) ? Q? (s, a) + H(s! );
// end for2
Q? (s, a) ?
Q? (s,a)
w
// end for1
Return Q? (s)
Figure 1: Pseudo-code for our API algorithm. The MDP !S, A, T, C, I" is assumed globally known. The general approach is inherited from [5], and is restated here for clarity.
Key differences are the use of Learn-Decision-List [28], as discussed in Section 3, and the
choice of action a in Draw-Training-Set (see Footnote 3).
instance (i.e. an initial state and a goal) from the planning domain by specifying both the
current world and the goal. We achieve this by letting P be the set of world predicates from
the classical domain together with a new set of goal predicates, one for each world predicate. Goal predicates are named by prepending a ?g? to the corresponding world predicate.
Thus, the MDP states are sets of world and goal facts involving some or all objects in O.
The objective is to reach MDP states where the goal facts are a subset of the world facts
(goal states). The state {on-table(a), on(a, b), clear(b), gclear(b)} is thus a goal state in
a blocks-world MDP, but would not be a goal state without clear(b). We represent this
objective by defining C to assign zero cost to actions taken in goal states and a positive
cost to actions in all other states. In addition, we take T to be the action simulator from
the planning domain (e.g. as defined by STRIPS rules), modified to treat goal states as
terminal and to preserve without change all goal predicates. With this cost function, a
low-cost policy must arrive at goal states as ?quickly? as possible. Finally, the initial state
distribution I can be any program that generates legal problem instances (MDP states) of
the planning domain?e.g. one might use a problem generator from a planning competition.
While here we assume and accurate T model is known, a more general reinforcementlearning context would require learning an approximate T , trading off exploitation of this
model with exploration to improve it.
Taxonomic Decision List Policies. We adapt the API method of Section 2 by using, for
Step 2, the policy-space language bias and learning method of our previous work on learning policies in relational domains from small problem solutions [28], briefly reviewed here.
In relational domains, useful rules often take the form ?apply action type a to any object in
set C?, e.g. ?unload any object that is at its destination?. In [19], decision lists of such rules
were used as a language bias for learning policies. We use such lists, and represent the sets
of objects needed using class expressions C written in taxonomic syntax [20], defined by
C ::= C0 | anything | ?C | (R C) | C ? C, with R ::= R0 | R ?1 | R ? R | R? .
Here, C0 is any one argument relation and R0 any binary relation from the predicates in P .
One argument relations denote the set of objects that they are true of, (R C) denotes the
image of the objects in class C under the binary relation R, and for the (natural) semantics of the other constructs shown, please refer to [28]. Given a state s and a concept C
expressed in taxonomic syntax, it is straightforward to compute, in time polynomial in the
sizes of s and C, the set of domain objects that are represented by C in s.
Restricting our attention to one-argument?action types5 , we write a policy as
!C1 :a1 , C2 :a2 , . . . , Cn :an ", where the Ci are taxonomic-syntax concepts and the ai are
action types. See Yoon et al. [28] for examples and details.
Our learner builds a decision-list of size-bounded rules by starting with the empty list and
greedily selecting a new rule to add, continuing until the list ?covers? all of the training
data. This procedure is described in Yoon et al. [28], where a heuristically guided beamsearch is used to greedily select the next rule to add. The only difference between the
learner in [28] and the one used here is the heuristic function, which incorporates Q-cost
? am )" in
? a1 ), . . . , Q(s,
information (unlike [28]). Given training example !s,? (s), Q(s,
D, we define the Q-advantage of taking action a instead of ?(s) in state s by ?(s, a) =
?
? a). We take the heuristic value of a concept-action rule to be the number
Q(s,?
(s)) ? Q(s,
of training examples where the rule ?fires? plus the cumulative Q-advantage that the rule
achieves on those training examples.6 Using Q-advantage rather than Q-cost focuses the
learner toward instances where large improvement over the previous policy is possible.
4
Relational Planning Experiments
Our experiments support three claims. 1) Using only the guidance of an (often weak)
domain-independent heuristic, API learns effective policies for entire classical planning
domains. 2) Each learned policy is a domain-specific planner that is fast and empirically
compares well to the state-of-the-art domain-independent planner FF [13]. 3) API can improve on previously published control knowledge and on that learned by previous systems.
Domains. We consider two deterministic domains with standard definitions and three
stochastic domains from Yoon et al. [28]?these are: BW(n), the n-block blocks world;
LW(l,t,p), the l location, t truck, p package logistics world; SBW(n), a stochastic variant of BW(n); SLW(l,c,t,p), the stochastic logistics world with c cars and t trucks; and
SPW(n), a version of SBW(n) with a paint action. We draw problem instances from each
domain by generating pairs of random initial states and goal conditions. The goal conditions specify block configurations involving all blocks in blocks worlds, and destinations
for all packages in logistics worlds.7
Throughout, we use the domain-independent FF heuristic [13].8 Each experiment specifies
a planning domain and an initial policy and then iterates API9 until ?no more progress? is
made. We evaluate each policy on 1000 random problem instances, recording the success
5
Multiple argument actions can be simulated at some cost with multiple single argument actions.
If the coverage term is not included, then covering a zero Q-advantage example is the same as
not covering it. But zero Q-advantage can be good (e.g. the previous policy is optimal in that state).
7
PSTRIPS domain definitions are at http://www.ece.purdue.edu/?givan/nips03-domains.html.
8
Space precludes a description of this complex and well studied planning heuristic here.
9
We use discount factor 1 and select large enough horizons to accurately rank most policies: 4?n
for BW(n) and SBW(n), 6?n for SPW(n), 12?p for LW(l,t,p) and SLW(l,c,t,p). Training set size is
6
SR
AL/H
BW(10)
SR
AL/H
BW(15)
LW(4,4,12)
1
1
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25
30
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iteration
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AL(S)/H
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iteration
(c)
Figure 2: Bootstrapping API with a domain-independent heuristic.
TL-BW-b in BW(10)
SR
AL/H
TL-BW-a in BW(10)
SR
TL-LW in LW(4,6,4)
AL(S)/H
SR
AL/H
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iteration
(b)
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iteration
(c)
Figure 3: Using TL-Plan control knowledge as initial policies.
Policy1 SR
SPW(10) Policy2 SR
Policy1 SR
SBW(10) Policy2 SR
SLW(4,3,3,4)Policy1 SR
Policy2 SR
1
1
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iteration
8
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2
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iteration
Figure 4: Using previously learned initial policies.
ratio SR (fraction of problems solved within the horizon) and normalized average solution
length AL/H (average plan length in successful trials divided by horizon), omitting AL/H
for very low SR. Initial-policy performance is plotted at iteration zero.
Bootstrapping from the Heuristic. We consider the domain-independent initial policy10
FF-Greedy, which acts using the FF heuristic with one-step look-ahead. Figures 2a and b
show SR and AL/H after each API iteration for BW(10) and BW(15). FF-Greedy is poor
in both domains. There is an initial period of no (apparent) progress, followed by rapid
improvement to nearly perfect SR. Examination of the learned BW(15) policies shows that
early iterations find important concepts and later iterations find a policy that achieves a
small SR; at that point, rapid improvement ensues. Figure 2c shows the SR and AL/H
for LW(4,4,12). FF-Greedy performs very well here; nevertheless, API yields compact
declarative policies of the same quality as FF-Greedy. We replicated these experiments in
the stochastic variants of these domains, with similar results (not shown for space reasons).
Initial Hand-Coded Policies. TL-Plan [3] uses human-coded domain-specific control
knowledge to solve classical planning problems. Here we use initial policies for API that
correspond to the domain-specific control knowledge appearing in [3].11 For the blocks
100 trajectories, and sampling width is always 1, which worked well even for stochastic domains. A
sampling width of 1 corresponds to a preference to draw a small number of trajectories from each of
a variety of problems rather than a larger number from each of relatively fewer training problems?in
either case, the learner must be robust to the noise resulting from stochastic effects.
10
What is considered ?domain independent? here is the means of constructing the policy.
11
We can not exactly capture the TL-Plan knowledge in our policy language. Instead, we write
policies that capture the knowledge but prune away some ?bad? actions that TL-Plan might consider.
world TL-Plan provides three sets of control knowledge of increasing quality?we use the
best and second best sets to get the policies TL-BW-a and TL-BW-b, respectively. For
logistics there is only one set of knowledge given, yielding the policy TL-LW.
Figures 3a?3c show the SR and AL/H for API when starting with TL-BW-a and TL-BWb in BW(10) and TL-LW in LW(4,4,12). In each case, API improves the human-coded
policies. Starting with TL-BW-a and TL-LW, which have perfect SR, API uncovers policies
that maintain SR but improve AL/H by approximately 6.3% and 13%, respectively. Starting
with TL-BW-b, which has SR of only 30%, API quickly uncovers policies with perfect SR.
There is a dramatic difference in the quality of FF-Greedy (iteration 0 of Figure 2a), TLBW-a, and TL-BW-b in BW(10); yet, for each initial policy, API finds policies of roughly
identical quality?requiring more iterations for lower quality initial policies.
Initial Machine-Learned Policies. In Yoon et al. [28], policies were learned from solutions to randomly drawn small problems for the three stochastic domains we test here,
among others. A significant range of policy qualities results, due to the random draw. Here,
we use API starting with some below-average policies from that work.12 Figures 4a-c show
results for SPW(10), SLW(4,3,3,4), and SBW(10). For each domain, API is shown to improve the SR for two arbitrarily selected, below-average, learned starting policies to nearly
1.0. API successfully exploits the previous, noisy learning to robustly obtain a good policy.
Comparing learned policies to FF. A
Table 1: FF vs. learned policies.
learned policy corresponds to a domainFF (in C)
API (Scheme)
specific planner for the target planning doDomains SR AL Time SR AL Time
main. Here we show that these policies
BW(10) 1
33 0.1s 0.99 25 1.5s
are competitive with FF, a state-of-the-art AI
BW(15)
0.96
58 2.7s 0.99 39 2.5s
planer, with respect to planning time and sucBW(20) 0.75 62 27.7s 0.98 55 3.7s
cess ratio. We selected a blocks-world polBW(30) 0.14 103 166.0s 0.99 86 2.8s
icy and logistics-world policy corresponding
LW(4,4,12)
1
42 0.0s 1
43 2.7s
to the learned policies (beyond iteration 0) in
LW(5,14,20) 1
73 0.4s 1
74 3.6s
Figures 2a and c with the best SR, breaking
ties with AL. We applied FF and the appropriate selected policy to each of 1000 new test
problems from each of the domains shown in Table 1. Planning cutoff times were set at
600, 300, and 100 seconds for BW(30), BW(20), and all other domains, respectively. Table 1 records the percent of problems solved within the time cutoff (SR), the average length
of successful trials (AL), and the average time for successful trials (Time) for both FF and
our two selected policies.
In blocks worlds with more than 10 blocks, the API policy improves on FF in every category, with scaling much better to 20 and 30 blocks. Using the same heuristic information
(in a different way), API uncovers policies that significantly outperform FF. FF?s heuristic is well suited to logistics worlds, eliminating search for these problems. Our method
performs equivalently, but for the slow prototype Scheme implementation.
5
Related Work
Typically, previous ?learning for planning? systems [22] learn from small-problem solutions to improve the efficiency and/or quality of planning. Two primary approaches are
to learn control knowledge for search-based planners, e.g. [23, 27, 10, 15, 1], and, more
closely related, to learn stand-alone control policies [17, 19, 28].
The former work is severely limited by the utility problem (see [21]), i.e., being ?swamped?
by low utility rules. Critically, our policy-language bias confronts this issue by preferring
simpler policies. Regarding the latter, our work is novel in using API to iteratively improve
12
For these stochastic domains we provide the heuristic (designed for deterministic domains) with
a deterministic STRIPS domain approximation (using the mostly likely outcome of each action).
policies, and leads to a more robust learner, as shown above. In addition, we leverage
a domain-independent planning heuristic to avoid the need for access to small problems.
Our learning approach is also not tied to having a base planner.
The most closely related work is relational reinforcement learning (RRL) [9], a form of online API that learns relational cost-function approximations. Q-cost functions are learned
in the form of relational decision trees (Q-trees) and are used to learn corresponding policies (P -trees). The RRL results clearly demonstrate the difficulty of learning cost-function
approximations in relational domains. Compared to P -trees, Q-trees tend to generalize
poorly and be much larger. RRL has not yet demonstrated scalability to problems as complex as those considered here?previous RRL blocks-world experiments include relatively
simple goals13 , which lead to cost functions that are much less complex than the ones here.
However, unlike RRL, our API assumes an unconstrained simulator and (for the FF heuristic) a world model, which must be provided or learned by additional techniques.
References
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13
The most complex blocks-world goal for RRL was to achieve on(A, B) in an n block environment. We consider blocks-world goals that involve all n blocks.
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1,603 | 2,457 | Information Bottleneck for
Gaussian Variables
Gal Chechik?
Amir Globerson?
Naftali Tishby
Yair Weiss
{ggal,gamir,tishby,yweiss}@cs.huji.ac.il
School of Computer Science and Engineering and
The Interdisciplinary Center for Neural Computation
The Hebrew University of Jerusalem, 91904, Israel
?
Both authors contributed equally
Abstract
The problem of extracting the relevant aspects of data was addressed through the information bottleneck (IB) method, by (soft)
clustering one variable while preserving information about another
- relevance - variable. An interesting question addressed in the
current work is the extension of these ideas to obtain continuous
representations that preserve relevant information, rather than discrete clusters. We give a formal definition of the general continuous
IB problem and obtain an analytic solution for the optimal representation for the important case of multivariate Gaussian variables.
The obtained optimal representation is a noisy linear projection to
eigenvectors of the normalized correlation matrix ?x|y ??1
x , which
is also the basis obtained in Canonical Correlation Analysis. However, in Gaussian IB, the compression tradeoff parameter uniquely
determines the dimension, as well as the scale of each eigenvector.
This introduces a novel interpretation where solutions of different
ranks lie on a continuum parametrized by the compression level.
Our analysis also provides an analytic expression for the optimal
tradeoff - the information curve - in terms of the eigenvalue spectrum.
1
Introduction
Extracting relevant aspects of complex data is a fundamental task in machine learning and statistics. The problem is often that the data contains many structures,
which make it difficult to define which of them are relevant and which are not in an
unsupervised manner. For example, speech signals may be characterized by their
volume level, pitch, or content; pictures can be ranked by their luminosity level,
color saturation or importance with regard to some task.
This problem was principally addressed by the information bottleneck (IB) approach
[1]. Given the joint distribution of a ?source? variable X and another ?relevance?
variable Y , IB operates to compress X, while preserving information about Y . The
variable Y thus implicitly defines what is relevant in X and what isn?t. Formally,
this is cast as the following variational problem
min L : L ? I(X; T ) ? ?I(T ; Y )
p(t|x)
(1)
where T represents the compression of X via the conditional distributions p(t|x),
while the information that T maintains on Y is captured by p(y|t). The positive
parameter ? determines the tradeoff between compression and preserved relevant
information, as the Lagrange multiplier for the constrained optimization problem
minp(t|x) I(X; T ) ? ? (I(T ; Y ) ? const).
The information bottleneck approach has been applied so far mainly to categorical
variables, with a discrete T that represents (soft) clusters of X. It has been proved
useful for a range of applications from documents clustering, to gene expression
analysis (see [2] for review and references). However, its general information theoretic formulation is not restricted, both in terms of the variables X and Y , as well
as in the compression variable T . It can be naturally extended to nominal and continuous variables, as well as dimension reduction techniques rather than clustering.
This is the goal of the current paper.
The general treatment of IB for continuous T yields the same set of self-consistent
equations obtained already in [1]. But rather than solving them for the distributions
p(t|x), p(t) and p(y|t) using the generalized Blahut-Arimoto algorithm as proposed
there, one can turn them into two coupled eigenvector problems for the logarithmic
p(x|t)
p(y|t)
functional derivatives ? log?t
and ? log?t
, respectively. Solving these equations,
in general, turns out to be a rather difficult challenge. As in many other cases,
however, the problem turns out to be analytically tractable when X and Y are
joint multivariate Gaussian variables, as shown in this paper.
The optimal compression in the Gaussian Information Bottleneck (GIB) is defined
in terms of the compression-relevance tradeoff, determined through the parameter
?. It turns out to be a noisy linear projection to a subspace whose dimension
is determined by the tradeoff parameter ?. The subspaces are spanned by the
basis vectors obtained in the well known Canonical Correlation Analysis (CCA)[3]
method, but the exact nature of the projection is determined in a unique way via
the tradeoff parameter ?. Specifically, as ? increases, additional dimensions are
added to the projection variable T , through a series of critical points (structural
phase transitions), while at the same time the relative magnitude of each basis
vector is rescaled. This process continues until all the relevant information about
Y is captured in T . This demonstrates how the IB formalism provides a continuous
measure of model complexity in information theoretic terms.
The idea of maximization of relevant information was also taken in the Imax framework [4, 5]. In that setting, there are two feed forward networks with inputs X a , Xb
and output neurons Ya , Yb . The output neuron Ya serves to define relevance to the
output of the neighboring network Yb . Formally, The goal is to tune the incoming
weights of both output neurons, such that their mutual information I(Y a ; Yb ) is
maximized. An important difference between Imax and the IB setting, is that in
the Imax setting, I(Ya ; Yb ) is invariant to scaling and translation of the Y ?s since
the compression achieved in the mapping Xa ? Ya is not modeled explicitly. In
contrast, the IB framework aims to characterize the dependence of the solution on
the explicit compression term I(T ; X), which is a scale sensitive measure when the
transformation is noisy. This view of compressed representation T of the inputs X
is useful when dealing with neural systems that are stochastic in nature and limited
in their response amplitudes and are thus constrained to finite I(T ; X).
2
Gaussian Information Bottleneck
We now formalize the problem of Information Bottleneck for Gaussian variables.
Let (X, Y ) be two jointly Gaussian variables of dimensions nx , ny and denote by
?x , ?y the covariance matrices of X, Y and by ?xy their cross-covariance matrix1 .
The goal of GIB is to compress the variable X via a stochastic transformation into
another variable T ? Rnx , while preserving information about Y . With Gaussian
X and Y , the optimal T is also jointly Gaussian with X and Y . The intuition is
that only second order correlations exist in the joint distribution p(X, Y ), so that
distributions of T with higher order moments do not carry additional information.
This can be rigorously shown using an application of the entropy power inequality
as in [6], and will be published elsewhere. Note that we do not explicitly limit the
dimension of T , since we will show that the effective dimension is determined by the
value of ?. Since every two random variables X, T with jointly Gaussian distribution
can be presented as T = AX + ?, where ? ? N (0, ?? ) is another Gaussian that is
independent of X, we formalize the problem as the minimization
min L ? I(X; T ) ? ?I(T ; Y )
A,??
(2)
over the noisy linear transformations parametrized by the transformation A and
noise covariance ?? . T is normally distributed T ? N (0, ?t ) with ?t = A?x AT +?? .
3
The optimal projection
A main result of this paper is the characterization of the optimal A,?? as a function
of ?
Theorem 3.1 The optimal projection T = AX + ? for a given tradeoff parameter
? is given by ?? = Ix and
?
?
? T
?
0 ; . . . ; 0T
0 ? ? ? ?1c ?
?
?
?
?
?
?
T
T
T
c
c ?
?
?
? ?T1 v1 , 0T ; . T. . ; 0 T ? ?1c ? ? ? ?2c
A=
(3)
?
v
;
?
v
;
0
;
.
.
.
,
0
?
?
?
?
?
1
2
1
2
2
3 ?
?
?
?
?
?
..
?
?
.
where {v1T , v2T , . . . , vnTx } are left eigenvectors of ?x|y ??1
x sorted by their correspond1
c
ing ascending eigenvalues ?1 , ?2 , . . . , ?nx , ?i = 1??i are critical ? values, ?i are
i )?1
coefficients defined by ?i ? ?(1??
, ri ? viT ?x vi , 0T is an nx dimensional row
?i ri
vector of zeros, and semicolons separate rows in the matrix A.
This theorem asserts that the optimal projection consists of eigenvectors of ? x|y ??1
x ,
combined in an interesting manner: For ? values that are smaller than the smallest
critical point ?1c , compression is more important than any information preservation
and the optimal solution is the degenerated one A ? 0. As ? is increased, it
goes through a series of critical points ?ic , at each of which another eigenvector
of ?x|y ??1
x is added to A. Even though the rank of A increases at each of these
transition points, it changes smoothly as a function of ? since at the critical point
?ic the coefficient ?i vanishes. Thus ? parameterizes a ?continuous rank? of the
projection.
To illustrate the form of the solution, we plot the landscape of the target function
L together with the solution in a simple problem where X ? R 2 and Y ? R. In
this case A has a single non-zero row, thus A can be thought of as a row vector
1
For simplicity we assume that ?x , ?y are full rank, otherwise X, Y can be reduced to
the proper dimensionality.
A.
B.
?5
?2.5
?2.5
2
?5
0
A
2
A
Figure 1. L as a function of all possible projections A, for A : R2 ? R, obtained numerically from Eq. 4. Darkred: low L values; light-yellow: large
L values. ?xy = [0.1 0.2], ?x = I2 . A.
For ? = 15, the optimal solution is the
degenerated solution A ? 0. B. For
? = 100, the eigenvector of ?x|y ??1
x
with a norm according to theorem 3.1
(superimposed) is optimal.
2.5
5
?5
0
2.5
?2.5
0
A1
2.5
5
5
?5
?2.5
0
A1
2.5
5
of length 2, that projects X to a scalar A : X ? R, T ? R. Figure 1 shows the
target function L as a function of the projection A. In this example, ?1 = 0.95,
thus ?1c = 20. Therefor, for ? = 15 (figure 1A) the zero solution is optimal, but for
? = 100 > ? c (figure 1B) the corresponding eigenvector is a feasible solution, and
the target function manifold contains two mirror minima. As ? increases from 0 to
?, these two minima, starting as a single unified minimum at zero, split at ? 1c , and
then diverge apart to ?.
We now turn to prove theorem 3.12 . We start by rewriting L using the formula
for the entropy of a d dimensional Gaussian variable h(X) = 21 log((2?e)d |?x |),
where | ? | denotes a determinant. Using the Schur complement formula to calculate
the covariance of the conditional variable T |Y we have ?t|y = ?t ? ?ty ??1
y ?yt =
T
A?x|y A + ?? , and the target function (up to a factor of 2) can be written as
L(A, ?? ) = (1??) log |A?x AT + ?? | ? log |?? | + ? log |A?x|y AT + ?? | .
(4)
Although L is a function of both the noise ?? and the projection A,?it can be easily
shown that for every pair (A, ?? ), there is another projection A? = D?1 V A where
? I) = L(A, ?? ) 3 . This allows us to simplify the calculations
?? = V DV T and L(A,
by replacing the noise covariance matrix ?? with the identity matrix.
To identify the minimum of L we now differentiate L w.r.t. to the projection A
?
using the algebraic identity ?A
log |ACAT | = (ACAT )?1 2AC which holds for any
symmetric matrix C. Equating this derivative to zero and rearranging, we obtain
necessary conditions for an internal minimum of L
?
?
?
?
(? ? 1)/? (A?x|y AT + Id )(A?x AT + Id )?1 A = A ?x|y ??1
.
(5)
x
Equation 5 shows that the multiplication of ?x|y ??1
x by A must reside in the span
of the rows of A. This means that A should be spanned by up to d eigenvectors of
?x|y ??1
x . We can therefore represent the projection A as a mixture A = W V where
the rows of V are left normalized eigenvectors of ?x|y ??1
x and W is a mixing matrix
that weights these eigenvectors. In the remainder of this section we characterize
the nature of the mixing matrix W .
Lemma 3.2 The optimal mixing matrix W is a diagonal matrix of the form
?s
?
s
?(1 ? ?1 ) ? 1 T
?(1 ? ?k ) ? 1 T T
W = diag ?
v1 , . . . ,
vk , 0 , . . . , 0 T ?
?1 r 1
?k r k
2
(6)
Further details of the proofs can be found in a technical report [7].
Although this theorem holds only for full rank ?? , it does not limit the generality of the
discussion since low rank matrices yield infinite values of L and are therefore suboptimal.
3
where {v1T , . . . , vkT } and {?1 , . . . , ?k } are k ? nx eigenvectors and eigenvalues of
c
c
?x|y ??1
x with ?1 , . . . , ?k ? ?.
Proof: We write V ?x|y ??1
x = DV where D is a diagonal matrix whose elements
are the corresponding eigenvalues, and denote by R the diagonal matrix whose i th
element is ri = viT ?x vi . When k = nx , we substitute A = W V into equation 5,
and use the fact that W is full rank to obtain
W T W = [?(I ? D) ? I](DR)?1 .
(7)
While this does not uniquely characterize W , we note that if we substitute A into
the target function in equation 4, and use properties of the eigenvalues, we have
L = (1 ? ?)
n
X
i=1
n
X
?
?
?
?
log ||wiT ||2 ri + 1 + ?
log ||wiT ||2 ri ?i + 1
(8)
i=1
where ||wiT ||2 is the ith element of the diagonal of W T W . This shows that L depends
only on the norm of the columns of W , and all matrices W that satisfy (7) yield
the same target function. We can therefore choose to take W to be the diagonal
matrix which is the square root of (7)
W =
p
[?(I ? D) ? I)](DR)?1
(9)
To prove the case of k < nx , consider a matrix W that is a k ?k matrix padded with
zeros, thus it mixes only the first k eigenvectors. In this case, calculation similar to
that above gives the solution A which has nx ? k zero rows. To complete the proof,
it remains to be shown that the above solution capture all extrema points. This
point is detailed in [7] due to space considerations.
We have thus characterized the set of all minima of L, and turn to identify which
of them achieve the global minima.
Corollary 3.3 The global minimum of L is obtained with all ?i satisfying ? > ?ic
Proof: Substituting the optimal W of equation 9 into equation 8 yields L =
Pk
1
i=1 (? ? 1) log ?i + log(1 ? ?i ) + f (?). Since 0 ? ? ? 1 and ? ? 1?? , L is
1
minimized by taking all the eigenvalues that satisfy ? > (1??i ) .
Taken together, these observations prove that for a given value of ?, the optimal
projection is obtained by taking all the eigenvectors whose eigenvalues ? i satisfy
1
? ? 1??
, and setting their norm according to A = W V . This completes the proof
i
of theorem 3.1.
4
The GIB Information Curve
The information bottleneck is targeted at characterizing the tradeoff between information preservation (accuracy of relevant predictions) and compression. Interestingly, much of the structure of the problem is reflected in the information curve,
namely the maximal value of relevant preserved information (accuracy), I(T ; Y ),
as a function of the complexity of the representation of X, measured by I(T ; X).
This curve is related to the rate-distortion function in lossy source coding, as well
as to the achievability limit in channel coding with side-information [8]. It is shown
to be concave in general [9], but its precise functional form depends on the joint
1
??1 = 1??
i
i
I(T;Y)/ ? log(? )
1
0
0
5
10
15
I(T;X)
20
25
Figure 2. GIB information curve obtained with
four eigenvalues ?i = 0.1,0.5,0.7,0.9. The information at the critical points are designated
by circles. For comparison, information curves
calculated with smaller number of eigenvectors are also depicted (all curves calculated for
? < 1000). The slope of the curve at each point
is the corresponding ? ?1 . The tangent at zero,
with slope ? ?1 = 1 ? ?1 , is super imposed on
the information curve.
distribution and can reveal properties of the hidden structure of the variables. Analytic forms for the information curve are known only for very special cases, such as
Bernoulli variables and some intriguing self-similar distributions. The analytic characterization of the Gaussian IB problem allows us to obtain a closed form expression
for the information curve in terms of the relevant eigenvalues.
To this end, we substitute the optimal projection A(?) into I(T ; X) and I(T ; Y )
and isolate I? (T ; Y ) as a function of I? (T ; X)
?n
!
nI
I
2I? (T ;X) Y
Y
1
1
nI
nI
nI
nI
I? (T ; Y ) = I? (T ; X) ?
(1??i ) + e
log
?i
(10)
2
i=1
i=1
where the products are over the first nI eigenvalues, since these obey the critical ?
PnI ?1
?
1??i
condition, with cnI ? I? (T ; X) ? cnI +1 and cnI = 12 i=1
log ?niI 1??
.
n
I
The GIB curve, illustrated in Figure 2, is continuous and smooth, but is built of
several of segments, since as I(T ; X) increases additional eigenvectors are used in
the projection. The derivative of the curve is given by ? ?1 , which can be easily
shown to be continuous and decreasing, yielding that the GIB information curve is
concave everywhere. At each value of I(T ; X) the curve is therefore bounded by a
tangent with a slope ? ?1 (I(T ; X)). Generally in IB, the data processing inequality
yields an upper bound on the slope at the origin, ? ?1 (0) < 1, in GIB we obtain a
tighter bound: ? ?1 (0) < 1??1 . The asymptotic slope of the curve is always zero, as
? ? ?, reflecting the law of diminishing return: adding more bits to the description
of X does not provide more accuracy about T . This interesting relation between
the spectral properties of the covariance matrices raises interesting questions for
special cases where more can be said about this spectrum, such as for patterns in
neural-network learning problems.
5
5.1
Relation To Other Works
Canonical Correlation Analysis and Imax
The GIB projection derived above uses weighted eigenvectors of the matrix
?1
?1
?x|y ??1
x = Ix ? ?xy ?y ?yx ?x . The same eigenvectors are also used in Canonical
correlations Analysis (CCA) [3], a statistical method that finds linear relations between two variables. CCA aims to find sets of basis vectors for the two variables that
maximize the correlation coefficient between the projections of the variables on the
?1
basis vectors. The CCA bases are the eigenvectors of the matrices ??1
y ?yx ?x ?xy
?1
?1
and ?x ?xy ?y ?yx , and the square roots of their corresponding eigenvalues are
termed canonical correlation coefficients. CCA was also shown to be a special case
of continuous Imax [4, 5].
Although GIB and CCA involve the spectral analysis of the same matrices, they have
some inherent differences. First of all, GIB characterizes not only the eigenvectors
but also their norm, in a way that that depends on the trade-off parameter ?. Since
CCA depends on the correlation coefficient between the compressed (projected)
versions of X and Y , which is a normalized measure of correlation, it is invariant
to a rescaling of the projection vectors. In contrast, for any value of ?, GIB will
choose one particular rescaling given by equation (4).
While CCA is symmetric (in the sense that both X and Y are projected), IB is non
symmetric and only the X variable is compressed. It is therefore interesting that
both GIB and CCA use the same eigenvectors for the projection of X.
5.2
Multiterminal information theory
The Information Bottleneck formalism was recently shown [9] to be closely related
to the problem of source coding with side information [8]. In the latter, two discrete
variables X, Y are encoded separately at rates Rx , Ry , and the aim is to use them
to perfectly reconstruct Y . The bounds on the achievable rates in this case were
found in [8] and can be obtained from the IB information curve.
When considering continuous variables, lossless compression at finite rates is no
longer possible. Thus, mutual information for continuous variables is no longer
interpretable in terms of encoding bits, but rather serves as an optimal measure of
information between variables. The IB formalism, although coinciding with coding
theorems in the discrete case, is more general in the sense that it reflects the tradeoff
between compression and information preservation, and is not concerned with exact
reconstruction. Such reconstruction can be considered by introducing distortion
measures as in [6] but is not relevant for the question of finding representations
which capture the information between the variables.
6
Discussion
We applied the information bottleneck method to continuous jointly Gaussian variables X and Y , with a continuous representation of the compressed variable T . We
derived an analytic optimal solution as well as a general algorithm for this problem
(GIB) which is based solely on the spectral properties of the covariance matrices
in the problem. The solution for GIB, characterized in terms of the trade-off parameter ?, between compression and preserved relevant information, consists of
eigenvectors of the matrix ?x|y ??1
x , continuously adding up as weaker compression
and more complex models are allowed. We provide an analytic characterization
of the information curve, which relates the spectrum to relevant information in
an intriguing manner. Besides its clean analytic structure, GIB offers a new way
for analyzing empirical multivariate data when only its correlation matrices can be
estimated. In thus extends and provides new information theoretic insight to the
classical Canonical Correlation Analysis method.
The IB optima are known to obey three self consistent equations, that can be
used in an iterative algorithm guaranteed to converge to a local optimum [1]. In
GIB, these iterations over the conditional distributions p(t|x), p(t) and p(y|t) can
be transformed into iterations over the projection parameter A. In this case, the
iterative IB algorithm turns into repeated projections on the matrix ?x|y ??1
x , as
used in power methods for eigenvector calculation. The parameter ? determines
the scaling of the vectors, such that some of the eigenvectors decay to zero, while
the others converge to their value defined in Theorem 3.1.
When handling real world data, the relevance variable Y often contains multiple structures that are correlated to X, although many of them are actually
irrelevant. The information bottleneck with side information (IBSI) [10] alleviates this problem using side information in the form of an irrelevance variable Y ? about which information is removed. IBSI thus aims to minimize
L = I(X; T ) ? ? (I(T ; Y + ) ? ?I(T ; Y ? )). This functional can be analyzed in the
case of Gaussian variables (GIBSI: Gaussian IB with side information), in a similar
way to the analysis of GIB presented above. This results in a generalized eigenvalue
problem involving the covariance matrices ?x|y+ and ?x|y? . The detailed solution
of this problem as a function of the tradeoff parameters remains to be investigated.
For categorical variables, the IB framework can be shown to be closely related to
maximum-likelihood in a latent variable model [11]. It would be interesting to
see whether the GIB-CCA equivalence can be extended and give a more general
understanding of the relation between IB and statistical latent variable models.
The extension of IB to continuous variables reveals a common principle behind regularized unsupervised learning methods ranging from clustering to CCA. It remains
an interesting challenge to obtain practical algorithms in the IB framework for dimension reduction (continuous T ) without the Gaussian assumption, for example
by kernelizing [12] or adding non linearities to the projections (as in [13]).
References
[1] N. Tishby, F.C. Pereira, and W. Bialek. The information bottleneck method. In Proc.
of 37th Allerton Conference on communication and computation, 1999.
[2] N. Slonim. Information Bottlneck theory and applications. PhD thesis, Hebrew University of Jerusalem, 2003.
[3] H. Hotelling. The most predictable criterion. Journal of Educational Psychology,,
26:139?142, 1935.
[4] S. Becker and G.E. Hinton. A self-organizing neural network that discovers surfaces
in random-dot stereograms. Nature, 355(6356):161?163, 1992.
[5] S. Becker. Mutual information maximization: Models of cortical self-organization.
Network: Computation in Neural Systems, pages 7?31, 1996.
[6] T. Berger abd R. Zamir. A semi-continuous version of the berger-yeung problem.
IEEE Transactions on Information Theory, pages 1520?1526, 1999.
[7] G. Chechik and A. Globerson. Information bottleneck and linear projections of gaussian processes. Technical Report 4, Hebrew University, May 2003.
[8] A.D. Wyner. On source coding with side information at the decoder. IEEE Trans.
on Info Theory, IT-21:294?300, 1975.
[9] R. Gilad-Bachrach, A. Navot, and N. Tishby. An information theoretic tradeoff between complexity and accuracy. In Proceedings of the COLT, Washington., 2003.
[10] G. Chechik and N. Tishby. Extracting relevant structures with side information. In
S. Becker, S. Thrun, and K. Obermayer, editors, Advances in Neural Information
Processing Systems 15, 2002.
[11] N. Slonim and Y. Weiss. Maximum likelihood and the information bottleneck. In
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[12] S. Mika, G. Ratsch, J. Weston, B. Scholkopf, A. Smola, and K. Muller. Invariant
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[13] A.J. Bell and T.J. Sejnowski. An information maximization approach to blind seperation and blind deconvolution. Neural Computation, 7:1129?1159, 1995.
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1,604 | 2,458 | An Autonomous Robotic System
For Mapping Abandoned Mines
D. Ferguson1 , A. Morris1 , D. H?ahnel2 , C. Baker1 , Z. Omohundro1 , C. Reverte1
S. Thayer1 , C. Whittaker1 , W. Whittaker1 , W. Burgard2 , S. Thrun3
1
The Robotics Institute
Carnegie Mellon University
Pittsburgh, PA
2
Computer Science Department
University of Freiburg
Freiburg, Germany
3
Computer Science Department
Stanford University
Stanford, CA
Abstract
We present the software architecture of a robotic system for mapping
abandoned mines. The software is capable of acquiring consistent 2D
maps of large mines with many cycles, represented as Markov random
?elds. 3D C-space maps are acquired from local 3D range scans, which
are used to identify navigable paths using A* search. Our system has
been deployed in three abandoned mines, two of which inaccessible to
people, where it has acquired maps of unprecedented detail and accuracy.
1
Introduction
This paper describes the navigation software of a deployed robotic system for mapping
subterranean spaces such as abandoned mines. Subsidence of abandoned mines poses a
major problem for society, as do ground water contaminations, mine ?res, and so on. Most
abandoned mines are inaccessible to people, but some are accessible to robots. Autonomy
is a key requirement for robots operating in such environments, due to a lack of wireless
communication technology for subterranean spaces.
Our vehicle, shown in Figure 1 (see [1] for a detailed hardware description) is equipped
with two actuated laser range ?nders. When exploring and mapping unknown mines, it
alternates short phases of motion guided by 2D range scans, with phases in which the
vehicle rests to acquire 3D range scans. An analysis of the 3D scans leads to a path that
is then executed, using rapidly acquired 2D scans to determine the robot?s motion relative
to the 3D map. If no such path is found a high-level control module adjusts the motion
direction accordingly.
Acquiring consistent large-scale maps without external geo-referencing through GPS is
largely considered an open research issue. Our approach relies on ef?cient statistical techniques for generating such maps in real-time. At the lowest level, we employ a fast scan
matching algorithm for registering successive scans, thereby recovering robot odometry.
Groups of scans are then converted into local maps, using Markov random ?eld representations (MRFs) to characterize the residual path uncertainty. Loop closure is attained
by adding constraints into those MRFs, based on a maximum likelihood (ML) estimator.
However, the brittleness of the ML approach is overcome by a ?lazy? data association
mechanism that can undo and redo past associations so as to maximize the overall map
consistency.
To navigate, local 3D scans are mapped into 2 12 D terrain maps, by analyzing surface gradients and vertical clearance in the 3D scans. The result is subsequently transformed into cost
Figure 1: The Groundhog robot is
a 1,500 pound custom-built vehicle
equipped with onboard computing, laser
range sensing, gas and sinkage sensors,
and video recording equipment. Its purpose is to explore and map abandoned
mines.
functions expressed in the robot?s three-dimensional con?guration space, by convolving
the 2 12 D terrain maps with kernels that describe the robot?s footprints in different orientations. Fast A* planning is then employed in con?guration space to generate paths executed
through PD control.
The system has been tested in a number of mines. Some of the results reported here
were obtained via manual control in mines accessible to people. Others involved fully
autonomous exploration, for which our robot operated fully self-guided for several hours
beyond the reach of radio communication.
2
2D Mapping
2.1 Generating Locally Consistent Maps
As in [6, 9], we apply an incremental scan matching technique for registering scans, acquired using a forward-pointed laser range ?nder while the vehicle is in motion. This
algorithm aligns scans by iteratively identifying nearby points in pairs of consecutive range
scans, and then calculating the relative displacement and orientation of these scans by minimizing the quadratic distance of these pairs of points [2]. This approach leads to the
recovery of two quantities: locally consistent maps and an estimate of the robot?s motion.
It is well-understood [3, 6], however, that local scan matching is incapable of achieving
globally consistent maps. This is because of the residual error in scan matching, which
accumulates over time. The limitation is apparent in the map shown in Figure 2a, which is
the result of applying local scan matching in a mine that is approximately 250 meters wide.
Our approach addresses this problem by explicitly representing the uncertainty in the map
and the path using a Markov random ?eld (MRF) [11]. More speci?cally, the data acquired
through every ?ve meters of consecutive robot motion is mapped into a local map [3].
Figure 3a shows such a local map. The absolute location of orientation of the k-th map will
be denoted by ?k = ( xk yk ?k )T ; here x and y are the Cartesian coordinates and ? is
the orientation. From the scan matcher, we can retrieve relative displacement information
of the form ?k,k?1 = ( ?xk,k?1 ?yk,k?1 ??k,k?1 )T which, if scan matching was errorfree, would enable us to recover absolute information via the following recursion (under
the boundary condition ?0 = (0, 0, 0)T )
?
!
xk?1 + ?xk,k?1 cos ?k,k?1 + ?yk,k?1 sin ?k?1
yk?1 ? ?xk,k?1 sin ?k,k?1 + ?yk,k?1 cos ?k?1 (1)
?k = f (?k?1 , ?k,k?1 ) =
?k?1 + ??k,k?1
However, scan matching is not without errors. To account for those errors, our approach
generalizes this recursion into a Markov random ?eld (MRF), in which each variable ? =
?1 , ?2 , . . . is a (three-dimensional) node. This MRF is de?ned through the potentials:
?(?k , ?k?1 )
=
exp ? 21 (?k ? f (?k?1 , ?k,k?1 ))T Rk,k?1 (?k ? f (?k?1 , ?k,k?1 )) (2)
Here Rk,k?1 is the inverse covariance of the uncertainty associated with the transition
?k,k?1 . Since the MRF is a linear chain without cycles, the mode of this MRF is the solution
to the recursion de?ned in (1). Figure 3b shows the MRF for the data collected in the
(a)
(b)
Figure 2: Mine map with incremental ML scan matching (left) and using our lazy data association
approach (right). The map is approximately 250 meters wide.
Bruceton Research Mine, over a distance of more than a mile. We note this representation
generalizes the one in [11], who represent posteriors by a local bank of Kalman ?lters.
2.2
Enforcing Global Consistency
The key advantage of the MRF representation is that it encompasses the residual uncertainty in local scan matching. This enables us to alter the shape of the map in accordance
with global consistency constraints. These constraints are obtained by matching local maps
acquired at different points in time (e.g., when closing a large cycle). In particular, if the
k-th map overlaps with some map j acquired at an earlier point in time, our approach localizes the robot relative to this map using once again local scan matching. As a result, it
recovers a relative constraint ?(?k , ?j ) between the coordinates of non-adjacent maps ?k
and ?j . This constraint is of the same form as the local constraints in (2), hence is represented by a potential. For any ?xed set of such potentials ? = {?(? k , ?j )}, the resulting
MRF is described through the following negative log-likelihood function
X
? log p(?) = const. + 21
(?k ? f (?j , ?k,j ))T Rk,j (?k ? f (?j , ?k,j ))
(3)
k,j
where ? = ?1 , ?2 , . . . is the set of all map poses, and f is de?ned in (1).
Unfortunately, the resulting MRF is not a linear chain any longer. Instead, it contains
cycles. The variables ? = ?1 , ?2 , . . . can be recovered using any of the standard inference
algorithms for inference on graphs with cycles, such as the popular loopy belief propagation
algorithm and related techniques [5, 14, 17]. Our approach solves this problem by matrix
inversion. In particular, we linearize the function f using a Taylor expansion:
f (?j , ?k,j )
? f (??j ) + Fk,j (?j ? ??j )
(4)
where ??j denotes a momentary estimate of the variables ?j (e.g., the solution of the
recursion (1) without the additional data association constraints). The matrix F k,j =
(a)
(b)
Figure 3: (a) Example of a local map.
(b) The Markov random ?eld: Each
node is the center of a local map, acquired when traversing the Bruceton Research Mine near Pittsburgh, PA.
??j f (??j , ?k,j ) is the Jacobean of f (?j , ?k,j ) at ??j :
?
?
1 0 ??xk,j sin ??k + ?yk,j cos ??k
Fk,j x = ? 0 1 ??xk,j cos ??k ? ?yk,j sin ??k ?
0 0
1
(5)
The resulting negative log-likelihood is given by
X
? log p(?) ? const. +
?1
(?k ? f (??j ) ? Fk,j (?j ? ??j ))T ?k,j
(?k ? f (??j ) ? Fk,j (?j ? ??j ))
1
2
k,j
is quadratic in the variables ? of the form const. + (A? ? a)T R (A? ? a), where A is a
diagonal matrix, a is a vector, and R is a sparse matrix that is non-zero for all elements j, k
in the set of potentials. The minimum of this function is attained at (AT RA)?1 AT Ra. This
solution requires the inversion of a sparse matrix. Empirically, we ?nd that this inversion
can be performed very ef?ciently using an inversion algorithm described in [15]; it only
requires a few seconds for matrices composed of hundreds of local map positions (and it
appears to be numerically more stable than the solution in [11, 6]). Iterative application of
this linearized optimization quickly converges to the mode of the MRF, which is the set of
locations and orientations ?. However, we conjecture that recent advances on inference in
loopy graphs can further increase the ef?ciency of our approach.
2.3 Lazy Data Association Search
Unfortunately, the approach described thus far leads only to a consistent map when the
additional constraints ?(?k , ?j ) obtained after loop closure are correct. These constraints
amount to a maximum likelihood solution for the challenging data association problem
that arises when closing a loop. When loops are large, this ML solution might be wrong?a
problem that has been the source of an entire literature on SLAM (simultaneous localization
and mapping) algorithms. Figure 4a depicts such a situation, obtained when operating our
vehicle in a large abandoned mine.
The current best algorithms apply proactive particle ?lter (PF) techniques to solve this
problem [4, 8, 12, 13]. PF techniques sample from the path posterior. When closing a loop,
random variations in these samples lead to different loop closures. As long as the correct
such closure is in the set of surviving particle ?lters, the correct map can be recovered.
In the context of our present system, this approach suffers from two disadvantages: it is
computationally expensive due to its proactive nature, and it provides no mechanism for
recovery should the correct loop closure not be represented in the particle set.
Our approach overcomes both of these limitations. When closing a loop, it always picks the
most likely data association. However, it also provides a mechanism to undo and redo past
data association decisions. The exact data association algorithm involves a step that monitors the likelihood of the most recent sensor measurement given the map. If this likelihood
falls below a threshold, data association constraints are recursively undone and replaced
by other constraints of decreasing likelihood (including the possibility of not generating a
constraint at all). The search terminates if the likelihood of the most recent measurement
(a)
(b)
con?ict
?
?
?
map after adjustment
?
start
Figure 4: Example of our lazy data association technique: When closing a large loop, the robot
?rst erroneously assumes the existence of a second, parallel hallway. However, this model leads to
a gross inconsistency as the robot encounters a corridor at a right angle. At this point, our approach
recursively searches for improved data association decisions, arriving at the map shown on the right.
exceeds the threshold [7]. In practice, the threshold test works well, since global inconsistencies tend to induce gross inconsistencies in the robot?s measurements at some point in
time.
The algorithm is illustrated in Figure 4. The left panel shows the ML association after
traversing a large loop inside a mine: At ?rst, it appears that the existence of two adjacent
corridors is more likely than a single one, according to the estimated robot motion. However, as the robot approaches a turn, a noticeable inconsistency is detected. Inconsistencies
are found by monitoring the measurement likelihood, using a threshold for triggering an
exception. As a result, our data association mechanism recursively removes past data association constraints back to the most recent loop closure, and then ?tries? the second most
likely hypothesis. The result of this backtracking step is shown in the right panel of Figure 4. The backtracking requires a fraction of a second, and with high likelihood leads to
a globally consistent map and, as a side-effect, to an improved estimate of the map coordinates ?. Figure 2b shows a proto-typical corrected map, which is globally consistent.
3
Autonomous Navigation
2D maps are suf?cient for localizing robots inside mines; however, they are insuf?cient to
navigate a robot due to the rugged nature of abandoned mines. Our approach to navigation
is based on 3D maps, acquired in periodic intervals while the vehicle suspends motion to
scan its environment. A typical 3D scan is shown in Figure 5a; others are shown in Figure 7.
3.1
2 21 D Terrain Maps
In a ?rst processing step, the robot projects local 3D maps onto 2 12 D terrain maps, such as
the one shown in Figure 5b. The gray-level in this map illustrates the degree at which the
map is traversable: the brighter a 2D location, the better suited it is for navigation.
The terrain map is obtained by analyzing all measurements hx, y, zi in the 3D scan
(where z is the vertical dimension). For each rectangular surface region {x min ; xmax } ?
{ymin ; ymax }, it identi?es the minimum z-value, denoted z. It then searches for the largest
z value in this region whose distance to z does not exceed the vehicle height (plus a safety
margin); this value will be called z?. The difference z? ? z is the navigational coef?cient:
it loosely corresponds to the ruggedness of the terrain under the height of the robot. If no
measurement is available for the target region {xmin ; xmax } ? {ymin ; ymax }, the region is
marked as unknown. For safety reasons, multiple regions {xmin ; xmax } ? {ymin ; ymax }
overlap when building the terrain map. The terrain map is subsequently convolved with
a narrow radial kernel that serves as a repellent potential ?eld, to keep the robot clear of
obstacles.
3.2 Con?guration Space Maps
The terrain map is used to construct a collection of maps that describe the robot?s con?guration space, or C-space [10]. The C-space is the three-dimensional space of poses that
(a)
(b)
(c)
Figure 5: (a) A local 3D model of the mine corridor, obtained by a scanning laser range ?nder. (b)
The corresponding 2 21 D terrain map extracted from this 3D snapshot: the brighter a location, the
easier it is to navigate. (c) Kernels for generating directional C-space maps from the 2 21 D terrain
map. The two black bars in each kernel correspond to the vehicle?s tires. Planning in these C-space
maps ensures that the terrain under the tires is maximally navigable.
the vehicle can assume; it comprises the x-y location along with the vehicle?s orientation
?. The C-space maps are obtained by convolving the terrain map with oriented kernels that
describe the robot?s footprint. Figure 5c shows some of these kernels: Most value is placed
in the wheel area of the vehicle, with only a small portion assigned to the area in between,
where the vehicle?s clearance is approximately 30 centimeters. The intuition of using such
a kernel is as follows: Abandoned mines often possess railroad tracks, and while it is perfectly acceptable to navigate with a track between the wheels, traversing or riding these
tracks causes unnecessary damage to the tires and will increase the energy consumption.
The result of this transformation is a collection of C-space maps, each of which applies to
a different vehicle orientation.
3.3
Corridor Following
Finally, A* search is employed in C-space to determine a path to an unexplored area. The
A* search is initiated with an array of goal points, which places the highest value at locations at maximum distance straight down a mine corridor. This approach ?nds the best path
to traverse, and then executes it using a PD controller.
If no such path can be found even within a short range (2.5 meters), the robot decides
that the hallway is not navigable and initiates a high-level decision to turn around. This
technique has been suf?cient for our autonomous exploration runs thus far (which involved
straight hallway exploration), but it does not yet provide a viable solution for exploring
multiple hallways connected by intersections (see [16] for recent work on this topic).
4
Results
The approach was tested in multiple experiments, some of which were remotely operated
while in others the robot operated autonomously, outside the reach of radio communication.
On October 27, 2002, Groundhog was driven under manual control into the Florence Mine
near Burgettstown, PA. Figure 6b shows a picture of the tethered and remotely controlled
vehicle inside this mine, which has not been entered by people for many decades. Its
partially ?ooded nature prevented an entry into the mine for more than approximately 40
meters. Maps acquired in this mine are shown in Figure 9.
On May 30, 2003, Groundhog successfully explored an abandoned mine using the fully
autonomous mode. The mine, known as the Mathies Mine near Pittsburgh, is part of a
large mine system near Courtney, PA. Existing maps for this mine are highly inaccurate,
and the conditions inside the mine were unknown to us. Figure 6a shows the robot as it
enters the mine, and Figure 7a depicts a typical 3D scan acquired in the entrance area.
(a)
(b)
Figure 6: (a) The vehicle as it enters the Mathies Mine on May 30, 2003. It autonomously descended
308 meters into the mine before making the correct decision to turn around due to a blockage inside
the mine. (b) The vehicle, as it negotiates acidic mud under manual remote control approximately 30
meters into the Florence Mine near Burgettstown, PA.
(a)
(b)
Figure 7: 3D local maps: (a) a typical corridor map that is highly navigable. (b) a map of a broken
ceiling bar that renders the corridor segment unnavigable. This obstacle was encountered 308 meters
into the abandoned Mathies Mine.
Figure 8: Fraction of the 2D mine map of the Mathies Mine, autonomously explored by the Groundhog vehicle. Also shown is the path of the robot and the locations at which it chose to take 3D scans.
The protruding obstacle shows up as a small dot-like obstacle in the 2D map.
(a)
(b)
(c)
Figure 9: (a) A small 2D map acquired by Groundhog in the Florence Mine near Burgettstown, PA.
This remotely-controlled mission was aborted when the robot?s computer was ?ooded by water and
mud in the mine. (b) View of a local 3D map of the ceiling. (c) Image acquired by Groundhog inside
the Mathies Mine (a dry mine).
After successfully descending 308 meters into the Mathies Mine, negotiating some rough
terrain along the way, the robot encountered a broken ceiling beam that draped diagonally
across the robot?s path. The corresponding 3D scan is shown in Figure 7b: it shows rubble
on the ground, along with the ceiling bar and two ceiling cables dragged down by the bar.
The robot?s A* motion planner failed to identify a navigable path, and the robot made the
appropriate decision to retreat. Figure 8 shows the corresponding 2D map; the entire map
is 308 meters long, but here we only show the ?nal section, along with the path and the
location at which the robot stop to take a 3D scan. An image acquired in this mine is
depicted in Figure 9c.
5
Conclusion
We have described the software architecture of a deployed system for robotic mine mapping. The most important algorithmic innovations of our approach are new, lazy techniques
for data association, and a fast technique for navigating rugged terrain. The system has
been tested under extreme conditions, and generated accurate maps of abandoned mines
inaccessible to people.
Acknowledgements
We acknowledge the contributions of the students of the class 16865 Mobile Robot Development
at CMU who helped build Groundhog. We also acknowledge the assistance provided by Bruceton
Research Mine (Paul Stefko), MSHA, PA-DEP, Workhorse Technologies, and the various people
in the mining industry who supported this work. Finally, we also gratefully acknowledge ?nancial
support by DARPA?s MARS program.
References
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1,605 | 2,459 | Ambiguous model learning made unambiguous
with 1/f priors
G. S. Atwal
Department of Physics
Princeton University
Princeton, NJ 08544
[email protected]
William Bialek
Department of Physics
Princeton University
Princeton, NJ 08544
[email protected]
Abstract
What happens to the optimal interpretation of noisy data when there
exists more than one equally plausible interpretation of the data? In a
Bayesian model-learning framework the answer depends on the prior expectations of the dynamics of the model parameter that is to be inferred
from the data. Local time constraints on the priors are insufficient to
pick one interpretation over another. On the other hand, nonlocal time
constraints, induced by a 1/f noise spectrum of the priors, is shown to
permit learning of a specific model parameter even when there are infinitely many equally plausible interpretations of the data. This transition
is inferred by a remarkable mapping of the model estimation problem
to a dissipative physical system, allowing the use of powerful statistical mechanical methods to uncover the transition from indeterminate to
determinate model learning.
1
Introduction
The estimation of a model underlying the production of noisy data becomes highly nontrivial when there exists more than one equally plausible model that could be responsible
for the output data. The viewing of ambiguous figures, such as the Necker cube [1], is
a classical problem of this type in the field of visual psychology. Pitch perception when
hearing a number of different harmonics is another example of ambiguous perception [2].
Previous studies [3] have reduced the problem of optimal interpretation of an ambiguous
stimulus to the problem of estimating a single variable which may vary in time ?(t), given
a time sequence of noisy data. Enforcing a prior belief that the local dynamics ?(t) should
not vary too rapidly embodies the observer?s knowledge that rapid variations in ?(t) are
unlikely in the natural world or in a given experiment. Such a prior prevents overfitting the
model estimate to the data as it arrives. The statistically optimal interpretation of the data
was then found to consist of ?(t) hopping randomly from one possible interpretation to
another. The rate of random switching between interpretations was found to be controlled
not by the noise level (e.g. in the neural hardware), as previously thought, but rather by
the observer?s prior hypotheses. This hopping persists indefinitely despite the fact that
the probability distribution of the incoming data remains the same. In such cases it is
impossible to learn a specific model parameter.
In this paper we introduce another prior over the dynamics of ?(t). We assume that fluctuations in ?(t) have a 1/f spectrum, as observed ubiquitously in nature. Such a prior is
shown to induce nonlocal time constraints on the trajectories of ?(t) and, unlike the local
constraints, can result in specific model learning in the case of ambiguous models. The fact
that 1/f priors can induce unambiguous model learning is the central result of this work.
The analyses of the long-time dynamics with nonlocal priors is permitted by a surprising
and remarkable mapping to a dissipative quantum system. This mapping not only guides
our intuition of the optimal trajectories of ?(t) but also permits the usage of powerful
statistical mechanical techniques. In particular, the renormalization group (RG) can be
employed to uncover the conditions in which there is a transition from non-specific model
learning to specific model learning.
2
Formalism
Suppose that we are given a series of N measurements {xt } at discrete times t. Then Bayes
rule gives us the conditional probability of {?t } giving rise to those data
P [{?t }|{xt }] =
P [{xt }|{?t }] P [{?t }]
,
P [{xt }]
(1)
where the probability of making the observations {xi } is given by summing up all the
possible models that may give rise to them,
Z
P ({xt }) = d? P [{xt }|{?t }]P [{?t }].
(2)
We further assume conditional independence of signals,
P [{xt }|{?t }] = P [x1 x2 ...xN |{?t }] =
N
Y
P [xt |?t ].
(3)
t=1
A natural step is then to consider how close our estimate of the model ?(t) lies to the true
underlying model ?(t), which we take to be stationary ?(t) = ?. We can think of these
probability distributions as Boltzmann distributions in which some effective potential acts
to hold ? close to ?
? ; thus we envision an energy landscape in the ? space with a minimum
at ?
?.
A more interesting, and generalized, question arises when we consider the global properties
of the extended energy landscape. In particular there may be M > 1 equally plausible
interpretations consistent with the input data1 in which case there exist degenerate minima
at ?m (m = 1, 2....M ),
P [xt |?1 ] = P [xt |?2 ] = ... = P [xt |?M ].
(4)
Therefore we may write Eq. (3) as
P [{xt }|{?t }] =
N
Y
t=1
M
Y
m=1
1/M
P [xt |?m ]
!
#
M
N
1 X X P [xt |?t ]
exp
ln
.
M m=1 t=1 P [xt |?m ]
"
(5)
On average, the term in square brackets is related to the Kullback-Leibler divergences between distributions conditional on ?(t) and distributions conditional on the true ?
? . If the
1
Of course it may be the case that some interpretations may be more plausible than others, resulting in a non uniform probability distribution over possible models. In this paper we illustrate the case
where all interpretations are equally likely, P [?m ] = 1/M .
time variation of ? is slow, we effectively collect many samples of x before ? changes, and
it makes sense to replace the sum over samples by its average:
Z
M X
N
M Z
X
P [xt |?t ]
1 X
P [x(t)|?(t)]
lim
ln
?
dt dxP [x(t)|?m ] ln
,
N ??
P [xt |?m ]
?0 m=1
P [x|?m ]
m=1 t=1
M Z
1 X
dtDKL [?m ||?(t)].
? ?
(6)
?0 m=1
where ?0 is the average time between observations, and we take the continuum limit.
2.1
Priors
We need to have some prior hypotheses about how ?(t) can vary in time, serving as our
prior probability distribution P [?(t)]. We introduce two different types of priors characterized by whether they constrain the local or nonlocal time dynamics,
P [?(t)] = Plocal [?(t)]Pnonlocal [?(t)].
(7)
To summarize our prior expectation that the local dynamics of ?(t) vary slowly, we assume
that the time derivative of ?(t) is chosen independently at each instant of time from a
Gaussian distribution,
"
2 #
Z
??
1
dt
.
(8)
Plocal [?(t)] ? exp ?
4D
?t
Note that this distribution corresponds to random walk with effective diffusion constant D.
Motivated by the ubiquitous occurrence of 1/f fluctuations in nature we chose to encapsulate the nonlocal dynamics by a Gaussian distribution with a 1/f power spectrum of noise,
conveniently expressed in Fourier coordinates ? as
Z
d? |?(?)|2
1
,
(9)
Pnonlocal [?(t)] ? exp ?
2
2? S(?)
where the spectral noise function takes the form
1
S(?) =
.
(10)
?|?|
Note that the spectrum must be even in ? since for any stationary process S(?) = S(??).
The parameter ? determines the strength of a priori belief in nonlocal dynamics, or as
we will see later, it can be equivalently viewed as a frictional constant determining the
dissipation of the time trajectories of ?(t). In the time-domain Eq. (9) becomes
"
2 #
Z
?(t) ? ?(t? )
?
?
Pnonlocal [?(t)] ? exp ?
dtdt
.
(11)
4?
t ? t?
Combining Eq. (8) and Eq. (11) we then obtain the total prior expectation of the probability
distribution over the time-dependence of the model parameter ?(t)
"
2
2 #
Z
Z
?
1
??
?
?(t)
?
?(t
)
P [?(t)] ? exp ?
dt
?
dtdt?
.
(12)
4D
?t
4?
t ? t?
Taken together, the local and non-local terms describe fluctuations in ? which are 1/f up
to a cutoff frequency, ?c ? D?. Returning to the Bayesian conditional probability Eq. (1)
we then obtain a path-integral expression
P [?(t)|{xi }] ? exp(?S[?(t)]),
(13)
where the action S[?(t)] is given by
#
"
2
2
Z
Z
??
dt? ?(t) ? ?(t? )
1
+?
+ Veff [?(t)] , (14)
S[?(t)] =
dt
4D ?t
4?
t ? t?
Veff [?(t)]
=
M
1 X
DKL [?m ||?(t)].
?0 M m=1
(15)
This is equivalent to the imaginary time path-integral for a quantum mechanical particle [4]
of mass 1/2D , with coordinates given by ?(t), moving in an effective potential Veff [?(t)]
and subject to (linear) frictional forces with a damping constant ?. This mapping provides
an extremely useful guide to our intuition for the probable trajectories of ?(t). Just as in the
analyses of particle dynamics in dissipative quantum mechanics [4] we anticipate that the
time-course of ?(t) may exhibit qualitatively different types of behavior depending on the
strength of the non-local terms. In addition, the equivalence to a physical system permits
exploitation of powerful techniques developed in the study of quantum mechanical systems
with infinite degrees of freedom.
In the following we consider the cases of m = 1 and m = 2 and use the RG transformations
to consider localization-delocalization transitions.
2.2
M=1 : One true interpretation of data
Now if ?(t) differs from ? by a small ??(t) we can Taylor expand the Kullback-Leibler
divergence to give a quadratic distance measure
DKL (?||?) =
1
F [?(t)]??(t)2 + O(??3 ),
2
where the metric is the Fisher information
2
Z
?P [x|?(t)]
1
.
F [?(t)] = dx
P [x|?(t)]
??(t)
(16)
(17)
Thus, close to the true parameter ? the potential energy term in Eq. (14) is simply a harmonic oscillator with stiffness given by the Fisher information. Guided by the mapping to a
dissipative quantum mechanical system we expect that if the initial distribution of ? already
happens to be closely centered around the correct value then the most likely trajectory will
be simply to move closer to the minima of the potential energy at ?1 .
The important point to note is that had we chosen just the local constraints on our priors
Eq. (8) then the trajectory of ?(t) would persistently fluctuate around ?1 , representing a
trade-off between avoiding overfitting the data and inertia of our estimate. In the quantum mechanical picture this corresponds to the zero point fluctuations around the minima.
Adding the dissipative term reduces the fluctuations around ?1 by an amount monotonically dependent on ?, thus improving on the optimal estimate.
A RG treatment of the single-well problem, within the harmonic approximation, renormalizes the Fisher information such that the curvature of the potential well increases for
all values of the ?, and thus the fixed point of the dynamics is simply the convergence
of ?(t) to reduced fluctuations around the true parameter ?1 . We explicitly carry out the
RG calculation in the more interesting case where we have two global minima in the next
section.
2.3
M=2 : Two equally possible interpretations of the data
In the case of two equally viable interpretations of the data, the potential energy term
becomes that of a double-well potential with degenerate minima at ?1 and ?2 and energy
barrier h
1
(DKL (?1 ||(?1 + ?2 )/2) + DKL (?2 ||?1 + ?2 )/2))
(18)
h=
2?0
20
V(?)
15
10
h
5
a1
0
-10
-5
a2
0
5
?
10
Figure 1: Potential energy landscape for ? where there exist two equally valid interpretations. Eq. (19)
Without any dissipative dynamics, the optimal estimate of ?(t) will switch between the
two minima, representing instanton trajectories of a quantum particle tunnelling through
the energy barrier backwards and forwards [3]. In contrast, it is well known that, at least in
some regimes, the problem with dissipation has a phase transition to a truly localized state.
Previous work has demonstrated such a dynamical phase transition in the strong-coupling
limit (i.e. large barrier height limit) using semi-classical approximations for the dynamics
[4,5,6], and in this section we will show that a perturbative RG treatment yields similar
results in the opposite weak-coupling limit.
For the sake of simplicity we employ the following simple quartic potential (see Fig.1),
although the results will be independent of its exact form,
V (?) =
2
h
?2 ? ?21 .
?41
(19)
The ? coordinates have been shifted such that ?1 = ??2 , and the height h of the energy
barrier located at ? = 0 sets the overall energy scale. It is useful to write the effective
action of Eq. (14) in dimensionless parameters
a=
?
,
?1
b = ??21 ,
c=
h
,
?
where ? = D/?21 is the energy/frequency scale 2
Z
Z
1
d?
1 2
S =
? + b|?| |a(?)|2 + c? dt V ? (a),
2
2? 2?
V ? (a)
=
(a2 ? 1)2 .
(20)
(21)
(22)
2
The constant of proportionality between energy and frequency is set to 1, akin to the common
physics computation setting of ?
h = 1.
By power counting in the first integral the dissipative term, at low frequencies, dominates
over the kinetic energy term. In the language of RG, the kinetic energy term is an irrelevant
operator and can thus be ignored if we now focus our attention to frequencies below some
cut-off ?. To determine the RG flow of the dimensionless coupling parameters the highfrequency components are integrated out from ? = ? ? d? to ? = ? to give a new effective
action S? over the low frequency modes ? < ?. To accomplish this the function ?(?) is
split
a(?) = a< (?)?(|?| < ? ? d?) + a> (?)?(? ? d? < |?| < ?),
(23)
and the new action is obtained by integrating over a> (?),
Z
Z =
Da exp[?S(a)],
Z Z
=
Da< Da> exp[?S(a< + a> )],
Z
? < )].
=
Da< exp[?S(a
(24)
Therefore,
? <) = b
S(a
2
Z
0
??d?
Z
d?
|?||a< (?)|2 + ln exp c? dtV ? (a< + ?> )
,
2?
a>
where the averaging is defined by
(
)
Z
Z
b ? d?
2
hAia> ? Da> exp ?
|?||a> (?)| A.
2 ??d? 2?
(25)
(26)
In the weak-coupling limit, we may expand the exponential term in Eq. (25) before performing the averaging,
Z
Z
?
?
exp[c? dtV (a< + a> )]
= 1 + c? dtV (a< + a> ) + ...
.
(27)
a>
a>
Terminating the expansion to first order in the potential represents a one-loop calculation
in field theories.
Making use of
a2> (t) a>
=
Z
?
??d?
1 d?
d? 1
?
,
? b|?|
?b ?
(28)
we find that the potential term renormalizes as
2 d?
(c?(a2 ? 1)2 )? ? (c?(a2 ? 1)2 )??d? ? (c?)? (a2< ? 1)2 + (3a2< ? 1)
, (29)
?b ?
where we have ignored terms including higher powers of d?/?. To recast the new lowerfrequency action into the same form as the original action the dimensionless coupling parameters must be renormalised. In particular, we observe that the dimensionless barrier
height c can either grow or shrink depending on the value of the dimensionless dissipation
b. Note that the coordinates must also be rescaled (also known as wavefunction renormalization) for the potential in Eq. (29) to maintain the same quartic form as in Eq. (22),
thereby inducing a rescaling of b. We concentrate here on the renormalized potential coupling term and find that, up to a constant,
6
d?
1?
,
(30)
c??d? = c? 1 +
?
?b
giving then the following differential RG flow equation
?
dc
b
=
? 1 c.
d ln ?
b
(31)
As the (dimensionless) barrier height c renormalizes towards lower frequencies we observe
two types of behavior depending on whether the parameter b is greater or smaller than the
critical value b? = 6/? (the actual numerical value may well be slightly altered by going
to higher orders in the perturbative expansion, but the important point to note that it is nonzero and thus gives rise to distinct dynamical phases). For b > b? the barrier height grows
without bounds and thus effectively traps ?(t) in one of the two minima, representing a
localized phase. This localization can be brought about by increasing the magnitude of ?,
the numerical prefactor of our dissipative nonlocal priors, and/or increasing ?1 the distance
between the two possible interpretations of the data. On the other hand, for b < b? the
potential becomes ineffective in localizing ?, and thus ? freely tunnels between the two
wells, representing indeterminancy of the correct true model parameter.
It is interesting to note that a flow equation, similar to Eq. (31), has been reported for the
opposite limit (strong-coupling) using the instanton method[5,6]. Arguably what we have
really shown is that even if one starts with weak coupling, so that it should be ?easy? to
jump from one interpretation to another, for b > b? we will flow to strong-coupling, at
which point known results about localization take over.
local
8
c
0
nonlocal
b*
b
Figure 2: Schematic RG flow of the potential energy coupling parameter for M ? 2. Note
that the flow-lines are not expected to be strictly vertical due to wavefunction renormalization.
The qualitative picture does not change when there are more than two possible model interpretations, M > 2. In fact, the case of M = ? has been studied [7] where the potential
energy landscape is taken to be sinusoidal, and it has been demonstrated that there again
exists a critical value b? which separates a localized phase from a nonlocalized phase. The
flow of the potential energy coupling constant c is shown in Fig.2 which is expected to be
qualitatively correct across the whole range 2 ? M ? ?.
3
Discussion
In summary, the optimal model estimate in the response of ambiguous signals always results in random perceptual switching when the priors only constrain the local dynamics.
We have shown that when we allow the possibility of 1/f noise in our priors then a specific
model is learnt amongst the many possible models.
The connection between estimation theory and statistical mechanics is well known. One
of the key results in statistical mechanics is that local interactions in one dimension can
never lead to a phase transition. Thus if we are interested in, for example, learning a
single parameter by making repeated observations, then there can be no phase transition
to certainty about the value of this parameter as long as our prior hypotheses about its
dynamics are equivalent to local models in statistical mechanics. Markov models, Gaussian
processes with rational spectra, and other common priors all fall in this local class.
The common occurrence of 1/f fluctuations in nature motivates the analyses of estimation
theory with such priors. Crucially, 1/f spectra do not correspond to local models. In fact
they correspond exactly to the addition of friction to the path integral describing a quantum
mechanical particle, a problem of general interest in condensed matter physics and more
recently in quantum computing. Here we note one important consequence of these priors,
namely that we can process data in a model which admits the possibility of time variation
for the underlying parameter, but nonetheless find that our best estimate of this parameter
is localized for all time to one of many equally plausible alternatives. It seems that 1/f
priors may provide a way to understand the emergence of certainty more generally as a
phase transition.
References
[1] G. H. Fisher, Perception & Psychophysics 4, 189 (1968)
[2] E. de Boer, Handbk. Sens. Physiol. 3, 479 (1976)
[3] W. Bialek and M. DeWeese, M. Phys. Rev. Lett. 74, 3079 (1995)
[4] A. O. Caldeira and A. J. Leggett, Phys. Rev. Lett. 46, 211 (1981)
[5] A. J. Bray and M. A. Moore, Phys. Rev. Lett 49, 1545 (1982)
[6] A. J. Leggett, S. Chakravarty, A. T. Dorsey, M. P. A. Fisher, A. Garg and W. Zwerger, Rev. Mod.
Phys. 59, 1 (1987)
[7] M. P. A. Fisher and W. Zwerger, Phys. Rev. Lett 32, 6190 (1985)
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1,606 | 246 | 84
Wilson and Bower
Computer Simulation of Oscillatory Behavior
in Cerebral Cortical Networks
Matthew A. Wilson and James M. Bower!
Computation and Neural Systems Program
Division of Biology, 216-76
California Institute of Technology
Pasadena, CA 9 1125
ABSTRACT
It has been known for many years that specific regions of the work-
ing cerebral cortex display periodic variations in correlated cellular
activity. While the olfactory system has been the focus of much of
this work, similar behavior has recently been observed in primary
visual cortex. We have developed models of both the olfactory
and visual cortex which replicate the observed oscillatory properties of these networks. Using these models we have examined the
dependence of oscillatory behavior on single cell properties and network architectures. We discuss the idea that the oscillatory events
recorded from cerebral cortex may be intrinsic to the architecture
of cerebral cortex as a whole, and that these rhythmic patterns
may be important in coordinating neuronal activity during sensory
processmg.
1
INTRODUCTION
An obvious characteristic of the general behavior of cerebral cortex, as evident in
EEG recordings, is its tendency to oscillate. Cortical oscillations have been observed
both in the electric fields generated by populations of cells (Bressler and Freeman
1 Please
address correspondence to James M. Bower at above address.
Computer Simulation or Oscillatory Behavior in Cerebral Cortical Networks
1980) as well as in the activity of single cells (Llinas 1988). Our previous efforts
to study this behavior involve the construction of realistic, large scale computer
simulations of one particular cortical area, the piriform (olfactory) cortex (Wilson
and Bower 1989). While the oscillatory behavior of this region has been known
for some time (Adrian 1942; Bressler and Freeman 1980), more recent findings
of oscillations within visual cortex (Eckhorn et al.,1988; Gray et at. 1989) have
generated increased interest in the role of oscillations in cerebral cortex in general.
It is particularly intriguing that although these cortical areas receive very different
kinds of sensory information, the periodic activity seen in both structures share a
common principle frequency component in the range of 30-60 Hz. At the same time,
however, the phase relationships of activity across each cortex differ. Piriform cortex
displays systematic phase shifts in field potential responses to afferent activation
(Freeman 1978; Haberly 1973), while correlations of neuronal activity in visual
cortex indicate no such systematic phase shifts (Gray et al. 1989).
In order to compare this oscillatory behavior in these two cortical systems, we have
developed a model of visual cortex by modifying the original piriform cortex model
to reflect visual cortical network features.
2
2.1
MODEL STRUCTURE
COMMON MODEL FEATURES
Each simulation has at its base the three basic cell types found throughout cerebral
cortex (Figure 1). The principle excitatory neuron, the pyramidal cell, is modeled
here as five coupled membrane compartments. In addition there are two inhibitory
neurons one principally mediating a slow K + inhibition and one mediating a fast CIinhibition. Both are modeled as a single compartment. Connections between modeled cells are made by axons with finite conduction velocities, but no explicit axonal
membrane properties other than delay are included. Synaptic activity is produced
by simulating the action-potential triggered release of presynaptic transmitter and
the resulting flow of transmembrane current through membrane channels. Each of
these channels is described with parameters governing the time course and amplitude of synaptically activated conductance changes. The compartmental models of
the cells integrate the transmembrane and axial currents to produce transmembrane
voltages. Excursions of the cell body membrane voltage above a specified threshold trigger action potentials. Details of the modeling procedures are described in
Wilson and Bower (1989).
Each model is intended to represent a 10 mm x 6 mm cortical region. The many millions of actual neurons in these areas are represented by 375 cells of the three types
for a total of 1125 cells. The input to each cortex is prvV'ided by 100 independent
fibers.
8S
86
Wilson and Bower
A
10 mm
D
B
rotItrally directed
_ciation fiber.
caudally directed
_elation flbe,.
_elldlon fibe,.
Figure 1: In the piriform cortex, input (A) and association fiber (B) projections
make distributed lateral contacts with cells over the extent of the cortex. In the
visual cortex model, input projections make local contact with cells over a 1 mm
radius in a point-to-point fashion (C) and association fibers connect to cells within
a limited radius (D).
While both the piriform and visual cortex models reflect these basic features of
cerebral cortical architecture, both also contain major structural simplifications.
The model referred to as "visual cortex", is particularly simplified. Our objective
was to reproduce cortical oscillations characteristic of visual cortex by modifying
those basic architectural features that differ between these two brain regions.
2.2
MODEL DIFFERENCES
The principle differences between the model of piriform and visual cortex involve
changes in the topography of input projections, and in the extent of intrinsic connections within each model. In piriform cortex, afferent input from the olfactory bulb
arrives via a tract ofaxons (LOT) projecting across the surface of the cortex (Fig.
lA) with no topographic relationship between the site of origin of individual LOT
axons in the olfactory bulb and their region of termination in the cortex (Haberly
1985). In contrast, projections from the lateral geniculate nucleus to visual cortex
are highly topographic, reflecting the retinotopic organization of many structures
in the visual system (Van Essen 1979). In piriform cortex, excitatory intrinsic association connections are sparse, distributed, and non-topographic, extending across
Computer Simulation of Oscillatory Behavior in Cerebral Cortical Networks
the entire cortex (Fig. Ie) (Haberly 1985). In the visual cortex, this association
fiber system is much more limited in extent (Gilbert 1983).
3
RESULTS
Space limitations do not allow a complete discussion of previous results modeling
piriform cortex. Readers are referred to Wilson and Bower (1989) for additional
details. Here, we will describe data obtained from the modified piriform cortex
model which replicate results from visual cortex.
1
2
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Figure 2: Comparison of auto and cross correlations from modeled (middle) and
actual (right) (modified from Gray et al. 1989) visual cortex. The left column shows
a diagram of the model with the stimulus region shaded. The numbers indicate the
location of the recording sites referred to in the auto (2-2) and cross (1-2) correlations. The correlations generated by presentation of a continuous and broken bar
stimulus are shown in the upper and lower panels respectively.
87
88
Wilson and Bower
Figure 2 shows a comparison of auto and cross correlations of neuronal spike activity taken from both simulated and actual (Gray et al. 1989) experimental data. In
each case the two recording sites in visual cortex are separated by approximately
6 mm. Total cross correlations in the modeled data were computed by averaging
correlations from 50 individual 500 msec trials. Within each trial simulated activity
was generated by providing input representing bars of light at different locations in
the visual field. In these cases the model produced oscillatory auto and cross correlations with peak energy in the 30-60 Hz range. As in the experimental data, this
effect is most clearly seen when the stimulus is a continuous bar of light activating
cells between the two recorded sites (fig. 2). A broken bar which does not stimulate
the intermediate region produces a weaker response (fig. 2), again consistent with
experimental evidence.
The oscillatory form of the the cross correlation function suggests coherent firing of
neurons at the two recorded locations. In order to determine the degree of synchrony
between modeled neurons, the difference in phase between the firing of cells in these
locations was estimated by measuring the offset of the dominant peak in the cross
correlation function. These values were consistent with measurements obtained
both through chi-square fitting of a modified sinc function and measurement of the
phase of the peak frequency component in the correlation function power spectra.
These measurements indicate phase shifts near zero ? 3 msec).
3.1
STIMULUS EFFECTS
As shown in figure 2, correlations are induced by the presence of a stimulus. However, in both experimental and simulated results these correlations cannot be accounted for through a simple stimulus locking effect. Shuffling the trials with respect
to each other prior to calculating cross correlation functions showed oscillations
which were greatly diminished or completely absent. At the same time, simulations
run in the absence of bar stimuli produced low baseline activity with no oscillations.
These results demonstrate that while the stimulus is necessary to induce oscillatory
behavior, the coherence between distant points is not due to the stimulus alone.
3.2
FREQUENCY
The visual cortex model generates oscillatory neural activity at a frequency in the
range of 30-60 Hz, consistent with actual data. As found in the model piriform
cortex, the frequency of these oscillations is primarily determined by the time course
of the fast feedback inhibitory input. Allowing inhibitory cells to inhibit other
inhibitory cells within a local region improved frequency locking and produced auto
and cross correlations with more pronounced oscillatory characteristics.
3.3
COHERENCE
In order to demonstrate the essential role of the association fiber system in establishing coherent activity, simulations were performed in which all long-range (> 1 mm)
Computer Simulation of Oscillatory Behavior in Cerebral Cortical Networks
association fibers were eliminated. Under these conditions the auto correlations at
each recording site continued to show strong oscillatory behavior, but oscillations
in the cross correlation function were completely eliminated. Increasing the range
of association fibers from 1 to 2 mm restored coherent oscillatory behavior. This
demonstrates that long-range association fibers are critical in establishing coherence
while local circuitry is sufficient for sustaining oscillations.
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Figure 3: Time course of cross correlation functions for relative association fiber
coupling strengths of 200 (left) and 300 (right). Upper traces display correlations
taken at successive 125 intervals over the 500 msec period. The bottom-most correlation function covers the entire 500 msec interval. The lower panels display the
power spectra of the overall correlation function.
89
90
Wilson and Bower
3.3.1
Association Fiber Delay
To examine the dependence of zero-phase coherence between distant sites on association fibers characteristics, the propagation velocity for spikes travelling between
pyramidal cells was reduced from a mean of 0.86 mls to 0.43 m/s. Under these conditions the phase shift in the cross correlation function for a continuous bar stimulus
remained less than 3 msec. This result indicates that the zero-phase coherence is
not a direct function of association fiber delays.
3.3.2
Coupling Strength
As shown in figure 3, increasing the degree of association fiber coupling by increasing
synaptic weights produced a transition from zero-phase coherence to a coherence
with an 8 msec phase shift. Intermediate shifts were not observed. Figure 3 also
illustrates the time course of coherence and phase relationships. There is a tendency
for the initial stimulus onset period (0-125 msec) to show zero-phase preference.
Later periods (> 125 msec) reflect the association coupling induced phase shift.
For weak coupling which produces zero-phase behavior, the correlation structure
decays over the 500 msec stimulus period. Increased coupling strength provides
more sustained coherence, as does the addition of mutual inhibition.
4
DISCUSSION
Analysis of the behavior of the models shows that several components are particularly important in establishing the different phase and frequency relationships. A
key factor in establishing zero-phase coherence appears to be the stimulation of a
cellular population which can activate, via association fibers, adjacent regions in a
symmetric fashion. In the case of the continuous bar, this intermediate region lies
in the center of the bar. This is consistent with experimental results which indicate
reduced coherence with bar stimuli which do not excite this region. The model also
indicates that frequency can be effectively modulated by inhibitory feedback. The
fact that inhibitory events with similar temporal properties are found throughout
the cerebral cortex suggests that oscillations in the 30-60 Hz range will be found in
a number of different cortical areas.
Interpreting phase coherence from correlation functions produced from the average
of many simulation trials pointed out the need to distinguish average phase effects
from instantaneous phase effects. Instantaneous phase implies that the statistics
of the correlation function taken at any point of any trial are consistent with the
statistics of the combined data. Average phase allows for systematic within-trial and
between-trial variability and is, therefore, a weaker assertion of actual coherence.
This distinction is particularly important for theories which rely on phase encoding
of stimulus information. Analysis of our model results indicates that the observed
phase relationships are an average rather than an instantaneous effect.
Based on previous observations of the behavior of the piriform cortex model, we
have proposed that high frequency oscillations may reflect the gating of intrinsic
Computer Simulation of Oscillatory Behavior in Cerebral Cortical Networks
network integration intervals. This modulatory role would serve to assure that
cells do not fire before they have received the necessary input to initiate another
round of cortical activity. While this is dearly only one possible functional role
for oscillations in piriform cortex, the model is being used to extend this idea to
processing in the visual cortex as well.
Acknowledgements
This research was supported by the NSF (EET-8700064), the ONR (Contract
N00014-88-K-0513), and the Lockheed Corporation.
References
Adrian, E.D. 1942. Olfactory reactions in the brain of the hedgehog. J. Physiol.
(Lond.) 100, 459-472.
Bressler, S.L. and W.J. Freeman. 1980. Frequency analysis of olfactory system
EEG in cat, rabbit and rat. Electroenceph. din. Neurophysiol. 50, 19-24.
Eckhorn, R., R. Bauer, Jordan, M. Brosch, W. Kruse, M. Munk, and H.J. Reitboeck.
1988. Coherent oscillations: A mechanism of feature linking in the visual cortex?
BioI. Cybern. 60, 121-130.
Freeman, W.J. 1978. Spatial properties of an EEG event in the olfactory bulb and
cortex. Electroenceph. din. Neurophysiol. 44,586-605.
Gilbert, C.D. 1983. Microcircuitry of the visual cortex. Ann. Rev. Neurosci.
6,217-247.
Gray, C.M., P. Konig, A.K. Engel, W. Singer. 1989. Oscillatory responses in cat
visual cortex exhibit inter-columnar synchronization which reflects global stimulus
properties. Nature 338, 334-337.
Haberly, L.B. 1985. Neuronal circuitry in olfactory cortex: anatomy and functional
implications. Chern. Senses 10, 219-238.
Haberly, L.B. 1973. Summed potentials evoked in opossum prepyriform cortex. J.
Neurophysiol. 36, 775-788.
Kammen, D.M., P.J. Holmes, and C. Koch. 1989. Cortical architecture and oscillations in neuronal networks: Feedback versus local coupling. In: Models of Brain
Function R.M.J. Cotterill, Ed. (Cambridge Univ. Press .)
Llinas, R. 1988. The intrinsic electrophysiological properties of mammalian neurons:
Insights into central nervous system function. Science 242:1654-1664.
Wilson, M.A. and J.M Bower. 1989. The simulation of large scale neuronal networks. In Methods in Neuronal Modeling: From Synapses to Networks C. Koch
and I. Segev, Eds. (MIT Press, Cambridge, MA.) pp. 291-334.
Van Essen, D.C. 1979. Visual areas of the mammalian cerebral cortex. Ann. Rev.
Neurosci. 2, 227-263.
91
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total:2 tendency:2 la:1 experimental:5 modulated:1 correlated:1 |
1,607 | 2,460 | Reconstructing MEG Sources
with Unknown Correlations
Maneesh Sahani
W. M. Keck Foundation Center
for Integrative Neuroscience,
UC, San Francisco, CA 94143-0732
[email protected]
Srikantan S. Nagarajan
Biomagnetic Imaging Laboratory,
Department of Radiology,
UC, San Francisco, CA 94143-0628
[email protected]
Abstract
Existing source location and recovery algorithms used in magnetoencephalographic imaging generally assume that the source activity at different brain locations is independent or that the correlation structure is
known. However, electrophysiological recordings of local field potentials show strong correlations in aggregate activity over significant distances. Indeed, it seems very likely that stimulus-evoked activity would
follow strongly correlated time-courses in different brain areas. Here,
we present, and validate through simulations, a new approach to source
reconstruction in which the correlation between sources is modelled and
estimated explicitly by variational Bayesian methods, facilitating accurate recovery of source locations and the time-courses of their activation.
1
Introduction
The brain?s neuronal activity generates weak magnetic fields (10 fT ? 1 pT). Magnetoencephalography (MEG) is a non-invasive technique for detecting and characterising
these magnetic fields. MEG sensors use super-conducting quantum interference devices
(SQUIDs) to measure the changes in the brain?s magnetic field on a millisecond time-scale.
When combined with electromagnetic source localisation, magnetic source imaging (MSI)
becomes a functional brain imaging method that allows us to characterise macroscopic
dynamic neural information processing.
In the past decade, the development of MSI source reconstruction algorithms has progressed significantly [1]. Currently, there are two general approaches to estimating MEG
sources: parametric methods and imaging methods [2]. With parametric methods, a few
current dipoles of unknown location and moment are assumed to represent the sources
of activity in the brain. In this case, solving the inverse problem requires a non-linear
optimisation to estimate the position and magnitude of an unknown number of dipoles.
With imaging methods, a grid of voxels is used to represent the entire brain volume. The
inverse problem is then to recover whole brain activation images, represented by the timedependent moment and magnitude of an elementary dipole source located at each voxel.
This formulation leads to a linear forward model. However, the ill-posed nature of the
problem leads to non-unique solutions which must be distinguished by prior information,
usually in the form of assumptions regarding the correlation between the sources.
In this paper, we formulate a general spatiotemporal imaging model for MEG data. Our formulation makes no assumptions about the correlation of the sources; instead, we estimate
the extent of the correlation by an evidence optimisation procedure within a variational
Bayesian framework [3].
1.1
MEG imaging
Many standard MEG devices measure the radial gradient of the magnetic field at a number,
db , of sensor locations (typically arranged on a segment of a sphere). Measurements made
at a single time can be formed into a db -dimensional vector b; an experiment yields a series
of N such samples, giving a db ? N data matrix B.
This measured field-gradient is affected by a number of different processes. The component we seek to isolate is stimulus- or event-related, and is presumably contributed to by
significant activity at a relatively small number of locations in the brain. This signal is
corrupted by thermal noise at the sensors, and by widespread spontaneous, unrelated brain
activity. For our purposes, these are both sources of noise, whose distributions are approximately normal [2] (in the case of the unrelated brain activity, the normality results from
the fact that any one sensor sees the sum of effects from a large number of locations). The
covariance matrix of this noise, ?, can be measured approximately by accumulating sensor
readings in a quiescent state; simulations suggest that the techniques presented here are
reasonably tolerant to mis-estimation of the noise level. Measurements are also affected
by other forms of interference associated with experimental electronics or bio-magnetic
activity external to the brain. We will not here treat such interference explicitly, instead
assuming that major sources have been removed by preprocessing the measured data, e.g.,
by using blind source separation methods [4].
To represent the significant brain sources, we divide the volume of the brain (or a subsection of that volume that contains the sources) into a number of voxels and then calculate the
lead-field matrix L that linearly relates the strength of a current dipole in each orientation
at each voxel, to the sensor measurements. For simplicity, we assume a spherical volume
conductor model, which permits analytical calculation of L independent of the tissue conductivity [2], and which is reasonably accurate for most brain regions [1]. (Non-uniform
volume conduction properties of the brain and surrounding tissues can be explicitly accounted for by elaborating the lead-field matrix calculation, but they do not otherwise affect
the analysis presented below.) In the simple model, only the two tangential components of
the current dipole which fall orthogonal to the radial direction contribute to b, and so the
source vector s has a dimension ds which is twice the number of voxels dv . The source
matrix S associated with the N field measurements has dimensions ds ? N . Thus the
probabilistic forward model for MEG measurements is given by
b ? N (Ls, ?)
(1)
Without considerable prior knowledge of the pattern of brain activation, the number of
possible degrees of freedom in the source vector, ds , will be far greater than the number of
measurements, db ; and so there is no unique maximum-likelihood estimate of s. Instead,
attempts at source recovery depend, either implicitly or explicitly, on the application of
prior knowledge about the source distribution. Most existing methods constrain the source
locations and/or activities in various ways: based on anatomical or fMRI data; by maximum
entropy, minimum L1 norm, weighted-minimum L2 norm or maximum smoothness priors;
or to achieve optimal resolution [1]. Most of these constraints can be formulated as priors
for maximum a posteriori estimation of the sources (although the original statements do not
always make such priors explicit). In addition, some studies have also included temporal
constraints on sources such as smoothness or phase-locking between sources [5].
Consider, for example, linear estimates of s given by ?s = F 0 b. The optimal estimate (in a
least-squares sense) is given by the Wiener filter:
?1
F = hbb0 i
?1
hbs0 i = hbb0 i
?1
h(Ls + n)s0 i = hbb0 i
L hss0 i ,
(2)
(where n ? N (0, ?) is a noise vector uncorrelated with s) and therefore requires knowledge of the source correlation matrix hss0 i.
One approach to source reconstruction, the minimum-variance adaptive beamformer (or
?beamformer? for short), can be viewed as an approximation to the Wiener filter in which
the correlation matrix of sensor measurements hbb0 i is estimated by the observed correlation BB 0 /N , and the sources at each location are taken to be uncorrelated [6]. If the
orientation of each source dipole is known or estimated independently (so that s contains
only one magnitude at each location), then the source correlation matrix hss0 i reduces to a
diagonal matrix of gain factors. For the beamformer, these factors are chosen to give a unit
?loop gain? for each source i.e. such that diag [F 0 L] = 1. It can be shown that the beamformer only yields accurate results when the number of active sources is few [7]. Thus, this
approach makes two assumptions about the sources: an explicit one of decorrelation and
an implicit one of sparse activation. Other techniques tend to make similar assumptions. A
related algorithm using Multiple Signal Classification (MUSIC) also assumes sparsity and
linear independence in the time-series of the sources [1]. Minimum-norm methods can also
be viewed as making specific assumptions about the source correlation matrix [8].
In sharp contrast to the assumed independence or known correlation of brain activity in
these algorithms, electrophysiological studies have shown pronounced and variable correlations in local potentials measured in different (sometimes widely separated) regions of the
brain, and indeed, have argued that these correlations reflect relevant aspects of brain processing [9, 10]. This simple observation has profound consequences for most current MEG
imaging algorithms. Not only are they unable to access this source of temporal information
about brain function (despite the temporal fidelity of the technique in other respects), but
they may also provide inaccurate source localisations or reconstructions by dint of their
incorrect assumptions regarding source correlation.
In this paper, we present a novel approach to source reconstruction. Our technique shares
with many of the methods described above the assumption of sparsity in source activation. However, it dispenses entirely with assumption of source independence. Instead, we
estimate the source correlation matrix from the data by hyperparameter optimisation.
2
Model
To parameterise the source correlation matrix in a manner tractable for learning, we assume
that the source activities s are formed by a linear combination, with weight matrix W , of
dz independent unit-variance normal pre-sources z,
z ? N (0, I) ,
s = W z;
(3)
so that learning the correlation matrix hss i = W W becomes equivalent to estimation of
the weights W .1 The sources are not really expected to have the Gaussian amplitude distribution that this construction implies. Instead, the assumption forms a convenient fiction,
making it easy to estimate the source correlation matrix. We show in simulations below
that estimation in this framework can indeed yield accurate estimates of the correlation
matrix even for non-normally distributed source activity. Once the correlation matrix has
been established, estimation using the Wiener filter of (2) provides the best linear estimate
of source activity (and would be the exact maximum a posteriori estimate if the sources
really were normally distributed).
0
0
1
This formulation is similar to that used in weighted minimum-norm methods, although there the
weights W are fixed, implying a pre-determined source correlation matrix.
The model of (3) parameterises the source correlation in a general way, subject to a maximum rank of dz . This rank constraint does not by itself favour sparsity in the source distribution, and could easily be chosen to be equal to ds . Instead, the sparsity emerges from
a hyperparameter optimisation similar to the automatic relevance determination (ARD) of
Mackay and Neal [11] (see also [12, 13]). Equation (3) defined a prior on s with parameters
W . We now add a hyperprior on W under which the expected power of both tangential
components at the vth voxel is determined by a hyperparameter ?v . For notational convenience we collect the ?v into a vector ? and introduce a ds ? dv indicator matrix J, with
Jiv = 1 if the ith source is located in the vth voxel and 0 otherwise. Thus, each column of
J contains exactly two unit entries, one for each tangential component of the corresponding
voxel dipole. Finally, we introduce a ds ? ds diagonal matrix A with Aii = (J?)i . Then
Wij ? N (0, A?1
ii ) .
(4)
Thus each ?v sets a prior distribution on the length of the two rows in the weight matrix
corresponding to source components at the vth voxel. As in the original ARD models, optimisation of the marginal likelihood or evidence, P (B | ?, L, ?), with respect to the ?v results in a number of the hyperparameters diverging to infinity. This imposes a zero-centred
delta-function prior on the corresponding row of W , in turn forcing the corresponding
source power to vanish. It is this optimisation, then, which introduces the sparsity.
Before passing to the optimisation scheme, we summarise the model introduced above
by the log joint probability it assigns to observations, pre-sources and weights (here, and
below, we drop the explicit conditioning on the fixed parameters L and ?)
1
log P (B, Z, W | ?) = ? (N log |2??| + Tr [(B ? LW Z)0 ??1 (B ? LW Z)])
2
1
1
? (N dz log(2?) + Tr [Z 0 Z]) ? (dz log |2?A?1 | + Tr [W 0 AW ]) (5)
2
2
3
Learning
R
Direct optimisation of the log marginal likelihood log dZ dW P (B, Z, W | ?) proves
to be intractable. Instead, we adopt the ?variational Bayes? (VB) framework of [3, 12].
VB is a form of the Expectation-Maximisation (EM) algorithm for maximum-likelihood
estimation. Given unknown distributions Qz (Z) and Qw (W ), Jensen?s inequality provides
a bound on the log-likelihood
Z
Qz (Z)Qw (W )
P (B, Z, W | ?)
Qz (Z)Qw (W )
? hlog P (B, Z, W | ?)iQz (Z)Qw (W ) + H(Qz ) + H(Qw )
log P (B | ?) = log
dZ dW
(where H(?) represents the Shannon entropy). This bound can then be optimised by alternate maximisations with respect to Qz , Qw and the hyperparameters ?. If, in place of the
factored distribution Qz (Z)Qw (W ) we had used a joint Q(Z, W ), this procedure would
be guaranteed to find a local maximum in the marginal likelihood (by analogy to EM). As
it is, the optimisation is only approximate, but has been found to yield good maxima in a
factor analysis model very similar to the one we consider here [12]. In our experiments,
a slight variant of the standard VB procedure, described below, improved further on the
accuracy of the solutions found.
Given estimates Qnz , Qnw and ?n at the nth step, the (n + 1)th iteration is given by:
Qn+1
(Z) ? exp hlog P (B, Z, W | ?n )iQn = N ?n+1
hW i0Qn L0 ??1 B, ?n+1
z
z
z
w
w
?1
0 0 ?1
with ?n+1
=
W
L
?
LW
+
I
,
z
Qn
w
n+1
n+1
n
Qw (W ) ? exp hlog P (B, Z, W | ? )iQn = N ?n+1
vec L0 ??1 B hZ 0 iQn+1
,
?
;
w
w
z
z
?1
with ?n+1
= hZZ 0 iQn+1
? L0 ??1 L + I ? An
,
w
z
h
i?1
and ?vn = dz J 0 diag hW iQn+1
hW i0Qn+1
(J 0 1)v ? ?v (J 0 diag ?n+1
)v .
w
w
w
v
where the normal distribution on Z implies a normal distribution on each column z; the
distribution on W is normal on vec (W ) 2 ; 1 is a vector of ones; and the diag [?] operator
returns the main diagonal of its argument as a vector.
Our experience is that better results can be obtained if the posterior expectation of ZZ 0 in
the Qw update is replaced by its value under the prior on Z, N I. This variant appears to
constrain the factored posterior to remain closer to the true joint distribution. It has the
additional benefit of simplifying both the notational and computational complexities of the
updates (for the latter, it reduces the complexity of the inversion needed to calculate ?w
from (ds dz )3 to d2s ). We can then rewrite the updates into a more compact form by using
this assumption, and by evaluating the expectations, to obtain
?n+1
= (W n 0 L0 ??1 LW n + Tr [L0 ??1 L0 ?nw ] I + I)
z
?1
?n+1
w
0
?1
n
?1
= (N L ? L + A )
n
?1
= (A )
n
?1
0
? (A ) L (N
(6a)
?1
n
?1
n
? + L(A ) L ) L(A )?1
(6b)
0
?1
0
?1
n n+1
W n+1 = ?n+1
w L ? BB ? LW ?z
0 ?1
?vn+1 = dz J 0 diag W n+1 W n+1 v ((J 0 1)v ? ?vn (J 0 diag ?n+1
)v ),
w
0 ?1
(6c)
(6d)
where W n = hW iQnw . The use of the matrix inversion lemma in (6b) exploits the diagonality of A to reduce the computational complexity of the algorithm with respect to ds .
The formulae of (6) are easily implemented and recover an estimate of W , and thus the
source correlation matrix, by iteration. The source activities can then be estimated by use
of the Wiener filter (2). The updates of (6) also demonstrate an important point concerning
the validity of our Gaussian model. Note that the measured data enter into the estimation
procedure only through their correlation BB 0 . In other words, the hyperparameter optimisation stage of our algorithm is only being used to model the data correlation, not their
amplitudes. As a result, the effects of incorrectly assuming a Gaussian source amplitude
distribution can be expected to remain relatively benign.
4
Simulations
Simulation studies provide an important tool for evaluating source recovery algorithms, in
that they provide ?sensor? data sets for which the correct answer (i.e. the true locations and
time-courses of the sources) is known. We report here the results of simulations carried out
using parameters similar to those that might be encountered in realistic recordings.
4.1
Methods
We simulated 100 1-s-long epochs of evoked response data. The sensor configuration was
taken from a real experiment: two sensor arrays, with 37 gradiometer coils each, were
2
for a discussion of the vec operator and the Kronecker product ? see e.g. [14]
located on either side of the head (see figure 1). Candidate source dipoles were located on
a grid with 1 cm spacing within a hemispherical brain volume with a radius of 8 cm, to give
a total of 956 possible source locations. Significant (above background) evoked activity
was simulated at 5 of these locations (see figure 1a), with random dipole orientations. The
evoked waveforms were similar in form to the evoked responses seen in many areas of the
brain (see figure 2a), and were strongly correlated between the five sites (figure 3a). The
two most lateral sites (one on each side), expressed bilateral primary sensory activation,
and had identical time-courses with the shortest latency. Another lateral site, on the left
side, had activity with the same waveform, but delayed by 50 ms. Two medial sites had
slower and more delayed activation profiles. The dipole orientation at each site was chosen
randomly in the plane parallel to the sensor tangent. Note that the amplitude distribution
of these sources is strongly non-Gaussian; we will see, however, that they can be recovered
successfully by the present technique despite its assumption of normality.
The simulated sensor recordings were corrupted by noise from two sources, both with
Gaussian distribution. Background activity in the brain was simulated with equal power
at every point on the grid of candidate sources, with a root-mean-square (RMS) amplitude
1.5 decades below that of the 5 significant sources. Although this background activity was
uncorrelated between brain locations, it resulted in correlated disturbances at the magnetic
sensors. Thermal noise in the sensors was uncorrelated, and had a similar magnitude (at
the sensors) to that of the background noise.
The novel Bayesian estimation technique was applied to the raw simulated sensor trace
rather than to epoch-averaged data. While in this simulation the evoked activity was identical in each trial, determining the correlation matrix from unaveraged data should, in the
more general case, make single-trial reconstructions more accurate. Once reconstructed,
the source timecourses were averaged, and are shown in figure 2. The number of presources dz , a free parameter in the algorithm, was set to 10. Sources associated with inverse
variance hyperparameters ?i above a threshold (here 1015 ) were taken to be inactive.
For comparison, we also reconstructed sources using the vector minimum-variance adaptive beamformer approach [15]. Note that this technique, along with many other existing
reconstruction methods, assumes that sources at different locations are uncorrelated and so
it should not be expected to perform well under the conditions of our simulation.
4.2
Results
Figure 1 shows the source locations and powers reconstructed by the novel Bayesian approach developed here (b) and by the beamformer (c). The Bayesian approach identified
the correct number of sources, at the correct locations and with approximately correct relative powers. By contrast, the beamformer approach, which assumes uncorrelated sources,
entirely failed to locate the sources of activity.
Figure 2b shows the average evoked-response reconstruction at each of the identified source
locations (with the simulated waveforms shown in panel a). The general time-course of the
activities has clearly been well characterised. The time-courses estimated by the vector
beamformer are shown in figure 2c. As beamformer localisation proved to be unreliable,
the time-courses shown are the reconstructions at the positions of the correct (simulated)
sources. Nonetheless, the strong correlations in the sources have corrupted the reconstructions. Note that the only difference between the time-courses shown in figure 2b and c is
premultiplication by the estimated source correlation matrix in b.
Finally, figure 3 shows the correlation coefficient matrices for the dipole amplitude timecourses of the active sources shown in figure 2. We see that the Bayesian approach finds a
reasonable approximation to the correct correlation structure. Again, however, the beamformer is unable to accurately characterise the correlation matrix.
Figure 1: Reconstructed source power. Each dot represents a single voxel, the size and shade of the
superimposed circles indicates the relative power of the corresponding source. Each column contains
two orthogonal projections of the same source distribution: (a) simulated sources, (b) reconstruction
by evidence optimisation, (c) beamformer reconstruction (powers have been compressed to make
smaller sources more visible)
source number
a
5
4
3
2
1
0
100
200
300
400
500
600
time within epoch (ms)
700
800
900
1000
0
100
200
300
400
500
600
time within epoch (ms)
700
800
900
1000
0
100
200
300
400
500
600
time within epoch (ms)
700
800
900
1000
source number
b
5
4
3
2
1
source number
c
5
4
3
2
1
Figure 2: Source waveforms at active locations. Sources are numbered from left to right in the brain.
The two traces for each location show the dipole components in two orthogonal directions. (a) simulated waveforms; (b) waveforms reconstructed by our novel algorithm; (c) waveforms reconstructed
by beamforming (at the simulated locations)
simulated correlations
b
reconstructed correlations
c
1
1
2
2
2
3
4
source
1
source
source
a
3
4
5
2
3
4
source
5
3
4
5
1
beamformer correlations
5
1
2
3
4
source
5
1
2
3
4
source
5
Figure 3: Source correlation coefficient matrices. Correlations were computed between epochaveraged dipole amplitude time-courses at each location. The size of each square indicates the magnitude of the corresponding coefficient (the maximum value being 1), with whites squares positive
and black squares negative. (a) simulated sources; (b) sources reconstructed by our novel algorithm;
(c) sources reconstructed by beamforming.
5
Conclusions
We have demonstrated a novel evidence-optimisation approach to the location and reconstruction of dipole sources contributing to MEG measurements. Unlike existing methods,
this new technique does not assume a correlation structure for the sources, instead estimating it from the data. As such, this approach holds great promise for high fidelity imaging
of correlated magnetic activity in the brain.
Acknowledgements
We thank Dr. Sekihara for useful discussions. This work is funded by grants from the
Whitaker Foundation and from NIH (1R01004855-01A1).
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1,608 | 2,461 | Out-of-Sample Extensions for LLE, Isomap,
MDS, Eigenmaps, and Spectral Clustering
Yoshua Bengio, Jean-Franc?ois Paiement, Pascal Vincent
Olivier Delalleau, Nicolas Le Roux and Marie Ouimet
D?epartement d?Informatique et Recherche Op?erationnelle
Universit?e de Montr?eal
Montr?eal, Qu?ebec, Canada, H3C 3J7
{bengioy,vincentp,paiemeje,delallea,lerouxni,ouimema}
@iro.umontreal.ca
Abstract
Several unsupervised learning algorithms based on an eigendecomposition provide either an embedding or a clustering only for given training points, with no straightforward extension for out-of-sample examples
short of recomputing eigenvectors. This paper provides a unified framework for extending Local Linear Embedding (LLE), Isomap, Laplacian
Eigenmaps, Multi-Dimensional Scaling (for dimensionality reduction)
as well as for Spectral Clustering. This framework is based on seeing
these algorithms as learning eigenfunctions of a data-dependent kernel.
Numerical experiments show that the generalizations performed have a
level of error comparable to the variability of the embedding algorithms
due to the choice of training data.
1
Introduction
Many unsupervised learning algorithms have been recently proposed, all using an eigendecomposition for obtaining a lower-dimensional embedding of data lying on a non-linear
manifold: Local Linear Embedding (LLE) (Roweis and Saul, 2000), Isomap (Tenenbaum,
de Silva and Langford, 2000) and Laplacian Eigenmaps (Belkin and Niyogi, 2003). There
are also many variants of Spectral Clustering (Weiss, 1999; Ng, Jordan and Weiss, 2002), in
which such an embedding is an intermediate step before obtaining a clustering of the data
that can capture flat, elongated and even curved clusters. The two tasks (manifold learning
and clustering) are linked because the clusters found by spectral clustering can be arbitrary
curved manifolds (as long as there is enough data to locally capture their curvature).
2
Common Framework
In this paper we consider five types of unsupervised learning algorithms that can be cast
in the same framework, based on the computation of an embedding for the training points
obtained from the principal eigenvectors of a symmetric matrix.
Algorithm 1
1. Start from a data set D = {x1 , . . . , xn } with n points in Rd . Construct a n ? n
?neighborhood? or similarity matrix M . Let us denote KD (?, ?) (or K for shorthand) the
data-dependent function which produces M by Mij = KD (xi , xj ).
? . Equivalently, this corre2. Optionally transform M , yielding a ?normalized? matrix M
?
?
?
?
sponds to generating M from a KD by Mij = KD (xi , xj ).
?.
3. Compute the m largest positive eigenvalues ?k and eigenvectors vk of M
4. The embedding of each example xi is the vector yi with yik the i-th element of the k -th
?
principal
? eigenvector vk of M . Alternatively (MDS and Isomap), the embedding is ei , with
eik = ?k yik . If the first m eigenvalues are positive, then ei ? ej is the best approximation
? ij using only m coordinates, in the squared error sense.
of M
In the following, we consider the specializations of Algorithm 1 for different unsupervised
learning algorithms. Let Si be the i-th row sum of the affinity matrix M :
X
Si =
Mij .
(1)
j
We say that two points (a, b) are k-nearest-neighbors of each other if a is among the k
nearest neighbors of b in D ? {a} or vice-versa. We denote by xij the j-th coordinate of
the vector xi .
2.1 Multi-Dimensional Scaling
Multi-Dimensional Scaling (MDS) starts from a notion of distance or affinity K that is
computed between each pair of training examples. We consider here metric MDS (Cox
and Cox, 1994). For the normalization step 2 in Algorithm 1, these distances are converted
to equivalent dot products using the ?double-centering? formula: !
X
? ij = ? 1 Mij ? 1 Si ? 1 Sj + 1
Sk .
(2)
M
2
2
n
n
n
k
?
The embedding eik of example xi is given by ?k vki .
2.2 Spectral Clustering
Spectral clustering (Weiss, 1999) can yield impressively good results where traditional
clustering looking for ?round blobs? in the data, such as K-means, would fail miserably. It
is based on two main steps: first embedding the data points in a space in which clusters are
more ?obvious? (using the eigenvectors of a Gram matrix), and then applying a classical
clustering algorithm such as K-means, e.g. as in (Ng, Jordan and Weiss, 2002). The affinity
matrix M is formed using a kernel such as the Gaussian kernel. Several normalization steps
have been proposed. Among the most successful ones, as advocated in (Weiss, 1999; Ng,
Jordan and Weiss, 2002), is the following:
? ij = pMij .
M
(3)
Si Sj
? are computed and K-means
To obtain m clusters, the first m principal eigenvectors of M
is applied on the unit-norm coordinates, obtained from the embedding yik = vki .
2.3 Laplacian Eigenmaps
Laplacian Eigenmaps is a recently proposed dimensionality reduction procedure (Belkin
and Niyogi, 2003) that has been proposed for semi-supervised learning. The authors use
an approximation of the Laplacian operator such as the Gaussian kernel or the matrix whose
element (i, j) is 1 if xi and xj are k-nearest-neighbors and 0 otherwise. Instead of solving
an ordinary eigenproblem, the following generalized eigenproblem is solved:
(S ? M )vj = ?j Svj
(4)
with eigenvalues ?j , eigenvectors vj and S the diagonal matrix with entries given by eq. (1).
The smallest eigenvalue is left out and the eigenvectors corresponding to the other small
eigenvalues are used for the embedding. This is the same embedding that is computed
with the spectral clustering algorithm from (Shi and Malik, 1997). As noted in (Weiss,
1999) (Normalization Lemma 1), an equivalent result (up to a componentwise scaling of
the embedding) can be obtained by considering the principal eigenvectors of the normalized
matrix defined in eq. (3).
2.4 Isomap
Isomap (Tenenbaum, de Silva and Langford, 2000) generalizes MDS to non-linear manifolds. It is based on replacing the Euclidean distance by an approximation of the geodesic
distance on the manifold. We define the geodesic distance with respect to a data set D, a
distance d(u, v) and a neighborhood k as follows:
X
? b) = min
D(a,
d(pi , pi+1 )
(5)
p
i
where p is a sequence of points of length l ? 2 with p1 = a, pl = b, pi ? D ?i ?
{2, . . . , l ? 1} and (pi ,pi+1 ) are k-nearest-neighbors. The length l is free in the minimiza? from which the embedding
tion. The Isomap algorithm obtains the normalized matrix M
is derived by transforming the raw pairwise distances matrix as follows: first compute the
? 2 (xi , xj ) of squared geodesic distances with respect to the data D, then
matrix Mij = D
apply to this matrix the distance-to-dot-product
transformation (eq. (2)), as for MDS. As in
?
MDS, the embedding is eik = ?k vki rather than yik = vki .
2.5 LLE
The Local Linear Embedding (LLE) algorithm (Roweis and Saul, 2000) looks for an embedding that preserves the local geometry in the neighborhood of each
P data point. First, a
sparse matrix of local predictive weights Wij is computed, such that j Wij = 1, Wij = 0
P
if xj is not a k-nearest-neighbor of xi and ( j Wij xj ?xi )2 is minimized. Then the matrix
M = (I ? W )0 (I ? W )
(6)
is formed. The embedding is obtained from the lowest eigenvectors of M , except for the
smallest eigenvector which is uninteresting because it is (1, 1, . . . 1), with eigenvalue 0.
Note that the lowest eigenvectors of M are the largest eigenvectors of M?? = ?I ? M to
fit Algorithm 1 (the use of ? > 0 will be discussed in section 4.4). The embedding is given
by yik = vki , and is constant with respect to ?.
3
From Eigenvectors to Eigenfunctions
To obtain an embedding for a new data point, we propose to use the Nystro? m formula (eq. 9)
(Baker, 1977), which has been used successfully to speed-up kernel methods computations
by focussing the heavier computations (the eigendecomposition) on a subset of examples.
The use of this formula can be justified by considering the convergence of eigenvectors
and eigenvalues, as the number of examples increases (Baker, 1977; Williams and Seeger,
2000; Koltchinskii and Gin?e, 2000; Shawe-Taylor and Williams, 2003). Intuitively, the
extensions to obtain the embedding for a new example require specifying a new column of
? , through a training-set dependent kernel function K
? D , in which one of
the Gram matrix M
the arguments may be required to be in the training set.
If we start from a data set D, obtain an embedding for its elements, and add more and
more data, the embedding for the points in D converges (for eigenvalues that are unique).
(Shawe-Taylor and Williams, 2003) give bounds on the convergence error (in the case of
kernel PCA). In the limit, we expect each eigenvector to converge to an eigenfunction for
the linear operator defined below, in the sense that the i-th element of the k-th eigenvector
converges to the application of the k-th eigenfunction to xi (up to a normalization factor).
R
Consider a Hilbert space Hp of functions with inner product hf, gip = f (x)g(x)p(x)dx,
with a density function p(x). Associate with
Z kernel K a linear operator Kp in Hp :
(Kp f )(x) =
K(x, y)f (y)p(y)dy.
(7)
We don?t know the true density p but we can approximate the above inner product and
linear operator (and its eigenfunctions) using the empirical distribution p?. An ?empirical?
Hilbert space Hp? is thus defined using p? instead of p. Note that the proposition below can be
applied even if the kernel is not positive semi-definite, although the embedding algorithms
we have studied are restricted to using the principal coordinates associated with positive
eigenvalues. For a more rigorous mathematical analysis, see (Bengio et al., 2003).
Proposition 1
?
Let K(a,
b) be a kernel function, not necessarily positive semi-definite, that gives rise to
? with entries M
? ij = K(x
? i , xj ) upon a dataset D = {x1 , . . . , xn }.
a symmetric matrix M
? vk = ?k vk . Let (fk , ?0 )
Let (vk , ?k ) be an (eigenvector,eigenvalue) pair that solves M
k
? p?fk )(x) = ?0 fk (x) for any x, with p?
be an (eigenfunction,eigenvalue) pair that solves (K
k
?
the empirical distribution over D. Let ek (x) = yk (x) ?k or yk (x) denote the embedding
associated with a new point x. Then
?0k
=
fk (x)
=
fk (xi )
=
yk (x)
=
yk (xi )
1
?k
n
? X
n
n
?
vki K(x,
xi )
?k i=1
?
nvki
n
fk (x)
1 X
?
? =
vki K(x,
xi )
?k i=1
n
= yik ,
ek (xi ) = eik
(8)
(9)
(10)
(11)
(12)
See (Bengio et al., 2003) for a proof and further justifications
? of the above formulae. The
generalized embedding for Isomap and MDS is ek (x) = ?k yk (x) whereas the one for
spectral clustering, Laplacian eigenmaps and LLE is yk (x).
Proposition 2
? D is positive semi-definite, then
In addition, if the data-dependent kernel K
r
n
fk (x) =
?k (x)
?k
where ?k (x) is the k-th component of the kernel PCA projection of x obtained from the
? D (up to centering).
kernel K
This relation with kernel PCA (Sch?olkopf, Smola and M?uller, 1998), already pointed out
in (Williams and Seeger, 2000), is further discussed in (Bengio et al., 2003).
4
Extending to new Points
Using Proposition 1, one obtains a natural extension of all the unsupervised learning algo? that gives
rithms mapped to Algorithm 1, provided we can write down a kernel function K
? on D, and can be used in eq. (11) to generalize the embedding. We
rise to the matrix M
consider each of them in turn below. In addition to the convergence properties discussed in
section 3, another justification for using equation (9) is given by the following proposition:
Proposition 3
If we define the fk (xi ) by eq. (10) and take a new point x, the value of fk (x) that minimizes
!2
n
m
X
X
0
?
K(x,
xi ) ?
? ft (x)ft (xi )
(13)
t
i=1
t=1
is given by eq. (9), for m ? 1 and any k ? m.
The proof is a direct consequence of the orthogonality of the eigenvectors v k . This proposition links equations (9) and (10). Indeed, we can obtain eq. (10) when trying to approximate
? at the data points by minimizing the cost
K
!2
n
m
X
X
0
? i , xj ) ?
K(x
?t ft (xi )ft (xj )
i,j=1
t=1
for m = 1, 2, . . . When we add a new point x, it is thus natural to use the same cost to
?
approximate the K(x,
xi ), which yields (13). Note that by doing so, we do not seek to
?
approximate K(x, x). Future work should investigate embeddings which minimize the
? but ignore the diagonal contributions.
empirical reconstruction error of K
4.1 Extending MDS
For MDS, a normalized kernel can be defined as follows, using a continuous version of the
double-centering eq. (2):
1
?
(14)
K(a,
b) = ? (d2 (a, b) ? Ex [d2 (x, b)] ? Ex0 [d2 (a, x0 )] + Ex,x0 [d2 (x, x0 )])
2
where d(a, b) is the original distance and the expectations are taken over the empirical data
D. An extension of metric MDS to new points has already been proposed in (Gower, 1968),
solving exactly for the embedding of x to be consistent with its distances to training points,
which in general requires adding a new dimension.
4.2 Extending Spectral Clustering and Laplacian Eigenmaps
Both the version of Spectral Clustering and Laplacian Eigenmaps described above are
based on an initial kernel K, such as the Gaussian or nearest-neighbor kernel. An equivalent normalized kernel is:
1
K(a, b)
?
K(a,
b) = p
n Ex [K(a, x)]Ex0 [K(b, x0 )]
where the expectations are taken over the empirical data D.
4.3 Extending Isomap
To extend Isomap, the test point is not used in computing the geodesic distance between
training points, otherwise we would have to recompute all the geodesic distances. A rea? b) in eq. (5), which only uses the training
sonable solution is to use the definition of D(a,
points in the intermediate points on the path from a to b. We obtain a normalized kernel by
?
applying the continuous double-centering of eq. (14) with d = D.
A formula has already been proposed (de Silva and Tenenbaum, 2003) to approximate
Isomap using only a subset of the examples (the ?landmark? points) to compute the eigenvectors. Using our notations, this formula is
1 X
? 2 (x0 , xi )] ? D
? 2 (xi , x)).
e0k (x) = ?
vki (Ex0 [D
(15)
2 ?k i
where Ex0 is an average over the data set. The formula is applied to obtain an embedding
for the non-landmark examples.
Corollary 1
The embedding proposed in Proposition 1 for Isomap (ek (x)) is equal to formula 15 (Land?
?
mark Isomap) when K(x,
y) is defined as in eq. (14) with d = D.
P
Proof: the proof relies on a property of the Gram matrix for Isomap: i Mij = 0, by construction. Therefore (1, 1, . . . 1)
Pis an eigenvector with eigenvalue 0, and all the other eigenvectors vk have the property i vki = 0 because of the orthogonality with (1, 1, . . . 1).
? 2 (x0 , xi )]? D
? 2 (x, xi )) = 2K(x,
?
? 2 (x0 , x00 )]?Ex0 [D
? 2 (x, x0 )]
Writing (Ex0 [D
xi )+Ex0 ,x00 [D
P
P
2
0
2
0
00
2
0
?
?
?
yields ek (x) = 2??
i vki K(x, xi ) + (Ex0 ,x00 [D (x , x )] ? Ex0 [D (x, x )])
i vki =
k
ek (x), since the last sum is 0.
4.4
Extending LLE
The extension of LLE is the most challenging one because it does not fit as well the framework of Algorithm 1: the M matrix for LLE does not have a clear interpretation in terms
of distance or dot product. An extension has been proposed in (Saul and Roweis, 2002),
but unfortunately it cannot be cast directly into the framework of Proposition 1. Their
embedding of a new point x is given by
n
X
yk (xi )w(x, xi )
(16)
yk (x) =
i=1
where w(x, xi ) is the weight of xi in the reconstruction of x by its k-nearest-neighbors in
the training set (if x = xj ? D, w(x, xi ) = ?ij ). This is very close to eq. (11), but lacks the
normalization by ?k . However, we can see this embedding as a limit case of Proposition 1,
as shown below.
? ? such that
We first need to define a kernel K
X
? ? (xi , xj ) = M
? ?,ij = (? ? 1)?ij + Wij + Wji ?
K
Wki Wkj
(17)
k
?0
for xi , xj ? D. Let us define a kernel K by
? 0 (xi , x) = K
? 0 (x, xi ) = w(x, xi )
K
? 0 (x, y) = 0 when neither x nor y is in the training set D. Let K
? 00 be defined by
and K
X
? 00 (xi , xj ) = Wij + Wji ?
K
Wki Wkj
k
? 00 (x, y) = 0 when either x or y isn?t in D. Then, by construction, the kernel K
?? =
and K
0
00
?
?
(? ? 1)K + K verifies eq. (17). Thus, we can apply eq. (11) to obtain an embedding of
a new point x, which yields
1 X
? 0 (x, xi ) + K
? 00 (x, xi )
y?,k (x) =
yik (? ? 1)K
?k i
? k ), and ?
? k being the k-th lowest eigenvalue of M . This rewrites into
with ?k = (? ? ?
??1 X
1 X
? 00 (x, xi ).
y?,k (x) =
yik w(x, xi ) +
yik K
?k
?k
???
???
i
i
Then when ? ? ?, y?,k (x) ? yk (x) defined by eq. (16).
Since the choice of ? is free, we can thus consider eq. (16) as approximating the use of the
? ? with a large ? in Proposition 1. This is what we have done in the experiments
kernel K
? ? verifying
described in the next section. Note however that we can find smoother kernels K
eq. (17), giving other extensions of LLE from Proposition 1. It is out of the scope of this
paper to study which kernel is best for generalization, but it seems desirable to use a smooth
kernel that would take into account not only the reconstruction of x by its neighbors x i , but
also the reconstruction of the xi by their neighbors including the new point x.
5
Experiments
We want to evaluate whether the precision of the generalizations suggested in the previous section is comparable to the intrinsic perturbations of the embedding algorithms.
The perturbation analysis will be achieved by considering splits of the data in three sets,
D = F ? R1 ? R2 and training either with F ? R1 or F ? R2 , comparing the embeddings
on F . For each algorithm described in section 2, we apply the following procedure:
10
x 10
4
x 10
4
20
8
6
15
4
10
2
5
0
0
-2
-4
7
0
x 10
0.05
0.1
0.15
0.2
-5
0.25
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
3
1
6
0.8
5
0.6
4
3
0.4
2
0.2
1
0
0
-1
-0. 2
-2
-3
0
0.02
0.04
0.06
0.08
0.1
0.12
0.14
-0. 4
0
0.05
0.1
0.15
0.2
0.25
Figure 1: Training set variability minus out-of-sample error, wrt the proportion of training
samples substituted. Top left: MDS. Top right: spectral clustering or Laplacian eigenmaps.
Bottom left: Isomap. Bottom right: LLE. Error bars are 95% confidence intervals.
1. We choose F ? D with m = |F | samples. The remaining n ? m samples in D/F
are split into two equal size subsets R1 and R2 . We train (obtain the eigenvectors)
over F ? R1 and F ? R2 . When eigenvalues are close, the estimated eigenvectors
are unstable and can rotate in the subspace they span. Thus we estimate an affine
alignment between the two embeddings using the points in F , and we calculate
the Euclidean distance between the aligned embeddings obtained for each s i ? F .
2. For each sample si ? F , we also train over {F ? R1 }/{si }. We apply the extension to out-of-sample points to find the predicted embedding of si and calculate
the Euclidean distance between this embedding and the one obtained when training with F ? R1 , i.e. with si in the training set.
3. We calculate the mean difference (and its standard error, shown in the figure)
between the distance obtained in step 1 and the one obtained in step 2 for each
sample si ? F , and we repeat this experiment for various sizes of F .
The results obtained for MDS, Isomap, spectral clustering and LLE are shown in figure 1
for different values of m. Experiments are done over a database of 698 synthetic face images described by 4096 components that is available at http://isomap.stanford.edu.
Qualitatively similar results have been obtained over other databases such as
Ionosphere (http://www.ics.uci.edu/?mlearn/MLSummary.html) and swissroll
(http://www.cs.toronto.edu/?roweis/lle/). Each algorithm generates a twodimensional embedding of the images, following the experiments reported for Isomap.
The number of neighbors is 10 for Isomap and LLE, and a Gaussian kernel with a standard
deviation of 0.01 is used for spectral clustering / Laplacian eigenmaps. 95% confidence
intervals are drawn beside each mean difference of error on the figure.
As expected, the mean difference between the two distances is almost monotonically increasing as the fraction of substituted examples grows (x-axis in the figure). In most cases,
the out-of-sample error is less than or comparable to the training set embedding stability:
it corresponds to substituting a fraction of between 1 and 4% of the training examples.
6
Conclusions
In this paper we have presented an extension to five unsupervised learning algorithms
based on a spectral embedding of the data: MDS, spectral clustering, Laplacian eigenmaps, Isomap and LLE. This extension allows one to apply a trained model to out-ofsample points without having to recompute eigenvectors. It introduces a notion of function
induction and generalization error for these algorithms. The experiments on real highdimensional data show that the average distance between the out-of-sample and in-sample
embeddings is comparable or lower than the variation in in-sample embedding due to replacing a few points in the training set.
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1,609 | 2,462 | Self-calibrating Probability Forecasting
Vladimir Vovk
Computer Learning Research Centre
Department of Computer Science
Royal Holloway, University of London
Egham, Surrey TW20 0EX, UK
[email protected]
Glenn Shafer
Rutgers School of Business
Newark and New Brunswick
180 University Avenue
Newark, NJ 07102, USA
[email protected]
Ilia Nouretdinov
Computer Learning Research Centre
Department of Computer Science
Royal Holloway, University of London
Egham, Surrey TW20 0EX, UK
[email protected]
Abstract
In the problem of probability forecasting the learner?s goal is to output,
given a training set and a new object, a suitable probability measure on
the possible values of the new object?s label. An on-line algorithm for
probability forecasting is said to be well-calibrated if the probabilities
it outputs agree with the observed frequencies. We give a natural nonasymptotic formalization of the notion of well-calibratedness, which we
then study under the assumption of randomness (the object/label pairs
are independent and identically distributed). It turns out that, although
no probability forecasting algorithm is automatically well-calibrated in
our sense, there exists a wide class of algorithms for ?multiprobability
forecasting? (such algorithms are allowed to output a set, ideally very
narrow, of probability measures) which satisfy this property; we call the
algorithms in this class ?Venn probability machines?. Our experimental
results demonstrate that a 1-Nearest Neighbor Venn probability machine
performs reasonably well on a standard benchmark data set, and one of
our theoretical results asserts that a simple Venn probability machine
asymptotically approaches the true conditional probabilities regardless,
and without knowledge, of the true probability measure generating the
examples.
1 Introduction
We are interested in the on-line version of the problem of probability forecasting: we observe pairs of objects and labels sequentially, and after observing the nth object xn the
goal is to give a probability measure pn for its label; as soon as pn is output, the label
yn of xn is disclosed and can be used for computing future probability forecasts. A good
review of early work in this area is Dawid [1]. In this introductory section we will assume
that yn ? {0, 1}; we can then take pn to be a real number from the interval [0, 1] (the
probability that yn = 1 given xn ); our exposition here will be very informal.
The standard view ( [1], pp. 213?216) is that the quality of probability forecasting systems
has two components: ?reliability? and ?resolution?. At the crudest level, reliability requires
that the forecasting system should not lie, and resolution requires that it should say something useful. To be slightly more precise, consider the first n forecasts pi and the actual
labels yi .
The P
most basic test is to compare the overall average forecast
Pn probability pn :=
n
n?1 i=1 pi with the overall relative frequency y n := n?1 i=1 yi of 1s among yi .
If pn ? y n , the forecasts are ?unbiased in the large?.
A more refined test would look at the subset of i for which pi is close to a given value p? ,
and compare the relative frequency of yi = 1 in this subset, say y n (p? ), with p? . If
y n (p? ) ? p? for all p? ,
(1)
the forecasts are ?unbiased in the small?, ?reliable?, ?valid?, or ?well-calibrated?; in later
sections, we will use ?well-calibrated?, or just ?calibrated?, as a technical term. Forecasting
systems that pass this test at least get the frequencies right; in this sense they do not lie.
It is easy to see that there are reliable forecasting systems that are virtually useless. For
example, the definition of reliability does not require that the forecasting system pay any
attention to the objects xi . In another popular example, the labels follow the pattern
?
1 if i is odd
yi =
0 otherwise.
The forecasts pi = 0.5 are reliable, at least asymptotically (0.5 is the right relative frequency) but not as useful as p1 = 1, p2 = 0, . . . ; the ?resolution? (which we do not define
here) of the latter forecasts is better.
In this paper we construct forecasting systems that are automatically reliable. To achieve
this, we allow our prediction algorithms to output sets of probability measures Pn instead
of single measures pn ; typically the sets Pn will be small (see ?5).
This paper develops the approach of [2?4], which show that it is possible to produce valid,
asymptotically optimal, and practically useful p-values; the p-values can be then used for
region prediction. Disadvantages of p-values, however, are that their interpretation is less
direct than that of probabilities and that they are easy to confuse with probabilities; some
authors have even objected to any use of p-values (see, e.g., [5]). In this paper we use the
methodology developed in the previous papers to produce valid probabilities rather than
p-values.
All proofs are omitted and can be found in [6].
2 Probability forecasting and calibration
From this section we start rigorous exposition. Let P(Y) be the set of all probability
measures on a measurable space Y. We use the following protocol in this paper:
M ULTIPROBABILITY FORECASTING
Players: Reality, Forecaster
Protocol:
FOR n = 1, 2, . . . :
Reality announces xn ? X.
Forecaster announces Pn ? P(Y).
Reality announces yn ? Y.
In this protocol, Reality generates examples zn = (xn , yn ) ? Z := X ? Y consisting
of two parts, objects xn and labels yn . After seeing the object xn Forecaster is required
to output a prediction for the label yn . The usual probability forecasting protocol requires
that Forecaster output a probability measure; we relax this requirement by allowing him to
output a family of probability measures (and we are interested in the case where the families
Pn become smaller and smaller as n grows). It can be shown (we omit the proof and even
the precise statement) that it is impossible to achieve automatic well-calibratedness, in our
finitary sense, in the probability forecasting protocol.
In this paper we make the simplifying assumption that the label space Y is finite; in many
informal explanations it will be assumed binary, Y = {0, 1}. To avoid unnecessary technicalities, we will also assume that the families Pn chosen by Forecaster are finite and have
no more than K elements; they will be represented by a list of length K (elements in the list
can repeat). A probability machine is a measurable strategy for Forecaster in our protocol,
where at each step he is required to output a sequence of K probability measures.
The problem of calibration is usually treated in an asymptotic framework. Typical asymptotic results, however, do not say anything about finite data sequences; therefore, in this
paper we will only be interested in the non-asymptotic notion of calibration. All needed
formal definitions will be given, but space limitations prevent us from including detailed
explanations and examples, which can be found in [6].
Let us first limit the duration of the game, replacing n = 1, 2, . . . in the multiprobability
forecasting protocol by n = 1, . . . , N for a finite horizon N . It is clear that, regardless of
formalization, we cannot guarantee that miscalibration, in the sense of (1) being violated,
will never happen: for typical probability measures, everything can happen, perhaps with a
small probability. The idea of our definition is: a prediction algorithm is well-calibrated if
any evidence of miscalibration translates into evidence against the assumption of randomness. Therefore, we first need to define ways of testing calibration and randomness; this
will be done following [7].
A game N -martingale is a function M on sequences of the form x1 , p1 , y1 , . . . , xn , pn , yn ,
where n = 0, . . . , N , xi ? X, pi ? P(Y), and yi ? Y, that satisfies
Z
M (x1 , p1 , y1 , . . . , xn?1 , pn?1 , yn?1 ) =
M (x1 , p1 , y1 , . . . , xn , pn , y)pn (dy)
Y
for all x1 , p1 , y1 , . . . , xn , pn , n = 1, . . . , N . A calibration N -martingale is a nonnegative
game N -martingale that is invariant under permutations:
M (x1 , p1 , y1 , . . . , xN , pN , yN ) = M (x?(1) , p?(1) , y?(1) , . . . , x?(N ) , p?(N ) , y?(N ) )
for any x1 , p1 , y1 , . . . , xN , pN , yN and any permutation ? : {1, . . . , N } ? {1, . . . , N }.
To cover the multiprobability forecasting protocol, we extend the domain of definition for
a calibration N -martingale M from sequences of the form x1 , p1 , y1 , . . . , xn , pn , yn ,
where p1 , . . . , pn are single probability measures on Y, to sequences of the form
x1 , P1 , y1 , . . . , xn , Pn , yn , where P1 , . . . , Pn are sets of probability measures on Y,
by
M (x1 , P1 , y1 , . . . , xn , Pn , yn ) :=
inf
p1 ?P1 ,...,pn ?Pn
M (x1 , p1 , y1 , . . . , xn , pn , yn ).
A QN -martingale, where Q is a probability measure on Z, is a function S on sequences of
the form x1 , y1 , . . . , xn , yn , where n = 0, . . . , N , xi ? X, and yi ? Y, that satisfies
Z
S(x1 , y1 , . . . , xn?1 , yn?1 ) =
S(x1 , y1 , . . . , xn?1 , yn?1 , x, y)Q(dx, dy)
Z
for all x1 , y1 , . . . , xn?1 , yn?1 , n = 1, . . . , N .
If a nonnegative QN -martingale S starts with S(?) = 1 and ends with S(x1 , y1 , . . . , yN )
very large, then we may reject Q as the probability measure generating individual examples
(xn , yn ). This interpretation is supported by Doob?s inequality. Analogously, if a game N martingale M starts with M (?) = 1 and ends with M (x1 , P1 , y1 , . . . , yN ) very large, then
we may reject the hypothesis that each Pn contains the true probability measure for yn . If
M is a calibration N -martingale, this event is interpreted as evidence of miscalibration.
(The restriction to calibration N -martingales is motivated by the fact that (1) is invariant
under permutations).
We call a probability machine F N -calibrated if for any probability measure Q on Z
and any nonnegative calibration N -martingale M with M (?) = 1, there exists a QN martingale S with S(?) = 1 such that
M (x1 , F (x1 ), y1 , . . . , xN , F (x1 , y1 , . . . , xN ), yN ) ? S(x1 , y1 , . . . , xN , yN )
for all x1 , y1 , . . . , xN , yN . We say that F is finitarily calibrated if it is N -calibrated for
each N .
3 Self-calibrating probability forecasting
Now we will describe a general algorithm for multiprobability forecasting. Let N be the
sets of all positive integer numbers. A sequence of measurable functions An : Zn ? Nn ,
n = 1, 2, . . . , is called a taxonomy if, for any n ? N, any permutation ? of {1, . . . , n}, any
(z1 , . . . , zn ) ? Zn , and any (?1 , . . . , ?n ) ? Nn ,
(?1 , . . . , ?n ) = An (z1 , . . . , zn ) =? (??(1) , . . . , ??(n) ) = An (z?(1) , . . . , z?(n) ).
In other words,
An : (z1 , . . . , zn ) 7? (?1 , . . . , ?n )
(2)
is a taxonomy if every ?i is determined by the bag *z1 , . . . , zn + and zi . We let |B| stand
for the number of elements in a set B. The Venn probability machine associated with (An )
is the probability machine which outputs the following K = |Y| probability measures py ,
y ? Y, at the nth step: complement the new object xn by the postulated label y; consider
the division of *z1 , . . . , zn +, where zn is understood (only for the purpose of this definition)
to be (xn , y), into groups (formally, bags) according to the values of An (i.e., zi and zj are
assigned to the same group if and only if ?i = ?j , where the ?s are defined by (2)); find the
empirical distribution py ? P(Y) of the labels in the group G containing the nth example
zn = (xn , y):
|{(x? , y ? ) ? G : y ? = y 0 }|
.
py ({y 0 }) :=
|G|
1
A Venn probability machine (VPM) is the Venn probability machine associated with some
taxonomy.
Theorem 1 Any Venn probability machine is finitarily calibrated.
1
By ?bag? we mean a collection of elements, not necessarily distinct. ?Bag? and ?multiset? are
synonymous, but we prefer the former term in order not to overload the prefix ?multi?.
It is clear that VPM depends on the taxonomy only through the way it splits the examples
z1 , . . . , zn into groups; therefore, we may specify only the latter when constructing specific
VPMs.
Remark The notion of VPM is a version of Transductive Confidence Machine (TCM)
introduced in [8] and [9], and Theorem 1 is a version of Theorem 1 in [2].
4 Discussion of the Venn probability machine
In this somewhat informal section we will discuss the intuitions behind VPM, considering
only the binary case Y = {0, 1} and considering the probability forecasts pi to be elements
of [0, 1] rather than P({0, 1}), as in ?1. We start with the almost trivial Bernoulli case,
where the objects xi are absent,2 and our goal is to predict, at each step n = 1, 2, . . . , the
new label yn given the previous labels y1 , . . . , yn?1 . The most naive probability forecast
is pn = k/(n ? 1), where k is the number of 1s among the first n ? 1 labels. (Often
?regularized? forms of k/(n ? 1), such as Laplace?s rule of succession (k + 1)/(n + 1),
are used.)
In the Bernoulli case there is only one natural VPM: the multiprobability forecast for yn is
{k/n, (k+1)/n}. Indeed, since there are no objects xn , it is natural to take the one-element
taxonomy An at each step, and this produces the VPM Pn = {k/n, (k + 1)/n}. It is clear
that the diameter 1/n of Pn for this VPM is the smallest achievable. (By the diameter of a
set we mean the supremum of distances between its points.)
Now let us consider the case where xn are present. The probability forecast k/(n ? 1)
for yn will usually be too crude, since the known population z1 , . . . , zn?1 may be very
heterogeneous. A reasonable statistical forecast would take into account only objects xi
that are similar, in a suitable sense, to xn . A simple modification of the Bernoulli forecast
k/(n ? 1) is as follows:
1. Split the available objects x1 , . . . , xn into a number of groups.
2. Output k 0 /n0 as the predicted probability that yn = 1, where n0 is the number
of objects among x1 , . . . , xn?1 in the same group as xn and k 0 is the number of
objects among those n0 that are labeled as 1.
At the first stage, a delicate balance has to be struck between two contradictory goals: the
groups should be as large as possible (to have a reasonable sample size for estimating probabilities); the groups should be as homogeneous as possible. This problem is sometimes
referred to as the ?reference class problem?; according to K?l?nc? [10], John Venn was the
first to formulate and analyze this problem with due philosophical depth.
The procedure offered in this paper is a simple modification of the standard procedure
described in the previous paragraph:
0. Consider the two possible completions of the known data
(z1 , . . . , zn?1 , xn ) = ((x1 , y1 ), . . . , (xn?1 , yn?1 ), xn ) :
in one (called the 0-completion) xn is assigned label 0, and in the other (called the
1-completion) xn is assigned label 1.
1. In each completion, split all examples z1 , . . . , zn?1 , (xn , y) into a number of
groups, so that the split does not depend on the order of examples (y = 0 for
the 0-partition and y = 1 for the 1-partition).
2
Formally, this correspond in our protocol to the situation where |X| = 1, and so xn , although
nominally present, do not carry any information.
2. In each completion, output k 0 /n0 as the predicted probability that yn = 1, where
n0 is the number of examples among z1 , . . . , zn?1 , (xn , y) in the same group as
(xn , y) and k 0 is the number of examples among those n0 that are labeled as 1.
In this way, we will have not one but two predicted probabilities that yn = 1; but in
practically interesting cases we can hope that these probabilities will be close to each other
(see the next section).
Venn?s reference class problem reappears in our procedure as the problem of avoiding overand underfitting. A taxonomy with too many groups means overfitting; it is punished by
the large diameter of the multiprobability forecast (importantly, this is visible, unlike the
standard approaches). Too few groups means underfitting (and poor resolution).
Important advantages of our procedure over the naive procedure are: our procedure is selfcalibrating; there exists an asymptotically optimal VPM (see ?6); we can use labels in
splitting examples into groups (this will be used in the next section).
5 Experiments
In this section, we will report the results for a natural taxonomy applied to the well-known
USPS data set of hand-written digits; this taxonomy is inspired by the 1-Nearest Neighbor
algorithm. First we describe the taxonomy, and then the way in which we report the results
for the VPM associated with this taxonomy.
Since the data set is relatively small (9298 examples in total), we have to use a crude
taxonomy: two examples are assigned to the same group if their nearest neighbors have
the same label; therefore, the taxonomy consists of 10 groups. The distance between two
examples is defined as the Euclidean distance between their objects (which are 16 ? 16
matrices of pixels and represented as points in R256 ).
The algorithm processes the nth object xn as follows. First it creates the 10 ? 10 matrix
A whose entry Ai,j , i, j = 0, . . . , 9, is computed by assigning i to xn as label and finding
the fraction of examples labeled j among the examples in the bag *z1 , . . . , zn?1 , (xn , i)+
belonging to the same group as (xn , i). The quality of a column of this matrix is its minimum entry. Choose a column (called the best column) with the highest quality; let the best
column be jbest . Output jbest as the prediction and output
?
?
min Ai,jbest , max Ai,jbest
i=0,...,9
i=0,...,9
as the interval for the probability that this prediction is correct. If the latter interval is [a, b],
the complementary interval [1 ? b, 1 ? a] is called the error probability interval.
Pn In Figure 1
we show the following three curves: the cumulative error curve En := i=1 erri , where
erri = 1 if an error (in the sense jbest 6= yi ) is made
Pnat step i and erri = 0 otherwise; the
cumulative lower error probability
curve
L
:=
n
i=1 li and the cumulative upper error
Pn
probability curve Un := i=1 ui , where [li , ui ] is the error probability interval output by
the algorithm for the label yi . The values En , Ln and Un are plotted against n. The plot
confirms that the error probability intervals are calibrated.
6 Universal Venn probability machine
The following result asserts the existence of a universal VPM. Such a VPM can be constructed quite easily using the histogram approach to probability estimation [11].
400
350
300
250
200
150
100
50
0
0
1000
2000
3000
4000
5000
6000
7000
8000
9000
10000
Figure 1: On-line performance of the 1-Nearest Neighbor VPM on the USPS data set
(9298 hand-written digits, randomly permuted). The dashed line shows the cumulative
number of errors En and the solid ones the cumulative upper and lower error probability
curves Un and Ln . The mean error EN /N is 0.0425 and the mean probability interval
(1/N )[LN , UN ] is [0.0407, 0.0419], where N = 9298 is the size of the data set. This
figure is not significantly affected by statistical variation (due to the random choice of the
permutation of the data set).
Theorem 2 Suppose X is a Borel space. There exists a VPM such that, if the examples are
generated from Q? ,
sup ?(Q(? | xn ), p) ? 0 (n ? ?)
p?Pn
in probability, where ? is the variation distance, Q(? | xn ) is a fixed version of the regular
conditional probabilities for yn given xn , and Pn are the multiprobabilities produced by
the VPM.
This theorem shows that not only all VPMs are reliable but some of them also have asymptotically optimal resolution. The version of this result for p-values was proved in [4].
7 Comparisons
In this section we briefly and informally compare this paper?s approach to standard approaches in machine learning.
Two most important approaches to analysis of machine-learning algorithms are Bayesian
learning theory and PAC theory (the recent mixture, the PAC-Bayesian theory, is part of
PAC theory in its assumptions). This paper is in a way intermediate between Bayesian
learning (no empirical justification for probabilities is required) and PAC learning (the goal
is to find or bound the true probability of error, not just to output calibrated probabilities).
An important difference of our approach from the PAC approach is that we are interested
in the conditional probabilities for the label given the new object, whereas PAC theory
(even in its ?data-dependent? version, as in [12?14]) tries to estimate the unconditional
probability of error.
Acknowledgments
We are grateful to Phil Dawid for a useful discussion and to the anonymous referees
for suggestions for improvement. This work was partially supported by EPSRC (grant
GR/R46670/01), BBSRC (grant 111/BIO14428), and EU (grant IST-1999-10226).
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1,610 | 2,463 | When Does Non-Negative Matrix Factorization
Give a Correct Decomposition into Parts?
David Donoho
Department of Statistics
Stanford University
Stanford, CA 94305
[email protected]
Victoria Stodden
Department of Statistics
Stanford University
Stanford, CA 94305
[email protected]
Abstract
We interpret non-negative matrix factorization geometrically, as the
problem of finding a simplicial cone which contains a cloud of data
points and which is contained in the positive orthant. We show that under
certain conditions, basically requiring that some of the data are spread
across the faces of the positive orthant, there is a unique such simplicial cone. We give examples of synthetic image articulation databases
which obey these conditions; these require separated support and factorial sampling. For such databases there is a generative model in terms
of ?parts? and NMF correctly identifies the ?parts?. We show that our
theoretical results are predictive of the performance of published NMF
code, by running the published algorithms on one of our synthetic image
articulation databases.
1
Introduction
In a recent article in Nature [4], Lee and Seung proposed the notion of non-negative matrix
factorization (NMF) as a way to find a set of basis functions for representing non-negative
data. They claimed that the notion is particularly applicable to image articulation libraries
made up of images showing a composite object in many articulations and poses. They
suggested (in the very title of the article) that when used in the analysis of such data, NMF
would find the intrinsic ?parts? underlying the object being pictured.
NMF is akin to other matrix decompositions which have been proposed previously, such
as positive matrix factorization (PMF) of Juvela, Lehtinen, and Paatero [3], [2] and various
minimum-volume transforms used in the analysis of remote-sensing data [1]. Numerous
applications of these methods have been attempted [6], [7], [9].
Despite all the literature and discussion of this method, two fundamental questions appear
not to have been posed clearly, let alone answered:
? Under what assumptions is the notion of non-negative matrix factorization welldefined, for example is the factorization in some sense unique?
? Under what assumptions is the factorization correct, recovering the ?right answer??
In this paper, we develop a geometric view of the setting underlying NMF factorization
and derive geometric conditions under which the factorization is essentially unique, so
NMF makes sense no matter what algorithm is being employed. We then consider those
conditions in the setting of image articulation libraries. We describe a class of image libraries which are created by an NMF-style generative model, where different parts have
separate support, and where all different combinations of parts are exhaustively sampled.
Our theory shows that, in such Separable Factorial Articulation Families, non-negative factorization is effectively unique. In such libraries, NMF will indeed successfully ?find the
parts?. We construct such a library, showing a stick figure with four limbs going through a
range of various motions, and verify that our theoretical analysis is predictive of the actual
performance of the Lee and Seung algorithm on this image library. Our viewpoint also
explains relations between NMF and other ideas for obtaining non-negative factorizations
and explains why uniqueness and stability may fail under other conditions.
We note that Plumbley [5] has in some sense already validated NMF for datasets which
are not only non-negative but which obey an independent components model. However,
in our view, this is actually a result about independent components analysis, not NMF. For
example, for the kinds of image articulation families where each part is viewed in one of
many positions, the underlying exclusion principle ? that a certain part can only be present
in one particular articulation ? guarantees that an ICA model does not apply. And this
parts-based setting is exactly the setting for NMF envisioned by Seung and Lee.
2
Non-Negative Matrix Factorization
NMF seeks to decompose a non-negative n ? p matrix X, where each row contains the p
pixel values for one of the n images, into
X = A?
(1)
where A is n ? r and ? is r ? p, and both A and ? have non-negative entries. The rows of
?, denoted (?j )rj=1 , are basis elements in Rp and the rows of A, (?i )ni=1 , belong to Rr and
can be thought of as coefficient sequences representing the images in that basis. Recalling
that the rows of X, (xi ), are individual images stored as row vectors, the representation
takes the form
r
X
i
x =
?ji ?j .
j=1
Indexing the pixels by k = 1, . . . , p, non-negativity of ?i and ?j can be written as:
?j (k) ? 0, j = 1, . . . , r, k = 1, . . . , p;
?ji ? 0, j = 1, . . . , r, i = 1, . . . , n.
(2)
It is clear that as a generative model, this approach makes sense; each of us can think of
some admittedly very simple imaging settings where the scene is composed out of ?standard
parts? in a variety of positions, where these are represented by the ?j and each image is
made by superposing some of those ?parts?. In this setting each part is either present or
absent, and the corresponding coefficient is thus positive or zero. An example of this kind
will be given in Section 4 below.
What is less clear is whether, when the generative model actually holds and we generate a
synthetic dataset based on that model, the NMF matrix factorization of the dataset will yield
underlying basis elements which have some connection to the true generative elements. In
this paper we investigate this question and exhibit conditions under which NMF will in fact
successfully recover the true generative elements.
3
Geometric Interpretation of the NMF Setting
We now describe a geometric viewpoint which will help explain the issues involved.
Each image in our database of images can be thought of as a point in a p-dimensional space,
whose p coordinates are given by the intensity values in each of the p pixels. The fact that
image data are non-negative means that every such point lies in the positive orthant P of
Rp .
The factorization X = A? says that there are vectors ?j in Rp such that all the data points
xi have a representation as non-negative linear combinations of the ?j . This algebraic
characterization has a geometric counterpart.
Definition. The simplicial cone generated by vectors ? = (?j )rj=1 is
X
? = ?? = {x : x =
?j ?j , ?j ? 0}.
j
The factorization (1) tells us geometrically that the (xi ) all lie in the simplicial cone ??
generated by the (?j ).
Now in general, for a given dataset (xi ), there will be many possible simplicial cones
containing the points in that dataset. Indeed, if ?? is a simplicial cone containing the data,
and ?? is another cone containing the first, so that
?? ? ?? ,
then the corresponding vectors ? = (?j ) also can furnish a representation of the dataset
(xi ). Now for any simplicial cone, there can always be another cone containing it strictly,
so there are an infinite number of factorizations X = A? with non-negative A, and various
? which are nontrivially different. Hence the constraint A ? 0 is not enough to lead to a
well-defined notion.
However, the geometric viewpoint we are developing does not so far include the positivity
constraint ? ? 0 on the generating vectors of the simplicial cone. Geometrically, this
constraint demands that the simplicial cone ?? lies inside the positive orthant P. Can we
obtain uniqueness with this extra constraint?
Not if the data values are strictly positive, so that
Xi,k ? > 0
?i, k.
(3)
Geometrically, this condition places the data points xi well inside the interior of the positive
orthant P. It is then evident by visual inspection that there will be many simplicial cones
containing the data. For example, P itself is a simplicial cone, and it contains the data
points. However, many other cones will also contain the data points. Indeed, for ? > 0
consider the collection of vectors ?? with individual vectors
??j = ej + ?1
where ej denotes the usual vector in the standard basis, and 1 denotes the vector of all
ones. Then, for ? < , the cone ??? also contains all the data points. Geometrically ???
is a dilation of the positive orthant that shrinks it slightly towards the main diagonal. Since
the positivity constraint (3) places all the data well inside the interior of the positive orthant,
for slight enough shrinkage it will still contain the data.
It follows from the geometric-algebraic correspondence that under the strict positivity condition (3), there are many distinct representations X = A? where A ? 0 and ? ? 0.
In short, we must look for situations where the data do not obey strict positivity in order to
have uniqueness.
4
An Example of Uniqueness
When we take the non-negativity constraint on the generating elements (the extreme rays
of the simplicial cone) into account, it can happen that there will only be one simplicial
cone containing the data. This is completely clear if the data somehow ?fill out? the positive
orthant. What is perhaps surprising is that uniqueness can hold even when the data only
?fill out? a proper subset of the positive orthant.
Here is an example of how that can occur. Consider the ?ice-cream cone?
C = {x : x0 1 ?
p
p ? 1||x||}
where p is again the dimensionality of the dataspace.
Lemma 1. There is a unique simplicial cone which both contains C and is itself contained
in the positive orthant.
Indeed that unique cone is P itself; no simplicial cone contained inside P contains all of
C!
To give a full proof, we introduce notions from the subject of convex duality [8]. Associated
with the primal domain of points x we have been dealing with so far, there is also the dual
domain of linear functionals ? acting on points x via ? 0 x. If we have a convex set C, its
dual C ? is defined as a collection of linear functionals which are positive on C:
C ? = {? : ? 0 x ? 0 ?x ? C}
The following facts are easily verified:
Lemma 2.
? If K is closed and convex then (K ? )? = K .
? The dual of a simplicial cone with p linearly independent generators, is another
simplicial cone with p generators.
? The positive orthant is self-dual: P ? = P .
? Duality reverses set inclusion:
B ? C =? C ? ? B ? .
(4)
We also need
Definition. Given a pointset (xi ), its conical hull is the simplicial hull generated by the
vectors (xi ) themselves.
Let X be the conical hull of a pointset. An abstraction of the NMF problem is:
Primal-Simplicial-Cone(r, X ) Find a simplicial cone with r generators contained in P
and containing X .
Consider now a problem in the dual domain, posed with reversed inclusions:
Dual-Simplicial-Cone(r, ?) Find a simplicial cone with r generators contained in ? and
containing P .
The two problems are indeed dual:
Lemma 3. Every solution to Primal-Simplicial-Cone(r, X ) is dual to a solution of DualSimplicial-Cone(r, X ? ), and vice-versa.
Proof. This is effectively the invocation of ?reversal of inclusion under duality? (4). Suppose we find a simplicial cone ? obeying
X ? ? ? P.
Then (4) says that
P ? ? ?? ? X ? ,
and so a solution to the primal solves the dual. In the other direction, if we find a simplicial
cone ?? obeying
P ? ? ?? ? X ?
then we have by (4)
(X ? )? ? (?? )? ? (P ? )? ;
? ?
we simply apply (K ) = K three times to see that a solution to the dual corresponds to a
solution to the primal. QED
Our motivation in introducing duality is to see something we couldn?t in the primal: we
can see that even if X is properly contained in P , there can be a unique simplicial hull for
X which lies inside P.
This follows from a simple observation about simplicial cones contained in convex cones.
Definition. An extreme ray of a convex cone ? is a ray Rx = {ax : a ? 0} where
x ? ? cannot be represented as a proper convex combination of two points x0 and x1
which belong to ? but not Rx .
For example, a simplicial cone with r linearly independent generators has r extreme rays;
each ray consists of all positive multiples of one generator.
Lemma 4. Suppose that ? and G are convex cones, that ? ? G ? Rr , that ? is a simplicial
cone with r generators and that G intersects ? in exactly r rays which are extreme rays of G.
Then (a) these rays are also extreme rays of ? and (b) no simplicial cone with r generators
?0 6= ? can satisfy ? ? ?0 ? G.
Proof. (a) Since the rays in question are extreme rays of G, which contains ?, they are also
extreme rays of ?. (b) Any simplicial cone ?0 with r generators and lying ?in between? ?
and G would have to also intersect G in the same r rays as ? does. Those r rays would also
have to be extreme rays for ?0 , because they are extreme rays for G, which by hypothesis
contains ?0 . But a simplicial cone with r generators is completely determined by its r
extreme rays. As ? and ?0 have the same extreme rays, ? = ?0 . QED
We can now prove Lemma 1. Recall the cone C defined above. Its dual is
C ? = {? : ? 0 1 ? ||?||}
Note (a) that every boundary ray of C ? is extreme; and (b) that C ? intersects P ? on the
n unit vectors ej . So by Lemma 4, P ? uniquely solves the Dual-Simplicial-Cone(n, C ? )
problem and P solves the Primal-Simplicial-Cone(n, C) problem uniquely. QED.
5
Uniqueness for Separable Factorial Articulation Families
We now describe families of articulated images which have at least a few ?realistic? features,
and which, because of the relevant convex geometry, offer an essentially unique NMF.
The families of images we have in mind consist of black-and-white images with P parts,
each exercised systematically through A articulations. As an illustration, Figure 1 shows
some sample images from the Swimmer dataset, which depicts a figure with four moving
parts (limbs), each able to exhibit four articulations (different positions).
Definition. A Separable Factorial Articulation Family is a collection X of points x
obeying these rules:
[R1] Generative Model. Each image x in the database has a representation
x=
P X
A
X
?q,a ?q,a
q=1 a=1
where the generators ?q,a ? Rp obey the non-negativity constraint ?q,a ? 0
along with the coefficients ?q,a ? 0. We speak of ?q,a as the q?th part in the
?a?-th articulation.
[R2] Separability. For each q, a there exists a pixel kq,a such that
?q0 ,a0 (kq,a ) = 1{a=a0 ,q=q0 }
(5)
I.e. each part/articulation pair?s presence or absence in the image is indicated by
a certain pixel associated to that pair.
[R3] Complete Factorial Sampling. The dataset contains all AP images in which the P
parts appear in all combinations of A articulations.
Figure 1: Sample images from the Swimmer database depicting four stick figures with four
limbs; the panels illustrate different articulations of the limbs.
The Swimmer dataset obeys these rules except for one disagreement: every image contains
an invariant region (the torso). As it turns out this is of small importance.
We note that assumption [R2] forces the generators ?q,a to be linearly independent, which
forces p > A?P . Consequently, the linear span of the generators is some subspace V ? Rp .
Theorem 1. Given a database obeying rules [R1]-[R3], there is a unique simplicial hull
with r = A ? P generators which contains all the points of the database, and is contained
in P ? V .
Since the generative model [R1] implies that a particular simplicial hull with a specific
choice of r generators contains the dataset, and a successful application of NMF also gives
a simplicial hull with r generators containing the dataset, and the theorem says these must
be the same hull, in this setting NMF recovers the generative model. Formally,
Corollary. Let X be generated by rules [R1]-[R3]. Any factorization obeying (1) and (2)
must recover the correct generators (?q,a ) modulo permutation of labels and rescaling.
6
Proof of Theorem 1.
We need to introduce the notion of duality relative to a vector space V ? Rp . In the case
of V ? Rp this is just the notion of duality already introduced. Suppose that we have a set
K ? V ; its relative dual K v is the set of linear functionals ? which, viewed as members of
Rp also belong to V , and which obey ? 0 x ? 0 for x ? K. In effect, the relative dual is the
ordinary dual taken within V rather than Rp . As a result, all the properties of Lemma 2 hold
for relative duality provided we talk about sets which are subsets of V ; e.g. (K v )v = K if
K is a closed convex subset of V .
Define PV = V ? P; this is a simplicial cone in V with r generators.
Let again X denote the conical hull of X = (xi ) and suppose that every (r?1)-dimensional
face of PV contains r ? 1 linearly independent points from X. Since the face of a cone
is a linear subspace, the face is uniquely determined by these r ? 1 points. The face is
part of a supporting hyperplane to PV which is also a supporting hyperplane to X . The
supporting hyperplane defines a point ? ? V which is in common between the duals PVv
and X v . Similar statements hold for all the r different (r?1)-faces of PV . But more is true.
Because of the linear independence mentioned above, the different supporting hyperplanes
in primal space correspond in fact to extreme rays in dual space ? extreme rays for both
PVv and X v . As this is true for all r of the (r ? 1)-dimensional faces, we are in a position
to apply Lemma 4 with G = X v and ? = PVv . This gives the conclusion that PVv is the
unique simplicial cone with r generators contained in X v and containing PVv . Theorem 1
then follows by duality.
It remains to establish the assumption about existence of r ? 1 linear independent points
on each (r ? 1)-face. The faces of PV are exactly the r different subspaces
Fq,a = {x ? V : ?q,a = 0}.
By the Complete Factorial Sampling assumption [R3], there are AP ?1 (A ? 1) points of X
in such a face. Define, for each (q 0 , a0 ) 6= (q, a),
?q0 ,a0 ;q,a = Ave{x ? X : ?q,a = 0, ?q0 ,a0 = 1}.
There are r ? 1 such terms, one for each part/articulation pair besides (q, a). By the Separability assumption [R2]:
?q0 ,a0 ;q,a (kq00 ,a00 ) = 1{q0 =q00 ,a0 =a00 } .
0
Hence the (?q0 ,a0 ;q,a : (q , a0 ) 6= (q, a)) are linearly independent. At the same time,
?q0 ,a0 ;q,a (kq,a ) = 0
so that each ?q0 ,a0 ;q,a ? Fq,a . Hence we have the required linearly independent subset in
each face. QED
7
Empirical Verification
We built the Swimmer image library of 256 32?32 images. Each image contains a ?torso?
of 12 pixels in the center and four ?arms? of 6 pixels that can be in one of 4 positions. All
combinations of all possible arm positions gives us 256 images. See Figure 1 for examples.
This collection of images has four ?parts?. It deviates slightly from the rules [R1]-[R5]
because there is an invariant region (the torso). Figure 2 shows that the 16 different
part/articulation pairs are properly resolved, but that the torso is not properly resolved.
Figure 2: NMF Generators recovered from Swimmer database. The 16 images shown
agree well with the known list of generators (4 ?limbs? in 4 positions each). The presence
of the torso (i.e. an invariant region) violates our conditions for a Factorial Separable
Articulation Library, and, not unexpectedly, ghosts of the torso contaminate several of the
reconstructed generators. Lee and Seung?s code [4] was used.
Acknowledgments
This work was partially supported by NSF grants DMS-0077261, DMS-0140698, and ANI008584 and a contract from DARPA ACMP. We would like to thank Aapo Hyv?arinen for
numerous helpful discussions.
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| 2463 |@word hyv:1 seek:1 sensed:1 decomposition:2 contains:14 series:1 recovered:1 surprising:1 written:1 must:3 realistic:1 happen:1 alone:1 generative:9 inspection:1 short:1 core:1 characterization:1 hyperplanes:1 plumbley:2 along:1 welldefined:1 consists:1 prove:1 ray:21 inside:5 introduce:2 x0:2 ica:1 indeed:5 themselves:1 actual:1 provided:1 underlying:4 panel:1 mass:1 what:5 kind:2 finding:1 guarantee:1 contaminate:1 every:5 exactly:3 stick:2 unit:1 grant:1 appear:2 positive:18 ice:1 despite:1 oxford:1 ap:2 abuse:1 black:1 factorization:20 range:1 obeys:1 unique:10 acknowledgment:1 intersect:1 empirical:1 remotely:1 thought:2 composite:1 cannot:1 interior:2 center:1 convex:10 rule:5 fill:2 stability:1 notion:7 coordinate:1 aerosol:2 suppose:4 modulo:1 speak:1 hypothesis:1 swimmer:5 element:5 particularly:1 database:9 cloud:2 unexpectedly:1 region:3 remote:2 envisioned:1 mentioned:1 seung:5 exhaustively:1 predictive:2 basis:5 completely:2 easily:1 resolved:2 darpa:1 various:3 represented:2 talk:1 intersects:2 chapter:1 articulated:1 separated:1 distinct:1 describe:3 tell:1 couldn:1 whose:1 stanford:6 posed:2 q00:1 say:3 statistic:2 think:1 itself:3 sequence:1 rr:2 relevant:1 elemental:1 r1:5 generating:2 object:3 help:1 derive:1 develop:1 illustrate:1 stat:2 pose:1 solves:3 recovering:1 pointset:2 implies:1 revers:1 direction:1 correct:3 hull:9 duals:1 violates:1 explains:2 require:1 arinen:1 decompose:1 strictly:2 hold:4 lying:1 earth:1 uniqueness:6 applicable:1 label:1 title:1 exercised:1 vice:1 successfully:2 clearly:1 always:1 rather:1 ej:3 shrinkage:1 asp:1 d15:2 corollary:1 validated:1 ax:1 seasonal:1 properly:3 fq:2 ave:1 sense:4 helpful:1 abstraction:1 a0:11 relation:1 going:1 pixel:7 issue:1 dual:16 superposing:1 denoted:1 spatial:1 construct:1 sampling:3 r5:1 look:1 few:1 composed:1 individual:2 geometry:1 recalling:1 investigate:1 extreme:14 primal:8 alaska:2 pmf:1 theoretical:2 ordinary:1 introducing:1 entry:1 subset:4 kq:3 successful:1 stored:1 answer:1 synthetic:3 fundamental:1 lee:5 contract:1 again:2 containing:10 positivity:4 style:1 rescaling:1 account:1 star:1 coefficient:3 matter:1 satisfy:1 view:2 closed:2 recover:2 clemens:1 ni:1 simplicial:39 yield:1 correspond:1 craig:1 basically:1 rx:2 published:2 explain:1 definition:4 involved:1 dm:2 proof:4 associated:2 recovers:1 sampled:1 dataset:10 recall:1 dimensionality:1 torso:6 actually:2 shrink:1 just:1 somehow:1 defines:1 indicated:1 perhaps:1 effect:1 requiring:1 verify:1 true:4 counterpart:1 contain:2 furnish:1 hence:3 q0:9 white:1 self:1 uniquely:3 evident:1 complete:2 motion:1 image:30 common:1 ji:2 volume:3 belong:3 interpretation:1 slight:1 interpret:1 a00:2 composition:1 versa:1 inclusion:3 moving:1 something:1 exclusion:1 recent:1 claimed:1 certain:3 minimum:2 employed:1 signal:1 full:1 multiple:1 rj:2 offer:1 malm:2 molecular:2 aapo:1 essentially:2 qed:4 source:1 extra:1 strict:2 subject:1 cream:1 member:1 nontrivially:1 presence:2 enough:2 variety:1 independence:1 idea:1 absent:1 whether:1 akin:1 algebraic:2 stodden:1 clear:3 factorial:7 transforms:2 generate:1 nsf:1 correctly:1 four:7 verified:1 imaging:1 geometrically:5 cone:46 letter:1 place:2 family:6 conical:3 correspondence:1 nonnegative:1 occur:1 constraint:7 scene:1 answered:1 span:1 separable:4 department:2 developing:1 combination:5 across:1 slightly:2 separability:2 invariant:3 indexing:1 taken:1 agree:1 previously:1 remains:1 turn:1 r3:4 fail:1 mind:1 dataspace:1 reversal:1 victoria:1 obey:5 limb:5 apply:3 disagreement:1 rp:9 existence:1 denotes:2 running:1 include:1 establish:1 question:3 already:2 usual:1 diagonal:1 exhibit:2 subspace:3 reversed:1 separate:1 thank:1 code:2 besides:1 juvela:3 illustration:1 statement:1 pvv:5 negative:15 proper:2 observation:1 datasets:1 orthant:11 supporting:4 situation:1 variability:1 august:2 intensity:1 nmf:22 atmospheric:2 david:1 introduced:1 pair:4 required:1 connection:1 able:1 suggested:1 below:1 articulation:19 ghost:1 built:1 force:2 pictured:1 representing:2 arm:2 library:8 numerous:2 identifies:1 created:1 negativity:3 hopke:2 deviate:1 literature:1 geometric:7 relative:4 permutation:1 generator:22 verification:1 article:2 principle:1 viewpoint:3 editor:1 systematically:1 row:5 supported:1 face:11 boundary:1 made:2 collection:4 far:2 transaction:1 functionals:3 reconstructed:1 dealing:1 xi:10 spectrum:2 why:1 dilation:1 nature:2 ca:2 obtaining:1 depicting:1 domain:3 spread:1 main:1 linearly:6 motivation:1 x1:1 depicts:1 position:7 pv:5 obeying:5 lie:4 invocation:1 theorem:4 specific:1 showing:2 sensing:2 r2:3 list:1 intrinsic:1 consist:1 exists:1 effectively:2 importance:1 demand:1 simply:1 visual:1 contained:9 partially:1 geoscience:1 corresponds:1 viewed:2 donoho:2 consequently:1 towards:1 absence:1 infinite:1 determined:2 except:1 acting:1 hyperplane:3 lemma:8 admittedly:1 duality:8 attempted:1 geophysical:2 formally:1 support:2 princeton:1 paatero:3 |
1,611 | 2,464 | Semi-Supervised Learning with Trees
Charles Kemp, Thomas L. Griffiths, Sean Stromsten & Joshua B. Tenenbaum
Department of Brain and Cognitive Sciences, MIT, Cambridge, MA 02139
{ckemp,gruffydd,sean s,jbt}@mit.edu
Abstract
We describe a nonparametric Bayesian approach to generalizing from
few labeled examples, guided by a larger set of unlabeled objects and
the assumption of a latent tree-structure to the domain. The tree (or a
distribution over trees) may be inferred using the unlabeled data. A prior
over concepts generated by a mutation process on the inferred tree(s)
allows efficient computation of the optimal Bayesian classification function from the labeled examples. We test our approach on eight real-world
datasets.
1
Introduction
People have remarkable abilities to learn concepts from very limited data, often just one
or a few labeled examples per class. Algorithms for semi-supervised learning try to match
this ability by extracting strong inductive biases from a much larger sample of unlabeled
data. A general strategy is to assume some latent structure T that underlies both the label
vector Y to be learned and the observed features X of the full data (unlabeled and labeled;
see Figure 1). The unlabeled data can be used to help identify the latent structure T , and an
assumption that Y is somehow ?smooth? with respect to T ? or in Bayesian terms, can be
assigned a strong prior conditional on T ? provides the inductive bias needed to estimate
Y successfully from very few labeled examples Yobs .
Different existing approaches can be understood within this framework. The closest to
our current work is [1] and its cousins [2-5]. The structure T is assumed to be a lowdimensional manifold, whose topology is approximated by a sparse neighborhood graph
defined over the data points (based on Euclidean distance between feature vectors in the X
matrix). The label vector Y is assumed to be smooth with respect to T ; [1] implements this
smoothness assumption by defining a Gaussian field over all complete labelings Y of the
neighborhood graph that expects neighbors to have the same label. This approach performs
well in classifying data with a natural manifold structure, e.g., handwritten digits.
The graphical model in Figure 1 suggests a more general strategy for exploiting other kinds
of latent structure T , not just low-dimensional manifolds. In particular, trees arise prominently in both natural and human-generated domains (e.g., in biology, language and information retrieval). Here we describe an approach to semi-supervised learning based on
mapping the data onto the leaf nodes of a rooted (and typically ultrametric) tree T .
The label vector Y is generated from a stochastic mutation process operating over branches
of T . Tree T can be inferred from unlabeled data using either bottom-up methods (agglomerative clustering) or more complex probabilistic methods. The mutation process defines
T
X
Y
Figure 1: A general approach to semi-supervised
learning. X is an observed object-feature matrix,
Y the hidden vector of true labels for these objects and Yobs a sparse vector of observed labels.
The unlabeled data in X assist in inferring Y by
allowing us to infer some latent structure T that
is assumed to generate both X and Y .
Yobs
a prior over all possible labelings of the unlabeled data, favoring those that maximize a
tree-specific notion of ?smoothness?. Figure 2 illustrates this Tree-Based Bayes (TBB)
approach. Each of the 32 objects in this dataset has two continuous features (x and y coordinates); X is a 32-by-2 matrix. Yobs contains four entries, two positive and two negative.
The shading in part (b) represents a probabilistic inference about Y : the darker an object?s
node in the tree, the more likely that its label is positive.
TBB classifies unlabeled data by integrating over all possible labelings of the domain that
are consistent with the observed labels Yobs , and is thus an instance of optimal Bayesian
concept learning [6]. Typically, optimal Bayes is of theoretical interest only [7], because
the sum over labelings is in general intractable and it is difficult to specify sufficiently
powerful and noise-resistant priors for real-world domains. Here, a prior defined in terms
of a tree-based mutation process makes the approach efficient and empirically successful.
The next section describes TBB, as well as a simple heuristic method, Tree Nearest Neighbor (TNN), which we show approximates TBB in the limit of high mutation rate. Section 3
presents experimental comparisons with other approaches on a range of datasets.
(a)
(b)
Figure 2: Illustration of the Tree-Based Bayesian approach to semi-supervised learning. (a)
We observe a set of unlabeled objects (small points) with some latent hierarchical structure
(gray ellipses) along with two positive and two negative examples of a new concept (black
and white circles). (b) Inferring the latent tree, and treating the concept as generated from
a mutation process on the tree, we can probabilistically classify the unlabeled objects.
2
Tree-Based Bayes (TBB)
We assume a binary classification problem with Y ? {?1, 1}n . We choose a label yi for
unlabeled object xi by computing p(yi = 1|Yobs , X) and thresholding at 0.5. Generalization to the multi-class case will be straightforward.
Ideally we would sum over all possible latent trees T :
X
p(yi = 1|Yobs , X) =
p(yi = 1|Yobs , T )p(T |Yobs , X)
T
(1)
First we consider p(yi = 1|Yobs , T ) and the classification of object xi given a particular
tree T . Section 2.2 discusses p(T |Yobs , X), the inference of tree T , and approaches to
approximating the sum over trees in Equation 1.
We predict object xi ?s label by summing over all possible complete labelings Y of the data:
X
p(yi = 1|Yobs , T ) =
p(yi = 1|Y )p(Y |Yobs , T )
(2)
Y
X p(yi = 1|Y )p(Yobs |Y, T )p(Y |T )
=
p(Yobs |T )
Y
P
p(yi = 1|Y )p(Yobs |Y )p(Y |T )
= Y P
Y p(Yobs |Y )p(Y |T )
(3)
(4)
In general, the likelihood p(Yobs |Y ) depends on assumptions about sampling and noise.
Typical simplifying assumptions are that the labeled objects were chosen randomly from
all objects in the domain, and that all observations are free of noise. Then p(Y obs |Y ) ? 1
if Yobs is consistent with Y and is zero otherwise.
Under these assumptions, Equation 4 becomes:
P
p(yi = 1|Yobs , T ) =
Y consistent with Yobs :yi =1 p(Y |T )
P
Y consistent with Yobs p(Y |T )
(5)
The probability that yi = 1 reduces to the weighted fraction of label vectors consistent with
Yobs that set yi = 1, with each label vector weighted by its prior under the tree, p(Y |T ).
When class frequencies are unbalanced, small training sets provide little scope for learning
if constructed using random sampling. Consider the problem of identifying genetic markers
for a disease that afflicts one person in 10,000. A training set for this problem might be
constructed by ?retrospective sampling,? e.g. taking data from 20 patients with the disease
and 20 healthy subjects. Randomly sampling subjects from the entire population would
mean that even a medium-sized training set would have little chance of including anyone
with the disease.
Retrospective sampling can be modeled by specifying a more complex likelihood
p(Yobs |Y ). The likelihood can also be modified to handle additional complexities, such
as learning from labeled examples of just a single class, or learning in the presence of label
noise. We consider none of these complexities here. Our experiments explore both random and retrospective sampling, but the algorithm we implement is strictly correct only for
noise-free learning under random sampling.
2.1
Bayesian classification with a mutation model
In many tree-structured domains it is natural to think of features arising from a history
of stochastic events or mutations. We develop a mutation model that induces a sensible
?smoothness? prior p(Y |T ) and enables efficient computation of Equation 5 via belief
propagation on a Bayes net. The model combines aspects of several previous proposals for
probabilistic learning with trees [8, 9, 10].
Let L be a feature corresponding to the class label. Suppose that L is defined at every
point along every branch, not just at the leaf nodes where the data points lie. Imagine L
spreading out over the tree from root to leaves ? it starts out at the root with some value
and could switch values at any point along any branch. Whenever a branch splits, both
lower branches inherit the value of L at the point immediately before the split.
Transitions between states of L are modeled using a continuous-time Markov chain with
infinitesimal matrix:
?? ?
Q=
? ??
The free parameter, ?, will be called the mutation rate. Note that the mutation process is
symmetric: mutations from -1 to 1 are just as likely as mutations in the other direction.
Other models of mutation could be substituted if desired. Generalization to the k-class
?
case is achieved by specifying a k by k matrix Q, with ?? on the diagonal and k?1
on the
off-diagonal.
Transition probabilities along a branch of length t are given by:
#
"
?2?t
?2?t
eQt =
1+e
2
1?e?2?t
2
1?e
2
1+e?2?t
2
(6)
That is, the probability that a parent and child separated by a branch of length t have
?2?t
different values of L is 1?e2 .
This mutation process induces a prior p(Y |T ) equal to the probability of generating the
label vector Y over leaves of T under the mutation process. The resulting distribution
favors labelings that are ?smooth? with respect to T . Regardless of ?, it is always more
likely for L to stay the same than to switch its value along a branch. Thus labelings that
do not require very many mutations are preferred, and the two hypotheses that assign the
same label to all leaf nodes receive the most weight. Because mutations are more likely to
occur along longer branches, the prior also favors hypotheses in which label changes occur
between clusters (where branches tend to be longer) rather than within clusters (where
branches tend to be shorter).
The independence assumptions implicit in the mutation model allow the right side of Equation 5 to be computed efficiently. Inspired by [9], we set up a Bayes net with the same
topology as T that captures the joint probability distribution over all nodes. We associate
with each branch a conditional probability table that specifies the value of the child conditioned on the value of the parent (based on Equation 6), and set the prior probabilities at
the root node to the uniform distribution (the stationary distribution of the Markov chain
specified by Q). Evaluating Equation 5 now reduces to a standard problem of inference in a
Bayes net ? we clamp the nodes in Yobs to their observed values, and compute the posterior
marginal probability at node yi . The tree structure makes this computation efficient and
allows specially tuned inference algorithms, as in [9].
2.2
A distribution over trees
We now consider p(T |Yobs , X), the second component of Equation 1. Using Bayes? theorem:
p(T |Yobs , X) ? p(Yobs , X|T )p(T )
(7)
We assume that each discrete feature in X is generated independently over T according
to the mutation model just outlined. Continuous features can be handled by an analogous
stochastic diffusion process in a continuous space (see for example [11]). Because the features are conditionally independent of each other and of Yobs given the tree, p(Yobs , X|T )
can be computed using the methods of the previous section.
To finish the theoretical development of the model it remains only to specify p(T ), a prior
over tree structures. Section 3.2 uses a uniform prior, but a Dirichlet Diffusion Tree prior
is another option [11].
2.3
Approximating the sum over trees
The sum over trees in Equation 1 is intractable for datasets of even moderate size. We
therefore consider two approximations. Markov Chain Monte Carlo (MCMC) techniques
have been used to approximate similar sums over trees in Bayesian phylogenetics [12], and
Section 3.2 applies these ideas to a small-scale example. Although theoretically attractive,
MCMC approaches are still expensive to use with large datasets. Section 3.1 follows a
simpler approach: we assume that most of the probability p(T |Yobs , X) is concentrated
on or near the most probable tree T ? and approximate Equation 1 as p(yi = 1|Yobs , T ? ).
The tree T ? can be estimated using more or less sophisticated means. In Section 3.1 we
use a greedy method ? average-link agglomerative clustering on the object-feature matrix
X, using Hamming or Euclidean distance in discrete or continuous domains, respectively.
In Section 3.2 we compare this greedy method to the best tree found in our MCMC runs.
Note that we ignore Yobs when building T ? , because we run many trials on each dataset
and do not want to compute a new tree for each value of Yobs . Since our data include many
features and few labeled objects, the contribution of Yobs is likely to be negligible.
2.4
Tree Nearest Neighbor (TNN)
A Bayesian formulation based on the mutation process provides a principled approach to
learning with trees, but there are simpler algorithms that instantiate similar intuitions. For
instance, we could build a one-nearest-neighbor classifier using the metric of distance in
the tree T (with ties resolved randomly). It is clear how this Tree Nearest Neighbor (TNN)
algorithm reflects the assumption that nearby leaves in T are likely to have the same label,
but it is not necessarily clear when and why this simple approach should work well.
An analysis of Tree-Based Bayes provides some insight here ? TBB and TNN become
equivalent when the ? parameter of TBB is set sufficiently high.
Theorem 1 For each ultrametric tree T , there is a ?0 such that TNN and TBB produce
identical classifications for all examples with a unique nearest neighbor when ? > ? 0 .
A
proof
is
available
at
http://www.mit.edu/?ckemp/papers/
treesslproof.pdf, but we give some intuition for the result here. Consider
the Bayes net described in Section 2.1 and suppose xi is an unlabeled object. The value
chosen for yi will depend on all the labels in Yobs , but the influence of any single label
decreases with distance in the tree from yi . Once ? becomes sufficiently high it can be
shown that yi is always determined uniquely by the closest labeled example in the tree.
Given this equivalence between the algorithms, TNN is the method of choice when a high
mutation rate is indicated. It is not only faster, but numerically more stable. For large
values of ?, the probabilities manipulated by TBB become very close to 0.5 and variables
that should be different may become indistinguishable within the limits of computational
precision. Our implementation of TBB therefore uses TNN when cross-validation indicates
that a sufficiently high value of ? is required.
3
3.1
Experiments
Trees versus Manifolds
We compared TBB and TNN with the Laplacian method of Belkin and Niyogi [4], an
approach that effectively assumes a latent manifold structure T . We also ran generic onenearest neighbor (NN) as a baseline.
The best performing method on a given dataset should be the algorithm that assumes the
right latent structure for that domain. We therefore tested the algorithms on several different
types of data: four taxonomic datasets (Beetles, Crustaceans, Salamanders and Worms,
with 192, 56, 30 and 286 objects respectively), two molecular biology sets (Gene Promoter
and Gene Splice, with sizes 106 and 3190), and two ?manifold? sets (Digits and Vowels,
with sizes 10,000 and 990).
The taxonomic datasets were expected to have a tree-like structure. Each set describes the external anatomy of a group of species, based on data available at http:
//biodiversity.uno.edu/delta/. One feature in the Beetles set, for example,
indicates whether a beetle?s body is ?strongly flattened, slightly flattened to moderately
convex, or strongly convex.? Since these taxonomic sets do not include class labels, we
chose features at random to stand in for the class label. We averaged across five such
choices for each dataset.
The molecular biology sets were taken from the UCI repository. The objects in both sets
are strings of DNA, and tree structures might also be appropriate here since these strings
arose through evolution. The manifold sets arose from human motor behaviors, and were
therefore expected to have a low-dimensional manifold structure. The Digits data are a
subset of the MNIST data, and the Vowels data are taken from the UCI repository.
Our experiments focused on learning from very small labeled sets. The number of labeled
examples was always set to a small multiple (m = 1, 2, 3, 5, or 10) of the total number
of classes. The algorithms were compared under random and retrospective sampling, and
training sets were always sampled with replacement. For each training-set size m, we averaged across 10 values of Yobs obtained by randomly sampling from the vector Y . Free
parameters for TBB (?) and Laplacian (number of nearest neighbors, number of eigenvectors) were chosen using randomized leave-one-out cross-validation.
Figure 3a shows the performance of the algorithms under random sampling for four representative datasets. TBB outperforms the other algorithms across the four taxonomic
sets (only Beetles and Crustaceans shown), but the differences between TBB and Nearest Neighbor are rather small. These results do suggest a substantial advantage for TBB
over Laplacian in tree-structured domains. As expected, this pattern is reversed on the
Digits set, but it is encouraging that the tree-based methods can still improve on Nearest
Neighbor even for datasets that are not normally associated with trees. Neither method
beats the baseline on the Vowels or the Gene Promoter sets, but TBB performs well on the
Gene Splice set, which suggests that it may find further uses in computational biology.
More dramatic differences between the algorithms appear under retrospective sampling
(Figure 3b). There is a clear advantage here for TBB on the taxonomic sets. TBB fares
better than the other algorithms when the class proportions in the training set do not match
the proportions in the population, and it turns out that many of the features in the taxonomic
datasets are unbalanced. Since the other datasets have classes of approximately equal size,
the results for retrospective sampling are similar to those for random sampling.
While not conclusive, our results suggest that TBB may be the method of choice on treestructured datasets, and is robust even for datasets (like Digits) that are not clearly treestructured.
3.2
MCMC over trees
Figure 3 shows that TBB can perform well on real-world datasets using only a single
tree. Working with a distribution over trees, although costly, could improve performance
when there is not sufficient data to strongly constrain the best tree, or when the domain is
not strongly tree-structured. Using a small synthetic example, we explored one such case:
learning from very sparse and noisy data in a tree-structured domain.
(a)
Beetles
Crustaceans
Gene Splice
Digits
35
TBB
NN
TNN
Laplacian
60
60
8
30
50
25
40
7
20
40
20
6
15
123
5
10
123
5
10
123
5
10
123
5
10
123
5
10
(b)
Error rate
40
30
20
10
123 5
10
Examples per class
60
50
60
40
55
30
50
20
45
10
40
123
5
10
50
40
30
20
10
123
5
10
Figure 3: Error rates for four datasets under (a) random and (b) retrospective sampling, as
a function of the number of labeled examples m per class. Mean standard error bars for
each dataset are shown in the upper right corner of the plot.
0.4
NN
TBB (modal)
0.35
Error rate
We generated artificial datasets consisting of 20 objects. Each dataset was based
on a ?true? tree T0 , with objects at the
leaves of T0 . Each object was represented by a vector of 20 binary features
generated by a mutation process over T0 ,
with high ?. Most feature values were
missing; the algorithms saw only 5 of
the 20 features for each object. For each
dataset, we created 20 test concepts from
the same mutation process. The algorithms saw m labeled examples of each
test concept and had to infer the labels of
the remaining objects. This experiment
was repeated for 10 random trees T0 .
TBB (agglom)
TBB (MCMC)
0.3
0.25
TBB (ideal)
0.2
Our MCMC approach was inspired by an
4
8
12
algorithm for reconstruction of phylogeNumber of labeled examples
netic trees [12], which uses MetropolisFigure 4: Error rates on sparse artificial data
Hastings over tree topologies with two
as a function of number of labels observed.
kinds of proposals: local (nearest neighbor interchange) and global (subtree
pruning and regrafting). Unlike the previous section, none of the trees considered (including the true tree T0 ) was ultrametric.
Instead, each branch in each tree was assigned a fixed length. This meant that any two trees
with the same hierarchical structure were identical, and we did not have to store trees with
the same topology but different branch lengths.
Figure 4 shows the mean classification error rate, based on 1600 samples after a burn-in of
400 iterations. Four versions of TBB are shown: ?ideal? uses the true tree T0 , ?MCMC?
uses model averaging over a distribution of trees, ?modal? uses the single most likely tree in
the distribution, and ?agglom? uses a tree built by average-link clustering. The ideal learner
beats all others because the true tree is impossible to identify with such sparse data. Using
MCMC over trees brings TBB substantially closer to the ideal than simpler alternatives that
ignore the tree structure (NN) or consider only a single tree (modal, agglom).
4
Conclusion
We have shown how to make optimal Bayesian concept learning tractable in a semisupervised setting by assuming a latent tree structure that can be inferred from the unlabeled data and defining a prior for concepts based on a mutation process over the tree. Our
Bayesian framework supports many possible extensions, including active learning, feature
selection, and model selection. Inferring the nature of the latent structure T ? rather than
assuming a manifold structure or a tree structure ? is a particularly interesting problem.
When little is known about the form of T , Bayesian methods for model selection could
be used to choose among approaches that assume manifolds, trees, flat clusters, or other
canonical representational forms.
Acknowledgments This project was supported by the DARPA CALO program and NTT Communication Science Laboratories. Our implementation of the Laplacian method was based on code
provided by Mikhail Belkin.
References
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19(9):1951?1601, 2002.
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1,612 | 2,465 | Link Prediction in Relational Data
Ben Taskar Ming-Fai Wong Pieter Abbeel Daphne Koller
{btaskar, mingfai.wong, abbeel, koller}@cs.stanford.edu
Stanford University
Abstract
Many real-world domains are relational in nature, consisting of a set of objects
related to each other in complex ways. This paper focuses on predicting the
existence and the type of links between entities in such domains. We apply the
relational Markov network framework of Taskar et al. to define a joint probabilistic model over the entire link graph ? entity attributes and links. The application
of the RMN algorithm to this task requires the definition of probabilistic patterns
over subgraph structures. We apply this method to two new relational datasets,
one involving university webpages, and the other a social network. We show that
the collective classification approach of RMNs, and the introduction of subgraph
patterns over link labels, provide significant improvements in accuracy over flat
classification, which attempts to predict each link in isolation.
1
Introduction
Many real world domains are richly structured, involving entities of multiple types that
are related to each other through a network of different types of links. Such data poses
new challenges to machine learning. One challenge arises from the task of predicting
which entities are related to which others and what are the types of these relationships. For
example, in a data set consisting of a set of hyperlinked university webpages, we might
want to predict not just which page belongs to a professor and which to a student, but also
which professor is which student?s advisor. In some cases, the existence of a relationship
will be predicted by the presence of a hyperlink between the pages, and we will have only
to decide whether the link reflects an advisor-advisee relationship. In other cases, we might
have to infer the very existence of a link from indirect evidence, such as a large number
of co-authored papers. In a very different application, we might want to predict links
representing participation of individuals in certain terrorist activities.
One possible approach to this task is to consider the presence and/or type of the link
using only attributes of the potentially linked entities and of the link itself. For example,
in our university example, we might try to predict and classify the link using the words on
the two webpages, and the anchor words on the link (if present). This approach has the
advantage that it reduces to a simple classification task and we can apply standard machine
learning techniques. However, it completely ignores a rich source of information that is
unique to this task ? the graph structure of the link graph. For example, a strong predictor
of an advisor-advisee link between a professor and a student is the fact that they jointly
participate in several projects. In general, the link graph typically reflects common patterns
of interactions between the entities in the domain. Taking these patterns into consideration
should allow us to provide a much better prediction for links.
In this paper, we tackle this problem using the relational Markov network (RMN) framework of Taskar et al. [14]. We use this framework to define a single probabilistic model
over the entire link graph, including both object labels (when relevant) and links between
objects. The model parameters are trained discriminatively, to maximize the probability
of the (object and) link labels given the known attributes (e.g., the words on the page, hyperlinks). The learned model is then applied, using probabilistic inference, to predict and
classify links using any observed attributes and links.
2
Link Prediction
A relational domain is described by a relational schema, which specifies a set of object
types and attributes for them. In our web example, we have a Webpage type, where each
page has a binary-valued attribute for each word in the dictionary, denoting whether the
page contains the word. It also has an attribute representing the ?class? of the webpage,
e.g., a professor?s homepage, a student?s homepage, etc.
To address the link prediction problem, we need to make links first-class citizens in our
model. Following [5], we introduce into our schema object types that correspond to links
between entities. Each link object ` is associated with a tuple of entity objects (o 1 , . . . , ok )
that participate in the link. For example, a Hyperlink link object would be associated with
a pair of entities ? the linking page, and the linked-to page, which are part of the link
definition. We note that link objects may also have other attributes; e.g., a hyperlink object
might have attributes for the anchor words on the link.
As our goal is to predict link existence, we must consider links that exist and links that
do not. We therefore consider a set of potential links between entities. Each potential link
is associated with a tuple of entity objects, but it may or may not actually exist. We denote
this event using a binary existence attribute Exists, which is true if the link between the
associated entities exists and false otherwise. In our example, our model may contain a
potential link ` for each pair of webpages, and the value of the variable `.Exists determines
whether the link actually exists or not. The link prediction task now reduces to the problem
of predicting the existence attributes of these link objects.
An instantiation I specifies the set of entities of each entity type and the values of all
attributes for all of the entities. For example, an instantiation of the hypertext schema is
a collection of webpages, specifying their labels, the words they contain, and which links
between them exist. A partial instantiation specifies the set of objects, and values for some
of the attributes. In the link prediction task, we might observe all of the attributes for all
of the objects, except for the existence attributes for the links. Our goal is to predict these
latter attributes given the rest.
3
Relational Markov Networks
We begin with a brief review of the framework of undirected graphical models or Markov
Networks [13], and their extension to relational domains presented in [14].
Let V denote a set of discrete random variables and v an assignment of values to V.
A Markov network for V defines a joint distribution over V. It consists of an undirected
dependency graph, and a set of parameters associated with the graph. For a graph G, a
clique c is a set of nodes Vc in G, not necessarily maximal, such that each Vi , Vj ? Vc
are connected by an edge in G. Each clique c is associated with a clique potential ? c (Vc ),
which is a non-negative function defined on the joint domain of Vc . Letting
Q C(G) be the
set of cliques, the Markov network defines the distribution P (v) = Z1 c?C(G) ?c (vc ),
where Z is the standard normalizing partition function.
A relational Markov network (RMN) [14] specifies the cliques and potentials between
attributes of related entities at a template level, so a single model provides a coherent distribution for any collection of instances from the schema. RMNs specify the cliques using the
notion of a relational clique template, which specify tuples of variables in the instantiation
using a relational query language. (See [14] for details.)
For example, if we want to define cliques between the class labels of linked pages,
we might define a clique template that applies to all pairs page1,page2 and link of types
Webpage, Webpage and Hyperlink, respectively, such that link points from page1 to
page2. We then define a potential template that will be used for all pairs of variables
page1.Category and page2.Category for such page1 and page2.
Given a particular instantiation I of the schema, the RMN M produces an unrolled
Markov network over the attributes of entities in I, in the obvious way. The cliques in the
unrolled network are determined by the clique templates C. We have one clique for each
c ? C(I), and all of these cliques are associated with the same clique potential ? C .
Taskar et al. show how the parameters of an RMN over a fixed set of clique templates
can be learned from data. In this case, the training data is a single instantiation I, where
the same parameters are used multiple times ? once for each different entity that uses
a feature. A choice of clique potential parameters w specifies a particular RMN, which
induces a probability distribution Pw over the unrolled Markov network.
Gradient descent over w is used to optimize the conditional likelihood of the target variables given the observed variables in the training set. The gradient involves a term which
is the posterior probability of the target variables given the observed, whose computation
requires that we run probabilistic inference over the entire unrolled Markov network. In
relational domains, this network is typically large and densely connected, making exact
inference intractable. Taskar et al. therefore propose the use of belief propagation [13, 17].
4
Subgraph Templates in a Link Graph
The structure of link graphs has been widely used to infer importance of documents in
scientific publications [4] and hypertext (PageRank [12], Hubs and Authorities [8]). Social
networks have been extensively analyzed in their own right in order to quantify trends in
social interactions [16]. Link graph structure has also been used to improve document
classification [7, 6, 15].
In our experiments, we found that the combination of a relational language with a probabilistic graphical model provides a very flexible framework for modeling complex patterns
common in relational graphs. First, as observed by Getoor et al. [5], there are often correlations between the attributes of entities and the relations in which they participate. For
example, in a social network, people with the same hobby are more likely to be friends.
We can also exploit correlations between the labels of entities and the relation type. For
example, only students can be teaching assistants in a course. We can easily capture such
correlations by introducing cliques that involve these attributes. Importantly, these cliques
are informative even when attributes are not observed in the test data. For example, if we
have evidence indicating an advisor-advisee relationship, our probability that X is a faculty
member increases, and thereby our belief that X participates in a teaching assistant link
with some entity Z decreases.
We also found it useful to consider richer subgraph templates over the link graph. One
useful type of template is a similarity template, where objects that share a certain graphbased property are more likely to have the same label. Consider, for example, a professor
X and two other entities Y and Z. If X?s webpage mentions Y and Z in the same context, it
is likely that the X-Y relation and the Y-Z relation are of the same type; for example, if Y
is Professor X?s advisee, then probably so is Z. Our framework accomodates these patterns
easily, by introducing pairwise cliques between the appropriate relation variables.
Another useful type of subgraph template involves transitivity patterns, where the presence of an A-B link and of a B-C link increases (or decreases) the likelihood of an A-C link.
For example, students often assist in courses taught by their advisor. Note that this type
of interaction cannot be accounted for just using pairwise cliques. By introducing cliques
over triples of relations, we can capture such patterns as well. We can incorporate even
more complicated patterns, but of course we are limited by the ability of belief propagation
to scale up as we introduce larger cliques and tighter loops in the Markov network.
We note that our ability to model these more complex graph patterns relies on our use
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Figure 1: (a) Relation prediction with entity labels given. Relational models on average performed
better than the baseline Flat model. (b) Entity label prediction. Relational model Neigh performed
significantly better. (c) Relation prediction without entity labels. Relational models performed better
most of the time, even though there are schools that some models performed worse.
of an undirected Markov network as our probabilistic model. In contrast, the approach of
Getoor et al. uses directed graphical models (Bayesian networks and PRMs [9]) to represent a probabilistic model of both relations and attributes. Their approach easily captures
the dependence of link existence on attributes of entities. But the constraint that the probabilistic dependency graph be a directed acyclic graph makes it hard to see how we would
represent the subgraph patterns described above. For example, for the transitivity pattern,
we might consider simply directing the correlation edges between link existence variables
arbitrarily. However, it is not clear how we would then parameterize a link existence variable for a link that is involve in multiple triangles. See [15] for further discussion.
5
Experiments on Web Data
We collected and manually labeled a new relational dataset inspired by WebKB [2]. Our
dataset consists of Computer Science department webpages from 3 schools: Stanford,
Berkeley, and MIT. A total of 2954 of pages are labeled into one of eight categories: faculty,
student, research scientist, staff, research group, research project, course and organization
(organization refers to any large entity that is not a research group). Owned pages, which
are owned by an entity but are not the main page for that entity, were manually assigned to
that entity. The average distribution of classes across schools is: organization (9%), student
(40%), research group (8%), faculty (11%), course (16%), research project (7%), research
scientist (5%), and staff (3%).
We established a set of candidate links between entities based on evidence of a relation
between them. One type of evidence for a relation is a hyperlink from an entity page or one
of its owned pages to the page of another entity. A second type of evidence is a virtual
link: We assigned a number of aliases to each page using the page title, the anchor text of
incoming links, and email addresses of the entity involved. Mentioning an alias of a page
on another page constitutes a virtual link. The resulting set of 7161 candidate links were
labeled as corresponding to one of five relation types ? Advisor (faculty, student), Member (research group/project, student/faculty/research scientist), Teach (faculty/research scientist/staff, course), TA (student, course), Part-Of (research group, research proj) ? or
?none?, denoting that the link does not correspond to any of these relations.
The observed attributes for each page are the words on the page itself and the ?metawords? on the page ? the words in the title, section headings, anchors to the page from
other pages. For links, the observed attributes are the anchor text, text just before the link
(hyperlink or virtual link), and the heading of the section in which the link appears.
Our task is to predict the relation type, if any, for all the candidate links. We tried two
settings for our experiments: with page categories observed (in the test data) and page
categories unobserved. For all our experiments, we trained on two schools and tested on
the remaining school.
Observed Entity Labels. We first present results for the setting with observed page categories. Given the page labels, we can rule out many impossible relations; the resulting
label breakdown among the candidate links is: none (38%), member (34%), part-of (4%),
advisor (11%), teach (9%), TA (5%).
There is a huge range of possible models that one can apply to this task. We selected a
set of models that we felt represented some range of patterns that manifested in the data.
Link-Flat is our baseline model, predicting links one at a time using multinomial logistic regression. This is a strong classifier, and its performance is competitive with other
classifiers (e.g., support vector machines). The features used by this model are the labels of
the two linked pages and the words on the links going from one page and its owned pages
to the other page. The number of features is around 1000.
The relational models try to improve upon the baseline model by modeling the interactions between relations and predicting relations jointly. The Section model introduces
cliques over relations whose links appear consecutively in a section on a page. This
model tries to capture the pattern that similarly related entities (e.g., advisees, members
of projects) are often listed together on a webpage. This pattern is a type of similarity
template, as described in Section 4. The Triad model is a type of transitivity template, as
discussed in Section 4. Specifically, we introduce cliques over sets of three candidate links
that form a triangle in the link graph. The Section + Triad model includes the cliques of
the two models above.
As shown in Fig. 1(a), both the Section and Triad models outperform the flat model, and
the combined model has an average accuracy gain of 2.26%, or 10.5% relative reduction in
error. As we only have three runs (one for each school), we cannot meaningfully analyze
the statistical significance of this improvement.
As an example of the interesting inferences made by the models, we found a studentprofessor pair that was misclassified by the Flat model as none (there is only a single
hyperlink from the student?s page to the advisor?s) but correctly identified by both the Section and Triad models. The Section model utilizes a paragraph on the student?s webpage
describing his research, with a section of links to his research groups and the link to his
advisor. Examining the parameters of the Section model clique, we found that the model
learned that it is likely for people to mention their research groups and advisors in the same
section. By capturing this trend, the Section model is able to increase the confidence of the
student-advisor relation. The Triad model corrects the same misclassification in a different
way. Using the same example, the Triad model makes use of the information that both the
student and the teacher belong to the same research group, and the student TAed a class
taught by his advisor. It is important to note that none of the other relations are observed in
the test data, but rather the model bootstraps its inferences.
Unobserved Entity Labels. When the labels of pages are not known during relations
prediction, we cannot rule out possible relations for candidate links based on the labels of
participating entities. Thus, we have many more candidate links that do not correspond to
any of our relation types (e.g., links between an organization and a student). This makes the
existence of relations a very low probability event, with the following breakdown among
the potential relations: none (71%), member (16%), part-of (2%), advisor (5%), teach (4%),
TA (2%). In addition, when we construct a Markov network in which page labels are not
observed, the network is much larger and denser, making the (approximate) inference task
much harder. Thus, in addition to models that try to predict page entity and relation labels
simultaneously, we also tried a two-phase approach, where we first predict page categories,
and then use the predicted labels as features for the model that predicts relations.
For predicting page categories, we compared two models. Entity-Flat model is multinomial logistic regression that uses words and ?meta-words? from the page and its owned
pages in separate ?bags? of words. The number of features is roughly 10, 000. The Neighbors model is a relational model that exploits another type of similarity template: pages
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Figure 2: (a) Average precision/recall breakeven point for 10%, 25%, 50% observed links. (b)
Average precision/recall breakeven point for each fold of school residences at 25% observed links.
with similar urls often belong to the same category or tightly linked categories (research
group/project, professor/course). For each page, two pages with urls closest in edit distance are selected as ?neighbors?, and we introduced pairwise cliques between ?neighboring? pages. Fig. 1(b) shows that the Neighbors model clearly outperforms the Flat model
across all schools, by an average of 4.9% accuracy gain.
Given the page categories, we can now apply the different models for link classification. Thus, the Phased (Flat/Flat) model uses the Entity-Flat model to classify the page
labels, and then the Link-Flat model to classify the candidate links using the resulting entity labels. The Phased (Neighbors/Flat) model uses the Neighbors model to classify
the entity labels, and then the Link-Flat model to classify the links. The Phased (Neighbors/Section) model uses the Neighbors to classify the entity labels and then the Section
model to classify the links.
We also tried two models that predict page and relation labels simultaneously. The
Joint + Neighbors model is simply the union of the Neighbors model for page categories
and the Flat model for relation labels given the page categories. The Joint + Neighbors
+ Section model additionally introduces the cliques that appeared in the Section model
between links that appear consecutively in a section on a page. We train the joint models
to predict both page and relation labels simultaneously.
As the proportion of the ?none? relation is so large, we use the probability of ?none? to
define a precision-recall curve. If this probability is less than some threshold, we predict
the most likely label (other than none), otherwise we predict the most likely label (including none). As usual, we report results at the precision-recall breakeven point on the test
data. Fig. 1(c) show the breakeven points achieved by the different models on the three
schools. Relational models, both phased and joint, did better than flat models on the average. However, performance varies from school to school and for both joint and phased
models, performance on one of the schools is worse than that of the flat model.
6
Experiments on Social Network Data
The second dataset we used has been collected by a portal website at a large university that
hosts an online community for students [1]. Among other services, it allows students to
enter information about themselves, create lists of their friends and browse the social network. Personal information includes residence, gender, major and year, as well as favorite
sports, music, books, social activities, etc. We focused on the task of predicting the ?friendship? links between students from their personal information and a subset of their links. We
selected students living in sixteen different residences or dorms and restricted the data to
the friendship links only within each residence, eliminating inter-residence links from the
data to generate independent training/test splits. Each residence has about 15?25 students
and an average student lists about 25% of his or her house-mates as friends.
We used an eight-fold train-test split, where we trained on fourteen residences and tested
on two. Predicting links between two students from just personal information alone is a
very difficult task, so we tried a more realistic setting, where some proportion of the links
is observed in the test data, and can be used as evidence for predicting the remaining links.
We used the following proportions of observed links in the test data: 10%, 25%, and 50%.
The observed links were selected at random, and the results we report are averaged over
five folds of these random selection trials.
Using just the observed portion of links, we constructed the following flat features: for
each student, the proportion of students in the residence that list him/her and the proportion
of students he/she lists; for each pair of students, the proportion of other students they have
as common friends. The values of the proportions were discretized into four bins. These
features capture some of the relational structure and dependencies between links: Students
who list (or are listed by) many friends in the observed portion of the links tend to have links
in the unobserved portion as well. More importantly, having friends in common increases
the likelihood of a link between a pair of students.
The Flat model uses logistic regression with the above features as well as personal
information about each user. In addition to individual characteristics of the two people, we
also introduced a feature for each match of a characteristic, for example, both people are
computer science majors or both are freshmen.
The Compatibility model uses a type of similarity template, introducing cliques between each pair of links emanating from each person. Similarly to the Flat model, these
cliques include a feature for each match of the characteristics of the two potential friends.
This model captures the tendency of a person to have friends who share many characteristics (even though the person might not possess them). For example, a student may be
friends with several CS majors, even though he is not a CS major himself. We also tried
models that used transitivity templates, but the approximate inference with 3-cliques often
failed to converge or produced erratic results.
Fig. 2(a) compares the average precision/recall breakpoint achieved by the different
models at the three different settings of observed links. Fig. 2(b) shows the performance
on each of the eight folds containing two residences each. Using a paired t-test, the Compatibility model outperforms Flat with p-values 0.0036, 0.00064 and 0.054 respectively.
7
Discussion and Conclusions
In this paper, we consider the problem of link prediction in relational domains. We focus
on the task of collective link classification, where we are simultaneously trying to predict
and classify an entire set of links in a link graph. We show that the use of a probabilistic
model over link graphs allows us to represent and exploit interesting subgraph patterns in
the link graph. Specifically, we have found two types of patterns that seem to be beneficial
in several places. Similarity templates relate the classification of links or objects that share
a certain graph-based property (e.g., links that share a common endpoint). Transitivity
templates relate triples of objects and links organized in a triangle. We show that the use of
these patterns significantly improve the classification accuracy over flat models.
Relational Markov networks are not the only method one might consider applying to the
link prediction and classification task. We could, for example, build a link predictor that
considers other links in the graph by converting graph features into flat features [11], as
we did in the social network data. As our experiments show, even with these features, the
collective prediction approach work better. Another approach is to use relational classifiers
such as variants of inductive logic programming [10]. Generally, however, these methods
have been applied to the problem of predicting or classifying a single link at a time. It is
not clear how well they would extend to the task of simultaneously predicting an entire link
graph. Finally, we could apply the directed PRM framework of [5]. However, as shown
in [15], the discriminatively trained RMNs perform significantly better than generatively
trained PRMs even on the simpler entity classification task. Furthermore, as we discussed,
the PRM framework cannot represent (in any natural way) the type of subgraph patterns
that seem prevalent in link graph data. Therefore, the RMN framework seems much more
appropriate for this task.
Although the RMN framework worked fairly well on this task, there is significant room
for improvement. One of the key problems limiting the applicability of approach is the
reliance on belief propagation, which often does not converge in more complex problems.
This problem is especially acute in the link prediction problem, where the presence of all
potential links leads to densely connected Markov networks with many short loops. This
problem can be addressed with heuristics that focus the search on links that are plausible
(as we did in a very simple way in the webpage experiments). A more interesting solution
would be to develop a more integrated approximate inference / learning algorithm.
Our results use a set of relational patterns that we have discovered to be useful in the
domains that we have considered. However, many other rich and interesting patterns are
possible. Thus, in the relational setting, even more so than in simpler tasks, the issue of
feature construction is critical. It is therefore important to explore the problem of automatic
feature induction, as in [3].
Finally, we believe that the problem of modeling link graphs has numerous other applications, including: analyzing communities of people and hierarchical structure of organizations, identifying people or objects that play certain key roles, predicting current and
future interactions, and more.
Acknowledgments. This work was supported by ONR Contract F3060-01-2-0564-P00002
under DARPA?s EELD program. P. Abbeel was supported by a Siebel Grad. Fellowship.
References
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1,613 | 2,466 | Hierarchical Topic Models and
the Nested Chinese Restaurant Process
David M. Blei
[email protected]
Thomas L. Griffiths
[email protected]
Michael I. Jordan
[email protected]
Joshua B. Tenenbaum
[email protected]
University of California, Berkeley
Berkeley, CA 94720
Massachusetts Institute of Technology
Cambridge, MA 02139
Abstract
We address the problem of learning topic hierarchies from data. The
model selection problem in this domain is daunting?which of the large
collection of possible trees to use? We take a Bayesian approach, generating an appropriate prior via a distribution on partitions that we refer
to as the nested Chinese restaurant process. This nonparametric prior allows arbitrarily large branching factors and readily accommodates growing data collections. We build a hierarchical topic model by combining
this prior with a likelihood that is based on a hierarchical variant of latent
Dirichlet allocation. We illustrate our approach on simulated data and
with an application to the modeling of NIPS abstracts.
1
Introduction
Complex probabilistic models are increasingly prevalent in domains such as bioinformatics, information retrieval, and vision. These domains create fundamental modeling challenges due to their open-ended nature?data sets often grow over time, and as they grow
they bring new entities and new structures to the fore. Current statistical modeling tools
often seem too rigid in this regard; in particular, classical model selection techniques based
on hypothesis testing are poorly matched to problems in which data can continue to accrue
and unbounded sets of often incommensurate structures must be considered at each step.
An important instance of such modeling challenges is provided by the problem of learning
a topic hierarchy from data. Given a collection of ?documents,? each of which contains a
set of ?words,? we wish to discover common usage patterns or ?topics? in the documents,
and to organize these topics into a hierarchy. (Note that while we use the terminology of
document modeling throughout this paper, the methods that we describe are general.) In
this paper, we develop efficient statistical methods for constructing such a hierarchy which
allow it to grow and change as the data accumulate.
We approach this model selection problem by specifying a generative probabilistic model
for hierarchical structures and taking a Bayesian perspective on the problem of learning
these structures from data. Thus our hierarchies are random variables; moreover, these
random variables are specified procedurally, according to an algorithm that constructs the
hierarchy as data are made available. The probabilistic object that underlies this approach
is a distribution on partitions of integers known as the Chinese restaurant process [1]. We
show how to extend the Chinese restaurant process to a hierarchy of partitions, and show
how to use this new process as a representation of prior and posterior distributions for topic
hierarchies.
There are several possible approaches to the modeling of topic hierarchies. In our approach,
each node in the hierarchy is associated with a topic, where a topic is a distribution across
words. A document is generated by choosing a path from the root to a leaf, repeatedly
sampling topics along that path, and sampling the words from the selected topics. Thus
the organization of topics into a hierarchy aims to capture the breadth of usage of topics
across the corpus, reflecting underlying syntactic and semantic notions of generality and
specificity. This approach differs from models of topic hierarchies which are built on the
premise that the distributions associated with parents and children are similar [2]. We
assume no such constraint?for example, the root node may place all of its probability mass
on function words, with none of its descendants placing any probability mass on function
words. Our model more closely resembles the hierarchical topic model considered in [3],
though that work does not address the model selection problem which is our primary focus.
2
Chinese restaurant processes
We begin with a brief description of the Chinese restaurant process and subsequently show
how this process can be extended to hierarchies.
2.1
The Chinese restaurant process
The Chinese restaurant process (CRP) is a distribution on partitions of integers obtained
by imagining a process by which M customers sit down in a Chinese restaurant with an
infinite number of tables.1 The basic process is specified as follows. The first customer sits
at the first table. The mth subsequent customer sits at a table drawn from the following
distribution:
p(occupied table i | previous customers) =
p(next unoccupied table | previous customers) =
mi
?+m?1
?
?+m?1
(1)
where mi is the number of previous customers at table i, and ? is a parameter. After M
customers sit down, the seating plan gives a partition of M items. This distribution gives
the same partition structure as draws from a Dirichlet process [4]. However, the CRP also
allows several variations on the basic rule in Eq. (1), including a data-dependent choice of
? and a more general functional dependence on the current partition [5]. This flexibility
will prove useful in our setting.
The CRP has been used to represent uncertainty over the number of components in a mixture model. In a species sampling mixture [6], each table in the Chinese restaurant is
associated with a draw from p(? | ?) where ? is a mixture component parameter. Each
data point is generated by choosing a table i from Eq. (1) and then sampling a value from
the distribution parameterized by ?i (the parameter associated with that table). Given a
data set, the posterior under this model has two components. First, it is a distribution over
seating plans; the number of mixture components is determined by the number of tables
which the data occupy. Second, given a seating plan, the particular data which are sitting at
each table induce a distribution on the associated parameter ? for that table. The posterior
can be estimated using Markov chain Monte Carlo [7]. Applications to various kinds of
mixture models have begun to appear in recent years; examples include Gaussian mixture
models [8], hidden Markov models [9] and mixtures of experts [10].
1
The terminology was inspired by the Chinese restaurants in San Francisco which seem to have
an infinite seating capacity. It was coined by Jim Pitman and Lester Dubins in the early eighties [1].
2.2
Extending the CRP to hierarchies
The CRP is amenable to mixture modeling because we can establish a one-to-one relationship between tables and mixture components and a one-to-many relationship between
mixture components and data. In the models that we will consider, however, each data
point is associated with multiple mixture components which lie along a path in a hierarchy.
We develop a hierarchical version of the CRP to use in specifying a prior for such models.
A nested Chinese restaurant process can be defined by imagining the following scenario.
Suppose that there are an infinite number of infinite-table Chinese restaurants in a city. One
restaurant is determined to be the root restaurant and on each of its infinite tables is a card
with the name of another restaurant. On each of the tables in those restaurants are cards that
refer to other restaurants, and this structure repeats infinitely. Each restaurant is referred to
exactly once; thus, the restaurants in the city are organized into an infinitely-branched tree.
Note that each restaurant is associated with a level in this tree (e.g., the root restaurant is at
level 1 and the restaurants it refers to are at level 2).
A tourist arrives in the city for a culinary vacation. On the first evening, he enters the root
Chinese restaurant and selects a table using Eq. (1). On the second evening, he goes to the
restaurant identified on the first night?s table and chooses another table, again from Eq. (1).
He repeats this process for L days. At the end of the trip, the tourist has sat at L restaurants
which constitute a path from the root to a restaurant at the Lth level in the infinite tree
described above. After M tourists take L-day vacations, the collection of paths describe a
particular L-level subtree of the infinite tree (see Figure 1a for an example of such a tree).
This prior can be used to model topic hierarchies. Just as a standard CRP can be used to
express uncertainty about a possible number of components, the nested CRP can be used
to express uncertainty about possible L-level trees.
3
A hierarchical topic model
Let us consider a data set composed of a corpus of documents. Each document is a collection of words, where a word is an item in a vocabulary. Our basic assumption is that
the words in a document are generated according to a mixture model where the mixing
proportions are random and document-specific. Consider a multinomial variable z, and an
associated set of distributions over words p(w | z, ?), where ? is a parameter. These topics (one distribution for each possible value of z) are the basic mixture components in our
model. The document-specific mixing proportions associated with these components are
denoted by a vector ?. Temporarily assuming K possible topics in the corpus, an assumption that we will soon relax, z thus ranges over K possible values and ? is a K-dimensional
PK
vector. Our document-specific mixture distribution is p(w | ?) = i=1 ?i p(w | z = i, ?i )
which is a random distribution since ? is random.
We now specify the following two-level generative probabilistic process for generating a
document: (1) choose a K-vector ? of topic proportions from a distribution p(? | ?), where
? is a corpus-level parameter; (2) repeatedly sample words from the mixture distribution
p(w | ?) for the chosen value of ?. When the distribution p(? | ?) is chosen to be a Dirichlet
distribution, we obtain the latent Dirichlet allocation model (LDA) [11]. LDA is thus a twolevel generative process in which documents are associated with topic proportions, and the
corpus is modeled as a Dirichlet distribution on these topic proportions.
We now describe an extension of this model in which the topics lie in a hierarchy. For the
moment, suppose we are given an L-level tree and each node is associated with a topic.
A document is generated as follows: (1) choose a path from the root of the tree to a leaf;
(2) draw a vector of topic proportions ? from an L-dimensional Dirichlet; (3) generate the
words in the document from a mixture of the topics along the path from root to leaf, with
mixing proportions ?. This model can be viewed as a fully generative version of the cluster
abstraction model [3].
Finally, we use the nested CRP to relax the assumption of a fixed tree structure. As we have
seen, the nested CRP can be used to place a prior on possible trees. We also place a prior on
the topics ?i , each of which is associated with a restaurant in the infinite tree (in particular,
we assume a symmetric Dirichlet with hyperparameter ?). A document is drawn by first
choosing an L-level path through the restaurants and then drawing the words from the L
topics which are associated with the restaurants along that path. Note that all documents
share the topic associated with the root restaurant.
1. Let c1 be the root restaurant.
2. For each level ` ? {2, . . . , L}:
(a) Draw a table from restaurant c`?1 using Eq. (1). Set c` to be the restaurant
referred to by that table.
3. Draw an L-dimensional topic proportion vector ? from Dir(?).
4. For each word n ? {1, . . . , N }:
(a) Draw z ? {1, . . . , L} from Mult(?).
(b) Draw wn from the topic associated with restaurant cz .
This model, hierarchical LDA (hLDA), is illustrated in Figure 1b. The node labeled T
refers to a collection of an infinite number of L-level paths drawn from a nested CRP.
Given T , the cm,` variables are deterministic?simply look up the `th level of the mth path
in the infinite collection of paths. However, not having observed T , the distribution of c m,`
will be defined by the nested Chinese restaurant process, conditioned on all the c q,` for
q < m.
Now suppose we are given a corpus of M documents, w1 , . . . , wM . The posterior on
the c?s is essentially transferred (via the deterministic relationship), to a posterior on the
first M paths in T . Consider a new document wM +1 . Its posterior path will depend,
through the unobserved T , on the posterior paths of all the documents in the original corpus.
Subsequent new documents will also depend on the original corpus and any new documents
which were observed before them. Note that, through Eq. (1), any new document can
choose a previously unvisited restaurant at any level of the tree. I.e., even if we have a
peaked posterior on T which has essentially selected a particular tree, a new document can
change that hierarchy if its words provide justification for such a change.
In another variation of this model, we can consider a process that flattens the nested CRP
into a standard CRP, but retains the idea that a tourist eats L meals. That is, the tourist eats
L times in a single restaurant under the constraint that he does not choose the same table
twice. Though the vacation is less interesting, this model provides an interesting prior. In
particular, it can be used as a prior for a flat LDA model in which each document can use
at most L topics from the potentially infinite total set of topics. We examine such a model
in Section 5 to compare CRP methods with selection based on Bayes factors.
4
Approximate inference by Gibbs sampling
In this section, we describe a Gibbs sampling algorithm for sampling from the posterior
nested CRP and corresponding topics in the hLDA model. The Gibbs sampler provides
a method for simultaneously exploring the parameter space (the particular topics of the
corpus) and the model space (L-level trees).
The variables needed by the sampling algorithm are: wm,n , the nth word in the mth document (the only observed variables in the model); cm,` , the restaurant corresponding to the
`th topic in document m; and zm,n , the assignment of the nth word in the mth document
?
4 3 2 1
2 1
?2
?1
4 3
?
?3
?
c1
c2
?
c3
z
2
?5
4 3
?
?6
w
cL
(a)
?
N M
8
1
?4
(b)
Figure 1: (a) The paths of four tourists through the infinite tree of Chinese restaurants (L =
3). The solid lines connect each restaurant to the restaurants referred to by its tables. The
collected paths of the four tourists describe a particular subtree of the underlying infinite
tree. This illustrates a sample from the state space of the posterior nested CRP of Figure 1b
for four documents. (b) The graphical model representation of hierarchical LDA with a
nested CRP prior. We have separated the nested Chinese restaurant process from the topics.
Each of the infinite ??s corresponds to one of the restaurants.
to one of the L available topics. All other variables in the model?? and ??are integrated
out. The Gibbs sampler thus assesses the values of zm,n and cm,` .
Conceptually, we divide the Gibbs sampler into two parts. First, given the current state
of the CRP, we sample the zm,n variables of the underlying LDA model following the
algorithm developed in [12], which we do not reproduce here. Second, given the values of
the LDA hidden variables, we sample the cm,` variables which are associated with the CRP
prior. The conditional distribution for cm , the L topics associated with document m, is:
p(cm | w, c?m , z) ? p(wm | c, w?m , z)p(cm | c?m ),
where w?m and c?m denote the w and c variables for all documents other than m. This
expression is an instance of Bayes? rule with p(wm | c, w?m , z) as the likelihood of the data
given a particular choice of cm and p(cm | c?m ) as the prior on cm implied by the nested
CRP. The likelihood is obtained by integrating over the parameters ?, which gives:
!
Q
(?)
(w)
(w)
L
Y
?(ncm,` ,?m + W ?)
w ?(ncm,` ,?m + ncm,` ,m + ?)
p(wm | c, w?m , z) =
,
Q
(w)
(?)
(?)
`=1
w ?(ncm,` ,?m + ?) ?(ncm,` ,?m + ncm,` ,m + W ?)
(w)
where ncm,` ,?m is the number of instances of word w that have been assigned to the topic
indexed by cm,` , not including those in the current document, W is the total vocabulary
size, and ?(?) denotes the standard gamma function. When c contains a previously unvisited
(w)
restaurant, ncm,` ,?m is zero.
Note that the cm must be drawn as a block. The set of possible values for cm corresponds
to the union of the set of existing paths through the tree, equal to the number of leaves,
with the set of possible novel paths, equal to the number of internal nodes. This set can be
enumerated and scored using Eq. (1) and the definition of a nested CRP in Section 2.2.
Structure
3 (7 6 5)
4 (6 6 5 5)
4 (6 6 6 4)
5 (7 6 5 5 4)
5 (6 5 5 5 4)
(a)
Leaf error
1
2
14% 4%
30% 2%
36% 0%
40% 16%
22% 16%
Other
12%
20%
12%
14%
12%
(b)
Figure 2: (a) Six sample documents from
a 100 document corpus using the three
level bars hierarchy described in Section 5 and ? skewed toward higher levels. Each document has 1000 words from
a 25 term vocabulary. (b) The correct hierarchy found by the Gibbs sampler on
this corpus.
5
0
70%
48%
52%
30%
50%
Figure 3: Results of estimating hierarchies
on simulated data. Structure refers to a three
level hierarchy: the first integer is the number
of branches from the root and is followed by
the number of children of each branch. Leaf
error refers to how many leaves were incorrect in the resulting tree (0 is exact). Other
subsumes all other errors.
Examples and empirical results
In this section we describe a number of experiments using the models described above.
In all experiments, we let the sampler burn in for 10000 iterations and subsequently took
samples 100 iterations apart for another 1000 iterations. Local maxima can be a problem
in the hLDA model. To avoid them, we randomly restart the sampler 25 times and take the
trajectory with the highest average posterior likelihood.
We illustrate that the nested CRP process is feasible for learning text hierarchies in hLDA
by using a contrived corpus on a small vocabulary. We generated a corpus of 100 1000word documents from a three-level hierarchy with a vocabulary of 25 terms. In this corpus,
topics on the vocabulary can be viewed as bars on a 5 ? 5 grid. The root topic places its
probability mass on the bottom bar. On the second level, one topic is identified with the
leftmost bar, while the rightmost bar represents a second topic. The leftmost topic has two
subtopics while the rightmost topic has one subtopic. Figure 2a illustrates six documents
sampled from this model. Figure 2b illustrates the recovered hierarchy using the Gibbs
sampling algorithm described in Section 4.
In estimating hierarchy structures, hypothesis testing approaches to model selection are
impractical since they do not provide a viable method of searching over the large space
of trees. To compare the CRP method on LDA models with a standard approach, we implemented the simpler, flat model described at the end of Section 3. We generated 210
corpora of 100 1000-word documents each from an LDA model with K ? {5, . . . , 25},
L = 5, a vocabulary size of 100, and randomly generated mixture components from a symmetric Dirichlet (? = 0.1). For comparison with the CRP prior, we used the approximate
Bayes factors method of model selection [13], where one chooses the model that maximizes p(data | K)p(K) for various K and an appropriate prior. With the LDA model, the
Bayes factors method is much slower than the CRP as it involves multiple runs of a Gibbs
sampler with speed comparable to a single run of the CRP sampler. Furthermore, with the
Bayes factors method one must choose an appropriate range of K. With the CRP prior,
the only free parameter is ? (we used ? = 1.0). As shown in Figure 4, the CRP prior was
more effective than Bayes factors in this setting. We should note that both the CRP and
Bayes factors are somewhat sensitive to the choice ?, the hyperparameter to the prior on
the topics. However, in simulated data, this hyperparameter was known and thus we can
provide a fair comparison.
In a similar experiment, we generated 50 corpora each from five different hierarchies using
5
10 15 20
true dimension
Bayes factors
25
found dimension
5
15
25
found dimension
5
15
25
CRP prior
5
10 15 20
true dimension
25
Figure 4: (Left) The average dimension found by a CRP prior plotted against the true
dimension on simulated data (the true value is jiggled to see overlapping points). For each
dimension, we generated ten corpora with a vocabulary size of 100. Each corpus contains
100 documents of 1000 words. (Right) Results of model selection with Bayes factors.
an hLDA model and the same symmetric Dirichlet prior on topics. Each corpus has 100
documents of 1000 words from a vocabulary of 100 terms. Figure 3 reports the results of
sampling from the resulting posterior on trees with the Gibbs sampler from Section 4. In
all cases, we recover the correct structure more than any other and we usually recover a
structure within one leaf of the correct structure. In all experiments, no predicted structure
deviated by more than three nodes from the correct structure.
Lastly, to demonstrate its applicability to real data, we applied the hLDA model to a text
data set. Using 1717 NIPS abstracts from 1987?1999 [14] with 208,896 words and a vocabulary of 1600 terms, we estimated a three level hierarchy as illustrated in Figure 5. The
model has nicely captured the function words without using an auxiliary list, a nuisance
that most practical applications of language models require. At the next level, it separated
the words pertaining to neuroscience abstracts and machine learning abstracts. Finally, it
delineated several important subtopics within the two fields. These results suggest that
hLDA can be an effective tool in text applications.
6
Summary
We have presented the nested Chinese restaurant process, a distribution on hierarchical
partitions. We have shown that this process can be used as a nonparametric prior for a
hierarchical extension to the latent Dirichlet allocation model. The result is a flexible,
general model for topic hierarchies that naturally accommodates growing data collections.
We have presented a Gibbs sampling procedure for this model which provides a simple
method for simultaneously exploring the spaces of trees and topics.
Our model has two natural extensions. First, we have restricted ourselves to hierarchies
of fixed depth L for simplicity, but it is straightforward to consider a model in which L
can vary from document to document. Each document is still a mixture of topics along
a path in a hierarchy, but different documents can express paths of different lengths as
they represent varying levels of specialization. Second, although in our current model a
document is associated with a single path, it is also natural to consider models in which
documents are allowed to mix over paths. This would be a natural way to take advantage
of syntactic structures such as paragraphs and sentences within a document.
Acknowledgements
We wish to acknowledge support from the DARPA CALO program, Microsoft Corporation, and NTT Communication Science Laboratories.
the, of,
a, to,
and, in,
is, for
neurons, visual,
cells, cortex,
synaptic, motion,
response, processing
cell,
neuron,
circuit,
cells,
input,
i,
figure,
synapses
chip,
analog,
vlsi,
synapse,
weight,
digital,
cmos,
design
algorithm, learning,
training, method,
we, new,
problem, on
recognition,
speech,
character,
word,
system,
classification,
characters,
phonetic
b,
x,
e,
n,
p,
any,
if,
training
hidden,
units,
layer,
input,
output,
unit,
x,
vector
control,
reinforcement,
learning,
policy,
state,
actions,
value,
optimal
Figure 5: A topic hierarchy estimated from 1717 abstracts from NIPS01 through NIPS12.
Each node contains the top eight words from its corresponding topic distribution.
References
?
[1] D. Aldous. Exchangeability and related topics. In Ecole
d??et?e de probabilit?es de Saint-Flour,
XIII?1983, pages 1?198. Springer, Berlin, 1985.
[2] E. Segal, D. Koller, and D. Ormoneit. Probabilistic abstraction hierarchies. In Advances in
Neural Information Processing Systems 14.
[3] T. Hofmann. The cluster-abstraction model: Unsupervised learning of topic hierarchies from
text data. In IJCAI, pages 682?687, 1999.
[4] T. Ferguson. A Bayesian analysis of some nonparametric problems. The Annals of Statistics,
1:209?230, 1973.
[5] J. Pitman. Combinatorial Stochastic Processes. Notes for St. Flour Summer School. 2002.
[6] J. Ishwaran and L. James. Generalized weighted Chinese restaurant processes for species sampling mixture models. Statistica Sinica, 13:1211?1235, 2003.
[7] R. Neal. Markov chain sampling methods for Dirichlet process mixture models. Journal of
Computational and Graphical Statistics, 9(2):249?265, June 2000.
[8] M. West, P. Muller, and M. Escobar. Hierarchical priors and mixture models, with application
in regression and density estimation. In Aspects of Uncertainty. John Wiley.
[9] M. Beal, Z. Ghahramani, and C. Rasmussen. The infinite hidden Markov model. In Advances
in Neural Information Processing Systems 14.
[10] C. Rasmussen and Z. Ghahramani. Infinite mixtures of Gaussian process experts. In Advances
in Neural Information Processing Systems 14.
[11] D. Blei, A. Ng, and M. Jordan. Latent Dirichlet allocation. Journal of Machine Learning
Research, 3:993?1022, January 2003.
[12] T. Griffiths and M. Steyvers. A probabilistic approach to semantic representation. In Proceedings of the 24th Annual Conference of the Cognitive Science Society, 2002.
[13] R. Kass and A. Raftery. Bayes factors. Journal of the American Statistical Association,
90(430):773?795, 1995.
[14] S. Roweis. NIPS abstracts, 1987?1999. http://www.cs.toronto.edu/ roweis/data.html.
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|
1,614 | 2,467 | Unsupervised context sensitive language
acquisition from a large corpus
Zach Solan, David Horn, Eytan Ruppin
Sackler Faculty of Exact Sciences
Tel Aviv University
Tel Aviv, Israel 69978
{rsolan,horn,ruppin}@post.tau.ac.il
Shimon Edelman
Department of Psychology
Cornell University
Ithaca, NY 14853, USA
[email protected]
Abstract
We describe a pattern acquisition algorithm that learns, in an unsupervised fashion, a streamlined representation of linguistic structures from a
plain natural-language corpus. This paper addresses the issues of learning structured knowledge from a large-scale natural language data set,
and of generalization to unseen text. The implemented algorithm represents sentences as paths on a graph whose vertices are words (or parts of
words). Significant patterns, determined by recursive context-sensitive
statistical inference, form new vertices. Linguistic constructions are represented by trees composed of significant patterns and their associated
equivalence classes. An input module allows the algorithm to be subjected to a standard test of English as a Second Language (ESL) proficiency. The results are encouraging: the model attains a level of performance considered to be ?intermediate? for 9th-grade students, despite having been trained on a corpus (CHILDES) containing transcribed
speech of parents directed to small children.
1
Introduction
A central tenet of generative linguistics is that extensive innate knowledge of grammar is
essential to explain the acquisition of language from positive-only data [1, 2]. Here, we
explore an alternative hypothesis, according to which syntax is an abstraction that emerges
from exposure to language [3], coexisting with the corpus data within the same representational mechanism. Far from parsimonious, the representation we introduce allows partial
overlap of linguistic patterns or constructions [4]. The incremental process of acquisition
of patterns is driven both by structural similarities and by statistical information inherent
in the data, so that frequent strings of similar composition come to be represented by the
same pattern. The degree of abstraction of a pattern varies: it may be high, as in the case of
a frame with several slots, each occupied by a member of an equivalence class associated
with it, or low, as in the extreme case of idioms or formulaic language snippets, where there
is no abstraction at all [5, 6]. The acquired patterns represent fully the original data, and,
crucially, enable structure-sensitive generalization in the production and the assimilation of
unseen examples.
Previous approaches to the acquisition of linguistic knowledge, such as n-gram Hidden
Markov Models (HMMs) that use raw data, aimed not at grammar induction but rather at
expressing the probability of a sentence in terms of the conditional probabilities of its constituents. In comparison, statistical grammar induction methods aim to identify the most
probable grammar, given a corpus [7, 8]. Due to the difficulty of this task, a majority of
such methods have focused on supervised learning [9]. Grammar induction methods that
do attempt unsupervised learning can be categorized into two classes: those that use corpora tagged with part-of-speech information, and those that work with raw, untagged data.
The former includes such recent work as alignment-based learning [10], regular expression
(?local grammar?) extraction [11], and algorithms that rely on the Minimum Description
Length (MDL) principle [12].
The present work extends an earlier study [13] which offered preliminary results demonstrating the feasibility of unsupervised learning of linguistic knowledge from raw data.
Here, we describe a new learning model and its implementation and extensive testing on a
large corpus of transcribed spoken language from the CHILDES collection [14] (the larger
corpora used in many other computational studies do not focus on children-directed language). Our new results suggest that useful patterns embodying syntactic and semantic
knowledge of language can indeed be extracted from untagged corpora in an unsupervised
manner.
2
The ADIOS model
The ADIOS (Automatic DIstillation Of Structure) model has two components: (1) a Representational Data Structure (RDS) graph, and (2) a Pattern Acquisition (PA) algorithm
that progressively refines the RDS in an unsupervised fashion. The PA algorithm aims to
detect significant patterns (SP): similarly structured sequences of primitives that recur in
the corpus. Each SP has an associated equivalence class (EC), which is a set of alternative
primitives that may fit into the slot in the SP to construct a given path through the graph
(see Figure 1a). The manner whereby the model supports generalization is exemplified in
Figure 1c. The algorithm requires neither prior classification of the primitives into syntactic categories, nor even a pre-setting of their scope: it can bootstrap itself from a corpus in
which all the words have been broken down into their constituent characters.
One of the few free parameters in the earlier version of the model, ADIOS1, was the
length L of the typical pattern the system was expected to acquire. Although presetting
the value of L sufficed to learn simple artificial grammars, it proved to be problematic for
natural language corpora. On the one hand, a small value of L led to over-generalization,
because of insufficient uniformity of ECs associated with short SPs (not enough context
sensitivity). On the other hand, using large values of L in conjunction with the ADIOS1
statistical learning algorithm did not lead to the emergence of well-supported SPs. The
ADIOS2 model addresses this issue by first identifying long significant paths (SPATH) in
the graph, then analyzing their k-gram statistics to identify short significant patterns SP.
2.1
Step 1: identifying a significant path
For each pathi (sequence of elements e1 ? e2 ? . . . ? ek ) longer than a given threshold,
the algorithm constructs a set P = {p1 , . . . , pm } of paths of the same length as pathi .
Each of the paths in P(pathi ) consists of the same non-empty prefix (some sequence of
graph edges), an equivalence class of vertices, and the same non-empty suffix (another
sequence of edges); as an example, consider the set of three paths starting with ?is? and
ending with theend of sentence symbol ?END? in Figure 1. Each such set is assigned a
.
score S(P) = j s(pathj ), with s(?) defined by eq. 1. This score assesses the likelihood
that P captures a significant regularity rather than a random fluctuation in the data. The set
with the maximal score in a given pass over the corpus is the SPATH.
within-sentence index
Sentence Number
(a)
(1)
node
where
104
(1)
BEGIN
cat
101
edge
(2)
101
(1)
is
102
103
(2)
(1)
(5)
(2)
(4)
101 (3)
that
(3)
a
102 (3)
104
103 (4)
104
the
?
(5) 102
(5)
103
(5)
(3)
and
dog
(4)
horse
(4)
(6)
(6)
(6)
END
(7)
(6)
cat
(b)
where
104
that
is
a
dog
?
104
horse
and
the
104
BEGIN
103
103
102
101
pattern 140: is that a _____ ?
(c)
cat
the
dog
cat
dog
horse
103
102
Equivalence Class 200:
eat
is
horse
Equivalence
Class 200
END
101
{cat, dog, horse}
sleep
ing
play
new equivalence
class 201
Figure 1: (a) A small portion of the RDS, which is a directed multi-graph, for a simple
corpus containing sentences #101 (is that a cat?) #102 (is that a dog?) #103 (and is
that a horse?) #104 (where is the dog?). Each sentence is depicted by a solid colored
line; edge direction is marked by arrows and is labeled by the sentence number and withinsentence index. The sentences in this example join a pattern is that a {dog, cat, horse}
?. (b). The abstracted pattern and the equivalence class associated with it are highlighted
(edges that belong to sequences not subsumed by this pattern, e.g., #104, are untouched).
(c) The identification of new significant patterns is done using the acquired equivalence
classes (e.g., #200). In this manner, the system ?bootstraps? itself, recursively distilling
more and more complex patterns. This kind of abstraction also supports generalization: the
original three sentences (shaded paths) form a pattern with two equivalence classes, which
can then potentially generate six new sentences (e.g., the cat is play-ing and the horse is
eat-ing).
s(pathi )
=
P (k) (pathi ) log P (k) (pathi )/P (2) (pathi )
P (k) (pathi )
=
P (e1 )P (e2 |e1 )P (e3 |e1 ? e2 )...P (ek |e1 ? e2 ? ... ? ek?1 ) (2)
=
P (e1 )P (e2 |e1 )P (e3 |e2 )...P (ek |ek?1 )
P
(2)
(pathi )
(1)
(3)
The algorithm estimates the probabilities of different paths from the respective k-gram
statistics (k being the length of the paths in the set under consideration), as per eq. 2. We
observe that P (1) (pathi ) corresponds to the ?first order? probability of choosing the set of
nodes e1 , . . . , ek without taking into account their sequential order along the path. Thus,
P (1) (pathi ) = P (e1 )P (e2 )P (e3 ) . . . P (ek ). In comparison, P (2) (see eq. 3) is a better
candidate for identifying significant strings, as opposed to mere sets of nodes, because it
takes into account the sequence of nodes along the path.
2.2
Step 2: identifying a significant pattern
Once the SPATH set is determined, the algorithm calculates the degree of cohesion cij
for each one of its member sub-paths, according to eq. 4. The k-gram matrix in eq. 4
accumulates all the statistics through order k th ? 1 of the SPATH embedded in the graph,
with the zeroth order statistics located at the diagonal. The sub-path with the highest cscore is now tagged as a Significant Pattern.
Our experience shows that the two-stage mechanism just described induces coherent equivalence classes, leading to the formation of meaningful short patterns. The new pattern is
added as a new vertex to the RDS graph, replacing the elements and edges it subsumes
(Figure 1(b)). Note that only those edges of the multi-graph that belong to the detected
pattern are rewired; edges that belong to sequences not subsumed by the pattern are left
intact. This highly context-sensitive method of pattern abstraction, which is unique to our
approach, allows ADIOS to achieve a high degree of representational parsimony without
sacrificing generalization power.
?
p(e1 )
p(e2 |e1 )
?
?
p(e3 |e1 e2 )
P =?
?
..
?
.
p(ek |e1 e2 ...ek?1 )
p(e1 |e2 )
p(e2 )
p(e3 |e2 )
..
.
p(ek |e2 e3 ...ek?1 )
Pij
cij = Pij log
Pi,j+1
p(e1 |e2 e3 )
p(e2 |e3 )
p(e3 )
..
.
p(ek |e3 e4 ...ek?1 )
for i > j
?
... p(e1 |e2 e3 ...ek )
... p(e2 |e3 e4 ...ek ) ?
... p(e3 |e4 e5 ...ek ) ?
?
?
..
?
.
...
p(ek )
(4)
During the pass over the corpus, the list of equivalence sets is updated continuously; new
significant patterns are found using the current equivalence classes. For each set of candidate paths, the algorithm tries to fit one or more equivalence classes from the pool it maintains. Because an element can a appear in several classes, the algorithm must check different combinations of equivalence classes. The winner combination is always the largest
class for which most of the members are found among the candidate paths in the set (the
ratio between the number of members that have been found among the paths and the total
number of members in the equivalence class is compared to a fixed threshold as one of the
configuration acceptance criteria). When not all the members appear in an existing set, the
algorithm creates a new equivalence class containing only those members that do. Thus,
as the algorithm processes more and more text, it bootstraps itself and enriches the RDS
graph structure with new SPs and their accompanying equivalence sets. The recursive nature of this process enables the algorithm to form more and more complex patterns, in a
hierarchical manner.
The relationships among the distilled patterns can be visualized in a tree format, with tree
depth corresponding to the level of recursion (e.g., Figure 2). Such a tree can be seen as
a blueprint for creating acceptable (?grammatical?) sequences of elements (strings). The
number of all possible string configurations can be estimated and compared to the number
of examples seen in the training corpus. The reciprocal of their ratio, ?, is the generalization
factor, which can be calculated for each pattern in the RDS graph (e.g., in Figure 1(c),
? = 0.33). Patterns whose significance score S and generalization factor ? are beneath
certain thresholds are rejected. The algorithm halts if it processes a given amount of text
without finding a new significant pattern or equivalence set (in real language acquisition
this process may never stop).
2.3
The test module
A collection of patterns distilled from a corpus can be seen as a kind of empirically determined construction grammar; cf. [5], p.63. The patterns can eventually become highly
abstract, thus endowing the model with an ability to generalize to unseen inputs. In pro-
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14384 (1)
13 17
Figure 2: Two typical patterns extracted from a subset of the CHILDES collection [14].
Hundreds of such patterns and equivalence classes (underscored) together constitute a concise representation of the raw data. Some of the phrases that can be described/generated
by patterns #16555 and #16543 are: let?s change her...; I thought you gonna change
her...; I was going to go to the.... None of these sentences appear in the training data,
illustrating the ability of ADIOS to generalize. The numbers in parentheses denote the
generalization factor ? of the patterns and their components (e.g., pattern #16555 generates 86% new strings, while pattern #16543 generates 75% new strings). The generation
process, which operates as a depth-first search of the tree corresponding to a pattern, is
illustrated on the left. For each non-terminal, the children are scanned from left to right;
for each equivalence class (underscored), one member is chosen. The scan continues from
the node corresponding to that member, with the elements reached at the terminal nodes
being written out.
duction, generalization is possible, for example, when two equivalence classes are placed
next to each other in a pattern, creating new paths among the members of the equivalence
classes. In comprehension, generalization can also ensue from partial activation of existing
patterns by novel inputs. This function is supported by the test module, designed to process
a novel sentence by forming its distributed representation in terms of activities of existing
patterns (a similar approach has been proposed for novel object and scene representation
in vision [15]). These values, which can be used to support grammaticality judgment, are
computed by propagating activation from bottom (the terminals) to top (the patterns) of the
RDS. The initial activities aj of the terminals ej are calculated given the novel stimulus
s1 , . . . , sk as follows:
P (sl , ej )
(5)
aj = max P (sl , ej ) log
l=1..k
P (sl )P (ej )
where P (sl , ej ) is the joint probability of sl and ej appearing in the same equivalence class,
and P (sl ) and P (ej ) are the probabilities of sl and ej appearing in any equivalence class.
For an equivalence class, the value propagated upwards is the strongest non-zero activation
of its members; for a pattern, it is the average weight of the children nodes, on the condition
that all the children were activated by adjacent inputs. Activity propagation continues until
it reaches the top nodes of the pattern lattice. When the algorithm encounters a novel word,
all the members of the terminal equivalence class contribute a value of = 0.01, which is
then propagated upwards as usual. This enables the model to make an educated guess as to
the meaning of the unfamiliar word, by considering the patterns that become active.
3
Empirical results
3.1
Working with real data: the CHILDES? parents
To illustrate the scalability of our method, we describe here briefly the outcome of applying
the PA algorithm to a subset of the CHILDES collection [14], which consists of transcribed
speech produced by, or directed at, children. The corpus we selected contained 300,000
sentences (1.3 million tokens) produced by parents. The following results were derived
from a snapshot of the algorithm?s state after 14 real-time days. Working at a rate of 250
patterns per day, the algorithm identified 3400 patterns and 3200 equivalence classes, representing the corpus in terms of these elements. The outcome (for some examples, see
Figure 2) was encouraging: the algorithm found intuitively significant SPs and produced
semantically adequate corresponding equivalence sets. The algorithm?s considerable ability to recombine and reuse constructions it learns is illustrated by the following examples,
in which a few of the sentences generated by ADIOS (left) are shown alongside sentences
from CHILDES described by the same compositions of patterns:
ADIOS
CHILDES (parents? speech)
what doe s Spot say ?
I don ?t think it ? s good !
it ? s gon ta go first .
there ? s a cup and there ? s some lamb s .
where doe s it go ?
that ? s good !
dog ? s gon ta eat first .
there ? s a table and there ? s some chair s .
3.2
Novel inputs
We have assessed the ability of the ADIOS model to deal with novel inputs by training it
on the CHILDES collection and then subjecting it to a grammaticality judgment test, in
the form of multiple choice questions used in English as Second Language (ESL) classes.
The particular test (http://www.forumeducation.net/servlet/pages/vi/mat/gram/dia001.htm)
has been administered to more than 10, 000 people in the G?oteborg (Sweden) education
system as a diagnostic tool when assessing students on upper secondary levels (that is,
children who typically had 9 years of school, but only 6-7 years of English; a test designed
for assessing proficiency of younger subjects in their native language would be more suitable, but is not available). The test consists of 100 three-choice questions; a score lower
than 50% is considered pre-intermediate, 50%?70% intermediate, and a score greater than
70% ? advanced, with 65% being the average score for the population mentioned. For each
of the three choices in a given question, our algorithm provided a grammaticality score.
The choice with the highest score was declared as the winner; if two choices received the
same top score, the answer was ?don?t know?. The algorithm?s performance in this test at
different stages of learning is plotted in Figure 3 versus the number of corpus sentences that
have been processed. Over the course of training, the proportion of questions that received
a definite answer grew (solid curve), while the proportion of correct answers remained
around 60% (dashed curve).
The best results were achieved with the ensemble of patterns distilled from two separate
runs (two different generalization factors were applied in each run: 0.01 and 0.05). As
a benchmark, we compared the performance of ADIOS in this test with that of a word
bi-gram model. The latter was tested using the same procedure as ADIOS, except that
significant patterns in the bi-gram model were defined as all the word pairs in the corpus
(we emphasize that there is no training phase in the bi-gram model, as all the ?patterns? are
already available in the raw data). ADIOS outperformed the bi-gram model by answering
60% of the questions with 60% hits, compared to 20% of the questions with only 45% hits
for the latter (note that chance performance in this test is 33%).
Figure 3: The performance of ADIOS2 in an ESL test based on grammaticality judgment,
plotted against the number of sentences (paths) scanned during training. The solid curve
represents the percentage of questions with a valid answer; the dashed curve shows the
percentage of correct answers.
4
Concluding remarks
The ADIOS model incrementally learns the (morpho)syntax of English from ?raw? input by distilling structural regularities (which can be thought of as constructions [16, 4])
from the accrued statistical co-occurrence and contextual cues. The resulting pattern-based
representations are more powerful than finite automata because of their potential for recursion. Their depth, however, is not unbounded (rather, it is driven by the demands of
the training data), a limitation that actually makes ADIOS a better candidate model for
psycholinguistics (cf. the human limitations on processing recursion [17]). The patterns
learned by ADIOS are also more powerful than context-free rewriting rules, because of
their conservative nature: members of an equivalence class are only ever considered as interchangeable in a specific context, a characteristic that distinguishes ADIOS from related
approaches [18, 10, 9]. On the one hand, this results in larger ? but not unmanageable ?
demands on memory (more patterns need to be stored); on the other hand, crucially, it leads
to efficient unsupervised probabilistic learning, and subsequent judicious use, of linguistic
knowledge.
The ultimate goal of this project is to address the entire spectrum of English syntax-related
phenomena (and, eventually, semantics, which, as the construction grammarians hold, is
intimately connected to syntax [16, 4]). With respect to some of these, the ADIOS model
is already known to behave reasonably: for example, subject-verb agreement (even longrange) is captured properly, due to the conservative structured pattern abstraction. While
providing empirical evidence that can be brought to bear on the poverty of the stimulus
argument for innateness, our work does not, of course, resolve completely the outstanding
issues. In particular, the treatment of many aspects of syntax such as anaphora, auxiliaries,
wh-questions, passive, control, etc. [19], awaits both further computational experimentation
and further theoretical work.
Acknowledgments. Supported by the US-Israel Binational Science Foundation, the Dan
David Prize Foundation, and the Horowitz Center for Complexity Science. We thank Todd
Siegel for helpful suggestions.
References
[1] N. Chomsky. Knowledge of language: its nature, origin, and use. Praeger, New York,
1986.
[2] S. Pinker. The Language Instinct: How the Mind Creates Language. William Morro,
New York, NY, 1994.
[3] P. J. Hopper. Emergent grammar. In M. Tomasello, editor, The new psychology of
language, pp. 155?175. Erlbaum, Mahwah, NJ, 1998.
[4] W. Croft. Radical Construction Grammar: syntactic theory in typological perspective. Oxford University Press, Oxford, 2001.
[5] R. W. Langacker. Foundations of cognitive grammar, volume I: theoretical prerequisites. Stanford University Press, Stanford, CA, 1987.
[6] A. Wray. Formulaic language and the lexicon. Cambridge University Press, Cambridge, UK, 2002.
[7] K. Lari and S. J. Young. The estimation of stochastic context-free grammars using
the Inside-Outside algorithm. Computer Speech and Language, 4:35?56, 1990.
[8] F. Pereira and Y. Schab`es. Inside-Outside reestimation from partially bracketed corpora. In Annual Meeting of the ACL, pp. 128?135, 1992.
[9] D. Klein and C. D. Manning. Natural language grammar induction using a
constituent-context model. In T. G. Dietterich, S. Becker, and Z. Ghahramani, ed.,
Advances in Neural Information Processing Systems 14, Cambridge, MA, 2002. MIT
Press.
[10] M. van Zaanen and P. Adriaans. Comparing two unsupervised grammar induction
systems: Alignment-based learning vs. EMILE. Report 05, School of Computing,
Leeds University, 2001.
[11] M. Gross. The construction of local grammars. In E. Roche and Y. Schab`es, ed.,
Finite-State Language Processing, pp. 329?354. MIT Press, Cambridge, MA, 1997.
[12] J. G. Wolff. Learning syntax and meanings through optimization and distributional
analysis. In Y. Levy, I. M. Schlesinger, and M. D. S. Braine, ed., Categories and
Processes in Language Acquisition, pp. 179?215. Lawrence Erlbaum, Hillsdale, NJ,
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[13] Z. Solan, E. Ruppin, D. Horn, and S. Edelman. Automatic acquisition and efficient
representation of syntactic structures. In S. Thrun, editor, Advances in Neural Information Processing, volume 15, Cambridge, MA, 2003. MIT Press.
[14] B. MacWhinney and C. Snow. The child language exchange system. Journal of
Computational Lingustics, 12:271?296, 1985.
[15] S. Edelman. Constraining the neural representation of the visual world. Trends in
Cognitive Sciences, 6:125?131, 2002.
[16] A. E. Goldberg. Constructions: A construction grammar approach to argument structure. University of Chicago Press, Chicago, 1995.
[17] M. C. MacDonald and M. H. Christiansen. Reassessing working memory: A comment on Just and Carpenter (1992) and Waters and Caplan (1996). Psychological
Review, 109:35?54, 2002.
[18] A. Clark. Unsupervised Language Acquisition: Theory and Practice. PhD thesis,
COGS, University of Sussex, 2001.
[19] I. A. Sag and T. Wasow. Syntactic theory: a formal introduction. CSLI Publications,
Stanford, CA, 1999.
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1,615 | 2,468 | No Unbiased Estimator of the Variance of
K-Fold Cross-Validation
Yoshua Bengio and Yves Grandvalet
Dept. IRO, Universit?e de Montr?eal
C.P. 6128, Montreal, Qc, H3C 3J7, Canada
{bengioy,grandvay}@iro.umontreal.ca
Abstract
Most machine learning researchers perform quantitative experiments to
estimate generalization error and compare algorithm performances. In
order to draw statistically convincing conclusions, it is important to estimate the uncertainty of such estimates. This paper studies the estimation
of uncertainty around the K-fold cross-validation estimator. The main
theorem shows that there exists no universal unbiased estimator of the
variance of K-fold cross-validation. An analysis based on the eigendecomposition of the covariance matrix of errors helps to better understand
the nature of the problem and shows that naive estimators may grossly
underestimate variance, as con?rmed by numerical experiments.
1
Introduction
The standard measure of accuracy for trained models is the prediction error (PE), i.e. the
expected loss on future examples. Learning algorithms themselves are often compared on
their average performance, which estimates expected value of prediction error (EPE) over
training sets. If the amount of data is large enough, PE can be estimated by the mean
error over a hold-out test set. The hold-out technique does not account for the variance
with respect to the training set, and may thus be considered inappropriate for the purpose
of algorithm comparison [4]. Moreover, it makes an inef?cient use of data which forbids
its application to small sample sizes. In this situation, one resorts to computer intensive
resampling methods such as cross-validation or bootstrap to estimate PE or EPE. We
focus here on K-fold cross-validation. While it is known that cross-validation provides an
unbiased estimate of EPE, it is also known that its variance may be very large [2]. This
variance should be estimated to provide faithful con?dence intervals on PE or EPE, and
to test the signi?cance of observed differences between algorithms. This paper provides
theoretical arguments showing the dif?culty of this estimation.
The dif?culties of the variance estimation have already been addressed [4, 7, 8]. Some
distribution-free bounds on the deviations of cross-validation are available, but they are
speci?c to locally de?ned classi?ers, such as nearest neighbors [3]. This paper builds upon
the work of Nadeau and Bengio [8], which investigated in detail the theoretical and practical merits of several estimators of the variance of cross-validation. Our analysis departs
from this work in the sampling procedure de?ning the cross-validation estimate. While [8]
considers K independent training and test splits, we focus on the standard K-fold cross-
validation procedure, with no overlap between test sets: each example is used once and
only once as a test example.
2
General Framework
Formally, we have a training set D = {z1 , . . . , zn }, with zi ? Z, assumed independently
sampled from an unknown distribution P . We also have a learning algorithm A : Z ? ? F
which maps a data set to a function. Here we consider symmetric algorithms, i.e. A is
insensitive to the ordering of examples in the training set D. The discrepancy between
the prediction and the observation z is measured by a loss functional L : F ? Z ? R.
For example one may take in regression L(f, (x, y)) = (f (x) ? y)2 , and in classi?cation
L(f, (x, y)) = 1f (x)6=y .
Let f = A(D) be the function returned by algorithm A on the training set D. In
application-based evaluation, the goal of learning is usually stated as the minimization
of the expected loss of f = A(D) on future test examples:
PE(D) = E[L(f, z)] ,
(1)
where the expectation is taken with respect to z ? P . To evaluate and compare learning algorithms [4] we care about the expected performance of learning algorithm A over
different training sets:
EPE(n) = E[L(A(D), z)] ,
(2)
where the expectation is taken with respect to D ? z independently sampled from P n ? P .
When P is unknown, PE and EPE have to be estimated, and it is crucial to assess the
uncertainty attached to this estimation. Although this point is often overlooked, estimating
c and EPE
[ requires caution, as illustrated here.
the variance of the estimates PE
2.1
Hold-out estimates of performance
c is given by the
The mean error over a hold-out test set estimates PE, and the variance of PE
usual variance estimate for means of independent variables. However, this variance estima[ the test errors are correlated when the training set is considered as
tor is not suited to EPE:
a random variable.
Figure 1 illustrates how crucial it is to take these correlations into account. The average
ratio (estimator of variance/empirical variance) is displayed for two variance estimators, in
an ideal situation where 10 independent training and test sets are available. The average of
?b1 /?, the naive variance estimator ignoring correlations, shows that this estimate is highly
down-biased, even for large sample sizes.
1
0.8
0.6
0.4
100
200
300
400
500
600
Figure 1: Average ratio (estimator of variance/empirical variance) on 100 000 experiments:
?b1 /? (ignoring correlations, lower curve) and ?b2 /? (taking into account correlations, upper
curve) vs. sample size n. The error bars represent ?2 standard errors on the average value.
Experiment 1 Ideal hold-out estimate of EPE.
We have K = 10 independent training sets D1 , . . . , DK of n independent examples
zi = (xi , yi ), where xi = (xi1 , . .p
. , xid )0 is a d-dimensional centered, unit covariance
Pd
Gaussian variable (d = 30), yi = 3/d k=1
p xik + ?i with ?i being independent, centered, unit variance Gaussian variables (the 3/d factor provides R2 ' 3/4). We also
have K independent test sets T1 , . . . , TK of size n sampled from the same distribution.
The learning algorithm consists in ?tting a line by ordinary least squares, and the
? =
[ = L
estimate of EPE is the average quadratic loss on test examples EPE
PK 1 P
1
k=1 n
zi ?Tk Lki , where Lki = L(A(Dk ), zi ).
K
PK P
1
? 2 , which
[ is ?b1 =
The ?rst estimate of variance of EPE
(Lki ? L)
Kn(Kn?1)
k=1
i
is unbiased provided there is no correlation between test errors. The second estimate is
PK P
1
?
?
?b2 = K(K?1)n
2
k=1
i,j (Lki ? L)(Lkj ? L), which estimates correlations.
Note that Figure 1 suggests that the naive estimator of variance ?b1 asymptotically converges
to the true variance. This can be shown by taking advantage of the results in this paper, as
a.s.
long as the learning algorithm converges (PE(D) ? limn?? EPE(n)), i.e. provided that
[ is due to the ?nite test size.
the only source of variability of EPE
2.2
K-fold cross-validation estimates of performance
In K-fold cross-validation [9], the data set D is ?rst chunked into K disjoint subsets (or
blocks) of the same size m = n/K (to simplify the analysis below we assume that n is a
multiple of K). Let us write Tk for the k-th such block, and Dk the training set obtained
by removing the elements in Tk from D. The estimator is
CV =
K
1 X 1 X
L(A(Dk ), zi ) .
K
m
k=1
(3)
zi ?Tk
Under stability assumptions on A, CV estimates PE(D) at least as accurately as the
training error [6]. However, as CV is an average of unbiased estimates of PE(D 1 ),
. . . , PE(DK ), a more general statement is that CV estimates unbiasedly EPE(n?m).
Note that the forthcoming analysis also applies to the version of cross-validation dedicated
to comparing algorithms, using matched pairs
K
1 X 1 X
?CV =
L(A1 (Dk ), zi ) ? L(A2 (Dk ), zi ) ,
K
m
k=1
zi ?Tk
and to the delete-m jackknife estimate of PE(D) debiasing the training error (see e.g. [5]):
?
!
n
K X
n
X
1X
1
1X
JK =
L(A(D), zi )?(K?1)
L(A(Dk ), zi ) ?
L(A(D), zi ) .
n i=1
K(n ? m)
n i=1
k=1 zi ?Dk
In what follows, CV, ?CV and JK will generically be denoted by ?
?:
?
?
=
n
K
1 X 1 X
1X
ei =
ei ,
n i=1
K
m
k=1
i?Tk
where, slightly abusing notation, i ? Tk means zi ? Tk and
?
? L(A(Dk ), zi )
L(A1 (Dk ), zi ) ? L(A2 (Dk ), zi )
?i ? Tk , ei =
? KL(A(D), z ) ? P
i
`6=k L(A(D` ), zi )
for ?
? = CV ,
for ?
? = ?CV ,
for ?
? = JK .
Note that ?
? is the average of identically distributed (dependent) variables. Thus, it asymptotically converges to a normally distributed variable, which is completely characterized by
its expectation E[?
?] and its variance Var[?
?].
3
Structure of the Covariance Matrix
P
The variance of ?
? is ? = n12 i,j Cov(ei , ej ) . By using symmetry over permutations of
the examples in D, we show that the covariance matrix has a simple block structure.
Lemma 1 Using the notation introduced in section 2.2, 1) all ei are identically distributed;
2) all pairs (ei , ej ) belonging to the same test block are jointly identically distributed; 3)
all pairs (ei , ej ) belonging to different test blocks are jointly identically distributed;
Proof: derived immediately from the permutation-invariance of P (D) and the symmetry
of A. See [1] for details and the proofs not shown here for lack of space.
Corollary 1 The covariance matrix ? of cross-validation errors e = (e1 , . . . , en )0 has
the simple block structure depicted in Figure 2: 1) all diagonal elements are identical
?i, Cov(ei , ei ) = Var[ei ] = ? 2 ; 2) all the off-diagonal entries of the K m ? m diagonal
blocks are identical ?(i, j) ? Tk2 : j 6= i, T (j) = T (i), Cov(ei , ej ) = ?; 3) all the
remaining entries are identical ?i ? Tk , ?j ? T` : ` 6= k, Cov(ei , ej ) = ?.
n
z
| {z }
}|
{
m
Figure 2: Structure of the covariance matrix.
Corollary 2 The variance of the cross-validation estimator is a linear combination of three
moments:
1 X
1
m?1
n?m
? =
Cov(ei , ej ) = ? 2 +
?+
?
(4)
2
n i,j
n
n
n
Hence, the problem of estimating ? does not involve estimating n(n + 1)/2 covariances,
but it cannot be reduced to that of estimating a single variance parameter. Three components intervene, which may be interpreted as follows when ?
? is the K-fold cross-validation
estimate of EPE:
1. the variance ? 2 is the average (taken over training sets) variance of errors for
?true? test examples (i.e. sampled independently from the training sets) when
algorithm A is fed with training sets of size m(K ? 1);
2. the within-block covariance ? would also apply to these ?true? test examples; it
arises from the dependence of test errors stemming from the common training set.
3. the between-blocks covariance ? is due to the dependence of training sets (which
share n(K ? 2)/K examples) and the fact that test block Tk appears in all the
training sets D` for ` 6= k.
4
No Unbiased Estimator of Var[?
?] Exists
Consider a generic estimator ?? that depends on the sequence of cross-validation errors
e = (e1 , e2 , . . . , en )0 . Assuming ?? is analytic in e, consider its Taylor expansion:
X
X
X
?? = ?0 +
?1 (i)ei +
?2 (i, j)ei ej +
?3 (i, j, k)ei ej ek + . . .
(5)
i
i,j
i,j,k
? = Var[?
We ?rst show that for unbiased variance estimates (i.e. E[ ?]
?]), all the ?i coef?cients must vanish except for the second order coef?cients ? 2,i,j .
Lemma 2 There is no universal unbiased estimator of Var[?
?] that involves the e i in a
non-quadratic way.
Proof: Take the expected value of ?? expressed as in (5), and equate it with Var[?
?] (4).
Since estimators that include moments other than the second moments in their expectation
are biased, we now focus on estimators which are quadratic forms of the errors, i.e.
X
?? = e0 We =
Wij ei ej .
(6)
i,j
Lemma 3 The expectation of quadratic estimators ?? de?ned as in (6) is a linear combination of only three terms
? = a(? 2 + ?2 ) + b(? + ?2 ) + c(? + ?2 ) ,
E[?]
(7)
where (a, b, c) are de?ned as follows:
?
Pn
?
?
ii ,
? a = Pi=1 W
P
?
K P
b =
k=1
i?Tk
j?Tk :j6=i Wij ,
?
P
P
PK P
?
?
c =
k=1
`6=k
i?Tk
j?T` Wij .
A ?trivial? representer of estimators with this expected value is
?? = as1 + bs2 + cs3 ,
(8)
where (s1 , s2 , s3 ) are the only quadratic statistics of e that are invariants to the within
blocks and between blocks permutations described in Lemma 1:
?
Pn 2
?
1
?
ei ,
?
? s1 = n i=1P
P
?
K P
1
s2 = n(m?1) k=1 i?Tk j?Tk :j6=i ei ej ,
(9)
?
PK P
P
P
?
?
1
? s =
3
k=1
`6=k
i?Tk
j?T` ei ej .
n(n?m)
Proof: in (6), group the terms that have the same expected values (from Corollary 1).
Theorem 1 There exists no universally unbiased estimator of Var[?
?].
? = Var[?
Proof: thanks to Lemma 2 and 3, it is enough to show that E[?]
?] has no solution
for quadratic estimators:
1
m?1
n?m
? = Var[?
E[?]
?] ? a(? 2 + ?2 ) + b(? + ?2 ) + c(? + ?2 ) = ? 2 +
?+
? .
n
n
n
Finding (a, b, c) satisfying this equality for all admissible values of (?, ? 2 , ?, ?) is impossible since it is equivalent to solving the following overdetermined system:
?
a
= n1 ,
?
?
b
= m?1
,
n
(10)
n?m
c
=
,
?
?
n
Q.E.D.
a+b+c = 0
5
Eigenanalysis of the covariance matrix
One way to gain insight on the origin of the negative statement of Theorem 1 is via the
eigenanalysis of ?, the covariance of e. This decomposition can be performed analytically
thanks to the very particular block structure displayed in Figure 2.
Lemma 4 Let vk be the binary vector indicating the membership of each example to test
block k. The eigenvalues of ? are as follows:
? ?1 = ? 2 ? ? with multiplicity n ? K and eigenspace orthogonal to {vk }K
k=1 ;
? ?2 = ? 2 + (m ? 1)? ? m? with multiplicity K ? 1 and eigenspace de?ned in
the orthogonal of 1 by the basis {vk }K
k=1 ;
? ?3 = ? 2 + (m ? 1)? + (n ? m)? with eigenvector 1.
Lemma 4 states that the vector e can be decomposed into three uncorrelated parts: n ? K
projections to the subspace orthogonal to {vk }K
k=1 , K ? 1 projections to the subspace
in
the
orthogonal
of
1,
and
one projection on 1.
spanned by {vk }K
k=1
A single vector example with n independent elements can be seen as n independent examples. Similarly, the uncorrelated projections of e can be equivalently represented by
respectively n ? K, K ? 1 and one uncorrelated one-dimensional examples.
In particular, for the projection on 1, with a single example, the sample variance is null,
resulting in the absence of unbiased variance estimator of ?3 . The projection of e on
the eigenvector n1 1 is precisely ?
?. Hence there is no unbiased estimate of V ar[?
?] = ?n3
when we have only one realization of the vector e. For the same reason, even with simple
parametric assumptions on e (such as e Gaussian), the maximum likelihood estimate of ?
is not de?ned. Only ? 1 and ?2 can be estimated unbiasedly. Note that this problem cannot
be addressed by performing multiple K-fold splits of the data set. Such a procedure would
not provide independent realizations of e.
6
Possible values for ? and ?
Theorem 1 states that no estimator is unbiased, and in its demonstration, it is shown that
the bias of any quadratic estimator is a linear combination of ?2 , ? 2 , ? and ?. Regarding
estimation, it is thus interesting to see what constraints restrict their possible range.
Lemma 5 For ?
? = CV and ?
? = ?CV, the following inequalities hold:
?
0
? ? ?
?2
1
1
2
2
? ? m (? + (m ? 1)?)
? ? n?m (? + (m ? 1)?) ?
0
? ? ? ?2
?
m
? n?m
?2 ? ? ? ?2 .
The admissible (?, ?) region is very large, and there is no constraint linking ? to ? 2 . Hence,
we cannot propose a variance estimate with universally small bias.
7
Experiments
The bias of any quadratic estimator is a linear combination of ?2 , ? 2 , ? and ?. The admissible values provided earlier suggest that ? and ? cannot be proved to be negligible
compared to ? 2 . This section illustrates that in practice, the contribution to the variance of
?
? due to ? and ? (see Equation (4)) can be of same order as the one due ? 2 . This con?rms
that the estimators of ? should indeed take into account the correlations of ei .
Experiment 2 True variance of K-fold cross-validation.
We repeat the experimental setup of Experiment 1, except that only one sample of size n is
available. Since cross-validation is known to be sensitive to the instability of algorithms,
in addition to this standard setup, we also consider another one with outliers:
The input xi = (xi1 , . . . , xid )0 is still 30-dimensional, but it is now a mixture of two centered Gaussian: let ti be a binary variable, withpP (ti = 1) = p = 0.95; ti = 1 ? xi ?
Pd
N (0, I), ti = 0 ? xi ? N (0, 100I); yi = 3/(d(p + 100(1 ? p))) k=1 xik + ?i ;
ti = 1 ? ?i ? N (0, 1/(p + 100(1 ? p))), ti = 0 ? ?i ? N (0, 100/(p + 100(1 ? p))).
We now look at the variance of K-fold cross-validation (K = 10), and decompose in the
three orthogonal components ? 2 , ? and ?. The results are shown in Figure 3.
0.25
4
2
?
?
?
0.2
?
?
0.15
?2
?
?
3
2
0.1
1
0.05
0
60
0
80 100 120 160 220 280 360 460 600
n?m
60
80 100 120 160 220 280 360 460 600
n?m
no outliers
outliers
Figure 3: Contributions of (? 2 , ?, ?) to total variance V ar[CV ] vs. n ? m.
Without outliers, the contribution of ? is very important for small sample sizes. For large
sample sizes, the overall variance is considerably reduced and is mainly caused by ? 2
because the learning algorithm returns very similar answers for all training sets. When
there are outliers, the contribution of ? is of same order as the one of ? 2 even when the
ratio of examples to free parameters is large (here up to 20). Thus, in dif?cult situations,
where A(D) varies according to the realization of D, neglecting the effect of ? and ? can
be expected to introduce a bias of the order of the true variance.
It is also interesting to see how these quantities are affected by the number of folds K. The
decomposition of ? in ? 2 , ? and ? (4) does not imply that K should be set either to n or
to 2 (according to the sign of ? ??) in order to minimize the variance of ?
?. Modifying
K affects ? 2 , ? and ? through the size and overlaps of the training sets D1 , . . . , DK , as
illustrated in Figure 4. For a ?xed sample size, the variance of ?
? and the contribution of ? 2 ,
? and ? vary smoothly with K (of course, the mean of ?
? is also affected in the process).
2.5
0.25
2
?
?
?
0.2
1.5
?
?
0.15
2
0.1
1
0.05
0.5
0
2 3 4 5 6 8 10 12 15 20 24 30 40 60120
K
no outliers
0
2 3 4 5 6 8 10 12 15 20 24 30 40 60120
K
outliers
Figure 4: Contributions of (? 2 , ?, ?) to total variance V ar[CV ] vs. K for n = 120.
8
Discussion
The analysis presented in this paper for K-fold cross-validation can be instantiated to several interesting cases. First, when having K independent training and test sets (K = 1
is the realistic case), the structure of hold-out errors resemble the one of cross-validation
errors, with ? = 0. Knowing that allows to build the unbiased estimate ?b2 given in 2.1:
knowing that ? = 0 removes the third equation of system (10) in the proof of Theorem 1.
Two-fold cross-validation has been advocated to perform hypothesis testing [4]. It is a
special case of K-fold cross-validation where the training blocks are mutually independent
since they do not overlap. However, this independence does not modify the structure of e
in the sense that ? is not null. The between-block correlation stems from the fact that the
training block D1 is the test block T2 and vice-versa.
Finally, Leave-one-out cross validation is another particular case, with K = n. The
structure of the covariance matrix is simpli?ed, without diagonal blocks. The estimation
dif?culties however remain: even in this particular case, there is no unbiased estimate of
variance. From the de?nition of b in Lemma 3, we have b = 0, and with m = 1 the linear
system (10) still admits no solution.
To summarize, it is known that K-fold cross-validation may suffer from high variability,
which can be responsible for bad choices in model selection and erratic behavior in the
estimated expected prediction error [2, 4, 8]. This paper demonstrates that estimating the
variance of K-fold cross-validation is dif?cult. Not only there is no unbiased estimate of
this variance, but we have no theoretical result showing that this bias should be negligible in the non-asymptotical regime. The eigenanalysis of the covariance matrix of errors
traces the problem back to the dependencies between test-block errors, which induce the
absence of redundant pieces of information regarding the average test error. i.e. the K-fold
cross-validation estimate. It is clear that this absence of redundancy is bound to provide
dif?culties in the estimation of variance.
Our experiments show that the bias incurred by ignoring test errors dependencies can be
of the order of the variance itself, even for large sample sizes. Thus, the assessment of the
signi?cance of observed differences in cross-validation scores should be treated with much
caution. The next step of this study consists in building and comparing variance estimators
dedicated to the very speci?c structure of the test-block error dependencies.
References
[1] Y. Bengio and Y. Grandvalet. No unbiased estimator of the variance of K-fold cross-validation.
Journal of Machine Learning Research, 2003.
[2] L. Breiman. Heuristics of instability and stabilization in model selection. The Annals of Statistics,
24(6):2350?2383, 1996.
[3] L. Devroye, L. Gy?or?, and G. Lugosi. A Probabilistic Theory of Pattern Recognition. Springer,
1996.
[4] T. G. Dietterich. Approximate statistical tests for comparing supervised classi?cation learning
algorithms. Neural Computation, 10(7):1895?1924, 1999.
[5] B. Efron and R. J. Tibshirani. An Introduction to the Bootstrap, volume 57 of Monographs on
Statistics and Applied Probability. Chapman & Hall, 1993.
[6] M. Kearns and D. Ron. Algorithmic stability and sanity-check bounds for leave-one-out crossvalidation. Neural Computation, 11(6):1427?1453, 1996.
[7] R. Kohavi. A study of cross-validation and bootstrap for accuracy estimation and model selection. In Proceedings of the Fourteenth International Joint Conference on Arti?cial Intelligence,
pages 1137?1143, 1995.
[8] C. Nadeau and Y. Bengio. Inference for the generalization error. Machine Learning, 52(3):239?
281, 2003.
[9] M. Stone. Cross-validatory choice and assessment of statistical predictions. Journal of the Royal
Statistical Society, B, 36(1):111?147, 1974.
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1,616 | 2,469 | Efficient Exact k-NN and Nonparametric
Classification in High Dimensions
Ting Liu
Computer Science Dept.
Carnegie Mellon University
Pittsburgh, PA 15213
[email protected]
Andrew W. Moore
Computer Science Dept.
Carnegie Mellon University
Pittsburgh, PA 15213
[email protected]
Alexander Gray
Computer Science Dept.
Carnegie Mellon University
Pittsburgh, PA 15213
[email protected]
Abstract
This paper is about non-approximate acceleration of high dimensional
nonparametric operations such as k nearest neighbor classifiers and the
prediction phase of Support Vector Machine classifiers. We attempt to
exploit the fact that even if we want exact answers to nonparametric
queries, we usually do not need to explicitly find the datapoints close
to the query, but merely need to ask questions about the properties about
that set of datapoints. This offers a small amount of computational leeway, and we investigate how much that leeway can be exploited. For
clarity, this paper concentrates on pure k-NN classification and the prediction phase of SVMs. We introduce new ball tree algorithms that on
real-world datasets give accelerations of 2-fold up to 100-fold compared
against highly optimized traditional ball-tree-based k-NN. These results
include datasets with up to 106 dimensions and 105 records, and show
non-trivial speedups while giving exact answers.
1
Introduction
Nonparametric models have become increasingly popular in the statistics communities and
probabilistic AI communities. They remain hampered by their computational complexity.
Spatial methods such as kd-trees [6, 17], R-trees [9], metric trees [18, 4] and ball trees [15]
have been proposed and tested as a way of alleviating the computational cost of such statistics without resorting to approximate answers. They have been used in many different
ways, and with a variety of tree search algorithms and with a variety of ?cached sufficient
statistics? decorating the internal leaves, for example in [14, 5, 16, 8].
The main concern with such accelerations is the extent to which they can survive high
dimensional data. Indeed, there are some datasets in this paper for which a highly optimized
conventional k nearest neighbor search based on ball trees is on average more expensive
than the naive linear search algorithm,but extracting the k nearest neighbors is often not
needed, even for a k nearest neighbor classifier. This paper is about the consequences of
the fact that none of these three questions have the same precise meaning: (a) ?What
are the k nearest neighbors of t?? (b) ?How many of the k nearest neighbors of t are
from the positive class?? and (c) ?Are at least q of the k nearest neighbors from the
positive class?? The computational geometry community has focused on question (a),
but uses of proximity queries in statistics far more frequently require (b) and (c) types of
computations. Further, in addition to traditional K-NN, the same insight applies to many
other statistical computations such as nonparametric density estimation, locally weighted
regression, mixture models, k-means and the prediction phase of SVM classification.
2
Ball trees
A ball tree is a binary tree in which each node represents a set of points, called
Points(Node). Given a dataset, the root node of a ball tree represents the full set of points
in the dataset. A node can be either a leaf node or a non-leaf node. A leaf node explicitly
contains a list of the points represented by the node. A non-leaf node does not explicitly
contain a set of points. It has two child nodes: Node.child1 and Node.child2, where
Points(Node.child1) ? Points(Node.child2) = ?
Points(Node.child1) ? Points(Node.child2) = Points(Node)
Points are organized spatially. Each node has a distinguished point called a pivot. Depending on the implementation, the pivot may be one of the datapoints, or it may be the centroid
of Points(Node). Each node records the maximum distance of the points it owns to its pivot.
Call this the radius of the node
Node.Radius = maxx?Points(Node) | Node.Pivot ? x |
Balls lower down the tree cover smaller volumes. This is achieved by insisting, at tree
construction time, that
x ? Points(Node.child1) ? | x ? Node.child1.Pivot | ? | x ? Node.child2.Pivot |
x ? Points(Node.child2) ? | x ? Node.child2.Pivot | ? | x ? Node.child1.Pivot |
Provided our distance function obeys the triangle inequality, this gives the ability to bound
the distance from a target point t to any point in any ball tree node. If x ? Points(Node)
then we can be sure that:
|x ? t| ? |t ? Node.Pivot| ? Node.Radius
(1)
|x ? t| ? |t ? Node.Pivot| + Node.Radius
(2)
Ball trees are constructed top-down. There are several ways to construct them, and practical
algorithms trade off the cost of construction (it would be useless to be O(R2 ) given a dataset
with R points, for example) against the tightness of the radius of the balls. [13] describes
one fast way of constructing a ball tree appropriate for computational statistics. If a ball
tree is balanced, then the construction time is O(CR log R), where C is the cost of a pointpoint distance computation (which is O(m) if there are m dense attributes, and O( f m) if
the records are sparse with only fraction f of attributes taking non-zero values).
2.1 KNS1: Conventional K nearest neighbor search with ball trees
In this paper, we call conventional ball-tree-based search [18] KNS1. Let a pointset PS be
a set of datapoints. We begin with the following definition:
Say that PS consists of the k-NN of t in pointset V if and only if
((| V |? k) ? (PS are the k-NN of t in V)) ? ((| V |< k) ? (PS = V ))
(3)
We now define a recursive procedure called BallKNN with the following inputs and output.
PSout = BallKNN(PSin , Node)
Let V = set of points searched so far, on entry. Assume PS in consists of the k-NN of
t in V. This function efficiently ensures that on exit, PSout consists of the k-NN of t in
V ? Points(Node).
?
i f | PSin |< k
(4)
Let Dsofar =
maxx?PSin | x ? t | i f | PSin |= k
Dsofar is the minimum distance within which points would become interesting to us.
Node.parent
max(|t ? Node.Pivot| ? Node.Radius, Dminp
) i f Node 6= Root
Let DNode
=
minp
max(|t ? Node.Pivot| ? Node.Radius, 0)
i f Node = Root
DNode
minp is the minimum possible distance from any point in Node to t.
(5)
Procedure BallKNN (PSin , Node)
begin
in
if (DNode
minp ? Dsofar ) then exit returning PS unchanged.
else if (Node is a leaf) PSout = PSin
?x ? Points(Node)
if (| x ? t |< Dsofar ) then
add x to PSout
if (| PSout |= k + 1) then
remove furthest neighbor from PSout ; update Dsofar
else if (Node is a non-leaf)
node1 = child of Node closest to t
node2 = child of Node furthest from t
PStemp = BallKNN(PSin , node1 )
PSout = BallKNN(PStemp , node2 )
end
A call of BallKNN({},Root) returns the k nearest neighbors of t in the Ball tree.
2.2 KNS2: Faster k-NN classification for skewed-class data
In several binary classification domains,one class is much more frequent than the other, For
example, in High Throughput Screening datasets, [19] it is far more common for the result
of an experiment to be negative than positive. In fraud detection or intrusion detection,
a non-attack is far more common than an attack. The new algorithm introduced in this
section, KNS2, is designed to accelerate k-NN based classification beyond the speedups
already available by using KNS1 (conventional ball-tree-based k-NN). KNS2 attacks the
problem by building two ball trees: Root pos is the root of a (small) ball tree built from all
the positive points in the dataset. Rootneg is the root of a (large) ball tree built from all
negative points.
Then, when it is time to classify a new target point t, we compute q, the number of k nearest
neighbors of t that are in the positive class, in the following fashion
? Step 1 ? ? Find positive?: Find the k nearest positive class neighbors of t (and
their distances to t) using conventional ball tree search.
? Step 2 ? ?Insert negative?: Do sufficient search of the negative tree to prove
that the number of positive datapoints among k nearest neighbors is q for some
value of q.
Step 2 is achieved using a new recursive search called NegCount. In order to describe
NegCount we need the following three definitions.
? The Dists Array. Dists is an array of elements Dists1 . . . Distsk consisting of the
distances to the k nearest positive neighbors of t, sorted in increasing order of
distance. We will also write Dists0 = 0 and Distsk+1 = ?.
? Pointsets. Define pointset V as the set of points in the negative balls visited so far.
? The Counts Array (n,C). Say that (n,C) summarize interesting negative points
for pointset V if and only if
1. ?i ? [0, n],
Ci =| V ? {x : Distsi ?| x ? t |< Distsi+1 } |
(6)
2. ?ni=0 Ci ? k, ?n?1
i=0 Ci < k. This simply declares that the length n of the C
array is as short as possible while accounting for the k members of V that are
nearest to t.
Step 2 of KNS2 is implemented by the recursive function
(nout ,Cout ) = NegCount(nin ,Cin , Node, Dists)
Assume that on entry that (nin ,Cin ) summarize interesting negative points for pointset V ,
where V is the set of points visited so far during the search. This algorithm efficiently
ensures that on exit (nout ,Cout ) summarize interesting negative points for V ? Points(Node).
Procedure NegCount (nin ,Cin , Node, Dists)
begin
nout := nin ; Cout := Cin
in
Let T = ?ni=0?1 Ciin
T is the total number of negative points closer than the nin th positive point
out
out
if (DNode
minp ? Distnin ) then exit and return(n ,C )
else if (Node is a leaf)
?x ? Points(Node)
Use binary search to find j ? [0, nout ], such that Distsj ?| x ? t |< Distsj+1
Cout
:= Cout
j
j + 1; T := T + 1
out
If T exceeds k, decrement nout until T = ?ni=0 ?1 Ciout < k.
Distsnout +1 := ?
if (nout = 0)exit and return(0, C out )
else if(Node is a non leaf)
node1 := child of Node closest to t
node2 := child of Node furthest from t
(ntemp ,Ctemp ) := NegCount(nin ,Cin , node1 , Dists)
if (ntemp = 0) exit and return (0, C out )
(nout ,Cout ) := NegCount(ntemp ,Ctemp , node2 , Dists)
end
We can stop the procedure when nout becomes 0 (which means all the k nearest neighbors
of t are in the negative class) or when we run out of nodes. The top-level call is
NegCount(k,C0 , NegTree.Root, Dists)
where C0 is an array of zeroes and Dists are defined in Equation 6 and obtained by applying
KNS1 to the (small) positive ball tree.
2.3 KNS3: Are at least q of the k nearest neighbors positive?
Unfortunately, space constraints prevent us from describing the details of KNS3. KNS3
removes KNS2?s constraint of an assumed skewedness in the class distribution, while introducing a new constraint: we answer the binary question ?are at least q nearest neighbors
positive?? (where the questioner must supply q). This is often the most statistically relevant question, for example during classification with known false positive and false negative
costs. KNS3 will be described fully in a journal-article length version of the paper 1 .
2.4 SVP1: Faster Radial Basis SVM Prediction
After an SVM [3] has been trained we hit the prediction phase. Given a batch of query
points q1 , q2 . . . qR we wish to classify each q j . Furthermore, in state-of-the-art training
algorithms such as SMO, training time is dominated by SVM evaluation [12]. q j should be
classified according to this rule:
ASUM(q j ) = ? ?i K(|q j ? xi |) , BSUM(q j ) = ? ?i K(|q j ? xi |)
(7)
i?posvecs
1 available
from www.autonlab.org
i?negvecs
Class(q j )
if ASUM(q j ) ? BSUM(q j ) ? ?b
=1
if ASUM(q j ) ? BSUM(q j ) < ?b
=0
Where the positive support vectors posvecs, the negative support vectors negvecs and the
weights {?i }, {?i } and constant term b are all obtained from SVM training.
We place the queries (not the support vectors) into a ball-tree. We can then apply the same
kinds of tricks as KNS2 and KNS3 in which we do not need to find the explicit values of
the ASUM and BSUM terms, but merely find balls in the tree in which we can prove all
query points satisfy one of the above inequalities.
To classify all the points in a node called Node we do the following:
1. Compute values (ASUMLO , ASUMHI ) such that we can be sure
?q j ? Node : ASUMLO ? ASUM(q j ) ? ASUMHI
(8)
without iterating over the queries in Node. This is achieved simply, for example if
q j ? Node we know
ASUM(q j )
=
?
?i K(|q j ? xi |)
?
?i K(|Node.pivot ? xi | + Node.Radius)
i?posvecs
?
i?posvecs
= ASUMLO
Similarly,
ASUM(q j )
=
?
?i K(|q j ? xi |)
?
?i K(max(|Node.pivot ? xi | ? Node.Radius, 0))
i?posvecs
?
i?posvecs
= ASUMHI
under the assumption that the kernel function is a decreasing function of distance.
This is true, for example, for Gaussian Radial Basis function kernels.
2. Similarly compute values (BSUMLO , BSUMHI ).
3. If ASUMLO ? BSUMHI ? ?b we have proved that all queries in Node should be
classified positively, and we can terminate this recursive call.
4. If ASUMHI ? BSUMLO < ?b we have proved that all queries in Node should be
classified negatively, and we can terminate this recursive call.
5. Else we recurse and apply the same procedure to the two children of Node, unless
Node is a leaf node in which case we must explicitly iterate over its members.
3
Experimental Results
Table 1 is a summary of the datasets in the empirical analysis.
Life Sciences: These were proprietary datasets (ds1 and ds2) similar to the publicly available Open Compound Database provided by the National Cancer Institute (NCI Open Compound Database, 2000). The two datasets are sparse. We also present results on datasets
derived from ds1, denoted ds1.10pca, ds1.100pca and ds2.100anchor by linear projection
using principal component analysis (PCA).
Link Detection: The first, Citeseer, is derived from the Citeseer web site (Citeseer,2002)
and lists the names of collaborators on published materials. The goal is to predict whether
J Lee ( the most common name) was a collaborator for each work based on who else is
listed for that work. We use J Lee.100pca to represent the linear projection of the data to
100 dimensions using PCA. The second link detection dataset is derived from the Internet
Movie Database (IMDB,2002) and is denoted imdb using a similar approach, but to predict
the participation of Mel Blanc (again the most common participant).
UCI/KDD data: We use three large datasets from KDD/UCI repository [2]. The datasets
can be identified from their names. They were converted to binary classification problems.
Each categorical input attribute was converted into n binary attributes by a 1-of-n encoding (where n is the attribute?s arity).The post-processed versions of these datasets are at
http://www.cs.cmu.edu/?awm/kns
1. Letter originally had 26 classes: A-Z. We performed binary classification using
the letter A as the positive class and ?Not A? as negative.
2. Movie is a dataset from[11]. The TREC-2001 Video Track organized by NIST
shot boundary Task. It is a 4 hours of video or 13 MPEG-1 video files at slightly
over 2GB of data.
3. Ipums (from ipums.la.97). We predict farm status, which is binary.
4. Kdd99(10%) has a binary prediction: Normal vs. Attack.
Table 1: Datasets
Dataset
Num.
records
ds1
26733
ds1.100pca 26733
ds2
88358
Letter
20000
Movie
38943
Ipums
70187
Num. Dimensions
6348
100
1100000
16
62
60
Num.
pos.
804
804
211
790
7620
119
Dataset
Num.
records
ds1.10pca
26733
ds2.100anchor 88358
J Lee.100pca 181395
Blanc Mel
186414
Kdd99(10%) 494021
Num. Dimensions
10
100
100
10
176
Num.
pos.
804
211
299
824
97278
For each dataset, we tested k = 9 and k = 101. For KNS3, we used q = dk/2e when
k = 9 and q = dpk/(n+p)e when k = 101, where p = Num.positive in the dataset and n =
Num.negative in the dataset. : a datapoint is classified as positive iff the majority of its k
nearest neighbors are positive. Each experiment performed 10-fold cross-validation. Thus,
each experiment required R k-NN classification queries (where R is the number of records
in the dataset) and each query involved the k-NN among 0.9R records. A naive implementation with no ball-trees would thus require 0.9R2 distance computations.These algorithms
are all exact. No approximations were used in the classifications.
Table 2 shows the computational cost of naive k-NN, both in terms of the number of distance computations and the wall-clock time on an unloaded 2 GHz Pentium. We then
examine the speedups of KNS1 (traditional use of Ball-trees) and our two new Ball-tree
methods (KNS2 and KNS3). It is notable that for some high dimensional datasets, KNS1
does not produce an acceleration over naive. KNS2 and KNS3 do, however, and in some
cases they are hundreds of times faster than KNS1. The ds2 result is particularly interesting
because it involves data in over a million dimensions. The first thing to notice is that conventional ball-trees (KNS1) were slightly worse than the naive O(R2 ) algorithm. In only
one case was KNS2 inferior to naive and KNS3 was always superior. On some datasets
KNS2 and KNS3 gave dramatic speedups.
Table 3 gives results for SVP1, the Ball-tree-based accelerator for SVM prediction2 In
general SVP1 appears to be 2-4 times faster than SV M light [12], with two far more dramatic
speedups in the case of two classification tasks where SVP1 quickly realizes that a large
node near the top of its query tree can be pruned as negative. As with previous results,
SVP1 is exact, and all predictions agree with SVM-Light. All these experiments used
Radial Basis kernels, with kernel width tuned for optimal test-set performance.
2 Because
training SVMs is so expensive, some of the results below used reduced training sets.
Table 2: Number of distance computations and wall-clock-time for Naive k-NN classification (2nd column). Acceleration for normal use of ball-trees in col, 2 (in terms of num.
distances and time). Accelerations of new methods KNS2 and KNS3 in other columns.
Naive times are independent of k.
NAIVE
dists
ds1
k=9
k=101
ds1.10pca k=9
k=101
ds1.100pca k=9
k=101
ds2
k=9
k=101
ds2.100- k=9
k=101
J Lee.100- k=9
k=101
Blanc Melk=9
k=101
Letter
k=9
k=101
Movie
k=9
k=101
Ipums
k=9
k=101
Kddcup99 k=9
(10%)
k=101
6.4 ? 10
8
6.4 ? 108
time
(secs)
4830
420
6.4 ? 10
8
2190
8.5 ? 10
9
105500
7.0 ? 10
9
24210
3.6 ? 10
10
142000
3.8 ? 10
10
44300
3.6 ? 10
8
290
1.4 ? 109
3100
4.4 ? 109
9520
2.7 ? 1011
1670000
KNS1
dists
speedup
1.6
1.0
11.8
4.6
1.7
0.97
0.64
0.61
15.8
10.9
2.6
2.2
3.0
2.9
8.5
3.5
16.1
9.1
195
69.1
4.2
4.2
time
speedup
1.0
0.7
11.0
3.4
1.8
1.0
0.24
0.24
14.3
14.3
2.4
1.9
3.0
3.1
7.1
2.6
13.8
7.7
136
50.4
4.2
4.2
KNS2
dists
speedup
4.7
1.6
33.6
6.5
7.6
1.6
14.0
2.4
185.3
23.0
28.4
12.6
47.5
7.1
42.9
9.0
29.8
10.5
665
144.6
574
187.7
time
speedup
3.1
1.1
21.4
4.0
7.4
1.6
2.8
0.83
144
19.4
27.2
11.6
60.8
33
26.4
5.7
24.8
8.1
501
121
702
226.2
KNS3
dists
speedup
12.8
10
71
40
23.7
16.4
25.6
28.7
580
612
15.6
37.4
51.9
203
94.2
45.9
50.5
33.3
1003
5264
4
3.9
time
speedup
5.8
4.2
20
6.1
29.6
6.8
3.0
3.3
311
248
12.6
27.2
60.7
134.0
25.5
9.4
22.4
11.6
515
544
4.1
3.9
Table 3: Comparison between SVM light and SVP1. We show the total number of distance
computations made during the prediction phase for each method, and total wall-clock time.
ds1
ds1.10pca
ds1.100pca
ds2.100pca
J Lee.100pca
Blanc Mel
Letter
Ipums
Movie
Kddcup99(10%)
4
SVM light
distances
6.4 ? 107
6.4 ? 107
6.4 ? 107
7.0 ? 108
6.4 ? 106
1.2 ? 108
2.6 ? 107
1.9 ? 108
1.4 ? 108
6.3 ? 106
SVP1
distances
1.8 ? 107
1.8 ? 107
2.3 ? 107
1.4 ? 108
2 ? 106
3.6 ? 107
1 ? 107
7.7 ? 104
4.4 ? 107
2.8 ? 105
SVM light
seconds
394
60
259
2775
31
61
21
494
371
69
SVP1
seconds
171
23
92
762
7
26
11
1
136
1
speedup
2.3
2.6
2.8
3.6
4.4
2.3
1.9
494
2.7
69
Comments and related work
Applicability of other proximity query work. For the problem of ?find the k nearest datapoints? (as opposed to our question of ?perform k-NN or Kernel classification?) in high
dimensions, the frequent failure of traditional ball trees to beat naive has lead to some innovative alternatives, based on random projections, hashing discretized cubes, and acceptance
of approximate answers. For example [7] gives a hashing method that was demonstrated
to provide speedups over a ball-tree-based approach in 64 dimensions by a factor of 2-5
depending on how much error in the approximate answer was permitted. Another approximate k-NN idea is in [1], one of the first k-NN approaches to use a priority queue of
nodes, in this case achieving a 3-fold speedup with an approximation to the true k-NN.
However, these approaches are based on the notion that any points falling within a factor of
(1 + ?) times the true nearest neighbor distance are acceptable substitutes for the true nearest neighbor. Noting in particular that distances in high-dimensional spaces tend to occupy
a decreasing range of continuous values [10], it remains an open question whether schemes
based upon the absolute values of the distances rather than their ranks are relevant to the
classification task. Our approach, because it need not find the k-NN to answer the relevant
statistical question, finds an answer without approximation. The fact that our methods are
easily modified to allow (1 + ?) approximation in the manner of [1] suggests an obvious
avenue for future research.
References
[1] S. Arya, D. Mount, N. Netanyahu, R. Silverman, and A. Wu. An optimal algorithm for approximate nearest neighbor searching fixed dimensions. Journal of the ACM, 45(6):891?923,
1998.
[2] S. D. Bay. UCI KDD Archive [http://kdd.ics.uci.edu]. Irvine, CA: University of California,
Dept of Information and Computer Science, 1999.
[3] C. Burges. A tutorial on Support Vector Machines for Pattern Recognition. Data Mining and
Knowledge Discovery, 2(2):955?974, 1998.
[4] P. Ciaccia, M. Patella, and P. Zezula. M-tree: An efficient access method for similarity search
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[5] K. Deng and A. W. Moore. Multiresolution Instance-based Learning. In Proceedings of the
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dept:4 tested:2 |
1,617 | 2,470 | Sparse Greedy Minimax Probability Machine
Classification
Thomas R. Strohmann
Department of Computer Science
University of Colorado, Boulder
[email protected]
Andrei Belitski
Department of Computer Science
University of Colorado, Boulder
[email protected]
Gregory Z. Grudic
Department of Computer Science
University of Colorado, Boulder
[email protected]
Dennis DeCoste
Machine Learning Systems Group
NASA Jet Propulsion Laboratory
[email protected]
Abstract
The Minimax Probability Machine Classification (MPMC) framework
[Lanckriet et al., 2002] builds classifiers by minimizing the maximum
probability of misclassification, and gives direct estimates of the probabilistic accuracy bound ?. The only assumptions that MPMC makes is
that good estimates of means and covariance matrices of the classes exist.
However, as with Support Vector Machines, MPMC is computationally
expensive and requires extensive cross validation experiments to choose
kernels and kernel parameters that give good performance. In this paper
we address the computational cost of MPMC by proposing an algorithm
that constructs nonlinear sparse MPMC (SMPMC) models by incrementally adding basis functions (i.e. kernels) one at a time ? greedily selecting the next one that maximizes the accuracy bound ?. SMPMC automatically chooses both kernel parameters and feature weights without using computationally expensive cross validation. Therefore the SMPMC
algorithm simultaneously addresses the problem of kernel selection and
feature selection (i.e. feature weighting), based solely on maximizing the
accuracy bound ?. Experimental results indicate that we can obtain reliable bounds ?, as well as test set accuracies that are comparable to state
of the art classification algorithms.
1 Introduction
The goal of a binary classifier is to maximize the probability that unseen test data will be
classified correctly. Assuming that the test data is generated from the same probability
distribution as the training data, it is possible to derive specific probability bounds for the
case that the decision boundary is a hyperplane. The following result due to Marshall and
Olkin [1] and extended by Bertsimas and Popescu [2] provides the theoretical basis for
assigning probability bounds to hyperplane classifiers:
1
sup
P r{aT z ? b} =
? 2 = inf aT t?b (t ? ?
z)T ??1
z) (1)
z (t ? ?
1 + ?2
E[z]=?
z,Cov[z]=?z
where a ? Rd , b are the hyperplane parameters, z is a random vector, and t is an ordinary
vector. Lanckriet et al (see [3] and [4]) used the above result to build the Minimax Probability Machine for binary classification (MPMC). From (1) we note that the only required
relevant information of the underlying probability distribution for each class is its mean
and covariance matrix. No other estimates and/or assumptions are needed, which implies
that the obtained bound (which we refer to as ?) is essentially distribution free, i.e. it holds
for any distribution with a certain mean and covariance matrix.
As with other classification algorithms such as Support Vector Machines (SVM) (see [5]),
the main disadvantage of current MPMC implementations is that they are computationally
expensive (same complexity as SVM), and require extensive cross validation experiments
to choose kernels and kernel parameter to give good performance on each data set. The
goal of this paper is to propose a kernel based MPMC algorithm that directly addresses
these computational issues.
Towards this end, we propose a sparse greedy MPMC (SMPMC) algorithm that efficiently
builds classifiers, while at the same time maintains the distribution free probability bound
of MPM type algorithms. To achieve this goal, we propose to use an iterative algorithm
which adds basis functions (i.e. kernels) one by one, to an initially ?empty? model. We
are considering basis functions that are induced by Mercer kernels, i.e. functions of the
following form f (z) = K? (z, zi ) (where zi is an input vector of the training data). Bases
are added in a greedy way: we select the particular zi that maximizes the MPMC objective
?. Furthermore, SMPMC chooses optimal kernel parameters that maximize this metric
(hence the subscript ? in K? ), including automatically weighting input features by ?j ?
0 for each kernel added, such that zi = (?1 z1 , ?2 z2 , ..., ?d zd ) for d dimensional data.
The proposed SMPMC algorithm automatically selects kernels and re-weights features (i.e.
does feature selection) for each new added basis function, by minimizing the error bound
(i.e. maximizing ?). Thus the large computational cost of cross validation (typically used
by SVM and MPMC) is avoided.
The paper is organized as follows: Section 2.1 reviews the standard MPMC; Section 2.2 describes the proposed sparse greedy MPMC algorithm (SMPMC); and Sections 2.3-2.4 show how we can use sparse MPMC to determine optimal kernel parameters. In section 3 we compare our results to the ones described in the original MPMC paper (see [4]), showing the probability bounds and the test set accuracies for different binary classification problems. The conclusion is presented
in section 4. Matlab source code for the SMPMC algorithm is available online:
http://nago.cs.colorado.edu/?strohman/papers.html
2 Classification model
In this section we develop a sparse version of the Minimax Probability Machine for binary classification. We show that besides a significant reduction in computational cost, the
SMPMC algorithm allows us to do automated kernel and feature selection.
2.1 Minimax Probability Machine for binary classification
We will briefly describe the underlying concepts of the MPMC framework as developed
by Lanckriet et al. (see [4]). The goal of MPMC is to find a decision boundary H(a, b) =
{z|aT z = b} such that the minimum probability ?H of classifying future data correctly is
maximized. If we assume that the two classes are generated from random vectors x and y,
we can express this probability bound just in terms of the means and covariances of these
random vectors:
?H =
inf
P r{aT x ? b ? aT y ? b}
(2)
x?(?
x,?x ),y?(?
y ,?y )
? , ?x , y
? , and ?x
Note that we do not make any distributional assumptions other than that x
are bounded. Exploiting a theorem from Marshall and Olkin [1], it is possible to rewrite
(2) as a closed form expression:
1
?H =
(3)
1 + m2
where
q
p
?) = 1
x?y
m = min aT ?x a + aT ?y a s.t. aT (?
(4)
a
The optimal hyperplane parameter a? is the vector that minimizes (4). The hyperplane
parameter b? can then be computed as:
p
aT? ?x a?
T
??
(5)
b? = a? x
m
T
A new data point znew is classified according to sign(a? znew ? b? ); if this yields +1,
znew is classified as belonging to class x, otherwise it is classified as belonging to class y.
2.2 Sparse MPM classification
One of the appealing properties of Support Vector Machines is that their models typically
rely only on a small fraction of the training examples, the so called support vectors. The
models obtained from the kernelized MPMC, however, use all of the training examples (see
[4]), i.e. the decision hyperplane will look like:
Ny
Nx
X
X
(x)
(y)
ai K(xi , z) +
ai K(yi , z) = b
(6)
(x)
i=1
(y)
where in general all ai , ai
i=1
6= 0.
This brings up the question whether one can construct sparse models for the MPMC where
(x)
(y)
most of the coefficients ai or ai are zero. In this paper we propose to do this by starting
with an initially ?empty? model and then adding basis functions one by one. As we will
see shortly, this approach is speeding up both learning and evaluation time while it is still
maintaining the distribution free probability bound of the MPMC.
Before we outline the algorithm we introduce some notation:
N
= Nx + Ny the total number of training examples
`
= (`1 , ..., `N )T ? {?1, 1}N the labels of the training data
(k)
(k)
(k)
`b
= (`b1 , ..., `bN )T ? RN output of the model after adding the kth basis function
(k)
a
= the MPMC hyperplane coefficients when adding the kth basis function
b(k) = the MPMC hyperplane offset when adding the kth basis function
~ b = (Kv (v, x1 ), ..., Kv (v, xN ), Kv (v, y1 ), ..., Kv (v, yN ))T
K
x
y
basis function evaluated on all training examples (empirical map)
~x
K
= (Kv (v, x1 ), ..., Kv (v, xNx ))T evaluated only on positive examples
v
~
Kyv = (Kv (v, y1 ), ..., Kv (v, yNy ))T evaluated only on negative examples
Note that `b(k) is a vector of real numbers (the distances of the training data to the hyperplane
before applying the sign function). v ? Rd is the training vector generating the basis
~ v 1 . We will simply write K
~ (k) , K
~ x(k) , K
~ y(k) for the kth basis function.
function K
~ for both the empirical map and the induced function. It
Note that we use the same symbol K
~ refers to.
will always be clear from the context what K
1
For the first basis we are solving the one dimensional MPMC:
q
q
~ x(1) ? K
~ y(1) ) = 1
m = min a? 2 (1) a + a? 2 (1) a s.t. a(K
a
~x
K
~y
K
(1)
(7)
(1)
~ x and ? 2 (1) are the mean and variance of the vector K
~ x (which is the first basis
where K
~x
K
function evaluated on all positive training examples).
Because of the constraint the feasible region contains just one value for a(1) :
~ x(1) ? K
~ y(1) )
a(1) = 1/(K
q
a? 2 (1) a
? ~ (1)
~
(8)
K
Kx
x
(1)
(1) ~ (1)
~ x(1) ?
q
= a(1) K
b
= a Kx ? q 2
? (1) +? (1)
2
a?
~ (1)
K
x
a+
a?
~ (1)
K
y
a
~
K
x
~
K
y
The first model then looks like:
~ (1) ? b(1)
`b(1) = a(1) K
(9)
All of the subsequent models use the previous estimation `b(k) as one input and the next
~ (k+1) as the other input. We set up the two dimensional classification problem:
basis K
(k) ~ (k+1)
x(k+1) = [`bx , K
] ? RNx ?2
x
(10)
(k) ~ (k+1)
(k+1)
b
y
= [`y , Ky
] ? RNy ?2
And solve the following optimization
q problem:
p
T
m = min a ?x(k+1) a + aT ?y(k+1) a s.t. aT (x(k+1) ? y(k+1) ) = 1
(11)
a
(k)
(k+1)
~x
is the 2-dimensional mean vector (`bx , K
)T and where ?x(k+1) is the
where
(k)
(k+1)
~x
2 ? 2 sample covariance matrix of the vectors `bx and K
.
(k+1) (k+1) T
(k+1)
) be the optimal
, a2
Let a
= (a1
q solution of (11). We set:
x(k+1)
T
a(k+1) ?x(k+1) a(k+1)
(12)
?q
b
=a
q
T
T
a(k+1) ?x(k+1) a(k+1) + a(k+1) ?y(k+1) a(k+1)
and obtain the next model as:
(k+1) b(k)
(k+1) ~ (k+1)
K
? b(k+1)
(13)
`b(k+1) = a1
` + a2
As stated above, one computational advantage of SMPMC is that we typically use only
a small number of training examples to obtain our final model (i.e. k << N ). Another
benefit is that we have to solve only one and two dimensional MPMC problems. As seen in
(8) the one dimensional solution is trivial to compute. An analysis of the two dimensional
problem shows that it can be reduced to the problem of finding the roots of a fourth order
polynomial. Polynomials of degree 4 still have closed form solutions (see e.g. [6]) which
can be computed efficiently. In the standard MPMC algorithm (see [4]), however, the
solution a for equation (4) has N dimensions and can therefore only be found by expensive
numerical methods.
(k+1)
(k+1)T
x(k+1)
It may seem that the values of ? = 1/(1 + m2 ) which we obtain from (11) are not true
for the whole model since we are considering only two dimensional problems and not all
of the k + 1 dimensions we have added so far through our basis functions. But it turns
out that the ?local? bound (from the 2D MPMC) is indeed equal to the ?global? bound
(when considering all k + 1 dimensions). We state this fact more formally in the following
theorem:
~ (1) + ... + ck K
~ (k) be the sparse MPMC model at the
Theorem 1: Let `b(k) = c0 + c1 K
(k+1)
(k+1)
kth iteration (k ? 1) and let a1
, a2
, b(k+1) be the solution of the two dimensional
(k+1) b(k)
(k+1) ~ (k+1)
(k+1)
b
MPMC: `
= a1
` + a2
K
? b(k+1) .
Then the values of ? for the two dimensional MPMC and for the k + 1 dimensional MPMC
are the same.
Proof: see Appendix
2.3 Selection of bases and Gaussian Kernel widths
In our experiments we are using the Gaussian kernel which looks like:
||u ? v||22
)
(14)
K? (u, v) = exp(?
2? 2
where ? is the so called kernel width. As mentioned before, one typically has to choose
? manually or determine it by cross validation (see [4]). The SMPMC algorithm greedily
selects a basis function ? out of a randomly chosen candidate set ? to maximize ? which is
equivalent to minimizing the value of m in (7) and (11). Before we state the optimization
problem for the one and two dimensional MPMC we rewrite (14) so that we can get rid of
the denominator:
K? (u, v) = exp(??||u ? v||22 ) ? ? 0
(15)
The optimization problem we solve for the first iteration is then:
q
q
~ x(1) ? K
~ y(1) ) = 1
min m(?) = min a? 2 (1) a + a? 2 (1) a s.t. a(K
?
a
~x
K
~y
K
(16)
~ x(1) , and
note that ? even though we did not state it explicitly ? the statistics ? 2~ (1) , ? 2~ (1) , K
Kx
~ y(1)
K
Ky
(and consequently the coefficient a) all depend on the kernel parameter ?.
The two dimensional problem that has to be solved for all subsequent iterations k ? 2 turns
into the following optimization problem
q for ?:
p
min m(?) = min aT ?x(k+1) a+ aT ?y(k+1) a s.t. aT (x(k+1) ?y(k+1) ) = 1 (17)
?
a
Again, x(k+1) , y(k+1) , ?x(k+1) , and ?y(k+1) all depend on the kernel parameter ? and from
these four statistics we can compute the minimizer a ? R2 analytically.
2.4 Feature selection
For doing feature selection with Gaussian kernels one has to replace the uniform kernel
width ? with a d dimensional vector ~? of kernel weightings:
Pd
(18)
K~? (u, v) = exp(? l=1 ?l (ul ? vl )2 ) (?l ? 0 l = 1, ..., d)
Note that the optimization problems (16) and (17) for the one respectively two dimensional
MPMC are now d dimensional instead of just one dimensional.
3 Experiments
In this section we describe the results we obtained for SMPMC on various classification
benchmarks. We used the same data sets as Lanckriet et al. in [4] for the standard MPMC.
The data sets were randomly divided into 90% training data and 10% test data and the
results were averaged over 50 runs for each of the five problems (see table 1). In all the
experiments listed in table 1 we used the feature selection algorithm (with the exception
of Sonar where width selection was used) and had a candidate set of size 5, i.e. at each
iteration the best basis out of 5 randomly chosen candidates was selected. The results we
obtained are comparable to the ones reported by Lanckriet et al [4]. Note that for all of the
data sets SMPMC uses significantly less basis functions than MPMC does which directly
translates into an accordingly smaller evaluation cost. The differences in training cost are
shown in table 2. The total training time for standard MPMC takes into account the 50-fold
cross validation and 10 candidates for the kernel parameter. We observe that for all of the
five data sets the training cost of sparse MPMC is only a fraction of the one for standard
MPMC.
The two plots in figure 1 show what typical learning curves for sparse MPMC look like.
As the number of basis function increases, both the bound ? and the test set accuracy start
Table 1: Bound ?, Test set accuracy (TSA), number of bases (B) for sparse and standard MPMC
SMPMC
Standard MPMC (Lanckriet et al.)
Dataset
?
TSA
?
TSA
B
B
Twonorm
Breast Cancer
Ionosphere
Pima Diabetes
Sonar
86.4 ? 0.1%
90.9 ? 0.1%
77.7 ? 0.2%
38.2 ? 0.1%
78.5 ? 0.2%
98.3 ? 0.4%
96.8 ? 0.3%
91.6 ? 0.5%
75.4 ? 0.7%
86.4 ? 1.0%
25
50
25
50
80
91.3 ? 0.1%
89.1 ? 0.1%
89.3 ? 0.2%
32.5 ? 0.2%
99.9 ? 0.1%
95.7 ? 0.5%
96.9 ? 0.3%
91.5 ? 0.7%
76.2 ? 0.6%
87.5 ? 0.9%
270
614
315
691
187
Table 2: training time (in seconds) for Matlab implementations of SMPMC and MPMC
# training
SMPMC
Standard MPMC (Lanckriet et al.)
Dataset
examples training time one optimization total training time
Twonorm
270
125.0
23.9
1199.2
614
188.5
122.4
6123.2
Breast Cancer
315
416.3
28.1
1404.3
Ionosphere
691
165.6
186.5
9324.2
Pima Diabetes
187
35.3
8.7
435.1
Sonar
to go up and after a while stabilize. The stabilization point usually occurs earlier when one
does full feature selection (a ? weight for each input dimension) instead of kernel width
selection (one uniform ? for all dimensions). We also experimented with different sizes
for the candidate set. The plots in figure 2 show what happens for 1, 5, and 10 candidates.
The overall behavior is that the test set accuracy as well as the ? value converge earlier for
larger candidate sets (but note that a larger candidate set also increases the computational
cost per iteration).
As seen in figure 1, feature selection gives usually better results in terms of the bound ?
and the test set accuracy. Furthermore, a feature selection algorithm should indicate which
features are relevant and which are not. We set up an experiment for the Twonorm data
(which has 20 input features) where we added 20 additional noisy features that were not
related to the output. The results are shown in figure 3 and demonstrate that the feature
selection algorithm obtained from SMPMC is able to distinguish between relevant and
irrelevant features.
4 Conclusion & future work
This paper introduces a new algorithm (Sparse Minimax Probability Machine Classification - SMPMC) for building sparse classification models that provide a lower bound on
the probability of classifying future data correctly. We have shown that the method of iteratively adding basis functions has significant computational advantages over the standard
MPMC, while it still maintains the distribution free probability bound ?. Experimental
results indicate that automated selection of kernel parameters, as well as automated feature
selection (weighting), both key characteristics of the SMPMC algorithm, result in error
rates that are competitive with those obtained by models where these parameters must be
tuned by computationally expensive cross validation.
Future research on sparse greedy MPMC will focus on establishing a theoretical framework
for a stopping criterion, when adding more basis functions (kernels) will not significantly
reduce error rates, and may lead to overfitting. Also, experiments have so far focused on using Gaussian kernels as basis functions. From the experience with other kernel algorithms,
it is known that other type of kernels (polynomial, tanh) can yield better results for certain
applications. Furthermore, our framework is not limited to Mercer kernels, and other types
Ionosphere
1
Sonar
0.9
0.8
0.9
0.7
0.8
0.6
0.7
0.5
0.6
? for WS
TSA for WS
? for FS
TSA for FS
0.5
0.4
0.3
0.2
0.3
0.2
? for WS
TSA for WS
? for FS
TSA for FS
0.4
0.1
0
5
10
15
20
25
0
0
10
20
30
basis functions
40
50
60
70
80
basis functions
Figure 1: Bound ? and Test Set accuracy (TSA) for width selection (WS) and feature selection (FS).
Note that the accuracies are all higher than the corresponding bounds.
0.4
0.78
Test set accuracy
0.76
? bound
0.35
0.74
0.3
0.72
0.25
0.7
0.2
0.68
1 candidate
5 candidates
10 candidates
0.66
0.64
1 candidate
5 candidates
10 candidates
0.15
0.1
0.62
0.05
0.6
0
2
4
6
8
10
12
14
16
18
0
2
4
6
8
10
12
14
16
18
20
basis functions
20
basis functions
Figure 2: Accuracy and bound for the Diabetes data set using 1,5 or 10 basis candidates per iteration.
Again, the ? bound is a true lower bound on the test set accuracy.
0.8
0.7
weight ?i
0.6
0.5
0.4
0.3
0.2
0.1
0
5
10
15
20
25
30
35
40
feature i
Figure 3: Average feature weighting for the Twonorm data set over 50 test runs. The first 20 features
are the original inputs, the last 20 features are additional noisy inputs
of basis functions are also worth investigating. Recent work by Crammer et al. [7] uses
boosting to construct a suitable kernel matrix iteratively. An interesting open question is
how this approach relates to sparse greedy MPMC.
References
[1] A. W. Marshall and I. Olkin. Multivariate chebyshev inequalities. Annals of Mathematical
Statistics, 31(4):1001?1014, 1960.
[2] I. Popescu and D. Bertsimas. Optimal inequalities in probability theory: A convex optimization
approach. Technical Report TM62, INSEAD, Dept. Math. O.R., Cambridge, Mass, 2001.
[3] G. R. G. Lanckriet, L. E. Ghaoui, C. Bhattacharyya, and M. I. Jordan. Minimax probability
machine. In T. G. Dietterich, S. Becker, and Z. Ghahramani, editors, Advances in Neural Information Processing Systems 14, Cambridge, MA, 2002. MIT Press.
[4] G. R. G. Lanckriet, L. E. Ghaoui, C. Bhattacharyya, and M. I. Jordan. A robust minimax approach
to classification. Journal of Machine Learning Research, 3:555?582, 2002.
[5] B. Sch?olkopf and A. Smola. Learning with Kernels. MIT Press, Cambridge, MA, 2002.
[6] William H. Beyer. CRC Standard Mathemathical Tables, page 12. CRC Press Inc., Boca Raton,
FL, 1987.
[7] K. Crammer, J. Keshet, and Y. Singer. Kernel design using boosting. In T. G. Dietterich,
S. Becker, and Z. Ghahramani, editors, Advances in Neural Information Processing Systems
15, Cambridge, MA, 2003. MIT Press.
Appendix: Proof of Theorem 1
We have to show that the values of m are equal for the two dimensional ?
MPMC and the k + 1
dimensional MPMC. We will just show the equivalence for the first term aT ?x a, an analogue
argumentation will hold for the second term.
For the two dimensional MPMC we have?
the following for the term under!the square root:
? (k+1) ?
?b
? 2(k)
(k) ~ (k+1)
? (k+1)
?
a1
`x K
b
`x
(k+1)
x
a1
a2
2
(k+1)
?K
?
(k+1)
(k)
a2
(k+1)
(19)
~
~
b
`x
K
x
x
=
(k+1) 2
[a1
(k+1) (k+1)
a2
?`(k) K
~ (k+1)
] ? 2(k) + 2a1
b
`x
bx
x
(k+1) 2
+ [a2
] ? 2~ (k+1)
Kx
Note that we can rewrite
~ x(1) + ... + ck K
~ x(k) , c0 + c1 K
~ x(1) + ... + ck K
~ x(k) )
? 2(k)
= Cov(c0 + c1 K
b
`x
=
?b
(k) ~ (k+1)
`
K
x
x
=
Pk
Pk
~x ,K
~x )
ci cj Cov(K
~ x(1) + ... + ck K
~ x(k) , K
~ x(k+1) )
Cov(c0 + c1 K
i=1
Pk
(i)
(j)
j=1
(20)
~x ,K
~x
c Cov(K
)
=
i=1 i
by using properties of the sample covariance (linearity, Cov(const, X) = 0).
(i)
(k+1)
For the k + 1 dimensional MPMC let us first determine the k + 1 coefficients:
(k+1)
~ x(1) + ... + ck K
~ x(k) ) + a(k+1) K
~ x(k+1) ? b(k+1)
`b(k+1) = a1
(c0 + c1 K
2
(k+1)
(1)
(k+1)
(k)
(k+1) ~ (k+1)
(k+1)
~
~
= a1
c1 Kx + ... + a1
c k Kx + a2
Kx
+ a1
c0 ? b(k+1)
The term under the square root then looks like:
? ? (k+1) ?
? (k+1) ?T ? ? 2 (1)
... ?K
?K
~ (1) K
~ (k)
~ (1) K
~ (k+1)
~
a1
c1
a1
c1
K
x
x
x
x
x
? ? ...
...
... ...
...
? ...
? ?
? ? (k+1) ?
? a(k+1) c ? ?
? (21)
... ? 2~ (k)
?K
? ?K~ x(k) K~ x(1)
~ (k) K
~ (k+1) ?
a1
ck
k
Kx
1
x
x
(k+1)
(k+1)
?K
... ?K
? 2~ (k+1)
a2
a2
~ (k+1) K
~ (1)
~ (k+1) K
~ (k)
Kx
x
x
x
x
Multiplying out (21) and substituting according to the equations in (20) yields exactly expression
T
(19) (which is the
p a ?x a term of the two dimensional MPM). Since this equivalence will hold
T
likewise for the a ?y a term in m, we have shown that m (and therefore ?) is equal for the two
dimensional and the k + 1 dimensional MPMC.
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1,618 | 2,471 | Minimising Contrastive Divergence in Noisy,
Mixed-mode VLSI Neurons
Hsin Chen, Patrice Fleury and Alan F. Murray
School of Engineering and Electronics
Edinburgh University
Mayfield Rd., Edinburgh
EH9 3JL, UK
{hc, pcdf, afm}@ee.ed.ac.uk
Abstract
This paper presents VLSI circuits with continuous-valued probabilistic behaviour realized by injecting noise into each computing
unit(neuron). Interconnecting the noisy neurons forms a Continuous Restricted Boltzmann Machine (CRBM), which has shown
promising performance in modelling and classifying noisy biomedical data. The Minimising-Contrastive-Divergence learning algorithm for CRBM is also implemented in mixed-mode VLSI, to
adapt the noisy neurons? parameters on-chip.
1
Introduction
As interests in interfacing electronic circuits to biological cells grows, an intelligent
embedded system able to classify noisy and drifting biomedical signals becomes important to extract useful information at the bio-electrical interface. Probabilistic
neural computation utilises probability to generalise the natural variability of data,
and is thus a potential candidate for underpinning such intelligent systems. To
date, probabilistic computation has been unable to deal with the continuous-valued
nature of biomedical data, while remaining amenable to hardware implementation. The Continuous Restricted Boltzmann Machine(CRBM) has been shown to
be promising in the modelling of noisy and drifting biomedical data[1][2], with
a simple Minimising-Contrastive-Divergence(MCD) learning algorithm[1][3]. The
CRBM consists of continuous-valued stochastic neurons that adapt their ?internal
noise? to code the variation of continuous-valued data, dramatically enriching the
CRBM?s representational power. Following a brief introduction of the CRBM, the
VLSI implementation of the noisy neuron and the MCD learning rule are presented.
2
Continuous Restricted Boltzmann Machine
Let si represent the state of neuron i, and wij the connection between neuron i and
neuron j. A noisy neuron j in the CRBM has the following form:
!
X
sj = ? j
wij si + ? ? Nj (0, 1) ,
(1)
i
1
1
0
0
?1
?1
0
1
?1
?1
(a)
0
1
(b)
Figure 1: (a)20 two-dimensional artificial training data (b)20-step reconstruction
by the CRBM after 30,000 epochs? fixed-step training
with ?j (xj ) = ?L + (?H ? ?L ) ?
1
1 + exp(?aj xj )
(2)
where Nj (0, 1) refers to a unit Gaussian noise with zero mean, ? a noise-scaling
constant, and ?j (?) the sigmoid function with asymptotes at ?H and ?L . Parameter
aj is the ?noise-control factor?, controlling the neuron?s output nonlinearity such
that a neuron j can learn to become near-deterministic (small aj ), continuousstochastic (moderate aj ), or binary-stochastic (large aj )[4][1].
A CRBM consists of one visible and one hidden layer of noisy neurons with interlayer connections defined by a weight matrix {W}. By minimizing the ?Contrastive
Divergence? between the training data and the one-step Gibbs sampled data [3],
the parameters {wij } and {aj } evolve according to the following equations [1]
?w?ij = ?w (hsi sj i ? hs?i s?j i)
?a
?a?j = 2 s2j ? s?j 2
aj
(3)
(4)
where s?i and s?j denote the one-step sampled state of neuron i and j respectively,
and h?i refers to the expectation over all training data. ?w and ?a denote the learning
rates for parameters {wij } and {aj }, respectively. Following [5], Eq.(3)and(4) are
further simplified to fixed-step directional learning, rather than variable accuratestep learning, as following.
?w?ij = ?w sign hsi sj i4 ? hs?i s?j i4
(5)
(6)
?a?j = ?a sign s2j 4 ? s?j 2 4
Note that the denominator 1/a2j in Eq.(4) is also absorbed and h?i4 indicates that
the expectation operator will be approximated by the average of four data as opposed to all training data. To validate the simplification above, a CRBM with 2
visible neurons and 4 hidden neurons was trained to model the two-dimensional
data distribution defined by 20 training data (Fig.1a), with ?w = 1.5, ?a = 15 for
visible neurons, and ?a = 1 for hidden neurons 1 . After 30,000 training updates,
the trained CRBM reconstructed the same data distribution (Fig.1b) from 200 initially random-distributed data, indicating that the simplification above reduces the
hardware complexity at the cost of a slightly slower convergence time.
1
constants ?H = ??L = 1 and ? = 0.2 for all neurons
si sr
wi
Vw
Mn1
Mp2
Vsi
Vsr
Vsi
Vsr
Vw
wr
M4
M5
Vw
Vsi
Vsr
I o1I o2I o3I o4
Vsr
Mn2
I o1I o2I o3I o4
Iout
Mp1
Io1
Io2
Io3
Io4
M3
M1
(a)
M2
M6
(b)
Figure 2: The circuits of the four-quadrant multiplier (a)one computing cell (b)full
circuit composed of two computing cell
3
Noisy neuron with variable nonlinearity
The circuits were fabricated on the AMS 0.6?m 2P3M CMOS process, which allows
a power supply voltage of five volts. Therefore, the states of neurons {s i } and the
corresponding weights {wij } are designed to be represented by voltage in [1.5, 3.5]V
and [0,5]V respectively, with both arithmetical zeros at 2.5V. As both si and wij
are real numbers, a four-quadrant multiplier is required to calculate wij si
3.1
Four-quadrant multiplier
While the Chible four-quadrant multiplier [6] has a simple architecture with a wide
input range, the reference zero of one of its inputs is process-dependent. Though
only relative values of weights matter for the neurons, the process-dependent reference becomes nontrivial if the same four-quadrant multiplier is used to implement
the MCD learning rule. We thus proposed a ?modified Chible multiplier? composed
of two computing cells, as shown in Fig.2, to allow external control of reference
zeros of both inputs.
Each computing cell contains two differential pairs biased by two complementary
branches, Mn1-Mn2 and Mp1-Mp2. (Io1 ?Io2 ) is thus proportional to (Vw ?Vth,n1 ?
nVth,n2 )(Vsi ? Vsr ) when Vw > (Vth,n1 + nVth,n2 ) 2 , and (Io3 ? Io4 ) proportional to
(n2 V dd ? Vw ? Vth,p1 ? nVth,p2 )(Vsr ? Vsi ) when Vw < (n2 V dd ? Vth,p1 ? nVth,p2 )[6].
Subject to careful design of the complementary biasing transistors[6], (Vth,n1 +
nVth,n2 ) ? (n2 V dd ? Vth,p1 ? nVth,p2 ) ? V dd/2. Combining the two differential
currents then gives
Io = (Io1 + Io3 ) ? (Io2 + Io4 ) = I(Vw ) ? (Vsi ? Vsr )
(7)
With wi input to one computing cell and wr to the other cell, as shown in Fig.2b,
M1-M6 generates an output current Iout ? (wi ? wr )(si ? sr ). The measured DC
characteristic from a fabricated chip is shown in Fig.4(a)
3.2
Noisy neuron
Fig.3 shows the circuit diagram of a noisy
P neuron. The four-quadrant multipliers
output a total current proportional to i wij si , while the differential pair, Mna and
2
n is the slope factor of MOS transistor, and Vth,x refers to the absolute value of
transistor Mx?s threshold voltage.
s1
Vaj
w1
Ib
isum
s2
w2
Vsr
io
_
+
Vx
Mbp1
ic1
Vsj
Mbp2
RL
ic2
Csj
in
si
wi
vni
Mna Mnb
Vnr
Vsigma
Figure 3: The circuit diagram of a noisy neuron
Mnb, transforms noise voltage vni into a noise current in = gm (vni ? Vnr ), where
Vsigma controls the transconductance gm and thus scales the noise current as ? in
Eq.(1). The current-to-voltage converter, composed of an operational amplifier and
an voltage-controlled active resistor[7], then sums all currents, outputting a voltage
Vx = Vsr ? isum ? R(Vaj ) to the sigmoid function.
The exponential nonlinearity of the sigmoid function is achieved by operating the
PMOS differential pair, Mbp1-Mbp2, in the lateral-bipolar mode [8], resulting in a
differential output current as following
io = ic1 ? ic2 = Ib ? ?(
Isum ? R(Vaj )
)
Vt
(8)
where ?(?) denotes the ?(?) with ?H = ??L = 1, and Vt = kT /q is the thermal
voltage. The resistor RL finally converts io into a output voltage vo = io RL + Vsr .
Eq.(8) implies that Vaj controls the feedback resistance of the I-V converter, and
consequently adapts the nonlinearity of the sigmoid function (which appears as aj
in Eq.(1)). With various Vaj , the measured DC characteristic (chip result) of the
sigmoid function is shown in Fig.4b.
Vsi=1.5
Vsi=1.75
Vsi=2.0
Vsi=2.25
Vsi=2.5
Vsi=2.75
Vsi=3.0
Vsi=3.25
Vsi=3.5
Iout (amps)
2.0?
1.0?
0.0
-1.0?
3.5
3.0
Vo (volts)
3.0?
2.5
Vaj=1.0
Vaj=1.4
Vaj=1.8
Vaj=2.2
Vaj=2.6
Vaj=3.0
2.0
-2.0?
-3.0?
0.0
1.5
2.5
Vw (volts)
(a)
5.0
-50.00? -25.00?
0.00
25.00?
50.00?
Isum (amps)
(b)
Figure 4: The measured DC characteristics of (a) four-quadrant multiplier
(b)sigmoid function with variable nonlinearity controlled by Vaj
(a)
(b)
Figure 5: (a)The measured output of a noisy neuron (upper trace) and the switching
signal (lower trace) that samples Vsj (b) Zooming-in of the second sample in(a)
Fig.5 shows the measured output of a noisy neuron (upper trace) with {si } sweeping
between 1.5 and 3.5V, {wi }=4V, Vaj =1.8V, and vni generated by LFSR (Linear
Feedback Shift Register) [9] with an amplitude of 0.4V. The {si } and {wi } above
forced the neuron?s output to sweep a sigmoid-shaped curve as Fig.4b, while the
input noise disturbed the curve to achieve continous-valued probabilistic output. A
neuron state Vsj was sampled periodically and held with negligible clock feedthrough
whenever the switch opened(went low).
4
Minimising-Contrastive-Divergence learning on chip
The MCD learning for the Product of Experts[3] has been successfully implemented
and reported in [10]. The MCD learning for CRBM is therefore implemented simply
by replacing the following two circuits. First, the four-quadrant multiplier described
in Sec.3.1 is substituted for the two-quadrant multiplier in [10] to enhance learning
flexibility; secondly, a pulse-coded learning circuit, rather than the analogue weightchanging circuit in [10], is employed to allow not only accurate learning steps but
also refresh of dynamically-held parameters.
4.1
MCD learning for CRBM
Fig.6 shows the block diagram of the VLSI implementation of the MCD learning
rules for the noisy neurons, along with the digital control signals. In learning mode
(LER/REF=1), the initial states si and sj are first sampled by clock signals CKsi
and CKsj , resulting in a current I+ at the output of four-quadrant multiplier.
After CK+ samples and holds I+ , the one-step reconstructed states s?i and s?j are
sampled by CKsip and CKsjp to produce another current I? . CKq then samples
and holds the output of the current subtracter Isub , which represents the difference
between initial data and one-step Gibbs sampled data. Repeating the above clocking
sequence for four cycles, four Isub are accumulated and averaged to derive Iave ,
representing hsi sj i4 ?hs?i s?j i4 in equation(5). Finally, Iave is compared to a reference
current to determine the learning direction DIR, and the learning circuit, triggered
by CKup , updates the parameter once. The dash-lined box represents the voltagelimiting circuit used only for parameter {aj }, whose voltage range should be limited
to ensure normal operation of the voltage-controlled active resistor in Fig.3. In
refresh mode (LER/REF=1), the signal REFR rather than DIR determines the
updating direction, maintaining the weight to a reference value.
CK si
CK q
Digital control
si
CK +
CK sip
q2
q1
I+
Isub
s? i
I
q3
CKsi
q4
Currentaccumulating/
averaging circuit
Iref
Iave
CK sj
CK+
-
+
Sign
Pulse-coded
learning circuit
Cw
CK sjp
s? j
CK up
Voltage
limiter
Vmu
LER/REF REFR
CKsip
CKsjp
DIR
sj
CKsj
Vcomp
CKq
q1
q4
CKup
Vmax Vmin
(a)
(b)
Figure 6: (a)The block diagram of VLSI implementation of MCD learning rules
described in Eq.(5)(6) (b)The digital control signals
The subtracter, accumulator and current comparator in Fig.6 are dominated by the
dynamic current mirror[11] and are the same as those used in [10]. The following
subsections therefore focus on the pulse-coded learning circuit and the measurement
results of on-chip MCD learning.
4.2
The pulse-coded learning circuit
The pulse-coded learning circuit consists of a pulse generator (Fig.7a) and the learning cell proposed in [12] (Fig.7b). The stepsize of the learning cell is adjustable
through VP and VN in Fig.7b [12]. However, transistor nonlinearities and process
variations do not allow different and accurate learning rates to be set for various
parameters in the same chip ({aj } and {wij } in our case). We therefore apply a
width-variable pulse to the enabling input (EN) of the learning cell, controlling the
learning step precisely by monitoring the pulse width off-chip. As the input capacitance of each learning cell is less than 0.1pF, one pulse generator can control all
the learning cells with the same learning rate. The simulation in Sec.2 implies that
only three pulse generators are required for ?w , ?av , and ?ah . The pulse generator
is therefore a simple way to achieve accurate control.
The pulse generator is largely a D-type flip-flop whose output Vpulse is initially
reset to low via reset. Vpulse then goes high on the rising edge of CKup , while the
Vpulse
D
Q
EN
VP
CK up
R
Q
Vd
Cw
C delay
Vmu
VN
INC/DEC
reset
(a)
(b)
Figure 7: The pulse-coded learning circuit composed of (a)a pulse generator and
(b)a learning cell proposed in [12]
Vcomp
Vmax
+
_
DIR
Vaj
Vmin
+
_
Figure 8: The voltage-limiting circuit
capacitor Cdelay prevents Vd from going from high to low instantly. Eventually,
Vpulse is reset to zero as soon as Vd is discharged. During the positive pulse, the
learning cell charges or discharges the voltage stored on Cw [12], according to the
directional input INC/DEC. Varying Vmu controls the pulse width accurately from
10ns (V? = 2.5V ) to 5us (V? = 0.9V ), amounting to learning stepsize from 1mV to
500mV as VN = 0.75V , VP = 4.29V , and Cw = 1pF .
4.3
Voltage-limiting circuit
Although Eq.(6) indicates that {aj } can be adapted with the same learning circuit
simply by substituting sj and s?j for si and s?i in Fig.6, the voltage Vaj should
be confined in [1,3]V, to ensure normal operation of the voltage-controlled active
resistor in Fig.3. A voltage-limiting circuit as shown in Fig.8 is thus designed to
limit the range of Vaj , defined by Vmax and Vmin through two voltage comparators.
As Vmax > Vaj > Vmi , DIR equals Vcomp , i.e. the MCD learning rule decides
the learning direction. However, DIR goes high to enforce decreasing Vaj when
Vaj > Vmax > Vmin , while DIR goes low to enforce increasing Vaj when Vmax >
Vmin > Vaj .
4.4
On-chip learning
Two MCD learning circuits, one for {wij } and the other for {aj }, have been fabricated successfully. Fig.9 shows the measured on-chip learning of both parameters
with (a) different learning rates (b) different learning directions. To ease testing,
si and s?i are fixed at 3.5V, while sj and s?j alternate between 1.5V and 3.5V, as
shown by the traces SJ and SJ P in Fig.9. With the reference zero being defined at
(a)
(b)
Figure 9: Measurement of parameter aj and wij learning in (a)different learning
rates (b)different directions
2.5V, the parameters should learn down when sj =3.5V and s?j =1.5V, and learn up
when sj =1.5V and s?j =3.5V.
In Fig.9a, both parameters were initially refreshed to 2.5V when signal LERREF is
low, and subsequently started to learn up and down in response to the changing SJ
and SJ P as LERREF goes high. As controlled by different pulse widths (PULSE1
and PULSE2), the two parameters were updated with different stepsizes (10mV and
34mV) but in the same direction. The trace of parameter aj shows digital noise
attributable to sub-optimal layout, and has been improved in a subsequent design.
In Fig.9b, both parameters were refreshed to 3.5V, a voltage higher than Vmax =3V
set for aj . Therefore, the learning circuit forces aj to decrease toward Vmax , while
wij remains learning up and down as Fig.9a.
5
Conclusion
Fabricated CMOS circuits have been presented and the implemention of noisy neural
computation that underlies the CRBM has been demonstrated. The promising measured results show that the CRBM is, as has been inferred in the past[1], amenable
to mixed-mode VLSI. This makes possible a VLSI system with continuous-valued
probabilistic behaviour and on-chip adaptability, adapting its ?internal noise? to
model the ?external noise? in its environment. A full CRBM system with two visible and four hidden neurons has thus been implemented to examine this concept.
The neurons in the proof-of-concept CRBM system are hard-wired to each other
and the multi-channel uncorrelated noise sources implemented by the LFSR [9]. A
scalable design will thus be an essential next step before pratical biomedical applications. Furthermore, the CRBM system may open the possibility of utilising VLSI
intrinsic noise for computation in the deep-sub-miron era.
References
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training algorithm,? IEE Proc. of Vision, Image and Signal Processing, vol. 150, no. 3,
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[3] G. E. Hinton, ?Training products of experts by minimizing contrastive divergence,? Neural
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[4] B. J. Frey, ?Continuous sigmoidal belief networks trained using slice sampling,? Advances in
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[12] G. Cauwenberghs, ?An analog vlsi recurrent neural network,? IEEE Tran. on Neural Networks, vol. 7, pp. 346?360, Mar. 1996.
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1,619 | 2,472 | A Recurrent Model of Orientation Maps
with Simple and Complex Cells
Paul Merolla and Kwabena Boahen
Department of Bioengineering
University of Pennsylvania
Philadelphia, PA 19104
{pmerolla,boahen} @seas.upenn.edu
Abstract
We describe a neuromorphic chip that utilizes transistor
heterogeneity, introduced by the fabrication process, to generate
orientation maps similar to those imaged in vivo. Our model
consists of a recurrent network of excitatory and inhibitory cells in
parallel with a push-pull stage. Similar to a previous model the
recurrent network displays hotspots of activity that give rise to
visual feature maps. Unlike previous work, however, the map for
orientation does not depend on the sign of contrast. Instead, signindependent cells driven by both ON and OFF channels anchor the
map, while push-pull interactions give rise to sign-preserving cells.
These two groups of orientation-selective cells are similar to
complex and simple cells observed in V1.
1
Orientation Maps
Neurons in visual areas 1 and 2 (V1 and V2) are selectively tuned for a number of
visual features, the most pronounced feature being orientation. Orientation
preference of individual cells varies across the two-dimensional surface of the
cortex in a stereotyped manner, as revealed by electrophysiology [1] and optical
imaging studies [2]. The origin of these preferred orientation (PO) maps is debated,
but experiments demonstrate that they exist in the absence of visual experience [3].
To the dismay of advocates of Hebbian learning, these results suggest that the initial
appearance of PO maps rely on neural mechanisms oblivious to input correlations.
Here, we propose a model that accounts for observed PO maps based on innate noise
in neuron thresholds and synaptic currents. The network is implemented in silicon
where heterogeneity is as ubiquitous as it is in biology.
2
Patterned Activity Model
Ernst et al. have previously described a 2D rate model that can account for the
origin of visual maps [4]. Individual units in their network receive isotropic
feedforward input from the geniculate and recurrent connections from neighboring
units in a Mexican hat profile, described by short-range excitation and long-range
inhibition. If the recurrent connections are sufficiently strong, hotspots of activity
(or ?bumps?) form periodically across space. In a homogeneous network, these
bumps of activity are equally stable at any position in the network and are free to
wander.
Introducing random jitter to the Mexican hat connectivity profiles breaks the
symmetry and reduces the number of stable states for the bumps. Subsequently, the
bumps are pinned down at the locations that maximize their net local recurrent
feedback. In this regime, moving gratings are able to shift the bumps away from
their stability points such that the responses of the network resemble PO maps.
Therefore, the recurrent network, given an ample amount of noise, can innately
generate its own orientation specificity without the need for specific hardwired
connections or visually driven learning rules.
2.1
Criticisms of the Bump model
We might posit that the brain uses a similar opportunistic model to derive and
organize its feature maps ? but the parallels between the primary visual cortex and
the Ernst et al. bump model are unconvincing. For instance, the units in their model
represent the collective activity of a column, reducing the network dynamics to a
firing-rate approximation. But this simplification ignores the rich temporal
dynamics of spiking networks, which are known to affect bump stability. More
fundamentally, there is no role for functionally distinct neuron types.
The primary criticism of the Ernst et al.?s bump model is that its input only consists
of a luminance channel, and it is not obvious how to replace this channel with ON
and OFF rectified channels to account for simple and complex cells. One possibility
would be to segregate ON-driven and OFF-driven cells (referred to as simple cells in
this paper) into two distinct recurrent networks. Because each network would have
its own innate noise profile, bumps would form independently. Consequently, there
is no guarantee that ON-driven maps would line up with OFF-driven maps, which
would result in conflicting orientation signals when these simple cells converge onto
sign-independent (complex) cells.
2.2
Simple Cells Solve a Complex Problem
To ensure that both ON-driven and OFF-driven simple cells have the same
orientation maps, both ON and OFF bumps must be computed in the same recurrent
network so that they are subjected to the same noise profile. We achieve this by
building our recurrent network out of cells that are sign-independent; that is both
ON and OFF channels drive the network. These cells exhibit complex cell-like
behavior (and are referred to as complex cells in this paper) because they are
modulated at double the spatial frequency of a sinusoidal grating input. The simple
cells subsequently derive their responses from two separate signals: an orientation
selective feedback signal from the complex cells indicating the presence of either an
ON or an OFF bump, and an ON?OFF selection signal that chooses the appropriate
response flavor.
Figure 1 left illustrates the formation of bumps (highlighted cells) by a recurrent
network with a Mexican hat connectivity profile. Extending the Ernst et al. model,
these complex bumps seed simple bumps when driven by a grating. Simple bumps
that match the sign of the input survive, whereas out-of-phase bumps are
extinguished (faded cells) by push-pull inhibition.
Figure 1 right shows the local connections within a microcircuit. An EXC
(excitatory) cell receives excitatory input from both ON and OFF channels, and
projects to other EXC (not shown) and INH (inhibitory) cells. The INH cell projects
back in a reciprocal configuration to EXC cells. The divergence is indicated in left.
ON-driven and OFF-driven simple cells receive input in a push-pull configuration
(i.e., ON cells are excited by ON inputs and inhibited by OFF inputs, and vise-versa),
while additionally receiving input from the EXC?INH recurrent network. In this
model, we implement our push-pull circuit using monosynaptic inhibitory
connections, despite the fact that geniculate input is strictly excitatory. This
simplification, while anatomically incorrect, yields a more efficient implementation
that is functionally equivalent.
ON Input
Luminance
OFF Input
left
right
EXC
EXC
Divergence
INH
INH
Simple Cells
Complex Cells
ON & OFF Input
ON
OFF
OFF
Space
Figure 1: left, Complex and simple cell responses to a sinusoidal grating input.
Luminance is transformed into ON (green) and OFF (red) pathways by retinal
processing. Complex cells form a recurrent network through excitatory and
inhibitory projections (yellow and blue lines, respectively), and clusters of activity
occur at twice the spatial frequency of the grating. ON input activates ON-driven
simple cells (bright green) and suppresses OFF-driven simple cells (faded red), and
vise-versa. right, The bump model?s local microcircuit: circles represent neurons,
curved lines represent axon arbors that end in excitatory synapses (v shape) or
inhibitory synapses (open circles). For simplicity, inhibitory interneurons were
omitted in our push-pull circuit.
2.3
Mathematical Description
?
The neurons in our network follow? the equation CV = ??n ?(t ? tn) + I syn ? I KCa ? I leak ,
where C is membrane capacitance, V is the temporal derivative of the membrane
voltage, ?(?) is the Dirac delta function, which resets the membrane at the times tn
when it crosses threshold, Isyn is synaptic current from the network, and Ileak is a
constant leak current. Neurons receive synaptic current of the form:
ON
I syn
= w+ I ON ? w? I OFF + wEE I EXC ? wEI I INH ,
EXC
I syn
= w+ ( I ON + I OFF ) + wEE I EXC ? wEI I INH + I back ,
OFF
INH
I syn
= w+ I OFF ? w? I ON + wEE I EXC ? wEI I INH , I syn
= wIE I EXC
where w+ is the excitatory synaptic strength for ON and OFF input synapses, w- is the
strength of the push-pull inhibition, wEE is the synaptic strength for EXC cell
projections to other EXC cells, wEI is the strength of INH cell projections to EXC
cells, wIE is the strength of EXC cell projections to INH cells, Iback is a constant input
current, and I{ON,OFF,EXC,INH} account for all impinging synapses from each of the four
cell types. These terms are calculated for cell i using an arbor function that consists
of a spatial weighting J(r) and a post-synaptic current waveform ?(t):
? J (i ? k ) ? ? (t ? t nk ) , where k spans all cells of a given type and n indexes their spike
k ,n
times. The spatial weighting function is described by J (i ? k ) = exp( ? i ? k ? ) , with ?
as the space constant. The current waveform, which is non-zero for t>0, convolves
a 1 t function with a decaying exponential: ? (t ) = (t ? c + ? 0 ) ?1 ? exp(? t ? e ) , where ?c is
the decay-rate, and ?e is the time constant of the exponential. Finally, we model
spike-rate adaptation with a calcium-dependent potassium-channel (KCa), which
integrates Ca triggered by spikes at times tn with a gain K and a time constant ?k, as
described by I KCa = ? K exp(tn ? t ? k ) .
n
3
Silicon Implementation
We implemented our model in silicon using the TSMC (Taiwan Semiconductor
Manufacturing Company) 0.25?m 5-metal layer CMOS process. The final chip
consists of a 2-D core of 48x48 pixels, surrounded by asynchronous digital circuitry
that transmits and receives spikes in real-time. Neurons that reach threshold within
the array are encoded as address-events and sent off-chip, and concurrently,
incoming address-events are sent to their appropriate synapse locations. This
interface is compatible with other spike-based chips that use address-events [5].
The fabricated bump chip has close to 460,000 transistors packed in 10 mm2 of
silicon area for a total of 9,216 neurons.
3.1
Circuit Design
Our neural circuit was morphed into hardware using four building blocks. Figure 2
shows the transistor implementation for synapses, axonal arbors (diffuser), KCa
analogs, and neurons. The circuits are designed to operate in the subthreshold
region (except for the spiking mechanism of the neuron). Noise is not purposely
designed into the circuits. Instead, random variations from the fabrication process
introduce significant deviations in I-V curves of theoretically identical MOS
transistors.
The function of the synapse circuit is to convert a brief voltage pulse (neuron spike)
into a postsynaptic current with biologically realistic temporal dynamics. Our
synapse achieves this by cascading a current-mirror integrator with a log-domain
low-pass filter. The current-mirror integrator has a current impulse response that
decays as 1 t (with a decay rate set by the voltage ?c and an amplitude set by A).
This time-extended current pulse is fed into a log-domain low-pass filter (equivalent
to a current-domain RC circuit) that imposes a rise-time on the post-synaptic current
set by ?e. ON and OFF input synapses receive presynaptic spikes from the off-chip
link, whereas EXC and INH synapses receive presynaptic spikes from local on-chip
neurons.
Synapse
Je
Diffuser
Ir
A
Ig
Jc
KCa Analog
Neuron
Jk
Vmem
Vspk
K
Figure 2: Transistor implementations are shown for a synapse, diffuser, KCa analog,
and neuron (simplified), with circuit insignias in the top-left of each box. The
circuits they interact with are indicated (e.g. the neuron receives synaptic current
from the diffuser as well as adaptation current from the KCa analog; the neuron in
turn drives the KCa analog). The far right shows layout for one pixel of the bump
chip (vertical dimension is 83?m, horizontal is 30 ?m).
The diffuser circuit models axonal arbors that project to a local region of space with
an exponential weighting. Analogous to resistive divider networks, diffusers [6]
efficiently distribute synaptic currents to multiple targets. We use four diffusers to
implement axonal projections for: the ON pathway, which excites ON and EXC cells
and inhibits OFF cells; the OFF pathway, which excites OFF and EXC cells and
inhibits ON cells; the EXC cells, which excite all cell types; and the INH cells, which
inhibits EXC, ON, and OFF cells. Each diffuser node connects to its six neighbors
through transistors that have a pseudo-conductance set by ?r, and to its target site
through a pseudo-conductance set by ?g; the space-constant of the exponential
synaptic decay is set by ?r and ?g?s relative levels.
The neuron circuit integrates diffuser currents on its membrane capacitance.
Diffusers either directly inject current (excitatory), or siphon off current (inhibitory)
through a current-mirror. Spikes are generated by an inverter with positive
feedback (modified from [7]), and the membrane is subsequently reset by the spike
signal. We model a calcium concentration in the cell with a KCa analog. K
controls the amount of calcium that enters the cell per spike; the concentration
decays exponentially with a time constant set by ?k. Elevated charge levels activate
a KCa-like current that throttles the spike-rate of the neuron.
3.2
Experimental Setup
Our setup uses either a silicon retina [8] or a National Instruments DIO (digital
input?output) card as input to the bump chip. This allows us to test our V1 model
with real-time visual stimuli, similar to the experimental paradigm of
electrophysiologists. More specifically, the setup uses an address-event link [5] to
establish virtual point-to-point connectivity between ON or OFF ganglion cells from
the retina chip (or DIO card) with ON or OFF synapses on the bump chip. Both the
input activity and the output activity of the bump chip is displayed in real-time
using receiver chips, which integrate incoming spikes and displays their rates as
pixel intensities on a monitor. A logic analyzer is used to capture spike output from
the bump chip so it can be further analyzed.
We investigated responses of the bump chip to gratings moving in sixteen different
directions, both qualitatively and quantitatively. For the qualitative aspect, we
created a PO map by taking each cell?s average activity for each stimulus direction
and computing the vector sum. To obtain a quantitative measure, we looked at the
normalized vector magnitude (NVM), which reveals the sharpness of a cell?s tuning.
The NVM is calculated by dividing the vector sum by the magnitude sum for each
cell. The NVM is 0 if a cell responds equally to all orientations, and 1 if a cell?s
orientation selectivity is perfect such that it only responds at a single orientation.
4
Results
We presented sixteen moving gratings to the network, with directions ranging from
0 to 360 degrees. The spatial frequency of the grating is tuned to match the size of
the average bump, and the temporal frequency is 1 Hz. Figure 3a shows a resulting
PO map for directions from 180 to 360 degrees, looking at the inhibitory cell
population (the data looks similar for other cell types). Black contours represent
stable bump regions, or equivalently, the regions that exceed a prescribed threshold
(90 spikes) for all directions. The PO map from the bump chip reveals structure that
resembles data from real cortex. Nearby cells tend to prefer similar orientations
except at fractures. There are even regions that are similar to pinwheels (delimited
by a white rectangle).
A PO is a useful tool to describe a network?s selectivity, but it only paints part of
the picture. So we have additionally computed a NVM map and a NVM histogram,
shown in Figure 3b and 3c respectively. The NVM map shows that cells with sharp
selectivity tend to cluster, particularly around the edge of the bumps. The histogram
also reveals that the distribution of cell selectivity across the network varies
considerably, skewed towards broadly tuned cells.
We also looked at spike rasters from different cell-types to gain insight into their
phase relationship with the stimulus. In particular, we present recordings for the
site indicated by the arrow (see Figure 3a) for gratings moving in eight directions
ranging from 0 to 360 degrees in 45-degree increments (this location was chosen
because it is in the vicinity of a pinwheel, is reasonably selective, and shows
considerable modulation in its firing rate). Figure 4 shows the luminance of the
stimulus (bottom sinusoids), ON- (cyan) and OFF-input (magenta) spike trains, and
the resulting spike trains from EXC (yellow), INH (blue), ON- (green), and OFFdriven (red) cell types for each of the eight directions. The center polar plot
summarizes the orientation selectivity for each cell-type by showing the normalized
number of spikes for each stimulus. Data is shown for one period.
Even though all cells-types are selective for the same orientation (regardless of
grating direction), complex cell responses tend to be phase-insensitive while the
simple cell responses are modulated at the fundamental frequency. It is worth
noting that the simple cells have sharper orientation selectivity compared to the
complex cells. This trend is characteristic of our data.
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
300
250
200
150
100
50
20
40
60
80
100
120
140
160
180
0
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Figure 3: (a) PO map for the inhibitory cell population stimulated with eight
different directions from 180 to 360 degrees (black represents no activity, contours
delineate regions that exceed 90 spikes for all stimuli). Normalized vector
magnitude (NVM) data is presented as (b) a map and (c) a histogram.
Figure 4: Spike rasters and polar plot for 8 directions ranging from 0 to 360 degrees.
Each set of spike rasters represent from bottom to top, ON- (cyan) and OFF-input
(magenta), INH (yellow), EXC (blue), and ON- (green) and OFF-driven (red). The
stimulus period is 1 sec.
5
Discussion
We have implemented a large-scale network of spiking neurons in a silicon chip that
is based on layer 4 of the visual cortex. The initial testing of the network reveals a
PO map, inherited from innate chip heterogeneities, resembling cortical maps. Our
microcircuit proposes a novel function for complex-like cells; that is they create a
sign-independent orientation selective signal, which through a push-pull circuit
creates sharply tuned simple cells with the same orientation preference.
Recently, Ringach et al. surveyed orientation selectivity in the macaque [9]. They
observed that, in a population of V1 neurons (N=308) the distribution of orientation
selectivity is quite broad, having a median NVM of 0.39. We have measured
median NVM?s ranging from 0.25 to 0.32. Additionally, Ringach et al. found a
negative correlation between spontaneous firing rate and NVM. This is consistent
with our model because cells closer to the center of the bump have higher firing
rates and broader tuning.
While the results from the bump chip are promising, our maps are less consistent
and noisier than the maps Ernst et al. have reported. We believe this is because our
network is tuned to operate in a fluid state where bumps come on, travel a short
distance and disappear (motivated by cortical imaging studies). But excessive
fluidity can cause non-dominant bumps to briefly appear and adversely shift the PO
maps. We are currently investigating the role of lateral connections between bumps
as a means to suppress these spontaneous shifts.
The neural mechanisms that underlie the orientation selectivity of V1 neurons are
still highly debated. This may be because neuron responses are not only shaped by
feedforward inputs, but are also influenced at the network level. If modeling is
going to be a useful guide for electrophysiologists, we must model at the network
level while retaining cell level detail. Our results demonstrate that a spike-based
neuromorphic system is well suited to model layer 4 of the visual cortex. The same
approach may be used to build large-scale models of other cortical regions.
References
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1,620 | 2,473 | Design of experiments via information theory ?
Liam Paninski
Center for Neural Science
New York University
New York, NY 10003
[email protected]
Abstract
We discuss an idea for collecting data in a relatively efficient manner. Our
point of view is Bayesian and information-theoretic: on any given trial,
we want to adaptively choose the input in such a way that the mutual information between the (unknown) state of the system and the (stochastic)
output is maximal, given any prior information (including data collected
on any previous trials). We prove a theorem that quantifies the effectiveness of this strategy and give a few illustrative examples comparing the
performance of this adaptive technique to that of the more usual nonadaptive experimental design. For example, we are able to explicitly calculate
the asymptotic relative efficiency of the ?staircase method? widely employed in psychophysics research, and to demonstrate the dependence of
this efficiency on the form of the ?psychometric function? underlying the
output responses.
1
Introduction
One simple model of experimental design (we have neurophysiological experiments in
mind, but our results are general with respect to the identity of the system under study)
is as follows. We have some set X of input stimuli, and some knowledge of how the
system should respond to every stimulus, x, in X. This knowledge is summarized in the
form of a prior distribution, p0 (?), on some space ? of models ?. A model is a set of
probabilistic input-output relationships: regular conditional distributions p(y|x, ?) on Y ,
the set of possible output responses, given each x in X. Thus the joint probability of
stimulus and response is:
Z
Z
p(x, y) = p(x, y, ?)d? = p0 (?)p(x)p(y|?, x)d?.
The ?design? of an experiment is given by the choice of input probability p(x). We want
to design our experiment ? choose p(x) ? optimally in some sense. One natural idea
would be to choose p(x) in such a way that we learn as much as possible about the underlying model, on average. Information theory thus suggests we choose p(x) to optimize the
?
A longer version of this paper, including proofs, has been submitted and is available at
http://www.cns.nyu.edu/?liam.
following objective function:
I({x, y}; ?) =
Z
p(x, y, ?) log
X?Y ??
p(x, y, ?)
p(x, y)p(?)
(1)
where I(.; .) denotes mutual information. In other words, we want to maximize the information provided about ? by the pair {x, y}, given our current knowledge of the model as
summarized in the posterior distribution given N samples of data:
pN (?) = p(?|{xi , yi }1?i?N ).
Similar ideas have seen application in a wide and somewhat scattered literature; for a partial
bibliography, see the longer draft of this paper at http://www.cns.nyu.edu/?liam. Somewhat surprisingly, we have not seen any applications of the information-theoretic objective
function (1) to the design of neurophysiological experiments (although see the abstract by
[7], who seem to have independently implemented the same idea in a simulation study).
The primary goal of this paper is to elucidate the asymptotic behavior of the a posteriori
density pN (?) when we choose x according to the recipe outlined above; in particular,
we want to compare the adaptive strategy to the more usual case, in which the stimuli are
drawn i.i.d. (non-adaptively) from some fixed distribution p(x). Our main result (section
2) states that, under acceptably weak conditions on the models p(y|?, x), the informationmaximization strategy leads to consistent and efficient estimates of the true underlying
model, in a natural sense. We also give a few simple examples to illustrate the applicability
of our results (section 3).
2
Main Result
First, we note that the problem as posed in the introduction turns out to be slightly easier
than one might have expected, because I({x, y}; ?) is linear in p(x). This, in turn, implies
that p(x) must be degenerate, concentrated on the points x where I is maximal. Thus,
instead of finding optimal distributions p(x), we need only find optimal inputs x, in the
sense of maximizing the conditional information between ? and y, given a single input x:
Z Z
p(y|x, ?)
I(y; ?|x) ?
pN (?)p(y|?, x) log R
.
p
(?)p(y|x, ?)
Y ?
? N
Our main result is a ?Bernstein-von Mises? - type theorem [12]. The classical form of
this kind of result says, basically, that if the posterior distributions are consistent (in the
sense that pN (U ) ? 1 for any neighborhood U of the true parameter ?0 ) and the relevant
likelihood ratios are sufficiently smooth on average, then the posterior distributions p N (?)
are asymptotically normal, with easily calculable asymptotic mean and variance. We adapt
this result to the present case, where x is chosen according to the information-maximization
recipe. It turns out that the hard part is proving consistency (c.f. section 4); we give the
basic consistency lemma (interesting in its own right) first, from which the main theorem
follows fairly easily.
Lemma 1 (Consistency). Assume the following conditions:
1. The parameter space ? is compact.
2. The loglikelihood log p(y|x, ?) is Lipschitz in ?, uniformly in x, with respect to
some dominating measure on Y .
3. The prior measure p0 assigns positive measure to any neighborhood of ?0 .
4. The maximal divergence supx DKL (?0 ; ?|x) is positive for all ? 6= ?0 .
Then the posteriors are consistent: pN (U ) ? 1 in probability for any neighborhood U of
?0 .
Theorem 2 (Asymptotic normality). Assume the conditions of Lemma 1, stengthened as
follows:
1. ? has a smooth, finite-dimensional manifold structure in a neighborhood of ? 0 .
2. The loglikelihood log p(y|x, ?) is uniformly C 2 in ?. In particular, the Fisher
information matrices
t
Z
p(y|x,
?
?)
p(y|x,
?
?)
I? (x) =
p(y|?, x),
p(y|x, ?)
p(y|x, ?)
Y
where the differential p? is taken with respect to ?, are well-defined and continuous
in ?, uniformly in (x, ?) in some neighborhood of ?0 .
3. The prior measure p0 is absolutely continuous in some neighborhood of ?0 , with
a continuous positive density at ?0 .
4.
maxC?co(I?0 (x)) det(C) > 0,
where co(I?0 (x)) denotes the convex closure of the set of Fisher information matrices I?0 (x).
Then
2
||pN ? N (?N , ?N
)|| ? 0
2
in probability, where ||.|| denotes variation distance, N (?N , ?N
) denotes the normal den2
sity with mean ?N and variance ?N , and ?N is asymptotically normal with mean ?0 and
2
variance ?N
. Here
2 ?1
(N ?N
) ? argmaxC?co(I?
0
(x))
det(C);
the maximum in the above expression is well-defined and unique.
Thus, under these conditions, the information maximization strategy works, and works
better than the i.i.d. x strategy (where the asymptotic variance ? 2 is inversely related to an
average, not a maximum, over x, and is therefore generically larger).
A few words about the assumptions are in order. Most should be fairly self-explanatory: the
conditions on the priors, as usual, are there to ensure that the prior becomes irrelevant in the
face of sufficient posterior evidence; the smoothness assumptions on the likelihood permit
the local expansion which is the source of asymptotic normality; and the condition on the
maximal divergence function supx DKL (?0 ; ?|x) ensures that distinct models ?0 and ? are
identifiable. Finally, some form of monotonicity or compactness on ? is necessary, mostly
to bound the maximal divergence function supx DKL (?0 ; ?|x) and its inverse away from
zero (the lower bound, again, is to ensure identifiability; the necessity of the upper bound,
on the other hand, will become clear in section 4); also, compactness is useful (though not
necessary) for adapting certain Glivenko-Cantelli bounds [12] for the consistency proof.
It should also be clear that we have not stated the results as generally as possible; we have
chosen instead to use assumptions that are simple to understand and verify, and to leave the
technical generalizations to the interested reader. Our assumptions should be weak enough
for most neurophysiological and psychophysical situations, for example, by assuming that
parameters take values in bounded (though possibly large) sets and that tuning curves are
not infinitely steep. The proofs of these three results are basically elaborations on Wald?s
consistency method and Le Cam?s approach to the Bernstein-von Mises theorem [12].
3
3.1
Applications
Psychometric model
As noted in the introduction, psychophysicists have employed versions of the informationmaximization procedure for some years [14, 9, 13, 6]. References in [13], for example, go
back four decades, and while these earlier investigators usually couched their discussion
in terms of variance instead of entropy, the basic idea is the same (note, for example,
that minimizing entropy is asymptotically equivalent to minimizing variance, by our main
theorem). Our results above allow us to precisely quantify the effectiveness of this stategy.
The standard psychometric model is as follows. The response space Y is binary, corresponding to subjective ?yes? or ?no? detection responses. Let f be ?sigmoidal?: a uniformly smooth, monotonically increasing function on the line, such that f (0) = 1/2,
limt??? f (t) = 0 and limt?? f (t) = 1 (this function represents the detection probability when the subject is presented with a stimulus of strength t). Let fa,? = f ((t ? ?)/a);
? here serves as a location (?threshold?) parameter, while a sets the scale (we assume a is
known, for now, although of course this can be relaxed [6]). Finally, let p(x) and p 0 (?) be
some fixed sampling and prior distributions, respectively, both with smooth densities with
respect to Lebesgue measure on some interval ?.
Now, for any fixed scale a, we want to compare the performance of the informationmaximization strategy to that of the i.i.d. p(x) procedure. We have by theorem 2 that
the most efficient estimator of ? is asymptotically unbiased with asymptotic variance
2
?1
?inf
,
o ? (N sup I?0 (x))
x
while the usual calculations show that the asymptotic variance of any efficient estimator
based on i.i.d. samples from p(x) is given by
Z
2
?iid ? (N
dp(x)I?0 (x))?1 ;
X
?2
?iid
?2
the key point, again, is that
is an average, while ?inf
o is a maximum, and hence
?iid ? ?inf o , with equality only in the exceptional case that the Fisher information I?0 (x)
is constant almost surely in p(x).
The Fisher information here is easily calculated here to be
I? =
(f?a,? )2
.
fa,? (1 ? fa,? )
We can immediately derive two easy but important conclusions. First, there is just one function f ? for which the i.i.d. sampling strategy is as asymptotically efficient as informationmaximization strategy; for all other f , information maximization is strictly more efficient.
The extremal function f ? is obtained by setting ?iid = ?inf o , implying that I?0 (x) is
constant a.e. [p(x)], and so f ? is the unique solution of the differential equation
1/2
df ?
= c f ? (t)(1 ? f ? (t))
,
dt
?
where the auxiliary constant c = I ? uniquely fixes the scale a. After some calculus, we
obtain
sin(ct) + 1
f ? (t) =
2
on the interval [??/2c, ?/2c] (and defined uniquely, by monotonicity, as 0 or 1 outside this
interval). Since the support of the derivative of this function is compact, this result is quite
dependent of the sampling density p(x); if p(x) places any of its mass outside of the interval
2
2
is always strictly greater than ?inf
[??/2c, ?/2c], then ?iid
o . This recapitulates a basic
theme from the psychophysical literature comparing adaptive and nonadaptive techniques:
when the scale of the nonlinearity f is either unknown or smaller than the scale of the
i.i.d. sampling density p(x), adaptive techniques are greatly preferable.
Second, a crude analysis shows that, as the scale a of the nonlinearity shrinks, the ratio
2
2
/?inf
?iid
o grows approximately as 1/a; this gives quantitative support to the intuition that
the sharper the nonlinearity with respect to the scale of the sampling distribution p(x), the
more we can expect the information-maximization strategy to help.
3.2
Linear-nonlinear cascade model
We now consider a model that has received increasing attention from the neurophysiology
community (see, e.g., [8] for some analysis and relevant references). The model is of
cascade form, with a linear stage followed by a nonlinear stage: the input space X is
a compact subset of d-dimensional Euclidean space (take X to be the unit sphere, for
concreteness), and the firing rate of the model cell, given input ~x ? X, is given by the
simple form
~ ~x >).
E(y|~x, ?) = f (< ?,
Here the linear filter ?~ is some unit vector in X 0 , the dual space of X (thus, the model
space ? is isomorphic to X), while the nonlinearity f is some nonconstant, nonnegative
function on [?1, 1]. We assume that f is uniformly smooth, to satisfy the conditions of
theorem 2; we also assume f is known, although, again, this can be relaxed. The response
space Y ? the space of possible spike counts, given the stimulus ~x ? can be taken to
be the nonnegative integers. For simplicity, let the conditional probabilities p(y|~x, ?) be
~ ~x >); the most convenient model, as
parametrized uniquely by the mean firing rate f (< ?,
~ ~x >). Finally, we assume
usual, is to assume that p(y|~x, ?) is Poisson with mean f (< ?,
that the sampling density p(x) is uniform on the unit sphere (this choice is natural for
several reasons, mainly involving symmetry; see, e.g., [2, 8]), and that the prior p 0 (?) is
positive and continuous (and is therefore bounded away from zero, by the compactness of
?).
The Fisher information for this model is easily calculated as
~ ~x >))2
(f 0 (< ?,
I? (x) =
P~x,? ,
~ ~x >)
f (< ?,
where f 0 is the usual derivative of the real function f and P~x,? is the projection operator
corresponding to ~x, restricted to the (d ? 1)-dimensional tangent space to the unit sphere
at ?. Theorem 2 now implies that
?1
f 0 (t)2 g(t)
2
?inf o ? N max
,
f (t)
t?[?1,1]
while
Z
?1
f 0 (t)2 g(t)
2
?iid
? N
dp(t)
,
f (t)
[?1,1]
where g(t) = 1 ? t2 , p(t) denotes the one-dimensional marginal measure induced on the
interval [?1, 1] by the uniform measure p(x) on the unit sphere, and ? 2 in each of these
two expressions multiplies the (d ? 1)-dimensional identity matrix.
2
2
Clearly, the arguments of subsection 3.1 apply here as well: the ratio ?iid
/?inf
o grows
roughly linearly in the inverse of the scale of the nonlinearity. The more interesting asymptotics here, though, are in d. This is because the unit sphere has a measure concentration
property [11]: as d ? ?, the measure p(t) becomes exponentially concentrated around
0. In fact, it is easy to show directly that, in this limit, p(t) converges in distribution to
the normal measure with mean zero and variance d?2 . The most surprising implication of
this result is seen for nonlinearities f such that f 0 (0) = 0, f (0) > 0; we have in mind, for
example, symmetric nonlinearities like those often used to model complex cells in visual
cortex. For these nonlinearities,
2
?inf
o
= O(d?2 ) :
2
?iid
that is, the information maximization strategy becomes infinitely more efficient than the
usual i.i.d. approach as the dimensionality of the spaces X and ? grows.
4
A Negative Example
Our next example is more negative and perhaps more surprising: it shows how the
information-maximation strategy can fail, in a certain sense, if the conditions of the consistency lemma are not met. Let ? be multidimensional, with coordinates which are ?independent? in a certain sense, and assume the expected information obtained from one
coordinate of the parameter remains bounded strictly away from the expected information
obtained from one of the other coordinates. For instance, consider the following model:
?
.5
?1 < x ? ??1 ,
?
?
?
f?1 ??1 < x ? 0,
p(1|x) =
?
.5
0 < x ? ?1 ,
?
?
f1
?1 < x ? 1
where 0 ? f?1 , f1 ? 1,
are known and ?1 < ??1
|f?1 ? .5| > |f1 ? .5|,
< 0 and 0 < ?1 < 1 are the parameters we want to learn.
Let the initial prior be absolutely continuous with respect to Lebesgue measure; this implies
that all posteriors will have the same property. Then, using the inverse cumulative probability transform and the fact that mutual information is invariant with respect to invertible
mappings, it is easy to show that the maximal information we can obtain by sampling from
the left is strictly greater than the maximal information obtainable from the right, uniformly
in N . Thus the information-maximization strategy will sample from x < 0 forever, leading
to a linear information growth rate (and easily-proven consistency) for the left parameter
and non-convergence on the right. Compare the performance of the usual i.i.d. approach
for choosing x (using any Lebesgue-dominating measure on the parameter space), which
leads to the standard root-N rate for both parameters (i.e., is strongly consistent in posterior
probability).
Note that this kind of inconsistency problem does not occur in the case of sufficiently
smooth p(y|x, ?), by our main theorem. Thus one way of avoiding this problem would
be to fix a finite sampling scale for each coordinate (i.e., discretizing). Below this scale,
no information can be extracted; therefore, when the algorithm hits this ?floor? for one
coordinate, it will switch to the other. However, it is possible to find other examples which
show that the lack of consistency is not necessarily tied to the discontinuous nature of the
conditional densities.
5
Directions
In this paper, we have presented a rigorous theoretical framework for adaptively designing experiments using an information-theoretic objective function. Most importantly, we
have offered some asymptotic results which clarify the effectiveness of adaptive experiment
design using the information-theoretic objective function (1); in addition, we expect that
our asymptotic approximations should find applications in approximative computational
schemes for optimizing stimulus choice during this type of online experiment. For example, our theorem 2 might suggest the use of a mixture-of-Gaussians representation as an
efficient approximation for the posteriors pN (?) [5].
It should be clear that we have left several important questions open. Perhaps the most
obvious such question concerns the use of non-information theoretic objective functions.
It turns out that many of our results apply with only modest changes if the experiment is
instead designed to minimize something like the Bayes mean-square error (perhaps defined only locally if ? has a nontrivial manifold structure), for example: in this case,
the results in sections 3.1 and 3.2 remain completely unchanged, while the statement of
our main theorem requires only slight changes in the asymptotic variance formula (see
http://www.cns.nyu.edu/?liam). Thus it seems our results here can add very little to any
discussion of what objective function is ?best? in general.
We briefly describe a few more open research directions below.
5.1
?Batch mode? and stimulus dependencies
Perhaps our strongest assumption here is that the experimenter will be able to freely choose
the stimuli on each trial. This might be inaccurate for a number of reasons: for example,
computational demands might require that experiments be run in ?batch mode,? with stimulus optimization taking place not after every trial, but perhaps only after each batch of k
stimuli, all chosen according to some fixed distribution p(x). Another common situation
involves stimuli which vary temporally, for which the system is commonly modelled as
responding not just to a given stimulus x(t), but also to all of its time-translates x(t ? ? ).
Finally, if there is some cost C(x0 , x1 ) associated with changing the state of the observational apparatus from the current state x0 to x1 , the experimenter may wish to optimize an
objective function which incorporates this cost, for example
I(y; ?|x1 ) C(x0 , x1 ).
Each of these situations is clearly ripe for further study. Here we restrict ourselves to
the first setting, and give a simple conjecture, based on the asymptotic results presented
above and inspired by results like those of [1, 4, 10]. First, we state more precisely the
optimization problem inherent in designing a ?batch? experiment: we wish to choose some
sequence, {xi }1?i?k , to maximize
I({xi , yi }1?i?k ; ?);
the main difference here is that {xi }1?i?k must be chosen nonadaptively, i.e., without sequential knowledge of the responses {yi }. Clearly, the order of any sequence of optimal
{xi }1?i?k is irrelevant to the above objective function; in addition, it should be apparent
that if no given piece of data (x, y) is too strong (for example, under Lipschitz conditions
like those in lemma 1) that any given elements of such an optimal sequence {x i }1?i?k
should be asymptotically independent. (Without such a smoothness condition ? for example, if some input x could definitively decide between some given ?0 and ?1 ? then no such
asymptotic independence statement can hold, since no more than one sample from such an
x would be necessary.) Thus, we can hope that we should be able to asymptotically approximate this optimal experiment by sampling in an i.i.d. manner from some well-chosen
p(x). Moreover, we can make a guess as to the identity of this putative p(x):
Conjecture (?Batch? mode). Under suitable conditions, the empirical distribution corre-
sponding to any optimal sequence {xi }1?i?k ,
p?k (x) ?
k
1X
?(xi ),
k i=1
converges weakly as k ? ? to S, the convex set of maximizers in p(x) of
E? log(det(Ex I? (x))).
(2)
Expression (2) above is an average over p(?) of terms proportional to the negative entropy
of the asymptotic Gaussian posterior distribution corresponding to each ?, and thus should
be maximized by any optimal approximant distribution p(x). (Note also that expression
(2) is concave in p(x), ensuring the tractability of the above maximization.) In fact, it is
not difficult, using the results of Clarke and Barron [3] to prove the above conjecture under
the conditions like those of Theorem 2, assuming that X is finite (in which case weak
convergence is equivalent to pointwise convergence); we leave generalizations for future
work.
Acknowledgements
We thank R. Sussman, E. Simoncelli, C. Machens, and D. Pelli for helpful conversations. This work
was partially supported by a predoctoral fellowship from HHMI.
References
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1,621 | 2,474 | A Model for Learning the Semantics of Pictures
V. Lavrenko, R. Manmatha, J. Jeon
Center for Intelligent Information Retrieval
Computer Science Department,
University of Massachusetts Amherst
{lavrenko,manmatha,jeon}@cs.umass.edu
Abstract
We propose an approach to learning the semantics of images which allows us to automatically annotate an image with keywords and to retrieve
images based on text queries. We do this using a formalism that models
the generation of annotated images. We assume that every image is divided into regions, each described by a continuous-valued feature vector.
Given a training set of images with annotations, we compute a joint probabilistic model of image features and words which allow us to predict the
probability of generating a word given the image regions. This may be
used to automatically annotate and retrieve images given a word as a
query. Experiments show that our model significantly outperforms the
best of the previously reported results on the tasks of automatic image
annotation and retrieval.
1
Introduction
Historically, librarians have retrieved images by first manually annotating them with keywords. Given a query, these annotations are used to retrieve appropriate pictures. Underlying this approach is the belief that the words associated (manually) with a picture essentially capture the semantics of the picture and any retrieval based on these keywords will,
therefore, retrieve relevant pictures. Since manual image annotation is expensive, there has
been great interest in coming up with automatic ways to retrieve images based on content.
Queries based on image concepts like color or texture have been proposed for retrieving
images by content but most users find it difficult to query using such visual attributes. Most
people would prefer to pose text queries and find images relevant to those queries. For
example, one should be able to pose a query like ?find me cars on a race track?. This is
difficult if not impossible with many of the current image retrieval systems and hence has
not led to widespread adoption of these systems. We propose a model which looks at the
probability of associating words with image regions. Single pixels and regions are often
hard to interpret. The surrounding context often simplifies the interpretation of regions as
a specific objects. For example, the association of a region with the word tiger is increased
by the fact that there is a grass region and a water region in the same image and should be
decreased if instead there is a region corresponding to the interior of an aircraft. Thus the
association of different regions provides context while the association of words with image
regions provides meaning. Our model computes a joint probability of image features over
different regions in an image using a training set and uses this joint probability to annotate
and retrieve images.
More formally, we propose a statistical generative model to automatically learn the semantics of images - that is, for annotating and retrieving images based on a training set of
images. We assume that an image is segmented into regions (although the regions could
simply be a partition of the image) and that features are computed over each of these regions. Given a training set of images with annotations, we show that probabilistic models
allow us to predict the probability of generating a word given the features computed over
different regions in an image. This may be used to automatically annotate and retrieve images given a word as a query. We show that the continuous relevance model - a statistical
generative model related to relevance models in information retrieval - allows us to derive
these probabilities in a natural way. The model proposed here directly associates continuous features with words and does not require an intermediate clustering stage. Experiments
show that the annotation performance of this continuous relevance model is substantially
better than any other model tested on the same data set. It is almost an order of magnitude
better (in terms of mean precision) than a model based on word-blob co-occurrence model,
more than two and a half times better than a state of the art model derived from machine
translation and 1.6 times as good as a discrete version of the relevance model. The model
also allows ranked retrieval in response to a text query and again performs much better than
any other model in this regard. Our model permits us to automatically associate semantics
(in terms of words) with pictures and is an important building step in performing automatic
object recognition.
2
Related Work
Recently, there has been some work on automatically annotating images by looking at
the probability of associating words with image regions. Mori et al. [9] proposed a Cooccurrence Model in which they looked at the co-occurrence of words with image regions
created using a regular grid. Duygulu et al [4] proposed to describe images using a vocabulary of blobs. First, regions are created using a segmentation algorithm like normalized
cuts. For each region, features are computed and then blobs are generated by clustering
the image features for these regions across images. Each image is generated by using a
certain number of these blobs. Their Translation Model applies one of the classical statistical machine translation models to translate from the set of keywords of an image to the
set of blobs forming the image. Jeon et al [5] instead assumed that this could be viewed as
analogous to the cross-lingual retrieval problem and used a cross-media relevance model
(CMRM) to perform both image annotation and ranked retrieval. They showed that the
performance of the model on the same dataset was considerably better than the models
proposed by Duygulu et al [4] and Mori et al. [9]. Blei and Jordan [3] extended the Latent
Dirichlet Allocation (LDA) Model and proposed a Correlation LDA model which relates
words and images. This model assumes that a Dirichlet distribution can be used to generate
a mixture of latent factors. This mixture of latent factors is then used to generate words and
regions. EM is again used to estimate this model. Blei and Jordan show a few examples
for labeling specific regions in an image.
The model proposed in this paper is called Continuous-space Relevance Model (CRM).
The model is closely related to models proposed by [3, 5], but there are several important
differences which we will highlight in the remainder of this section.
On the surface, CRM appears to be very similar to one of the intermediate models considered by Blei and Jordan [3]. Specifically, their GM-mixture model employs a nearly
identical dependence structure among the random variables involved. However, the topological structure of CRM is quite different from the one employed by [3]. GM-mixture
assumes a low-dimensional topology, leading to a fully-parametric model where 200 or so
?latent aspects? are estimated using the EM algorithm. To contrast that, CRM makes no assumptions about the topological structure, and leads to a doubly non-parametric approach,
where expectations are computed over every individual point in the training set. In that
regard, CRM appears very similar to the cross-media relevance model (CMRM) [5], which
is also doubly non-parametric. There are two significant differences between CRM and
CMRM. First, CMRM is a discrete model and cannot take advantage of continuous features. In order to use CMRM for image annotation we have to quantize continuous feature
position,size,
texture,shape,
color, ...
tiger =
w1
grass =
w2
sun =
w3
g1
P(w|J)
J
P(g|J)
g2
g3
P(r|g)
r1
=
r2
=
r3
=
Figure 1: A generative model of annotated images. Words wj in the annotation are i.i.d.
sampled from the underlying multinomial. Image pixels are produced by first picking a
set of i.i.d. feature vectors {g1 . . .gn }, then generating image regions {r1 . . .rn } from the
feature vectors, and finally stacking the regions on top of each other.
vectors into a discrete vocabulary (similarly to the co-ocurrence and translation [4] models). CRM, on the other hand, directly models continuous features. The second difference
is that CMRM relies on clustering of the feature vectors into blobs. Annotation quality
of the CMRM is very sensitive to clustering errors, and depends on being able to a-priori
select the right cluster granularity: too many clusters will result in exptreme sparseness of
the space, while too few will lead us to confuse different objects in the images. CRM does
not rely on clustering and consequently does not suffer from the granularity issues.
We would like to stress that the difference between CRM and previously discussed models
is not merely conceptual. In section 4 we will show that CRM performs significantly better
than all previosly proposed models on the tasks of image annotation and retrieval. To ensure
a fair comparison, we use exactly the same data set and same feature representations as
were used in [3, 4, 5, 9].
3
A Model of Annotated Images
The purpose of this section is to introduce a statistical formalism that will allow us to
model a relationship between the contents of a given image and the annotation of that
image. We will describe an approach to learning a joint probability disdribution P (r, w)
over the regions r of some image and the words w in its annotation. Knowing the joint
distribution P (r, w) is the key to solving two important real-world problems:
1. Image Annotation. Suppose we are given a new image for which no annotation
is provided. That is, we know r, but do not know w. Having a joint distribution
allows us to compute a conditional likelihood P (w|r) which can then be used to
guess the most likely annotation w for the image in question. The new annotation
can be presented to a user, indexed, or used for retrieval purposes.
2. Image Retrieval. Suppose we are given a collection of un-annotated images and
a text query wqry consisting of a few keywords. Knowing the joint model of
images and annotations, we can compute the query likelihood P (wqry |rJ ) for
every image J in the dataset. We can then rank images in the collection according
to their likelihood of having the query as annotation, resulting in a special case of
the popular Language Modeling approach to Information Retrieval [6].
The remainder of this section is organized as follows. In section 3.1 we discuss our choice
of representation for images and their annotations. Section 3.2 presents a generative framework for relating image regions with image annotations. Section 3.3 provides detailed
estimates for the components of our model.
3.1
Representation of Images and Annotations
Let C denote the finite set of all possible pixel colors. We assume that C includes one
?transparent? color c0 , which will be handy when we have to layer image regions. As
a matter of convenience, we assume that all images are of a fixed size W ? H.1 This
assumption allows us to represent any image as an element of a finite set R = C W ?H . We
assume that each image contains several distinct regions {r1 . . .rn }. Each region is itself
an element of R and contains the pixels of some prominent object in the image, all pixels
around the object are set to be transparent. For example, in Figure 1 we have a hypothetical
picture containing three prominent objects: a tiger, the sun and some grass. Each object is
represented by its own region: r1 for the sun, r2 for the grass, and r3 for the tiger. The final
image is the result of stacking or layering the regions on top of each other, as shown on the
right side of Figure 1.
In our model of images, a central part will be played by a special function G which maps
image regions r ? R to real-valued vectors g ? IRk . The value G(r) represents a set of
features, or characteristics of an image region. The features could reflect the position of
an object region, its relative size, a crude reflection of shape, as well as predominant colors
and textures. For example, in Figure 1 the region r1 (sun) is a round object, located in the
upper-right portion of the image, yellowish in color with a smooth texture. When we model
image generation we will treat the output of G as a generator or a ?recipe? for producing
a certain type of image. For example, a feature vetor g1 = G(r1 ) can be thought of as a
generator for any image region resembling a sun-like object in the upper-left corner.
Finally, an annotation for a given image is a set of words {w1 . . .wm } drawn from some
finite vocabulary V. We assume that the annotation describes the objects represented by
regions {r1 . . .rn }. However, contrary to prior work [4, 3] we do not assume an underlying
one-to-one correspondence between the objects in the image annotation and words in the
annotation. Instead, we are interested in modeling a joint probability for observing a set of
image regions {r1 . . .rn } together with the set of annotation words {w1 . . .wm }.
3.2 A Model for Generating Annotated Images
Suppose T is the training set of annotated images, and let J be an element of T . According to the previous section J is represented as a set of image regions rJ = {r1 . . .rn }
along with the corresponding annotation wJ = {w1 . . .wm }. We assume that the process
that generated J is based on three distinct probability distributions. First, we assume that
the words in wJ are an i.i.d. random sample from some underlying multinomial distribution PV (?|J). Second, the regions rJ are produced from a corresponding set of generator
vectors g1 . . .gn according to a process PR (ri |gi ) which is independent of J. Finally, the
generator vectors g1 . . .gn are themselves an i.i.d. random sample from some underlying
multi-variate density function PG (?|J).
Now let rA = {r1 . . .rnA } denote the regions of some image A, which is not in the training set T . Similarly, let wB = {w1 . . .wnB } be some arbitrary sequence of words. We
would like to model P (rA , wA ), the joint probability of observing an image defined by rA
together with annotation words wB . We hypothesize that the observation {rA , wB } came
from the same process that generated one of the images J ? in the training set T . However,
we don?t know which process that was, and so we compute an expectation over all images
J?T . The overall process for jointly generating wB and rA is as follows:
1. Pick a training image J ? T with some probability PT (J)
2. For b = 1 . . . nB :
(a) Pick the annotation word wb from the multinomial distribution PV (?|J).
3. For a = 1 . . . nA :
(a) Sample a generator vector ga from the probability density PG (?|J).
(b) Pick the image region ra according to the probability PR (ra |ga )
1
The assumptions of finite colormap and fixed image size can easily be relaxed but require arguments that are beyond the scope of this paper.
Figure 1 shows a graphical dependency diagram for the generative process outlined above.
We show the process of generating a simple image consisting of three regions and a corresponding 3-word annotation. Note that the number of words in the annotation n B does not
have to be the same as the number of image regions nA . Formally, the probability of a joint
observation {rA , wB } is given by:
nB
nA Z
X
Y
Y
P (rA , wB ) =
PT (J)
PV (wb |J)
PR (ra |ga )PG (ga |J)dga
(1)
J?T
b=1
a=1
IRk
3.3 Estimating Parameters of the Model
In this section we will discuss simple but effective estimation techniques for the four components of the model: PT , PV , PG and PR . PT (J) is the probability of selecting the
underlying model of image J to generate some new observation r, w. In the absence of
any task knowledge we use a uniform prior PT (J) = 1/NT , where NT is the size of the
training set.
PR (r|g) is a global probability distribution responsible for mapping generator vectors
g?IRk to actual image regions r?R. In our case for every image region r there is only
one corresponding generator g = G(r), so we can assume a particularly simple form for
the distribution PR :
1/Ng if G(r) = g
PR (r|g) =
(2)
0
otherwise
0
0
where Ng is the number of all regions r in R such that G(r ) = g. For the scope of
the current paper we do not attempt to reliably estimate Ng , instead we assume it to be a
constant independent of g.
PG (?|J) is a density function responsible for generating the feature vectors g1 . . .gn , which
are later mapped to image regions rJ according to PR . We use a non-parametric kernelbased density estimate for the distribution PG . Assuming rJ = {r1 . . .rn } to be the set of
regions of image J we estimate:
n
1X
1
p
PG (g|J) =
exp (g ? G(ri ))> ??1 (g ? G(ri ))
(3)
k
k
n i=1 2 ? |?|
Equation (3) arises out of placing a Gaussian kernel over the feature vector G(r i ) of every
region of image J. Each kernel is parametrized by the feature covariance matrix ?. As
a matter of convenience we assumed ? = ??I, where I is the identity matrix. ? playes
the role of kernel bandwidth: it determines the smoothness of PG around the support point
G(ri ). The value of ? is selected empirically on a held-out portion of the training set T .
PV (?|J) is the multinomial distribution that is assumed to have generated the annotation
wJ of image J?T . We use a Bayesian framework for estimating PV (?|J). Let IP V be
the simplex of all multinomial distributions over V. We assume a Dirichlet prior over IP V
that has parameters {?pv : v?V}. Here ? is a constant, selected empirically, and pv is the
relative frequency of observing the word v in the training set. Introducing the observation
wJ results in a Dirichlet posterior over IP V with parameters {?pv + Nv,J : v?V}. Here
Nv,J is the number of times v occurs in the observation wJ . Computing the expectation
over this Dirichlet posterior gives us the following Bayesian estimate for PV :
?pv + Nv,J
P
PV (v|J) =
(4)
? + v0 Nv0 ,J
4
Experimental Results
To provide a meaningful comparison with previously-reported results, we use, without any
modification, the dataset provided by Duygulu et al.[4] 2 . This allows us to compare the
2
Available at http://www.cs.arizona.edu/people/kobus/ research/data/eccv 2002
Models
#words with recall ? 0
Co-occurence Translation CMRM
19
49
66
Results on 49 best words, as in[1, 5]
Mean per-word Recall
0.34
0.48
Mean per-word Precision
0.20
0.40
Results on all 260 words
Mean per-word Recall
0.02
0.04
0.09
Mean per-word Precision
0.03
0.06
0.10
CRM
107 +62%
0.70 +46%
0.59 +48%
0.19 +111%
0.16 +60 %
Table 1: Comparing recall and precision of the four models on the task of automatic image annotation. Our model (CRM) substantially outperforms all other models. Percent
improvements are over the best previously-reported results (CMRM).
performance of models in a strictly controlled manner. The dataset consists of 5,000 images
from 50 Corel Stock Photo cds. Each cd includes 100 images on the same topic. Each
image contains an annotation of 1-5 keywords. Overall there are 371 words. Prior to
modeling, every image in the dataset is pre-segmented into regions using general-purpose
algorithms, such as normalized cuts [11]. We use pre-computed feature vector G(r) for
every segmented region r. The feature set consists of 36 features: 18 color features, 12
texture features and 6 shape features. For details of the features refer to [4]. Since we
directly model the generation of feature vectors, there is no need to quantize feature data,
as was done in [1, 4, 5]. We divided the dataset into 3 parts - with 4,000 training set images,
500 evaluation set images and 500 images in the test set. The evaluation set is used to find
system parameters. After fixing the parameters, we merged the 4,000 training set and 500
evaluation set images to make a new training set. This corresponds to the training set of
4500 images and the test set of 500 images used by Duygulu et al [4].
4.1
Results: Automatic Image Annotation
In this section we evaluate the performance of our model on the task of automatic image
annotation. We are given an un-annotated image J and are asked to automatically produce
an annotation wauto . The automatic annotation is then compared to the held-out human
annotation wJ . We follow the experimental methodology used by[4, 5]. Given a set of
image regions rJ we use equation (1) to arrive at the conditional distribution P (w|rJ ). We
take the top 5 words from that distribution and call them the automatic annotation of the
image in question. Then, following [4], we compute annotation recall and precision for
every word in the testing set. Recall is the number of images correctly annotated with a
given word, divided by the number of images that have that word in the human annotation.
Precision is the number of correctly annotated images divided by the total number of images
annotated with that particular word (correctly or not). Recall and precision values are
averaged over the set of testing words.
We compare the annotation performance of the four models: the Co-occurrence Model
[9], the Translation Model [4], CMRM [5] and the model proposed in this paper (CRM).
We report the results on two sets of words: the subset of 49 best words which was used
by[4, 5], and the complete set of all 260 words that occur in the testing set. Table 1 shows
the performance on both word sets. The figures clearly show that the model presented here
(CRM) substabtially outperforms the other models and is the only one of the four capable of
producing reasonable mean recall and mean precision numbers when every word in the test
set is used. In Figure2 we provide sample annotations for the two best models in the table,
CMRM and CRM, showing that the model in this paper is considerably more accurate.
Figure 2: The generative model based on contiuous features (CRM) that is proposed here
performs substantially better than the discrete cross-media relevance model (CMRM) for
annotating images in the test set.
Query length
Number of queries
Relevant images
CMRM
CRM
CMRM
CRM
1 word
2 words
3 words
179
386
178
1675
1647
542
Precision after 5 retrieved images
0.1989
0.1306
0.1494
0.2480 +25% 0.1902 +45% 0.1888 +26%
Mean Average Precision
0.1697
0.1642
0.2030
0.2353 +39% 0.2534 +54% 0.3152 +55%
4 words
24
67
0.2083
0.2333 +12%
0.2765
0.4471 +61%
Table 2: Comparing our model to the Cross-Media Relevance Model (CMRM) on the task
of image retrieval. Our model outperforms the CMRM model by a wide margin on all
query sets. Boldface figures mark improvements that are statistically significant according
to sign test with a confidence of 99% (p-value < 0.01).
4.2
Results: Ranked Retrieval of Images
In this section we turn our attention to the problem of ranked retrieval of images. In the
retrieval setting we are given a text query wqry and a testing collection of un-annotated
images. For each testing image J we use equation (1) to get the conditional probability
P (wqry |rJ ). All images in the collection are ranked according to the conditional likelihood P (wqry |rJ ). This can be thought of as a special case of the popular Langauge Modeling approach to Information Retrieval, proposed by Ponte and Croft[6]. In our retrieval
experiments we do our best to reproduce the same settings that were used by Jeon et.al[5]
in their work. Following[5], we use four sets of queries, constructed from all 1-, 2-, 3- and
4-word combinations of words that occur at least twice in the testing set. An image is considered relevant to a given query if its manual annotation contains all of the query words.
As our evaluation metrics we use precision at 5 retrieved images and non-interpolated average precision3 , averaged over the entire query set. Precision at 5 documents is a good
measure of performance for a casual user who is interested in retrieving a couple of relevant items without looking at too much junk. Average precision is more appropriate for a
professional user who wants to find a large proportion of relevant items.
Table 2 shows the performance of our model on the four query sets, contrasted with performance of the CMRM[5] baseline on the same data. Baseline performance figures are
quoted directly from the tables in[5]. We observe that our model substantially outperforms
the CMRM baseline on every query set. Improvements in average precision are particularly
impressive, our model outperforms the baseline by 40 - 60 percent. All improvements on
1-, 2- and 3-word queries are statistically significant based on a sign test with a p-value of
3
Average precision is the average of precision values at the ranks where relevant items occur.
Figure 3: Example: top 5 images retrieved in responce to text query ?cars track?
0.01. We are also very encouraged by the precision our model shows at 5 retrieved images:
precision values around 0.2 suggest that an average query always has a relevant image in
the top 5. Figure 3 shows top 5 images retrieved in response to the text query ?cars track?.
5
Conclusions and Future Work
We have proposed a new statistical generative model for learning the semantics of images.
We showed that this model works significantly better than a number of other models for
image annotation and retrieval. Our model works directly on the continuous features. Future work will include the extension of this work to larger datasets (both training and test
data). We believe this is needed both for better coverage and an evaluation of how such
algorithms extend to large data sets. Improved feature sets may also lead to substantial
improvements in performance.
6
Acknowledgments
We thank Kobus Barnard for making their dataset [4] available. This work was supported in
part by the Center for Intelligent Information Retrieval, by the National Science Foundation
under grant NSF IIS-9909073 and by SPAWARSYSCEN-SD under grants N66001-991-8912 and N66001-02-1-8903. Jiwoon Jeon is partially supported by the Government
of Korea. Any opinions, findings and conclusions or recommendations expressed in this
material are the author(s) and do not necessarily reflect those of the sponsor.
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1,622 | 2,475 | A Functional Architecture for Motion
Pattern Processing in MSTd
Scott A. Beardsley
Dept. of Biomedical Engineering
Boston University
Boston, MA 02215
[email protected]
Lucia M. Vaina
Dept. of Biomedical Engineering
Boston University
Boston, MA 02215
[email protected]
Abstract
Psychophysical studies suggest the existence of specialized
detectors for component motion patterns (radial, circular, and
spiral), that are consistent with the visual motion properties of cells
in the dorsal medial superior temporal area (MSTd) of non-human
primates. Here we use a biologically constrained model of visual
motion processing in MSTd, in conjunction with psychophysical
performance on two motion pattern tasks, to elucidate the
computational mechanisms associated with the processing of widefield motion patterns encountered during self-motion. In both tasks
discrimination thresholds varied significantly with the type of
motion pattern presented, suggesting perceptual correlates to the
preferred motion bias reported in MSTd. Through the model we
demonstrate that while independently responding motion pattern
units are capable of encoding information relevant to the visual
motion tasks, equivalent psychophysical performance can only be
achieved using interconnected neural populations that
systematically inhibit non-responsive units. These results suggest
the cyclic trends in psychophysical performance may be mediated,
in part, by recurrent connections within motion pattern responsive
areas whose structure is a function of the similarity in preferred
motion patterns and receptive field locations between units.
1
In trod u ction
A major challenge in computational neuroscience is to elucidate the architecture of
the cortical circuits for sensory processing and their effective role in mediating
behavior. In the visual motion system, biologically constrained models are playing
an increasingly important role in this endeavor by providing an explanatory
substrate linking perceptual performance and the visual properties of single cells.
Single cell studies indicate the presence of complex interconnected structures in
middle temporal and primary visual cortex whose most basic horizontal connections
can impart considerable computational power to the underlying neural population
[1, 2]. Combined psychophysical and computational studies support these findings
Figure 1: a) Schematic of the graded motion pattern (GMP) task. Discrimination
pairs of stimuli were created by perturbing the flow angle (?) of each 'test' motion
(with average dot speed, vav), by ??p in the stimulus space spanned by radial and
circular motions. b) Schematic of the shifted center-of-motion (COM) task.
Discrimination pairs of stimuli were created by shifting the COM of the ?test?
motion to the left and right of a central fixation point. For each motion pattern the
COM was shifted within the illusory inner aperture and was never explicitly visible.
and suggest that recurrent connections may play a significant role in encoding the
visual motion properties associated with various psychophysical tasks [3, 4].
Using this methodology our goal is to elucidate the computational mechanisms
associated with the processing of wide-field motion patterns encountered during
self-motion. In the human visual motion system, psychophysical studies suggest the
existence of specialized detectors for the motion pattern components (i.e., radial,
circular and spiral motions) associated with self-motion [5, 6]. Neurophysiological
studies reporting neurons sensitive to motion patterns in the dorsal medial superior
temporal area (MSTd) support the existence of such mechanisms [7-10], and in
conjunction with psychophysical studies suggest a strong link between the patterns
of neural activity and motion-based perceptual performance [11, 12].
Through the combination of human psychophysical performance and biologically
constrained modeling we investigate the computational role of simple recurrent
connections within a population of MSTd-like units. Based on the known visual
motion properties within MSTd we ask what neural structures are computationally
sufficient to encode psychophysical performance on a series of motion pattern tasks.
2
M o t i o n pa t t e r n d i sc r i m i n a t i o n
Using motion pattern stimuli consistent with previous studies [5, 6], we have
developed a set of novel psychophysical tasks designed to facilitate a more direct
comparison between human perceptual performance and the visual motion
properties of cells in MSTd that have been found to underlie the discrimination of
motion patterns [11, 12]. The psychophysical tasks, referred to as the graded motion
pattern (GMP) and shifted center-of-motion (COM) tasks, are outlined in Fig. 1.
Using a temporal two-alternative-forced-choice task we measured discrimination
thresholds to global changes in the patterns of complex motion (GMP task), [13],
and shifts in the center-of-motion (COM task). Stimuli were presented with central
fixation using a constant stimulus paradigm and consisted of dynamic random dot
displays presented in a 24o annular region (central 4o removed). In each task, the
stimulus duration was randomly perturbed across presentations (440?40 msec) to
control for timing-based cues, and dots moved coherently through a radial speed
Figure 2: a) GMP thresholds across 8 'test' motions at two mean dot speeds for two
observers. Performance varied continuously with thresholds for radial motions (?=0,
180o) significantly lower than those for circular motions (?=90,270o), (p<0.001;
t(37)=3.39). b) COM thresholds at three mean dot speeds for two observers. As with
the GMP task, performance varied continuously with thresholds for radial motions
significantly lower than those for circular motions, (p<0.001; t(37)=4.47).
gradient in directions consistent with the global motion pattern presented.
Discrimination thresholds were obtained across eight ?test? motions corresponding
to expansion, contraction, CW and CCW rotation, and the four intermediate spiral
motions. To minimize adaptation to specific motion patterns, opposing motions
(e.g., expansion/ contraction) were interleaved across paired presentations.
2.1
Results
Discrimination thresholds are reported here from a subset of the observer population
consisting of three experienced psychophysical observers, one of which was na?ve
to the purpose of the psychophysical tasks. For each condition, performance is
reported as the mean and standard error averaged across 8-12 thresholds.
Across observers and dot speeds GMP thresholds followed a distinct trend in the
stimulus space [13], with radial motions (expansion/contraction) significantly lower
than circular motions (CW/CCW rotation), (p<0.001; t(37)=3.39), (Fig. 2a). While
thresholds for the intermediate spiral motions were not significantly different from
circular motions (p=0.223, t(60)=0.74), the trends across 'test' motions were well fit
within the stimulus space (SB: r>0.82, SC: r>0.77) by sinusoids whose period and
phase were 196 ? 10o and -72 ? 20o respectively (Fig. 1a).
When the radial speed gradient was removed by randomizing the spatial distribution
of dot speeds, threshold performance increased significantly across observers
(p<0.05; t(17)=1.91), particularly for circular motions (p<0.005; t(25)=3.31), (data
not shown). Such performance suggests a perceptual contribution associated with
the presence of the speed gradient and is particularly interesting given the fact that
the speed gradient did not contribute computationally relevant information to the
task. However, the speed gradient did convey information regarding the integrative
structure of the global motion field and as such suggests a preference of the
underlying motion mechanisms for spatially structured speed information.
Similar trends in performance were observed in the COM task across observers and
dot speeds. Discrimination thresholds varied continuously as a function of the 'test'
motion with thresholds for radial motions significantly lower than those for circular
motions, (p<0.001; t(37)=4.47) and could be well fit by a sinusoidal trend line (e.g.
SB at 3 deg/s: r>0.91, period = 178 ? 10 o and phase = -70 ? 25o), (Fig. 2b).
2.2
A local or global task?
The consistency of the cyclic threshold profile in stimuli that restricted the temporal
integration of individual dot motions [13], and simultaneously contained all
directions of motion, generally argues against a primary role for local motion
mechanisms in the psychophysical tasks. While the psychophysical literature has
reported a wide variety of ?local? motion direction anisotropies whose properties
are reminiscent of the results observed here, e.g. [14], all would predict equivalent
thresholds for radial and circular motions for a set of uniformly distributed and/or
spatially restricted motion direction mechanisms. Together with the computational
impact of the speed gradient and psychophysical studies supporting the existence of
wide-field motion pattern mechanisms [5, 6], these results suggest that the threshold
differences across the GMP and COM tasks may be associated with variations in the
computational properties across a series of specialized motion pattern mechanisms.
3
A computational model
The similarities between the motion pattern stimuli used to quantify human
perception and the visual motion properties of cells in MSTd suggests that MSTd
may play a computational role in the psychophysical tasks. To examine this
hypothesis, we constructed a population of MSTd-like units whose visual motion
properties were consistent with the reported neurophysiology (see [13] for details).
Across the population, the distribution of receptive field centers was uniform across
polar angle and followed a gamma distribution ?(5,6) across eccenticity [7]. For
each unit, visual motion responses followed a gaussian tuning profile as a function
of the stimulus flow angle G( ?), (?i=60?30o; [10]), and the distance of the stimulus
COM from the unit?s receptive field center Gsat(xi, yi, ?s=19o), Eq. 1, such that its
preferred motion response was position invariant to small shifts in the COM [10]
and degraded continuously for large shifts [9].
Within the model, simulations were categorized according to the distribution of
preferred motions represented across the population (one reported in MSTd and a
uniform control). The first distribution simulated an expansion bias in which the
density of preferred motions decreased symmetrically from expansions to contraction [10]. The second distribution simulated a uniform preference for all motions
and was used as a control to quantify the effects of an expansion bias on
psychophysical performance. Throughout the paper we refer to simulations
containing these distributions as ?Expansion-biased? and ?Uniform? respectively.
3.1
Extracting perceptual estimates from the neural code
For each stimulus presentation, the ith unit?s response was calculated as the average
firing rate, Ri, from the product of its motion pattern and spatial tuning profiles,
(
)
Ri = Rmax G min[? ? ?i ] ,? ti G sati (x? xi , y ? y i ,? s ) + P (? = 12 )
(1)
where Rmax is the maximum preferred stimulus response (spikes/s), min[ ] refers to
the minimum angular distance between the stimulus flow angle ? and the unit?s
preferred motion ?i, Gsat is the unit?s spatial tuning profile saturated within the
central 5?3o, ?ti and ?s are the standard deviations of the unit?s motion pattern and
Figure 3: Model vs. psychophysical performance for independently responding
units. Model thresholds are reported as the average (?1 S.E.) across five simulated
populations. a) GMP thresholds were highest for contracting motions and lowest for
expanding motions across all Expansion-biased populations. b) Comparable trends
in performance were observed for COM thresholds. Comparison with the Uniform
control simulations in both tasks (2000 units shown here) indicates that thresholds
closely followed the distribution of preferred motions simulated within the model.
spatial tuning profiles respectively, (xi,yi) is the spatial location of the unit?s
receptive field center, (x,y) is the spatial location of the stimulus COM, and
P(?=12) is the background activity simulated as an uncorrelated Poisson process.
The psychophysical tasks were simulated using a modified center-of-gravity
approach to decode estimates of the stimulus properties, i.e. flow angle (?^) and
COM location in the visual field (x? , y? ) , from the neural population
?
? ? xi Ri ? y i Ri
v ?
, i
, ? ? i Ri ?
? Ri
i
?
?
i
i
?
? i
(x?, y? , ??) = ?? i? R
(2)
v
where ?i is the unit vector in the stimulus space (Fig. 1a) corresponding to the
unit?s preferred motion. For each set of paired stimuli, psychophysical judgments
were made by comparing the estimated stimulus properties according to the
discrimination criteria, specified in the psychophysical tasks. As with the
psychophysical experiments, discrimination thresholds were computed using a leastsquares fit to percent correct performance across constant stimulus levels.
3.2
Simulation 1: Independent neural responses
In the first series of simulations, GMP and COM thresholds were quantified across
three populations (500, 1000, and 2000 units) of independently responding units for
each simulated distribution (Expansion-biased and Uniform). Across simulations,
both the range in thresholds and their trends across ?test? motions were compared
with human psychophysical performance to quantify the effects of population size
and an expansion biased preferred motion distribution on model performance.
Over the psychophysical range of interest (?p ? 7o), GMP thresholds for contracting
motions were at chance across all Expansion-biased populations, (Fig. 3a). While
thresholds for expanding motions were generally consistent with those for human
observers, those for circular motions remained significantly higher for all but the
largest populations. Similar trends in performance were observed for the COM task,
(Fig. 3b). Here the range of COM thresholds was well matched with human
performance for simulations containing 1000 units, however, the trends across
motion patterns remained inconsistent even for the largest populations.
Figure 4: Proposed recurrent connection profile between motion pattern units. a)
Across the motion pattern space connection strength followed an inverse gaussian
profile such that the ith unit (with preferred motion ?i) systematically inhibited units
with anti-preferred motions centered at 180+?i. b) Across the visual field connection
strength followed a difference-of-gaussians profile as a function of the relative
distance between receptive field centers such that spatially local units are mutually
excitatory (?Re=10o) and more distant units were mutually inhibitory (?Ri=80o).
For simulations containing a uniform distribution of preferred motions, the
threshold range was consistent with human performance on both tasks, however, the
trend across motion patterns was generally flat. What variability did occur was due
primarily to the discrete sampling of preferred motions across the population.
Comparison of the discrimination thresholds for the Expansion-biased and Uniform
populations indicates that the trend across thresholds was closely matched to the
underlying distributions of preferred motions. This result in due in part to the nearequal weighting of independently responding units and can be explained to a first
approximation by the proportional increase in the signal-to-noise ratio across the
population as a function of the density of units responsive to a given 'test' motion.
3.3
Simulation 2: An interconnected neural structure
In a second series of simulations, we examined the computational effect of adding
recurrent connections between units. If the distribution of preferred motions in
MSTd is in fact biased towards expansions, as the neurophysiology suggests, it
seems unlikely that independent estimates of the visual motion information would
be sufficient to yield the threshold profiles observed in the psychophysical tasks.
We hypothesize that a simple fixed architecture of excitatory and/or inhibitory
connections is sufficient to account for the cyclic trends in discrimination
thresholds. Specifically, we propose that a recurrent connection profile whose
strength varies as a function of (a) the similarity between preferred motion patterns
and (b) the distance between receptive field centers, is computationally sufficient to
recover the trends in GMP/COM performance (Fig. 4),
wij = S R e
?
( xi ? x j )2 + ( yi ? y j )2
2 ? R2 e
?
SR
e
2
?
?(min[ ?i ? ? j ])2
( xi ? x j )2 + ( yi ? y j )2
2
2 ? Ri
? S? e
2? I2
(3)
Figure 5: Model vs. psychophysical performance for populations containing
recurrent connections (?I=80o). As the number of units increased for Expansionbiased populations, discrimination thresholds decreased to psychophysical levels
and the sinusoidal trend in thresholds emerged for both the (a) GMP and (b) COM
tasks. Sinusoidal trends were established for as few as 1000 units and were well fit
(r>0.9) by sinusoids whose periods and phases were (193.8 ? 11.7o, -70.0 ? 22.6o)
and (168.2 ? 13.7o, -118.8 ? 31.8o) for the GMP and COM tasks respectively.
where wij is the strength of the recurrent connection between ith and jth units, (xi,yi)
and (xj,yj) denote the spatial locations of their receptive field centers, ?Re (=10o) and
?Ri (=80o) together define the spatial extent of a difference-of-gaussians interaction
between receptive field centers, and SR and S? scale the connection strength. To
examine the effects of the spread of motion pattern-specific inhibition and
connection strength in the model, ?I, S?, and SR were considered free parameters.
Within the parameter space used to define recurrent connections (i.e., ?I, S? and SR),
Monte Carlo simulations of Expansion-biased model performance (1000 units)
yielded regions of high correlation on both tasks (with respect to the psychophysical
thresholds, r>0.7) that were consistent across independently simulated populations.
Typically these regions were well defined over a broad range such that there was
significant overlap between tasks (e.g., for the GMP task (SR=0.03), ?I=[45,120o],
S?=[0.03,0.3] and for the COM task (?I=80o), S? = [0.03,0.08], SR = [0.005, 0.04]).
Fig. 5 shows averaged threshold performance for simulations of interconnected units
drawn from the highly correlated regions of the (?I, S?, SR) parameter space. For
populations not explicitly examined in the Monte Carlo simulations connection
strengths (S?, SR) were scaled inversely with population size to maintain an
equivalent level of recurrent activity. With the incorporation of recurrent
connections, the sinusoidal trend in GMP and COM thresholds emerged for
Expansion-biased populations as the number of units increased. In both tasks the
cyclic threshold profiles were established for 1000 units and were well fit (r>0.9) by
sinusoids whose periods and phases were consistent with human performance.
Unlike the Expansion-biased populations, Uniform populations were not
significantly affected by the presence of recurrent connections (Fig. 5). Both the
range in thresholds and the flat trend across motion patterns were well matched to
those in Section 3.2. Together these results suggest that the sinusoidal trends in
GMP and COM performance may be mediated by the combined contribution of the
recurrent interconnections and the bias in preferred motions across the population.
4
D i s c u s s i on
Using a biologically constrained computational model in conjunction with human
psychophysical performance on two motion pattern tasks we have shown that the
visual motion information encoded across an interconnected population of cells
responsive to motion patterns, such as those in MSTd, is computationally sufficient
to extract perceptual estimates consistent with human performance. Specifically, we
have shown that the cyclic trend in psychophysical performance observed across
tasks, (a) cannot be reproduced using populations of independently responding units
and (b) is dependent, in part, on the presence of an expanding motion bias in the
distribution of preferred motions across the neural population.
The model?s performance suggests the presence of specific recurrent structures
within motion pattern responsive areas, such as MSTd, whose strength varies as a
function of the similarity between preferred motion patterns and the distance
between receptive field centers. While such structures have not been explicitly
examined in MSTd and other higher visual motion areas there is anecdotal support
for the presence of inhibitory connections [8]. Together, these results suggest that
robust processing of the motion patterns associated with self-motion and optic flow
may be mediated, in part, by recurrent structures in extrastriate visual motion areas
whose distributions of preferred motions are biased strongly in favor of expanding
motions.
Acknowledgments
This work was supported by National Institutes of Health grant EY-2R01-07861-13
to L.M.V.
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| 2475 |@word neurophysiology:5 middle:1 seems:1 integrative:1 simulation:13 contraction:4 extrastriate:1 cyclic:5 series:4 com:22 comparing:1 reminiscent:1 distant:1 visible:1 hypothesize:1 designed:1 medial:2 discrimination:13 v:2 cue:1 ith:3 contribute:1 location:5 preference:2 five:1 constructed:1 direct:1 fixation:2 burr:1 behavior:1 examine:2 anisotropy:1 underlying:3 matched:3 circuit:1 lowest:1 what:2 rmax:2 developed:1 finding:1 temporal:5 ti:2 gravity:1 scaled:1 control:4 unit:35 underlie:1 grant:1 engineering:2 timing:1 local:4 encoding:2 firing:1 quantified:1 examined:3 suggests:5 range:6 averaged:2 acknowledgment:1 yj:1 saito:1 area:6 significantly:9 radial:10 refers:1 suggest:8 cannot:1 tootell:1 gilbert:1 equivalent:3 center:12 independently:6 duration:1 qian:1 vaina:4 spanned:1 population:30 variation:1 elucidate:3 play:2 decode:1 substrate:1 hypothesis:1 pa:1 trend:19 particularly:2 malach:1 observed:6 role:6 region:4 inhibit:1 removed:2 highest:1 dynamic:1 mstd:16 various:1 represented:1 stemmler:1 forced:1 distinct:1 effective:1 ction:1 monte:2 sc:2 whose:10 emerged:2 encoded:1 interconnection:1 favor:1 reproduced:1 propose:1 interconnected:5 product:1 interaction:1 adaptation:1 relevant:2 moved:1 recurrent:15 measured:1 eq:1 strong:1 indicate:1 quantify:3 direction:4 closely:2 correct:1 centered:1 human:12 biological:1 leastsquares:1 sati:1 considered:1 predict:1 matthew:1 major:1 purpose:1 polar:1 sensitive:1 largest:2 impart:1 anecdotal:1 gaussian:2 modified:1 snowden:1 conjunction:3 encode:1 indicates:2 dependent:1 sb:2 unlikely:1 typically:1 explanatory:1 wij:2 constrained:4 spatial:8 integration:1 field:14 never:1 sampling:1 broad:1 stimulus:22 inhibited:1 primarily:1 few:1 randomly:1 harel:1 simultaneously:1 ve:1 gamma:1 individual:1 national:1 phase:4 consisting:1 opposing:1 maintain:1 interest:1 circular:11 investigate:1 highly:1 saturated:1 capable:1 trod:1 re:2 increased:3 modeling:1 newsome:2 deviation:1 subset:1 uniform:9 reported:7 perturbed:1 randomizing:1 varies:2 combined:2 density:2 bu:2 graziano:1 together:4 continuously:4 na:1 central:4 containing:4 suggesting:1 account:1 sinusoidal:5 explicitly:3 observer:8 recover:1 contribution:2 minimize:1 degraded:1 judgment:1 yield:1 anton:1 carlo:2 niebur:1 cybernetics:1 detector:2 against:1 associated:7 ask:1 illusory:1 higher:2 methodology:1 response:5 burnod:1 strongly:1 anderson:1 angular:1 biomedical:2 correlation:1 horizontal:1 facilitate:1 effect:4 consisted:1 sinusoid:3 spatially:3 i2:1 during:2 self:4 criterion:1 demonstrate:1 argues:1 motion:116 percent:1 novel:1 superior:2 rotation:2 specialized:3 functional:1 perturbing:1 celebrini:2 cerebral:1 linking:1 significant:2 refer:1 tuning:4 outlined:1 consistency:1 dot:9 similarity:4 cortex:2 inhibition:1 yi:5 minimum:1 ey:1 paradigm:1 period:4 signal:1 annular:1 paired:2 schematic:2 impact:1 basic:1 vision:3 poisson:1 wurtz:2 achieved:1 cell:6 background:1 decreased:2 biased:11 unlike:1 sr:8 usher:1 flow:5 inconsistent:1 extracting:1 presence:6 symmetrically:1 intermediate:2 spiral:4 variety:1 xj:1 fit:5 architecture:3 inner:1 regarding:1 shift:3 generally:3 inhibitory:3 shifted:3 neuroscience:5 estimated:1 discrete:1 affected:1 four:1 threshold:39 drawn:1 angle:5 inverse:1 reporting:1 throughout:1 comparable:1 interleaved:1 followed:6 display:1 encountered:2 yielded:1 activity:3 strength:8 occur:1 optic:1 incorporation:1 ri:9 lucia:1 flat:2 speed:13 min:3 structured:1 according:2 combination:1 across:35 increasingly:1 primate:1 biologically:4 explained:1 restricted:2 invariant:1 computationally:4 mutually:2 mechanism:8 gaussians:2 eight:1 responsive:5 alternative:1 existence:4 responding:5 graded:2 r01:1 psychophysical:32 coherently:1 spike:1 receptive:9 primary:2 gradient:6 cw:2 link:1 distance:5 simulated:8 extent:1 code:1 providing:1 ratio:1 mediating:1 neuron:2 anti:1 supporting:1 variability:1 varied:4 pair:2 specified:1 connection:19 gmp:16 established:2 tanaka:1 pattern:38 scott:1 perception:1 challenge:1 shifting:1 power:1 overlap:1 inversely:1 created:2 mediated:3 extract:1 health:1 literature:1 relative:1 contracting:2 interesting:1 proportional:1 sufficient:5 consistent:9 systematically:2 playing:1 uncorrelated:1 excitatory:2 supported:1 free:1 jth:1 bias:5 institute:1 wide:3 distributed:1 ccw:2 cortical:1 calculated:1 sensory:1 made:1 correlate:1 preferred:21 aperture:1 deg:1 global:4 xi:7 morrone:1 robust:1 expanding:4 expansion:16 complex:2 did:3 spread:1 noise:1 profile:11 convey:1 categorized:1 fig:10 referred:1 experienced:1 position:1 msec:1 perceptual:7 weighting:1 remained:2 specific:3 r2:1 adding:1 duffy:2 boston:4 neurophysiological:1 visual:19 contained:1 vav:1 chance:1 harris:1 ma:2 goal:1 endeavor:1 presentation:3 towards:1 considerable:1 change:1 specifically:2 uniformly:1 beardsley:2 koechlin:1 support:3 dorsal:2 dept:2 correlated:1 |
1,623 | 2,476 | All learning is local:
Multi-agent learning in global reward games
Yu-Han Chang
MIT CSAIL
Cambridge, MA 02139
[email protected]
Tracey Ho
LIDS, MIT
Cambridge, MA 02139
[email protected]
Leslie Pack Kaelbling
MIT CSAIL
Cambridge, MA 02139
[email protected]
Abstract
In large multiagent games, partial observability, coordination, and credit
assignment persistently plague attempts to design good learning algorithms. We provide a simple and efficient algorithm that in part uses
a linear system to model the world from a single agent?s limited perspective, and takes advantage of Kalman filtering to allow an agent to
construct a good training signal and learn an effective policy.
1
Introduction
Learning in a single-agent stationary-environment setting can be a hard problem, but relative to the multi-agent learning problem, it is easy. The multi-agent learning problem has
been approached from a variety of approaches, from game theory to partially observable
Markov decision processes. The solutions are often complex. We take a different approach
in this paper, presenting a simplifying abstraction and a reward filtering technique that allows computationally efficient and robust learning in large multi-agent environments where
other methods may fail or become intractable.
In many multi-agent settings, our learning agent does not have a full view of the world.
Other agents may be far away or otherwise obscured. At the very least, our learning agent
usually does not have a a complete representation of the internal states of the other agents.
This partial observability creates problems when the agent begins to learn about the world,
since it cannot see how the other agents are manipulating the environment and thus it cannot
ascertain the true world state. It may be appropriate to model the observable world as a
non-stationary Markov Decision Process (MDP). A separate problem arises when we train
multiple agents using a global reward signal. This is often the case in cooperative games
in which all the agents contribute towards attaining some common goal. Even with full
observability, the agents would need to overcome a credit assignment problem, since it may
be difficult to ascertain which agents were responsible for creating good reward signals. If
we cannot even observe what the other agents are doing, how can we begin to reason about
their role in obtaining the current reward?
Consider an agent in an MDP, learning to maximize a reward that is a function of its observable state and/or actions. There are many well-studied learning techniques to do this
[Sutton and Barto, 1999]. The effects of non-stationarity, partial observability, and global
rewards can be thought of as replacing the true reward signal with an alternate signal that is
a non-stationary function of the original reward. Think of the difference between learning
with a personal coach and learning in a large class where feedback is given only on collective performance. This causes problems for an agent that is trying to use the collective
?global? reward signal to learn an optimal policy. Ideally the agent can recover the original
?personal reward signal? and learn using that signal rather than the global reward signal.
We show that in many naturally arising situations of this kind, an effective approach is for
an individual agent to model the observed global reward signal as the sum of its own contribution (which is the personal reward signal on which it should base its learning) and a
random Markov process (which is the amount of the observed reward due to other agents
or external factors). With such a simple model, we can estimate both of these quantities efficiently using an online Kalman filtering process. Many external sources of reward (which
could be regarded as noise) can be modeled as or approximated by a random Markov process, so this technique promises broad applicability. This approach is more robust than
trying to learn directly from the global reward, allowing agents to learn and converge faster
to an optimal or near-optimal policy.
2
Related Work
This type of problem has been approached in the past using a variety of techniques. For
slowly varying environments, Szita et al. [2002] show that Q-learning will converge as long
as the variation per time step is small enough. In our case, we attempt to tackle problems
where the variation could be larger. Choi et al. [1999] investigate models in which there
are ?hidden modes?. When the environment switches between modes, all the rewards may
be altered. This works if we have fairly detailed domain knowledge about the types of
modes we expect to encounter. For variation produced by the actions of other agents in
the world, or for truly unobservable environmental changes, this technique would not work
as well. Auer et al. [1995] show that in arbitrarily varying environments, we can craft a
regret-minimizing strategy for playing repeated games. The results are largely theoretical
in nature and can yield fairly loose performance bounds, especially in stochastic games.
Rather than filtering the rewards as we will do, Ng et al. [1999] show that a potential function can be used to shape the rewards without affecting the learned policy while possibly
speeding up convergence. This assumes that learning would converge in the first place,
though possibly taking a very long time. Moreover, it requires domain knowledge to craft
this shaping function. Wolpert and Tumer [1999] provide a framework called COIN, or
collective intelligence, for analyzing distributed reinforcement learning. They stress the
importance of choosing utility functions that lead to good policies. Finally, McMahan et
al. [2003] discuss learning in the scenario in which the opponent gets to choose the agent?s
reward function.
The innovative aspect of our approach is to consider the reward signal as merely a signal
that is correlated with our true learning signal. We propose a model that captures the
relationship between the true reward and the noisy rewards in a wide range of problems.
Thus, without assuming much additional domain knowledge, we can use filtering methods
to recover the underlying true reward signal from the noisy observed global rewards.
3
Mathematical model
The agent assumes that the world possesses one or more unobservable state variables that
affect the global reward signal. These unobservable states may include the presence of other
agents or changes in the environment. Each agent models the effect of these unobservable
state variables on the global reward as an additive noise process bt that evolves according
to bt+1 = bt + zt , where zt is a zero-mean Gaussian random variable with variance ?w .
The global reward that it observes if it is in state i at time t is gt = r(i) + bt , where r
is a vector containing the ideal training rewards r(i) received by the agent at state i. The
standard model that describes such a linear system is:
gt = Cxt + vt ,
xt = Axt?1 + wt ,
vt ? N (0, ?2 )
wt ? N (0, ?1 )
In our case, we desire estimates of xt = [rtT bt ]T . We impart our domain knowledge into
the model by specifying the estimated variance and covariance of the components of x t .
In our case, we set ?2 = 0 since we assume no observation noise when we experience
rewards; ?1 (j, j) = 0, j 6= |S| + 1, since the rewards are fixed and do not evolve over
time; ?1 (|S| + 1, |S| + 1) = ?w since the noise term evolves with variance ?w . The system
matrix is A = I, and the observation matrix is C = [0 0 . . . 1i . . . 0 0 1] where the 1i
occurs in the ith position when our observed state is state i.
Kalman filters [Kalman, 1960] are Bayes optimal, minimum mean-squared-error estimators
for linear systems with Gaussian noise. The agent applies the following causal Kalman
filtering equations at each time step to obtain maximum likelihood estimates for b and the
individual rewards r(i) for each state i given all previous observations. First, the estimate
x
? and its covariance matrix P are updated in time based on the linear system model:
x
?0t
= A?
xt?1
(1)
Pt0
= APt?1 AT + ?1
(2)
Then these a priori estimates are updated using the current time period?s observation g t :
Kt
x
?t
Pt
= Pt0 C T (CPt0 C T + ?2 )?1
= x
?0t + Kt (gt ? C x
?0t )
= (I ? Kt C)Pt0
(3)
(4)
(5)
As shown, the Kalman filter also gives us the estimation error covariance Pt , from which
we know the variance of the estimates for r and b. We can also compute the likelihood
of observing gt given the model and all the previous observations. This will be handy
for evaluating the fit of our model, if needed. We could also create more complicated
models if our domain knowledge shows that a different model would be more suitable. For
example, if we wanted to capture the effect of an upward bias in the evolution of the noise
process (perhaps to model the fact that all the agents are learning and achieving higher
rewards), we could add another variable u, initialized such that u0 > 0, modifying x to be
x = [r T b u]T , and changing our noise term update equation to bt+1 = bt + ut + wt . In
other cases, we might wish to use non-linear models that would require more sophisticated
techniques such as extended Kalman filters.
For the learning mechanism, we use a simple tabular Q-learning algorithm [Sutton and
Barto, 1999], since we wish to focus our attention on the reward signal problem. Q-learning
keeps a ?Q-value? for each state-action pair, and proceeds using the following update rule:
Qt (s, a) = (1 ? ?)Qt?1 (s, a) + ?(r + ? min
Qt (s0 , a0 )) ,
0
a
(6)
where 0 < ? < 1 is parameter that controls the learning rate, r is the reward signal used
for learning at time t given s and a, 0 < ? ? 1 is the discount factor, and s, a, and s 0
are the current state, action, and next state of the agent, respectively. Under fairly general
conditions, in a stationary MDP, Q-learning converges to the optimal policy, expressed as
?(s) = argmaxa Q(s, a) .
1
+5
2
+10
3
...
4
24
...
25
Figure 1: This shows the dynamics of our 5x5 grid world domain. The states correspond
to the grid locations, numbered 1,2,3,4,...,24,25. Actions move the agent N,S,E, or W,
except in states 6 and 16, where any action takes the agent to state 10 and 18, respectively,
shown by the curved arrows in the figure at left. The optimal policy is shown at center,
where multiple arrows at one state denotes indifference between the possibilities. A policy
learned by our filtering agent is shown at right.
4
The filtering learning agent
Like any good student, the filtering learning agent chooses to accept well-deserved praise
from its teacher and ignore over-effusive rewards. The good student does not update his behavior at every time step, but only upon observing relevant rewards. The question remains:
How does an agent decide upon the relevance of the rewards it sees? We have proposed a
model in which undeserved rewards over time are captured by a Markov random process b.
Using observations from previous states and actions, an agent can approach this question
from two perspectives. In the first, each time the agent visits a particular state i, it should
gain a better sense of the evolution of the random variable b between its last visit and its
current visit. It is important to note that rewards are received frequently, thus allowing frequent updating of b. Secondly, given an estimate of bt upon visiting state i at time t, it has
a better idea of the value of bt+1 when it visits state i0 at time t + 1, since we assume bt
evolves slowly over time. These are the ideas captured by the causal Kalman filter, which
only uses the history of past states and observations to provides estimates of r(i) and b.
The agent follows this simple algorithm:
1. From initial state i0 , take some action a, transition to state i, and receive reward
signal g0 . Initialize x
?0 (i0 ) = g0 and x
?0 (|S| + 1) = b0 = 0, since b0 = 0.
2. Perform a Kalman update using equations 1-5 to compute the current vector of
estimates x
?, which includes a component that is the reward estimate r?(i0 ), which
will simply equal g this time.
3. From the current state i at time t, take another action with some mix of exploration
and exploitation; transition to state j, receiving reward signal gt . If this is the first
visit to state i, initialize x
?t (i) = gt ? ?bt?1 .
4. Perform a Kalman update using equations 1-5 to compute the current vector of
estimates x
?, which includes a component that is the reward estimate r?(i).
5. Update the Q-table using r?(i) in place of r in equation 6; return to Step 3.
The advantage of the Kalman filter is that it requires a constant amount of memory ? at no
time does it need a full history of states and observations. Instead, it computes a sufficient
statistic during each update, x and P , which consists of the maximum likelihood estimate
of r and b, and the covariance matrix of this estimate. Thus, we can run this algorithm
online as we learn, and its speed does not deteriorate over time. Its speed is most tied to
3.5
20
250
3
200
15
2.5
150
2
10
100
1.5
5
1
50
0
0.5
0
0
?50
0
500
1000
1500
2000
2500
3000
?5
0
500
1000
1500
2000
2500
3000
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
4
x 10
Figure 2: (Left) As the agent is attempting to learn, the reward signal value (y-axis) changes
dramatically over time (x-axis) due to the noise term. While the true range of rewards in
this grid world domain only falls between 0 and 20, the noisy reward signal ranges from
-10 to 250, as shown in the graph at left. (Center) Given this noisy signal, the filtering agent
is still able to learn the true underlying rewards, converging to the correct relative values
over time, as shown in the middle graph. (Right) The filtering learning agent (bold line)
accrues higher rewards over time than the ordinary Q-learner (thin line), since it is able to
converge to an optimal policy whereas the non-filtering Q-learner remains confused.
the number of observation states that we choose to use, since the Kalman update (Eqn. 3)
needs to perform a matrix inversion of size |S| ? |S|. However, since our model assumes
the agent only has access to a limited, local observation space within the true global state
space, this computation remains feasible.
5
Empirical results
If the world dynamics exactly match the linear model we provide the Kalman filter, then
this method will provably converge to the correct reward value estimates and the find the
optimal policy under conditions similar to those guaranteeing Q-learning?s eventual convergence. However, we would rarely expect the world to fit this grossly simplified model.
The interesting question concerns situations in which the actual dynamics are clearly different from our model, and whether our filtering agent will still learn a good policy. This
section examines the efficacy of the filtering learning agent in several increasingly difficult
domains: (1) a single agent domain in which the linear system describes the world perfectly, (2) a single agent domain where the noise is manually adjusted without following
the model, (3) a multi-agent setting in which the noise term is meant to encapsulate presence of other agents in the environment, and (4) a more complicated multi-agent setting
that simulates an mobile ad-hoc networking domain in which mobile agent nodes try to
maximize total network performance.
For ease of exposition, all the domains we use are variants of the popular grid-world domain
shown in Figure 1 [Sutton and Barto, 1999]. The agent is able to move North, South, East,
or West, and most transitions give the agent zero reward, except all actions from state 6
move the agent directly to state 10 with a reward of 20, and all actions from state 16 move
the agent directly to state 18 with a reward of 10. Bumps into the wall cost the agent -1 in
reward and move the agent nowhere. We use a discount factor of 0.9.
To demonstrate the basic feasibility of our filtering method, we first create a domain that
follows the linear model of the world given in Section 3 perfectly. That is, in each time
step, a single agent receives its true reward plus some noise term that evolves as a Markov
random process. To achieve this, we simply add a noise term to the grid world domain
given in Figure 1. As shown in Figure 2, an agent acting in this domain will receive a large
range of reward values due to the evolving noise term. In the example given, sometimes
this value ranges as high as 250 even though the maximum reward in the grid world is
4
4
3.5
3.5
3
3
2.5
2.5
2
2
1.5
1.5
1
1
0.5
0.5
0
0
?0.5
?1
0
1
2
3
4
5
6
7
8
9
10
4
x 10
?0.5
0
1
2
3
4
5
6
7
8
9
10
4
x 10
Figure 3: (Left) Filtering agents are able to distinguish their personal rewards from the
global reward noise, and thus able to learn optimal policies and maximize their average
reward over time in a ten-agent grid-world domain. (Right) In contrast, ordinary Q-learning
agents do not process the global reward signal and can become confused as the environment
changes around them. Graphs show average rewards (y-axis) within 1000-period windows
for each of the 10 agents in a typical run of 10000 time periods (x-axis).
20 ? the noise term contributes 230 to the reward signal! A standard Q-learning agent
does not stand a chance at learning anything useful using this reward signal. However, the
filtering agent can recover the true reward signal from this noisy signal and use that to learn.
Figure 2 shows that the filtering agent can learn the underlying reward signals, converging
to these values relatively quickly. The graph to the right compares the performance of the
filtering learner to the normal Q-learner, showing a clear performance advantage.
The observant reader may note that the learned rewards do not match the true rewards
specified by the grid world. Specifically, they are offset by about -4. Instead of mostly
0 rewards at each state, the agent has concluded that most states produce reward of -4.
Correspondingly, state 6 now produces a reward of about 16 instead of 20. Since Q-learning
will still learn the correct optimal policy subject to scaling or translation of the rewards, this
is not a problem. This oddity is due to the fact that our model has a degree of freedom in
the noise term b. Depending on the initial guesses of our algorithm, the estimates for the
rewards may be biased. If most of the initial guesses for the rewards underestimated the
true reward, then the learned value will be correspondingly lower than the actual true value.
In fact, all the learned values will be correspondingly lower by the same amount.
To further test our filtering technique, we next evaluate its performance in a domain that
does not conform to our noise model perfectly, but which is still a single agent system.
Instead of an external reward term that evolves according to a Gaussian noise process, we
adjust the noise manually, introducing positive and negative swings in the reward signal
values at arbitrary times. The results are similar to those in the perfectly modeled domain,
showing that the filtering method is fairly robust.
The most interesting case occurs when the domain noise is actually caused by other agents
learning in the environment. This noise will not evolve according to a Gaussian process,
but since the filtering method is fairly robust, we might still expect it to work. If there are
enough other agents in the world, then the noise they collectively generate may actually
tend towards Gaussian noise. Here we focus on smaller cases where there are 6 or 10
agents operating in the environment. We modify the grid world domain to include multiple
simultaneously-acting agents, whose actions do not interfere with each other, but whose
reward signal now consists of the sum of all the agents? personal rewards, as given in the
basic single agent grid world of Figure 1.
We again compare the performance of the filtering learner to the ordinary Q-learning algorithm. As shown in Figure 3, most of the filtering learners quickly converge to the optimal
R
1
S
S
0
1
2
3
4
5
6
7
8
9
10
4
x 10
Figure 4: (Left) A snapshot of the 4x4 adhoc-networking domain. S denotes the sources, R
is the receiver, and the dots are the learning agents, which act as relay nodes. Lines denote
current connections. Note that nodes may overlap. (Right) Graph shows average rewards
(y-axis) in 1000-period windows as filtering (bold line) and ordinary (thin line) agents try
to learn good policies for acting as network nodes. The filtering agent is able to learn a
better policy, resulting in higher network performance (global reward). Graph shows the
average for each type of agent over 10 trial runs of 100000 time periods (x-axis) each.
policy. Three of the 10 agents converge to a suboptimal policy that produces slightly lower
average rewards. However, this artifact is largely due to our choice of exploration rate,
rather than a large error in the estimated reward values. The standard Q-learning algorithm
also produces decent results at first. Approximately half of the agents find the optimal policy, while the other half are still exploring and learning. An interesting phenomenon occurs
when these other agents finally find the optimal policy and begin receiving higher rewards.
Suddenly the performance drops drastically for the agents who had found the optimal policy first. Though seemingly strange, this provides a perfect example of the behavior that
motivates this paper. When the other agents learn an optimal policy, they begin affecting
the global reward, contributing some positive amount rather than a consistent zero. This
changes the world dynamics for the agents who had already learned the optimal policy and
causes them to ?unlearn? their good behavior.
The unstable dynamics of the Q-learners could be solved if the agents had full observability,
and we could learn using the joint actions of all the agents, as in the work of Claus and
Boutilier [1998]. However, since our premise is that agents have only a limited view of
the world, the Q-learning agents will only exhibit convergence to the optimal policy if they
converge to the optimal policy simultaneously. This may take a prohibitively long time,
especially as the number of agents grows.
Finally, we apply our filtering method to a more realistic domain. Mobilized ad-hoc networking provides an interesting real-world environment that illustrates the importance of
reward filtering due to its high degree of partial observability and a reward signal that depends on the global state. In this domain, there are a number of mobile nodes whose task
is to move in such a way as to optimize the connectivity (performance) of the network.
Chang et al. [2003] cast this as a reinforcement learning problem. As the nodes move
around, connections form between nodes that are within range of one another. These connections allow packets to be transmitted between various sources and receivers scattered
among the nodes. The nodes are limited to having only local knowledge of their immediate
neighboring grid locations (rather than the numbered state locations as in the original grid
world), and thus do not know their absolute location on the grid. They are trained using
a global reward signal that is a measure of total network performance, and their actions
are limited functions that map their local state to N, S, E, W movements. We also limit
their transmission range to a distance of one grid block. For simplicity, the single receiver
is stationary and always occupies the grid location (1,1). Source nodes move around randomly, and in our example here, there are two sources and eight mobile agent nodes in a
4x4 grid. This setup is shown in Figure 4, and the graph shows a comparison of an ordinary
Q-learner and the filtering learner, plotting the increase in global rewards over time as the
agents learn to perform their task as intermediate network nodes. The graph plots average
performance over 10 runs, showing the benefit of the filtering process.
6
Limitations and extensions
The Kalman filtering framework seems to work well in these example domains. However,
there are some cases where we may need to apply more sophisticated techniques. In all the
above work, we have assumed that the reward signal is deterministic ? each state, action
pair only produces a single reward value. There are some domains in which we?d like to
model the reward as being stochastic, such as the multi-armed bandit problem. When the
stochasticity of the rewards approximates Gaussian noise, we can use the Kalman framework directly. In equation 1, v was set to exhibit zero mean and zero variance. However,
allowing some variance would give the model an observation noise term that could reflect
the stochasticity of the reward signal.
Finally, in most cases the Kalman filtering method provides a very good estimate of r over
time. However, since we cannot guarantee an exact estimate of the reward values when the
model is not an exact representation of the world, the agent may make the wrong policy
decision sometimes. However, even if the policy is sub-optimal, the error in our derived
value function is at least bounded by 1??
, as long as the |r(i) ? r?(i)| < ?i, and ? is
again the discount rate. In the majority of cases, the estimates are good enough to lead the
agent to learning a good policy.
Conclusion and future work. This paper provides the general framework for a new approach to solving large multi-agent problems using a simple model that allows for efficient
and robust learning using well-studied tools such as Kalman filtering. As a practical application, we are working on applying these methods to a more realistic version of the mobile
ad-hoc networking domain.
References
[Auer et al., 1995] P. Auer, N. Cesa-Bianchi, Y. Freund, and R. Schapire. Gambling in a rigged
casino: the adversarial multi-armed bandit problem. In Proceedings of the 36th Annual Symposium
on Foundations of Computer Science, 1995.
[Chang et al., 2003] Y. Chang, T. Ho, and L. P. Kaelbling. Reinforcement learning in mobilized
ad-hoc networks. MIT AI Lab Memo AIM-2003-025, 2003.
[Choi et al., 1999] S. Choi, D. Yeung, and N. Zhang. Hidden-mode Markov decision processes. In
IJCAI Workshop on Neural, Symbolic, and Reinforcement Methods for Sequence Learning, 1999.
[Claus and Boutilier, 1998] Caroline Claus and Craig Boutilier. The dynamics of reinforcement
learning in cooperative multiaent systems. In Proceedings of the 15th AAAI, 1998.
[Kalman, 1960] R. E. Kalman. A new approach to linear filtering and prediction problems. Transactions of the American Society of Mechanical Engineers, Journal of Basic Engineering, 1960.
[McMahan et al., 2003] H. McMahan, G. Gordon, and A. Blum. Planning in the presence of cost
functions controlled by an adversary. In Proceedings of the 20th ICML, 2003.
[Ng et al., 1999] Andrew Y. Ng, Daishi Harada, and Stuart Russell. Policy invariance under reward
transformations: theory and application to reward shaping. In Proc. 16th ICML, 1999.
[Sutton and Barto, 1999] Richard S. Sutton and Andrew G. Barto. Reinforcement Learning: An
Introduction. MIT Press, 1999.
[Szita et al., 2002] Istvan Szita, Balimt Takacs, and Andras Lorincz. e-mdps: Learning in varying
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1,624 | 2,477 | Sparseness of Support Vector Machines?Some
Asymptotically Sharp Bounds
Ingo Steinwart
Modeling, Algorithms, and Informatics Group, CCS-3, Mail Stop B256
Los Alamos National Laboratory
Los Alamos, NM 87545, USA
[email protected]
Abstract
The decision functions constructed by support vector machines (SVM?s)
usually depend only on a subset of the training set?the so-called support
vectors. We derive asymptotically sharp lower and upper bounds on the
number of support vectors for several standard types of SVM?s. In particular, we show for the Gaussian RBF kernel that the fraction of support
vectors tends to twice the Bayes risk for the L1-SVM, to the probability
of noise for the L2-SVM, and to 1 for the LS-SVM.
1
Introduction
Given a training set T = ((x1 , y1 ), . . . , (xn , yn )) with xi ? X, yi ? Y := {?1, 1}
standard support vector machines (SVM?s) for classification (cf. [1], [2]) solve
n
arg min ?kf k2H +
f ?H
b?R
1X
L yi (f (xi ) + b) ,
n i=1
(1)
where H is a reproducing kernel Hilbert space (RKHS) of a kernel k : X ?X ? R (cf. [3],
[4]), ? > 0 is a free regularization parameter and L : R ? [0, ?) is a convex loss function.
Common choices for L are the hinge loss function L(t) := max{0, 1?t}, the squared hinge
loss function L(t) := (max{0, 1 ? t})2 and the least square loss function L(t) := (1 ? t)2 .
The corresponding classifiers are called L1-SVM, L2-SVM and LS-SVM, respectively.
Common choices of kernels are the Gaussian RBF k(x, x0 ) = exp(?? 2 kx ? x0 k22 ) for
x, x0 ? Rd and fixed ? > 0 and polynomial kernels k(x, x0 ) = (hx, x0 i+c)m for x, x0 ? Rd
and fixed c ? 0, m ? N.
If (fT,? , bT,? ) ? H ? R denotes a solution of (1) we have
n
fT,? =
1 X
yi ?i? k(xi , .)
2? i=1
(2)
for suitable coefficients ?1? , . . . , ?n? ? R (cf. [5]). Obviously, only the samples xi with
?i? 6= 0 have an impact on fT,? . These samples are called support vectors. The fewer
support vectors fT,? has the faster it can be evaluated. Moreover, it is well known that
the number of support vectors #SV (fT,? ) of the representation of fT,? (cf. Section 3
for a brief discusssion) also has a large impact on the time needed to solve (1) using the
dual problem. Therefore, it is of high interest to know how many support vectors one
can expect for a given classification problem. In this work we address this question by
establishing asymptotically lower and upper bounds on the number of support vectors for
typical situations.
The rest of the paper is organized as follows: in Section 2 we introduce some technical
notions and recall recent results in the direction of the paper. In Section 3 our results are
presented and discussed, and finally, in Section 4 their proofs can be found.
2
Notations and known results
The standard assumption in classification is that the training set T consists of i.i.d. pairs
drawn from an unknown distribution P on X ?Y . For technical reason we assume throughout this paper that X is a compact metric space, e.g. a bounded, closed subset of R d . A
Bayes decision function (cf. [6]) fP : X ? Y is a function that PX -a.s. equals 1 and ?1
on C1 := {x ? X : P (1|x) > 1/2} and C?1 := {x ? X : P (?1|x) > 1/2}, respectively.
The corresponding classification error RP of such a function is called the Bayes risk of P .
Recall, that the Bayes risk is the smallest possible classification error.
A RKHS H is called universal if H is k.k? -dense in the space of continuous functions
C(X). The best known example of a universal kernel is the Gaussian RBF kernel (cf. [7]).
Let us recall some results of the recent paper [8]. To simplify the statements, let us assume
that P has no discrete components, i.e. PX ({x}) = 0 for all x ? X. Furthermore, let L be
a continuous convex loss function satisfying some minor regularity conditions. Then it was
shown for universal RKHS?s and stritly positive nullsequences (?n ) satisfying a regularity
condition that the following statements hold for all ? > 0 and n ? ?:
P n T ? (X ? Y )n : #SV (fT,?n ) ? (RP ? ?)n ? 1 .
(3)
In particular, this result holds for L1-SVM?s. Furthermore, for L being also differentiable
(e.g. L2-SVM?s and LS-SVM?s) it was proved
P n T ? (X ? Y )n : #SV (fT,?n ) ? (S P ? ?)n ? 1 ,
(4)
where S P := PX ({x ? X : 0 < P (1|x) < 1}) denotes the probability of the set of points
where noise occurs. Obviously, we always have S P ? 2RP and for noisy non-degenerate
P , that is for P with
PX x ? X : P (1|x) 6? {0, 1/2, 1} > 0
this relation becomes a strict inequality. We shall prove in the next section that (3) can
be significantly improved for the L1-SVM. We shall also show that this new lower bound
is also an upper bound under moderate conditions on P and H. Furthermore, we prove
that (4) is asymptotically optimal for the L2-SVM and show that it can be significantly
improved for the LS-SVM.
3
New bounds
We begin with lower and upper bounds for the L1-SVM. Recall, that the problem (1) for
this classifier can be reformulated as
n
P
minimize
?hf, f i + n1
?i
for f ? H, b ? R, ? ? Rn
i=1
(5)
subject to
yi f (xi ) + b ? 1 ? ?i ,
i = 1, . . . , n
?i ? 0,
i = 1, . . . , n .
Instead of solving (5) directly, one usually solves the dual optimization problem (cf. [4])
maximize
subject to
n
P
i=1
n
P
?i ?
1
4?
n
P
yi yj ?i ?j k(xi , xj )
for ? ? Rn
i,j=1
(6)
yi ?i = 0,
i=1
0 ? ?i ?
1
n,
i = 1, . . . , n .
(?1? , . . . , ?n? )
? R denotes a solution of (6) then fT,? can be computed by (2). Note that
If
the representaion of fT,? is not unique in general, i.e. using other algorithms for solving (5)
can lead to possibly sparser representations. However, in contrast to the general case the
representation (2) of fT,? is P n -almost surely (a.s.) unique if the kernel is universal and
P has no discrete components (cf. [8]). Since our results for the L1-SVM hold for general
kernels we always assume that fT,? is found by (6). Finally, for a loss function L and a
RKHS H we write
RL,P,H := inf RL,P (f + b) ,
f ?H
b?R
where RL,P (f ) := E(x,y)?P L yf (x) . Note, that fT,?n +bT,?n cannot achieve an L-risk
better than RL,P,H , if H is the RKHS used in (1). Now, our first result is:
Theorem 3.1 Let k be a continuous kernel on X and P be a probability measure on X ?Y
with no discrete components. Then for the L1-SVM using a regularization sequence (? n )
with ?n ? 0 and n?2n / log n ? ? and all ? > 0 we have
P n T ? (X ? Y )n : #SV (fT,?n ) ? (RL,P,H ? ?)n ? 1 .
Remark 3.2 If k is a universal kernel we have RL,P,H = 2RP (cf. Ste7) and thus Theorem
3.1 yields the announced improvement of (3). For non-universal kernels we even have
RL,P,H > 2RP in general.
Remark 3.3 For specific kernels the regularity condition n?2n / log n ? ? can be weakened. Namely, for the Gaussian RBF kernel on X ? Rd it can be substituted by
n?n |log ?n |?d?1 ? ?. Only slightly stronger conditions are sufficient for C ? -kernels.
The interested reader can prove such conditions by establishing (9) using the results of [9].
Remark 3.4 If H is finite dimensional and n > dim H the representation (2) of fT,?n can
be simplified such that only at most dim H kernel evaluations are neccessary. However,
this simplification has no impact on the time needed for solving (6).
In order to formulate an upper bound on #SV (fT,?n ) recall that a function is called analytic if it can be locally represented by its Taylor series. Let L be a loss function, H
be a RKHS over X and P be a probability measure on X ? Y . We call the pair (H, P )
non-trivial (with respect to L) if
RL,P,H < inf RL,P (b) ,
b?R
i.e. the incorporation of H has a non-trivial effect on the L-risk of P . If H is universal we
have RL,P,H = inf{RL,P (f ) f : X ? R} (cf. [9]) and therefore (H, P ) is non-trivial
if P has two non-vanishing classes, i.e. PX (C1 ) > 0 and PX (C?1 ) > 0. Furthermore, we
denote the open unit ball of Rd by BRd . Now our upper bound is:
Theorem 3.5 Let H be the RKHS of an analytic kernel on BRd . Furthermore, let X ? BRd
be a closed ball and P be a noisy non-degenerate probability measure on X ? Y such that
PX has a density with respect to the Lebesgue measure on X and (H, P ) is non-trivial.
Then for the L1-SVM using a regularization sequence (?n ) which tends sufficiently slowly
to 0 we have
#SV (fT,?n )
? RL,P,H
n
in probability.
Probably the most restricting condition on P in the above theorem is that PX has to have a
density with respect to the Lebesgue measure. Considering the proof this condition can be
slightly weakened to the assumption that every d?1-dimensional subset of X has measure
zero. Although it would be desirable to exclude only probability measures with discrete
components it is almost obvious that such a condition cannot be sufficient for d > 1 (cf. [10,
p.32]). The assumption that P is noisy and non-degenerate is far more less restrictive since
neither completely noise-free P nor noisy problems with only ?coin-flipping? noise often
occur in practice. Finally, the condition that (H, P ) is non-trivial is more or less implicitly
assumed whenever one uses nontrivial classifiers.
Example 3.6 Theorem 3.5 directly applies to polynomial kernels. Note, that the limit
RL,P,H depends on both P and the choice of the kernel.
Example 3.7 Let k be a Gaussian RBF kernel with RKHS H and X be a closed ball of
Rd . Moreover, let P and (?n ) be according to Theorem 3.5. Recall, that k is universal and
hence (H, P ) is non-trivial iff P has two non-vanishing classes. Since k is also analytic on
Rd we find
#SV (fT,?n )
? 2 RP .
n
Therefore, (4) shows that in general this L1-SVM produces sparser decision functions than
the L2-SVM and the LS-SVM based on a Gaussian RBF kernel (cf. also Theorem 3.11).
Remark 3.8 A variant of the L1-SVM that is often considered in theoretical papers is
based
Pn on the optimization problem (5) with a-priori fixed b := 0. Besides the constraint
i=1 yi ?i = 0, which no longer appears, the corresponding dual problem is identical to
(6). Hence it is easily seen that Theorem 3.1 also holds for this classifier. Moreover, for
this modification Theorem 3.5 can be simplified. Namely, the assumption that P is noisy
and non-degenerate is superfluous (cf. [8, Prop. 33] to guarantee (14)). In particular, for a
Gaussian RBF kernel and noise-free problems P we then obtain
#SV (fT,?n )
? 0,
(7)
n
i.e. the number of support vectors increases more slowly than linearly. This motivates the
often claimed sparseness of SVM?s.
The following theorem shows that the lower bound (4) on #SV (fT,?n ) for the L2-SVM is
often asymptotically optimal. This result is independent of the used optimization algorithm
since we only consider universal kernels and measures with no discrete components.
Theorem 3.9 Let H be the RKHS of an analytic and universal kernel on BRd . Furthermore, let X ? BRd be a closed ball and P be a probability measure on X ? Y with
RP > 0 such that PX has a density with respect to the Lebesgue measure on X and
(H, P ) is non-trivial. Then for the L2-SVM using using a regularization sequence (? n )
which tends sufficiently slowly to 0 we have
#SV (fT,?n )
? SP
n
in probability.
Remark 3.10 For the L2-SVM with fixed offset b := 0 the assumption RP > 0 in the
above theorem is superfluous (cf. proof of Theorem 3.9 and proof of [8, Prop. 33]). In
particular, for a Gaussian RBF kernel and noise-free problems P we obtain (7), i.e. for
noise-free problems this classifier also tends to produce sparse solutions in the sense of
Remark 3.8.
Our last result shows that LS-SVM?s often tend to use almost every sample as a support
vector:
Theorem 3.11 Let H be the RKHS of an analytic and universal kernel on BRd . Furthermore, let X ? BRd be a closed ball and P be a probability measure on X ? Y such that
PX has a density with respect to the Lebesgue measure on X and (H, P ) is non-trivial.
Then for the LS-SVM using a regularization sequence (?n ) which tends sufficiently slowly
to 0 we have
#SV (fT,?n )
?1
n
in probability.
Remark 3.12 Note, that unlike the L1-SVM and the L2-SVM (with fixed offset) the LSSVM does not tend to produce sparse decision functions for noise-free P . This still holds
if one fixes the offset for L2-SVM?s, i.e. one considers regularization networks (cf. [11]).
The reason for the different behaviours is the margin as already observed in [12]: the
assumptions on H and P ensure that only a very small fraction of samples xi can be
mapped to ?1 by fT,?n (cf. also Remark 4.1). For the L2-SVM this asymptotically ensures
that most of the samples are mapped to values outside the margin, i.e. yi fT,?n (xi ) > 1,
(cf. the properties of Bn \ A? in the proof of Theorem 3.9) and it is well-known that such
samples cannot be support vectors. In contrast to this the LS-SVM has the property that
every point not lying on the margin is a support vector. Using the techniques of our proofs
it is fairly easy to see that the same reasoning holds for the hinge loss function compared
to ?modified hinge loss functions with no margin?.
4
Proofs
Let L be a loss function and T be a training set. For a function f : X ? R we denote the
empirical L-risk of f by
n
1X
RL,T (f + b) :=
L yi (f (xi ) + b) .
n i=1
Proof of Theorem 3.1: Let (fT,?n , bT,?n , ? ? ) ? H ? R ? Rn and ?? ? Rn be solutions
of (5) and (6) for the regulariztion parameter ?n , respectively. Since there is no duality gap
between (5) and (6) we have (cf. [4]):
n
n
n
1 X
1X ? X ?
? =
?i ?
yi yj ?i? ?j? k(xi , xj )
(8)
?n hfT,?n , fT,?n i +
n i=1 i
4?
n i,j=1
i=1
By (2) this yields
n
n
n
X
1X ?
1X ?
?i ? 2?n hfT,?n , fT,?n i +
?i =
?i? .
n i=1
n i=1
i=1
Furthermore, recall that ?n ? 0 and n?2n / log n ? ? implies
n
1X ?
? = RL,T (fT,?n + bT,?n ) ? RL,P,H
n i=1 i
(9)
in probability for n ? ? (cf. [9]) and hence for all ? > 0 the probability of
n
X
?i? ? RL,P,H ? ?
(10)
i=1
tends to 1 for n ? ?. Now let us assume that our training set satisfies (10). Since
?i? ? 1/n we then find
RL,P,H ? ? ?
n
X
?i? ?
X 1
1
= #SV (fT,?n )
n
n
?
?i 6=0
i=1
which finishes the proof.
For our further considerations we need to consider the optimization problem (1) with respect to P , i.e. we treat the (solvable, see [8]) problem
(fP,? , bP,? ) := arg min ?kf k2H + RL,P (f + b) .
f ?H
b?R
(11)
Proof of Theorem 3.5: Since H is the RKHS of an analytic kernel every function f ? H is
analytic. Using the holomorphic extension of a non-constant f ? H we see (after a suitable
complex linear coordinate change, cf. [10, p. 31f]) that for c ? R and x1 , . . . , xd?1 ? R the
equation f (x1 , . . . , xd?1 , xd ) = c has at most j solutions xd , where j ? 0 is locally (with
respect to x1 , . . . , xd?1 ? R) constant . By a simple compactness argument we hence find
PX {x ? X : f (x) = c} > 0
?
f (x) = c PX -a.s.
(12)
for all f ? H and all c ? R. Now, let us suppose that
PX {x ? X : fP,? (x) + bP,? = fP (x)} > 0
(13)
for some ? > 0, where fP denotes the Bayes decision function. Then we may assume
without loss of generality that PX {x ? X : fP,? (x) + bP,? = 1} > 0 holds. By (12) this
leads to fP,? (x) + bP,? = 1 PX -a.s. However, since RL,P (fP,? + bP,? ) ? RL,P,H for
? ? 0 (cf. [9]) we see that fP,? cannot be constant for small ? since (H, P ) was assumed
to be non-trivial. Therefore (13) cannot hold for small ? > 0 and hence we may assume
without loss of generality that
PX {x ? X : |fP,? (x) + bP,? ? fP (x)| = 0} = 0
holds for all ? > 0. We define A? (?) := x ? X : |fP,? (x) + bP,? ? fP (x)| ? ?
for ?, ? > 0. Our above considerations show that for all ? > 0 there exists a ? > 0
with PX (A? (?)) ? ?. We write ?? := 21 sup{? > 0 : PX (A? (?)) ? ?}. We first show
that there exists no sequence ?n ? ? 6= 0 with ??n ? 0. Let us assume the converse.
Then there exists a subsequence with (fP,?nj , bP,?nj ) ? (fP,? , bP,? ) weakly and we have
lim supj?? A3??n (?nj ) ? A0 (?). By the construction we have PX (A3??n (?nj )) ? ?
j
j
and hence PX (lim supj?? A3??n (?nj )) ? ? by the Lemma of Fatou. This gives the
j
??
contradiction PX (A0 (?)) ? ?. Thus, the increasing function ? 7? m(?) := inf{??? : ?
?} satisfies m(?) > 0 for all ? > 0. We fix a T = ((x1 , y1 ), . . . , (xn , yn )) with
kfT,?n + bT,?n ? fP,?n ? bP,?n k? ? ?n ,
(14)
RL,T (fT,? + bT,? ) ? RL,P (fP,? + bP,? ) ? ?
(15)
n
n
n
n
4
3
and {i : xi ? A?n (n))} ? 2?n. If m (?n )?n n ? ? the results of [9] and [8] ensure,
that the probability of such a T converges to 1 for n ? ?. Moreover, by (8) we find
2?n hfT,?n , fT,?n i + RL,T (fT,?n + bT,?n ) =
n
X
i=1
?i? .
(16)
Since fT,?n + bT,?n and fP,?n + bP,?n minimize the regularized risks, (15) implies
?n kfT,?n k2H + RL,T (fT,?n + bT,?n ) ? ?n kfP,?n k2H ? RL,P (fP,?n + bP,?n ) ? ? . (17)
Furthermore, if n ? ? we have
(18)
?n kfP,?n k2H + RL,P (fP,?n + bP,?n ) ? RL,P,H
2
(cf. [9]) and therefore we obtain ?n kfT,?n kH + RL,T (fT,?n + bT,?n ) ? RL,P,H ? 2?
for large n. Now, (15), (17) and (18) implies ?n hfT,?n , fT,?n i ? 3? for large n. Hence
(16) yields
n
X
(19)
RL,P,H + 5? ?
?i?
i=1
if n is sufficiently large. Now let us suppose that we have a sample (xi , yi ) of T with
xi 6? A?n (n). Then we have |fP,?n (xi ) + bP,?n ? fP (xi )| > ?n and hence fT,?n (xi ) +
bT,?n 6= ?1 by (14). By [4, p. 107] this means either ?i? = 0 or ?i? = 1/n. Therefore, by
(19) we find
n
n
X
X
1
RL,P,H + 5? ?
?i? ?
?i? = {i : xi 6? A?n (n) and ?i? 6= 0}
n
i=1
xi 6?A?n (n)
Since we have at most 2?n samples in A?n (n) we finally obtain
1
#SV (fT,?n ) ? RL,P,H + 7? .
n
Now the assertion follows by Theorem 3.1.
Remark 4.1 The proof of Theorem 3.5 is based on a kind of paradox: recall that it was
shown in [8] that
fT,?n + bT,?n ? fP
on x ? X : P (1|x) 6? {0, 1/2, 1} in probability. However, the assumption on both H
and P ensures that for typical T the sets
x ? X : |fT,?n (x) + bT,?n ? fP (x)| ? ?
become arbitrarily small for ? ? 0. We will apply these seemingly contradicting properties
in the following proofs, too.
Proof of Theorem 3.9: Let N := {x ? X : 0 < P (1|x) < 1} be the subset of X where P
is noisy. Furthermore, let A? (n) be defined as in the proof of Theorem 3.5. We write
B? (n) :=
x ? C1 \ N : fP,?n (x) + bP,?n ? 1 ? ?
? x ? C?1 \ N : fP,?n (x) + bP,?n ? ?1 + ? .
By [8, Thm. 22]) for all n ? 1 there exists a ? > 0 with PX (B? (n)) ? PX (X \ N ) ? ?.
We define ?n := 21 sup{? > 0 : PX (A? (n)) ? ? and PX (B? (n)) ? PX (X \ N ) ? ?}. Let
us fix a training set T = ((x1 , y1 ), . . . , (xn , yn )) with
kfT,?n + bT,?n ? fP,?n ? bP,?n k? ? ?n ,
{i : xi ? B? (n) \ A? (n)} ? n PX (X \ N ) ? 3? .
n
Again, the probability of such T converges to 1 for n ? ? whenever (?n ) converges
sufficiently slowly to 0. In view of (4) it suffices to show that every sample xi ? B? (n) \
A?n (n) cannot be a support vector. Given an xi ? B? (n) \ A?n (n) we may assume without
loss of generality that xi ? C1 . Then xi ? B? (n) implies fP,?n (xi )+bP,?n ? 1??n while
xi 6? A?n (n) yields |fP,?n (xi )+bP,?n ?1| > ?n . Hence we find fP,?n (xi )+bP,?n > 1+?n
and thus fT,?n (xi )+bT,?n > 1. By the Karush-Kuhn-Tucker conditions of the primal/dual
optimization problem of the L2-SVM (cf. [4, p. 105]) this shows that xi is not a support
vector.
Proof of Theorem 3.11: Let A? (n) and ?n be defined as in the proof of Theorem 3.5. Without loss of generality we may assume ?n ? (0, 1/2). Let us define C0 := {x ? X :
P (1|x) = 1/2} and
Dn = x ? C0 : |fP,?n (x) + bP,?n | ? 1/2 .
By [8, Thm. 22] we may assume without loss of generality that PX (Dn ) ? PX (C0 ) ? ?
for all n ? 1. Now, let us fix a training set T = ((x1 , y1 ), . . . , (xn , yn )) with
kfT,?n + bT,?n ? fP,?n ? bP,?n k? ? ?n
{i : xi ? A? (n)} ? 2 ? n
n
{i : xi ? Dn } ? n PX (C0 ) ? 2? .
Again, the probability of such T converges to 1 for n ? ? whenever (?n ) converges
sufficiently slowly to 0. Now let us consider a sample xi ? (X \ A?n (n)) ? C1 of T .
Then we have |fP,?n (xi ) + bP,?n ? 1| > ?n and hence fT,?n (xi ) + bT,?n 6= 1. By [8,
Cor. 32] this shows that xi is a support vector. Obviously, the same holds true for samples
xi ? (X \ A?n (n)) ? C?1 . Finally, for samples xi ? Dn we have |fT,?n (xi ) + bT,?n | ?
1/2 + ?n < 1 and hence these samples are always support vectors.
Acknowledgments
I would like to thank D. Hush and C. Scovel for helpful comments.
References
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[10] R.M. Range. Holomorphic Functions and Integral Representations in Several Complex Variables. Springer, 1986.
[11] F. Girosi, M. Jones, and T. Poggio. Regularization theory and neural networks architectures. Neural Computation, 7:219?269, 1995.
[12] A. Kowalczyk. Sparsity of data representation of optimal kernel machine and leaveone-out estimator. In T.K. Leen, T.G. Dietterich, and V. Tresp, editors, Advances in
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1,625 | 2,478 | Multiple Instance Learning via
Disjunctive Programming Boosting
Stuart Andrews
Department of Computer Science
Brown University, Providence, RI, 02912
[email protected]
Thomas Hofmann
Department of Computer Science
Brown University, Providence, RI, 02912
[email protected]
Abstract
Learning from ambiguous training data is highly relevant in many
applications. We present a new learning algorithm for classification
problems where labels are associated with sets of pattern instead
of individual patterns. This encompasses multiple instance learning as a special case. Our approach is based on a generalization
of linear programming boosting and uses results from disjunctive
programming to generate successively stronger linear relaxations of
a discrete non-convex problem.
1
Introduction
In many applications of machine learning, it is inherently difficult or prohibitively
expensive to generate large amounts of labeled training data. However, it is often
considerably less challenging to provide weakly labeled data, where labels or annotations y are associated with sets of patterns or bags X instead of individual patterns
x ? X. These bags reflect a fundamental ambiguity about the correspondence of
patterns and W
the associated label which can be expressed logically as a disjunction
of the form: x?X (x is an example of class y). In plain English, each labeled bag
contains at least one pattern (but possibly more) belonging to this class, but the
identities of these patterns are unknown.
A special case of particular relevance is known as multiple instance learning [5]
(MIL). In MIL labels are binary and the ambiguity is asymmetric in the sense that
bags with negative labels are always of size one. Hence the label uncertainty is
restricted to members of positive bags. There are many interesting problems where
training data of this kind arises quite naturally, including drug activity prediction
[5], content-based image indexing [10] and text categorization [1]. The ambiguity
typically arises, because of polymorphisms allowing multiple representations, e.g. a
molecule which can be in different conformations, or because of a part/whole am-
biguity, e.g. annotations may be associated with images or documents where they
should be attached to objects in an image or passages in a document. Notice also
that there are two intertwined objectives: the goal may be to learn a pattern-level
classifier from ambiguous training examples, but sometimes one may be primarily
interested in classifying new bags without necessarily resolving the ambiguity for
individual patterns.
A number of algorithms have been developed for MIL, including special purpose
algorithms using axis-parallel rectangular hypotheses [5], diverse density [10, 14],
neural networks [11], and kernel methods [6]. In [1] two versions of a maximummargin learning architecture for solving the multiple instance learning problem have
been presented. Because of the combinatorial nature of the problem, a simple
optimization heuristic was used in [1] to learn discriminant functions. In this paper,
we take a more principled approach by carefully analyzing the nature of the resulting
optimization problem and by deriving a sequence of successively stronger relaxations
that can be used to compute lower and upper bounds on the objective. Since it
turns out that exploiting sparseness is a crucial aspect, we have focused on a linear
programming formulation by generalizing the LPBoost algorithm [7, 12, 4] we call
the resulting method Disjunctive Programming Boosting (DPBoost).
2
Linear Programming Boosting
LPBoost is a linear programming approach to boosting, which
P aims at learning
ensemble classifiers of the form G(x) = sgn F (x) with F (x) = k ?k hk (x), where
hk : <d ? {?1, 1}, k = 1, . . . , n are the so-called base classifiers, weak hypotheses,
or features and ?k ? 0 are combination weights. The ensemble margin of a labeled
example (x, y) is defined as yF (x).
Given a set of labeled training examples {(x1 , y1 ), . . . , (xm , ym )}, LPBoost formulates the supervised learning problem using the 1-norm soft margin objective
min
?, ?
n
X
?k + C
m
X
?i
s.t. yi F (xi ) ? 1 ? ?i , ?i ? 0, ?i, ?k ? 0, ?k .
(1)
i=1
k=1
Here C > 0 controls the tradeoff between the Hinge loss and the L1 regularization
term. Notice that this formulation remains meaningful even if all training examples
are just negative or just positive [13].
Following [4] the dual program of Eq. (1) can be written as
max
u
m
X
i=1
ui ,
s.t.
m
X
ui yi hk (xi ) ? 1, ?k,
0 ? ui ? C, ?i .
(2)
i=1
It is useful to take a closer look at the KKT complementary conditions
!
m
X
ui (yi F (xi ) + ?i ? 1) = 0, and ?k
ui yi hk (xi ) ? 1 = 0.
(3)
i=1
Since the optimal values of the slack variables are implicitly determined by ? as
?i (?) = [1 ? yi F (xi )]+ , the first set of conditions states that ui = 0 whenever
yi F (xi ) > 1. Since ui can be interpreted as the ?misclassification? cost, this implies
that only instances with tight margin constraints may have non-vanishing
associated
Pm
costs. The second set of conditions ensures that ?k = 0, if i=1 ui yi hk (xi ) < 1,
which states that
P a weak hypothesis hk is never included in the ensemble, if its
weighted score i ui yi hk (xi ) is strictly below the maximum score of 1. So a typical
LPBoost solution may be sparse in two ways: (i) Only a small number of weak
hypothesis with ?k > 0 may contribute to the ensemble and (ii) the solution may
only depend on a subset of the training data, i.e. those instances with ui > 0.
LPBoost exploits the sparseness of the ensemble by incrementally selecting columns
from the simplex tableau and optimizing the smaller tableau. This amounts to
finding in each round a hypothesis hk for which the constraint in Eq. (2) is violated,
adding it to the ensemble and re-optimizing the tableau with the selected columns.
As a column selection heuristic the authors of [4] propose to use the
P magnitude of
the violation, i.e. pick the weak hypothesis hk with maximal score i ui yi hk (xi ).
3
Disjunctive Programming Boosting
In order to deal with pattern ambiguity, we employ the disjunctive programming
framework [2, 9]. In the spirit of transductive large margin methods [8, 3], we
propose to estimate the parameters ? of the discriminant function in a way that
achieves a large margin for at least one of the patterns in each bag. Applying this
principle, we can compile the training data into a set of disjunctive constraints on
?. To that extend, let us define the following polyhedra
(
)
X
Hi (x) ? (?, ?) : yi
?k hk (x) + ?i ? 1 , Q ? {(?, ?) : ?, ? ? 0} . (4)
k
Then we can formulate the following disjunctive program:
min
?,?
n
X
?k + C
m
X
?i ,
s.t. (?, ?) ? Q ?
Hi (x) .
(5)
i x?Xi
i=1
k=1
\ [
Notice that if |Xi | ? 2 then the constraint imposed by Xi is highly non-convex,
since it is defined via a union of halfspaces. However, for trivial bags with |Xi | = 1,
the resulting constraints are the same as in Eq. (1). Since we will handle these two
cases quite differently in the sequel, let us introduce index sets I = {i : |Xi | ? 2}
and J = {j : |Xj | = 1}.
A suitable way to define a relaxation to this non-convex optimization problem is
to replace the disjunctive set in Eq. (5) by its convex hull. As shown in [2], a
whole hierarchy of such relaxations can be built, using the fundamental fact that
cl-conv(A) ? cl-conv(B) ? cl-conv(A ? B), where cl-conv(A) denotes the closure of
the convex hull of the limiting points of A. This means a tighter convex relaxation
is obtained, if we intersect as many sets as possible, before taking their convex hull.
Since repeated intersections of disjunctive sets with more than one element each
leads to an combinatorial blow-up in the number of constraints, we propose to intersect every ambiguous disjunctive constraint with every non-ambiguous constraint
as well as with Q. This is also called a parallel reduction step [2]. It results in the
following convex relaxation of the constraints in Eq. (5)
?
?
??
\
[
\
?Hi (x) ? Q ?
cl-conv ?
Hj (xj )?? ,
(?, ?) ?
(6)
i?I
x?Xi
j?J
where we have abused the notation slightly and identified Xj = {xj } for bags with
one pattern. The rationale in using this relaxation is that the resulting convex
optimization problem is tractable and may provide a reasonably accurate approximation to the original disjunctive program, which can be further strengthened by
using it in combination with branch-and-bound search.
There is a lift-and-project representation of the convex hulls in Eq. (6), i.e. one
can characterize the feasible set as a projection of a higher dimensional polyhedron
which can be explicitly characterized [2].
Proposition 1. Assume a set of non-empty
linear constraints Hi ? {z : Ai z ?
S
i
b } 6= ? is given. Then z ? cl-conv i Hi if and only if there exist z j and ? j ? 0
such that
X
X
z=
zj ,
? j = 1, Aj z j ? ? j bj .
j
j
Proof. [2]
Let us pause here briefly and recapitulate what we have achieved so far. We have
derived a LP relaxation of the original disjunctive program for boosting with ambiguity. This relaxation was obtained by a linearization of the original non-convex
constraints. Furthermore, we have demonstrated how this relaxation can be improved using parallel reduction steps.
Applying this linearization to every convex hull in Eq. (6) individually, notice that
one needs to introduce duplicates ?x , ? x of the parameters ? and slack
P variables ?,
for every x ? Xi . In addition to the constraints ?kx , ?ix , ?jx , ?ix ? 0 and x?Xi ?ix = 1
the relevant constraint set for ambiguous bag Xi for i ? I of the resulting LP can
be written as
X
?x ? Xi : yi
?kx hk (x) + ?ix ? ?ix ,
(7a)
k
?x ? Xi , ?j ? J : yj
X
?kx hk (xj ) + ?jx ? ?ix ,
(7b)
X
(7c)
k
?k, ?j ? I ? J : ?k =
x?Xi
?kx ,
?j =
X
?jx .
x?Xi
The first margin constraint in Eq. (7a) is the one associated with the specific pattern
x, while the second set of margin constraints in Eq. (7b) stems from the parallel
reduction performed with unambiguous bags. One can calculate the dual LP of
the above relaxation, the derivation of which can be found in the appendix. The
resulting program has a more complicated bound structure on the u-variables and
the following crucial constraints involving the data
X
X
?i, ?x ? Xi : yi uxi hk (x) +
yj uxj hk (xj ) ? ?ik ,
?ik = 1 .
(8)
j?J
i?I
However, the size of the resulting problem is significant. As a result of linearization
and parallel reductions, the number of parameters in the primal LP is now O(q ? n +
q ?r), where q, r ? m denote the number of patterns in ambiguous and unambiguous
bags, compared to O(n + m) of the standard LPBoost. The number of constraints
(variables in the dual) has also been inflated significantly from O(m) to O(q?r+p?n)),
where p ? q is the number of ambiguous bags.
In order to maintain the spirit of LPBoost in dealing efficiently with a large-scale
linear program, we propose to maintain the column selection scheme of selecting
one or more ?kx in every round. Notice that the column
P selection can not proceed
independently because of the equality constraints x?Xi ?kx = ?k for all Xi ; in
particular, ?kx > 0 implies ?k > 0, so that ?kz > 0 for at least some z ? Xi for each
Xi , i ? I. We hence propose to simultaneously add all columns {?kx : x ? Xi , i ? I}
involving the same weak hypothesis and to prune those back after each boosting
round in order to exploit the expected sparseness of the solution. In order to select
a feature hk , we compute the following score
?
?
X
X
S(k) =
??ik ? 1, ??ik ? max ?yi uxi hk (x) +
yj uxj hk (xj )? .
(9)
x
i
j?J
Notice that due to the block structure of the tableau, working with a reduced set of
columns also eliminates a large number of inequalities (rows). However, the large
set of q ? r inequalities for the parallel reductions is still prohibitive.
In order to address this problem, we propose to perform incremental row selection
in an outer loop. Once we have converged to a column basis for the current relaxed
LP, we add a subset of rows corresponding to the most useful parallel reductions.
One can use the magnitude of the margin violation as a heuristic to perform this
row selection. Hence we propose to use the following score
X
T (x, j) = ?ix ? yj
?kx hk (xj ), where x ? Xi , i ? I, j ? J
(10)
k
This means that for current values of the duplicated ensemble weights ?kx , one
selects the parallel reduction margin constraint associated with ambiguous pattern
x and unambiguous pattern j that is violated most strongly.
Although the margin constraints imposed by unambiguous training instances
(xj , yj ) are redundant after we performed the parallel reduction step in Eq. (6),
we add them to the problem, because this will give us a better starting point with
respect to the row selection process, and may lead to a sparser solution. We hence
add the following constraints to the primal
X
yj
?k hk (xj ) + ?j ? 1, ?j ? J ,
(11)
k
which will introduce additional dual variables uj , j ? J. Notice that in the worst
case where all inequalities imposed by ambiguous training instances Xi are vacuous,
this will make sure that one recovers the standard LPBoost formulation on the
unambiguous examples. One can then think of the row generation process as a way
of deriving useful information from ambiguous examples. This information takes
the form of linear inequalities in the high dimensional representation of the convex
hull and will sequentially reduce the version space, i.e. the set of feasible (?, ?) pairs.
Algorithm 1 DPBoost Algorithm
1: initialize H = ?, C = {?i : i ? I ? J}, R = {ux
i : x ? Xi , i ? I} ? {uj : j ? J}
1
2: uj = |J|
, uxi = 0, ?i = 0
3: repeat
4:
repeat
5:
column selection: select hk 6? H with maximal S(k)
6:
H = H ? {hk }
7:
C = C ? {?k } ? {?kx : ?x ? Xi , ?i ? I}
8:
solve LP (C, R)
9:
until max S(k) <
10:
row selection: select a set S of pairs (x, j) 6? R with maximal T (x, j) > 0
11:
R = R ? {uxj : (x, j) ? S}, C = C ? {?jx : (x, j) ? S}
12:
solve LP (C, R)
13: until max T (x, j) <
90
80
70
60
50
90
80
70
60
50
1
3
5
7
90
80
70
60
50
1
3
5
7
1
3
5
7
Figure 1: (Left) Normalized intensity plot used to generate synthetic data sets.
(Right) Performance relative to the degree of label ambiguity. Mean and standard
deviation of the pattern-level classification accuracy plotted versus ?, for perfectknowledge (solid), perfect-selector (dotted), DPboost (dashed), and naive (dashdot) algorithms. The three plots correspond to data sets of size |I| = 10, 20, 30.
4
Experiments
We generated a set of synthetic weakly labeled data sets to evaluate DPboost on a
small scale. These were multiple-instance data sets, where the label uncertainty was
asymmetric; the only ambiguous bags (|Xi | > 1) were positive. More specifically, we
generated instances x ? [0, 1] ? [0, 1] sampled uniformly at random from the white
(yi = 1) and black (yi = ?1) regions of Figure 1, leaving the intermediate gray
area as a separating margin. The degree of ambiguity was controlled by generating
ambiguous bags of size k ? Poisson(?) having only one positive and k ? 1 negative
patterns. To control data set size, we generated a pre-specified number of ambiguous
bags, and the same number of singleton unambiguous bags.
As a proof of concept benchmark, we compared the classification perfomance of
DPboost with two other LPboost variants: perfect-knowledge, perfect-selector, and
naive algorithms. All variants use LPboost as their base algorithm and have slightly
different preprocessing steps to accomodate the MIL data sets. The first corresponds
to the supervised LPboost algorithm; i.e. the true pattern-level labels are used.
Since this algorithm does not have to deal with ambiguity, it will perform better
than DPboost. The second uses the true pattern-level labels to prune the negative
examples from ambiguous bags and solves the smaller supervised problem with
LPboost as above. This algorithm provides an interesting benchmark, since its
performance is the best we can hope for from DPboost. At the other extreme, the
third variant assumes the ambiguous pattern labels are equal to their respective
bag labels. For all algorithms, we used thresholded ?RBF-like? features.
Figure 2 shows the discriminant boundary (black line), learned by each of the four
algorithms for a data set generated with ? = 3 and having 20 ambiguous bags
(i.e. |I| = 20, no. ambig. = 71, no. total = 91). The ambiguous patterns are
marked by ?o?, unambiguous ones ?x?, and the background is shaded to indicate
the value of the ensemble F (x) (clamped to [?3, 3]). It is clear from the shading that
the ensemble has a small number of active features for DPboost, perfect-selector
and perfect-knowledge algorithms. For each classifier, we report the pattern-level
classification accuracy for a uniform grid (21 x 21) of points. The sparsity of the dual
variables was also verified; less than 20 percent of the dual variables and reductions
were active.
We ran 5-fold cross-validation on the synthetic data sets for ? = 1, 3, 5, 7 and for
data sets having |I| = 10, 20, 30. Figure 1 (right side) shows the mean pattern-level
classification accuracy with error bars showing one standard deviation, as a function
3
2
1
0
?1
?2
?3
Figure 2: Discriminant boundaries learned by naive (accuracy = 53.3 %), DPboost
(85.3 %), perfect-selector (86.6 %) and perfect-knowledge (92.7 %) algorithms.
of the parameter ?.
5
Conclusion
We have presented a new learning algorithm for classification problems where labels
are associated with sets of pattern instead of individual patterns. Using synthetic
data, the expected behaviour of the algorithm has been demonstrated. Our current
implementation could not handle large data sets, and so improvements, followed by
a large-scale validation and comparison to other algorithms using benchmark MIL
data sets, will follow.
Acknowledgments
David Musicant for making his CPLEX MEX interface available online. Also, to
Ioannis Tsochantaridis and Keith Hall, for useful discussion and advice. This work
was sponsored by an NSF-ITR grant, award number IIS-0085836.
References
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atsch, Sebastian Mika, Bernhard Sch?
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Appendix
The primal variables are ?k , ?kx , ?i , ?ix , ?jx , and ?ix . The dual variables are ux and
uxj for the margin constraints, and ?ik , ?i , and ?i for the equality constraints on ?k ,
? and ?, respectively.
The Lagrangian is given by
?
?
X
X
X
X X
L=
?k + C ?
?i +
?j ? ?
uxi
i
k
?
X X X
i
?
x?Xi
X
?ik
?k ?
x?Xi
i
?kx hk (xj )
?kx
X
yj
x?Xi
+
?jx
?
k
X
x?Xi
X X X
i
uxj
j
i,k
?
j
X
?
? kx ?kx
?
k
!
?
?i ?
i
X X
i
?i
x?Xi
yi
??ix ?ix ?
?ix
X
k
!
X
x?Xi
+
X
?i
?
X
x?Xi
1?
i
?ix
!
X X X
i
?kx hk (x) + ?ix ? ?ix
X
x?Xi
?ij
?j ?
i,j
??jx ?jx ?
j
i
x?Xi
i
?i ? uxi +
X
uxj ,
uxi ? C,
uxj ? ?ij ,
?ij ? C
i
j
yi uxi hk (x) +
X
X
j
yj uxj hk (xj ) ? ?ik ,
X
i
!
X
?jx
x?Xi
X X
Taking derivatives w.r.t. primal variables, leads to the following dual
X
max
?i
s.t.
?ix
!
?ik = 1
??ix ?ix .
!
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1,626 | 2,479 | Human and Ideal Observers for Detecting Image
Curves
Alan Yuille
Department of Statistics & Psychology
University of California Los Angeles
Los Angeles CA
[email protected]
Fang Fang
Psychology, University of Minnesota
Minneapolis MN 55455
[email protected]
Paul Schrater
Psychology, University of Minnesota
Minneapolis MN 55455
[email protected]
Daniel Kersten
Psychology, University of Minnesota
Minneapolis MN 55455
[email protected]
Abstract
This paper compares the ability of human observers to detect target image curves with that of an ideal observer. The target curves are sampled from a generative model which specifies (probabilistically) the geometry and local intensity properties of the curve. The ideal observer
performs Bayesian inference on the generative model using MAP estimation. Varying the probability model for the curve geometry enables us
investigate whether human performance is best for target curves that obey
specific shape statistics, in particular those observed on natural shapes.
Experiments are performed with data on both rectangular and hexagonal
lattices. Our results show that human observers? performance approaches
that of the ideal observer and are, in general, closest to the ideal for conditions where the target curve tends to be straight or similar to natural
statistics on curves. This suggests a bias of human observers towards
straight curves and natural statistics.
1 Introduction
Detecting curves in images is a fundamental visual task which requires combining local
intensity cues with prior knowledge about the probable shape of the curve. Curves with
strong intensity edges are easy to detect, but those with weak intensity edges can only be
found if we have strong prior knowledge of the shape, see figure (1) But, to the best of
our knowledge, there have been no experimental studies which test the ability of human
observers to perform curve detection for semi-realistic stimuli with locally ambiguous intensity cues or to explore how the difficulty of the task varies with the geometry of the
curve.
This paper formulates curve detection as Bayesian inference. Following Geman and Jedynak [6] we define probability distributions PG (.) for the shape geometry of the target
curve and Pon (.), Poff (.) for the intensity on and off the curve. Sampling this model gives
us semi-realistic images defined on either rectangular or hexagonal grids. The human ob-
Almost impossible
Intermediate
Easy
Figure 1: It is plausible that the human visual system is adapted to the shape statistics of curves
and paths in images like these. Left panel illustrates the trade-off between the reliability of intensity
measurements and priors on curve geometry. The tent is easy to detect because of the large intensity
difference between it and the background, so little prior knowledge about its shape is required. But
detecting the goat (above the tent) is harder and seems to require prior knowledge about its shape.
Centre panel illustrates the experimental task of tracing a curve (or road) in clutter. Right panel shows
that the first order shape statistics from 49 object images (one datapoint per image) are clustered
round P (straight) = 0.64 (with P (lef t) = 0.18 and P (right) = 0.18) for both rectangular and
hexagonal lattices, see [1].
server?s task is to detect the target curve and to report it by tracking it with the (computer)
mouse. Human performance is compared with that of an ideal observer which computes
the target curve using Bayesian inference (implemented by a dynamic programming algorithm). The ideal observer gives a benchmark against which human performance can be
measured.
By varying the probability distributions PG , Pon .Poff we can explore the ability of the
human visual system to detect curves under a variety of conditions. For example, we can
vary PG and determine what changes in Pon .Poff are required to maintain a pre-specified
level of detection performance.
In particular, we can investigate how human performance depends on the geometrical distribution PG of the curves. It is plausible that the human visual system has adapted to the
statistics of the natural world, see figure (1), and in particular to the geometry of salient
curves. Our measurements of natural image curves, see figure (1), and studies by [16],
[10], [5] and [2], show distributions for shape statistics similar to those found for image
intensities statistics [11, 9, 13]. We therefore investigate whether human performance approaches that of the ideal when the probability distributions PG is similar to that for curves
in natural images.
This investigation requires specifying performance measures to determine how close human performance is to the ideal (so that we can quantify whether humans do better or
worse relative to the ideal for different shape distributions PG ). We use two measures of
performance. The first is an effective order parameter motivated by the order parameter
theory for curve detection [14], [15] which shows that the detectability of target curves, by
an ideal observer, depends only on an order parameter K which is a function of the probability distributions characterizing the problem. The second measure computes the value
of the posterior distribution for the curves detected by the human and the ideal and takes
the logarithm of their ratio. (For theoretical reasons this is expected to give a performance
measure similar to the effective order parameter).
The experiments are performed by human observers who are required to trace the target
curve in the image. We simulated the images first on a rectangle grid and then on a hexagonal grid to test the generality of the results. In these experiments we varied the probability
distributions of the geometry PG and the distribution Pon of the intensity on the target curve
to allow us to explore a range of different conditions (we kept the distribution P off fixed).
In section (2) we briefly review previous psychophysical studies on edge detection. Sec-
0.25
0.2
P
0.15
P on
Probailty
0.1
0.05
0
0
4
8
12
16
Intesiy
Figure 2: Left panel: the tree structure superimposed on the lattice. Centre panel: a pyramid structure used in the simulations on the rectangular grid. Right panel: Typical distributions of Pon , Poff
tion (3) describes our probabilistic model and specifies the ideal observer. In section (4), we
describe the order parameter theory and define two performance measures. Sections (5,6)
describe experimental results on rectangular and hexagonal grids respectively in terms of
our two performance measures.
2 Previous Work
Previous psychophysical studies have shown conditions for which the human visual system is able to effectively group contour fragments when embedded in an array of distracting fragments [3, 8]. Most of these studies have focused on the geometrical aspects of
the grouping process. For example, it is known that the degree to which a target contour
?pops out? depends on the degree of similarity of the orientation of neighboring fragments
(typically gabor patches) [3], and that global closure facilitates grouping [8].
Recently, several researchers have shown that psychophysical performance for contour
grouping may be understood in terms of the statistical properties of natural contours [12, 5].
For example, Geisler [5] has shown that human contour detection for line segments can be
quantitatively predicted from a local grouping rule derived from measurements of local
edge statistics.
However, apart from studies that manipulate the contrast of gabor patch tokens [4], there
has been little work on how intensity and contour geometry information is combined by the
visual system under conditions that begin to approximate those of natural contours. In this
paper we attempt to fill this gap by using stimuli sampled from a generative model which
enables us to quantitatively characterize the shape and intensity information available for
detecting curves and compare human performance with that of an ideal detector.
3 The Probabilistic Model for Data Generation
We now describe our model in detail. Following [6], we formulate target curve detection as
tree search, see figure (2), through a Q-nary tree. The starting point and initial direction is
specified and there are QN possible distinct paths down the tree. A target curve hypothesis
consists of a set of connected straight-line segments called segments. We can represent a
path by a sequence of moves {ti } on the tree. Each move ti belongs to an alphabet {a? } of
size Q. For example, the simplest case sets Q = 3 with an alphabet a 1 , a2 , a3 corresponding to the decisions: (i) a1 ? go straight (0 degrees), (ii) a2 ? go left (-5 degrees), or (iii)
a3 ? go right (+ 5 degrees). This determines a path x 1 , . . . , xN in the image lattice where
xi , xi+1 indicate the start and end points of the ith segment. The relationship between the
two representations is given by xi+1 = xi + w(xi ? xi?1 , ti ), where w(xi ? xi?1 , ti ) is
a vector of approximately fixed magnitude (choosen to ensure that the segment ends on a
pixel) and whose direction depends on the angle of the move t i relative to the direction of
the previous segment xi ? xi?1 . In this paper we restrict Q = 3.
We put a prior probability on the geometry of paths down the tree. This is of form
QN
P ({ti }) = i=1 P (ti ). We will always require that the probabilities to go left or right
are equal and hence we can specify the distribution by the probability P (straight) that the
curve goes straight. Our analysis of image curve statistics suggests that P (straight) =
0.64 for natural images, see figure (1).
We specify the probability models Pon , Pof f for the image intensity on and off to be
of Poisson form defined over the range (1, ..., 16), see figure (2). This reduced range
means that the distributions are expressed as Pon (I = n) = (1/Kon )e??on ?non /n! and
Pof f (I = n) = (1/Kof f )e??of f ?nof f /n!, where Kon , Kof f are normalization factors. We
fix ?of f = 8.0 and will vary ?on . The quantity ?on ? ?of f is a measure of the local intensity contrast of the target contour and so we informally refer to it as the signal-to-noise
ratio (SNR).
The Ideal Observer estimates the target curve trajectory by MAP estimation (which we
compute using dynamic programming). As described in [6], MAP estimation corresponds
to finding the path {ti } with filter measurements {yi } which maximizes the (scaled) loglikelihood ratio, or reward function,
r({ti }, {yi }) =
N
X
1
{log P (Y |X) + log P (X) ?
log U (ti )}
N
i=1
N
N
1 X
1 X
log{Pon (yi )/Pof f (yi )} +
log{PG (ti )/U (ti )},
(1)
N i=1
N i=1
PN
where U (.) is the uniform distribution (i.e. U (t) = 1/3 ?t) and so i=1 log U (ti ) =
?N log 3 which is a constant. The length of the curve is N = 32 in our experiments.
=
We implement this model on both rectangular and hexagonal lattices (the hexagonal latttices equate for contrast at borders, and are visually more realistic). The tree representation
used by Geman and Jedynak must be modified when we map onto these lattices. For a
rectangular lattice, the easiest way to do this involves defining a pyramid where paths start
at the apex and the only allowable ?moves? are: (i) one step down, (ii) one step down
and one step left, and (iii) one step down and one step right. This can be represented by
xi+1 = xi +w(ti ) where ti ? {?1, 0, 1} and w(?1) = ?~i?~j, w(0) = ?~j, w(1) = +~i?~j
(where ~i, ~j are the x, y directions on the lattice).
A similar procedure is used on the hexagonal lattice. But for certain geometry probabilities we observed that the sampled curves had ?clumping? where the path consists of a
large number of zig-zags. This was sometimes confusing to the human observers. So we
implemented a higher-order Markov model which explicitly forbade zig-zags. We show
experimental results for both the Clumping and No-Cluming models.
To obtain computer simulations of target curves in background clutter we proceed in two
stages. In the first stage, we stochastically sample from the distribution P G (t) to produce
a target curve in the pyramid (starting at the apex and moving downwards). In the second
stage, we must sample from the likelihood function to generate the image. So if a pixel
x is on or off the target curve (which we generated in the first stage) then we sample the
intensity I(x) from the distribution Pon (I) or Pof f (I) respectively.
4 Order Parameters and Performance Measures
Yuille et al [14],[15] analyzed the Geman and Jedynak model [6] to determine how the
ability to detect the target curve depended on the geometry Pg and the intensity properties
Pon .Poff . The analysis showed that the ability to detect the target curve behaves as e ?KN ,
where N is the length of the curve and K is an order parameter. The larger the value of K
then the easier it is to detect the curve.
The order parameter is given by K = D(Pon ||Pof f ) + D(PG ||U ) ? log Q [15], where U is
the uniform distribution. If K > 0 then detecting the target curve is possible but if K < 0
then it becomes impossible to find it (informally, it becomes like looking for a needle in a
haystack).
The order parameter illustrates the trade-off between shape and intensity cues and determines which types of curves are easiest to detect by an ideal observer. The intensity cues
are quantified by D(Pon ||Pof f ) and the shape cues by D(PG ||U ). The easiest curves to
detect are those which are straight lines (i.e. D(PG ||U ) takes its largest possible value).
The hardest curves to detect are those for which the geometry is most random. The stronger
the intensity cues (i.e. the bigger D(Pon ||Pof f )) then, of course, the easier the detection
becomes.
So when comparing human performance to ideal observers we have to take into account
that some types of curves are inherent easier to detect (i.e. thay have larger K). Human
observers are good at detecting straight line curves but so are ideal obervers. We need
performance measures to quantify the relative effectiveness of human and ideal observers.
Otherwise, we will not be able to conclude that human observers are biased towards particular curve shapes (such as those occuring in natural images).
We now define two performance measures to quantify the relative effectivenes of human
and ideal observers. Our first measure is based on the hypothesis that human observers
have an ?effective order parameter?. In other words, their performance on the target curve
tracking task behaves like e?N KH where KH is an effective order parameter which difference from the true order parameter K might reflect a human bias towards straight lines or
ecological shape priors. We estimate the effective order parameters by fixing P G , Poff and
adjusting Pon until the observers achieve a fixed performance level of at most 5 errors on
I
H
a path of length 32. This gives distributions Pon
, Pon
for the ideal and human observers
H
I
H
I
respectively. Then we set KH = K ? D(Pon ||Pof f ) + D(Pon
||Pof f ), where Pon
, Pon
are
the distributions used by the human and the ideal (respectively) to achieve similar performance.
H
I
Our first performance measure is the difference ?K = D(Pon
||Pof f ) ? D(Pon
||Pof f )
between the effective and the true order parameters.
But order parameter analysis should be regarded with caution for the curve detection task
used in our experiments. The experimental criterion that the target path be found with 5
or less errors, see section (5), was not included in the theoretical analysis [14],[15]. Also
some small corrections need to be made to the order parameters due to the nature of the
rectangular grid, see [15] for computer calculations of the size of these corrections. These
two effects ? the error criterion and the grid correction ? means that the order parameters
are only approximate for these experimental conditions.
This motivates a second performance measure where we calculate the value of the posterior
probability (proportional to the exponential of r in equation (1)) for the curve detected by
the human and the ideal observer (for identical distributions PG , Pon , Poff ). We measure
the logarithm of the ratio of these values. (A theoretical relationship can be shown between
these two measures).
5 Experimental Results on Rectangular Grid
To assess human performance on the road tracking task, we first had a set of 7 observers find
the target curve in a tree defined by a rectangular grid figure (3)A. The observer tracked the
Stimulus
A
Rectangular grid
Ideal
B
Stimulus
Human
Ideal
Human
C
Figure 3: A. Rectangular Grid Stimulus (Left), Example Path: Ideal (Center), Example Path: Human
(Right). B & C. Hexagonal Grid Stimulus (Left), Example Path: Ideal (Center), Example Path: Human (Right). Panel C shows an example of a path with higher order constraints to prevent ?clumping?.
There were a number of other differences between the rectangular and hexagonal grid psychophysics,
including rectangle samples were slightly smaller than the hexgaons, and feedback was presented to
the observers without (rectangular) or with background (hexagonal), and the lowest p(straight) was
0.0 for rectangular and0.1 for hexagonal grids.
contour by starting at the far left corner and making a series of 32 key presses that moved
the observer?s tracked contour either left, right, or straight at each key press. Each contour
estimate was scored by counting the number of positions the observer?s contour was off the
true path. Each observer had a training period in which the observer was shown examples
of contours produced from the four different geometry distributions and practiced tracing
in noise.
During an experimental session, the geometry distribution was fixed at one the four possible values and observers were told which geometry distribution was being used to generate
the contours. The parameter ?on of Pon was varied using an adaptive procedure until the
human observer managed to repeatedly detect the target curve with at most five misclassified pixels. This gave a threshold of ?on ? ?of f for each probability distribution defined
by P (straight). This threshold could be compared to that of the Ideal Observer (obtained
by using dynamic programming to estimate the ideal, also allowing for up to five errors).
The process was repeated several times for the four geometry distribution conditions.
The thresholds for 7 observers and the ideal observer are shown in figure 4. These thresholds can be used to calculate our first performance measure (?K) and determine how
effectively observers are using the available image information at each P (straight).
The results are illustrated in figure (4)B where the human data was averaged over seven
subjects. They show that humans perform best for curves with P (straight) = 0.66 which
is closest to the natural priors, see figure (1). Conversely, ?K is biggest for the curves with
P (straight) = 0.0, which is the condition that differs most from the natural statistics.
We next compute our second performance measure (for which P on , Pof f , PG are the same
for the ideal and the human observer). The average difference of this performance measure
for the each geometry distribution is an alternative way how well observers are using the intensity information as a function of geometry, with a zero difference indicating optimal use
of the information. The results are shown in figure (4)C. Notice that the best performance
is achieved with P (straight) = 0.9.
Observe that the two performance measures give different answers for this experiment.
We conclude that our results are consistent either with a bias to ecological statistics or to
straight lines. But the rectangular lattice
6 Experiments on Hexagonal Lattices
In these experiments we used a hexagonal lattice because, for the human observers, the
contrast at the edges corresponding to a left, straight, or right move is the same (in contrast
to the rectangular grid, in which left and right moves only share a corner). We also use the
same values of Pon , Poff , P (straight) for the humans and the ideal.
Figure 4: A-C. Psychophysical results on rectangular grid. A. Threshold ?on ? ?of f plotted against
P (straight). The top seven curves are the results of the seven subjects. The bottom curve is for
the ideal observer. B. The difference between human and ideal K order parameters. C. The average
reward difference between ideal and human observers. D-I shows psychophyscial results on a hexagonal grid. D-F are for the Clumping condition, and G-I for the No Clumping condition for which
high order statistics prevented sharp turns that result in ?clumps?.
We performed experiments on the hexagonal lattice under four different probabilities for
the geometry. These were specified by P (straight) = 0.10, 0.33, 0.66, 0.90 (in other
words, the straightest curves will be sampled when P (straight) = 0.90 and the least
straight from P (straight) = 0.10). For reasons described previously, we did the experiment in two conditions. (1) allowing zig-zags ?Clumping?, (2) forbidding zig-zags ?NoClumping?. We show examples of the stimuli, the ideal results (indicated by dotted path),
and the human results (indicated by dotted path) for the Clumping amd No-Clumping cases
in figure (4B & C), respectively.
The threshold SNR results for Clumping and No Clumping are summarized in figures (4D
& G. The average ?K = Khuman ? Kideal results for Clumping and No Clumping are
summarized in figure (4E & H). The average reward difference, ?r = r ideal ? rhuman ,
results for Clumping and No Clumping are summarized in figure (4F & I).
Both performance measures give consistent results for the Clumping data suggesting that
humans are best when detecting the straightest lines (P (straight) = 0.9). But the situation
is more complicated for the No Clumping case where human observers show preferences
for P (straight) = 0.9 or P (straight) = 0.66.
7 Summary and Conclusions
The results of our experiments suggest that humans are most effective at detecting curves
which are straight or which obey ecological statistics. But further experiments are needed
to clarify this. Our two performance measures were not always consistent, particularly for
the rectangular grid (we are analyzing this discrepency theoretically). The first measure
suggested a bias towards ecological statistics on the rectangular grid and for No Clumping
stimuli on the hexagonal grid. The second measure showed a bias towards curves with
P (straight) = 0.9 on the rectangular and hexagonal grids.
To our knowledge, this is the first experiment which tests the performance of human observers for detecting target curves by comparison with that of an ideal observer with ambiguous intensity data. Our novel experimental design and stimuli may cause artifacts due
to the rectangular and hexagonal grids. Further experiments may need to ?quantize? curves
more carefully and reduce the effect of the grids.
Further experiments performed on a larger number of subjects may be able to isolated
more precisely the strategy that human observers employ. Do they, for example, make use
of a specific geometry prior based on empirical edge statistics [16], [10]. If so, this might
account for the bias towards straigthness and natural priors observed in the experiments
reported here.
Acknowledgments
Supported by NIH RO1 EY11507-001, EY02587, EY12691 and, EY013875-01A1, NSF
SBR-9631682, 0240148.
References
[1] Brady, M. J. (1999). Psychophysical investigations of incomplete forms and forms with background. Ph. D., University of Minnesota.
[2] Elder J.H. and Goldberg R.M.. Ecological Statistics of Gestalt Laws for the Perceptual Organization of Contours. Journal of Vision, 2, 324-353. 2002.
[3] Field, D. J., Hayes, A., & Hess, R. F. Contour integration by the human visual system: evidence
for a local ?association field?. Vision Res, 33, (2), 173-93. 1993.
[4] Field, D. J., Hayes, A., & Hess, R. F. The roles of polarity and symmetry in the perceptual
grouping of contour fragments. Spat Vis, 13, (1), 51-66.2000.
[5] Geisler W.S. , Perry J.S. , Super B.J. and Gallogly D.P. . Edge co-occurrence in natural images
predicts contour grouping performance. Vision Res, 41, (6), 711-24. 2001.
[6] Geman D. and Jedynak B. . ?An active testing model for tracking roads from satellite images?.
IEEE Trans. Pattern Anal. Mach. Intell., 18, 1-14, 1996.
[7] Hess, R., & Field, D. . Integration of contours: new insights. Trends Cogn Sci, 3, (12), 480486.1999.
[8] Kovacs, I., & Julesz, B. A closed curve is much more than an incomplete one: effect of closure
in figure-ground segmentation. Proc Natl Acad Sci U S A, 90, (16), 7495-7. 1993.
[9] Lee A.B., Huang J.G., and Mumford D.B., ?Random collage model for natural images?, Int?l J.
of Computer Vision, Oct. 2000.
[10] Ren X. and Malik J. . ?A Probabilistic Multi-scale Model for Contour Completion Based on
Image Statistics?. In Proceedings ECCV. 2002
[11] Ruderman D.L. and Bialek W. , ?Statistics of natural images: scaling in the woods?, Phy. Rev.
Letter, 73:814-817, 1994.
[12] Sigman, M., Cecchi, G. A., Gilbert, C. D., & Magnasco, M. O. . On a common circle: natural
scenes and Gestalt rules. Proc Natl Acad Sci U S A, 98, (4), 1935-40. 2001.
[13] Wainwright M.J. and Simoncelli E.P., ?Scale mixtures of Gaussian and the statistics of natural
images?, NIPS, 855-861, 2000.
[14] Yuille A.L. and Coughlan J.M. . ?Fundamental Limits of Bayesian Inference: Order Parameters
and Phase Transitions for Road Tracking? . IEEE PAMI. Vol. 22. No. 2. February. 2000.
[15] Yuille A.L. , Coughlan J.M., Wu Y-N. and Zhu S.C. . ?Order Parameters for Minimax Entropy
Distributions: When does high level knowledge help?? IJCV. 41(1/2), pp 9-33. 2001.
[16] Zhu S.C. . ?Embedding Gestalt Laws in Markov Random Fields ? A theory for shape modeling
and perceptual organization?. IEEE PAMI, Vol. 21, No.11, pp1170-1187, Nov, 1999.
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1,627 | 248 | 226
Mann
The Effects of Circuit Integration on a Feature
Map Vector Quantizer
Jim lVIann
MIT Lincoln Laboratory
244 Wood St.
Lexington, ~IA 02173
email: [email protected]
ABSTRACT
The effects of parameter modifications imposed by hardware constraints on a self-organizing feature map algorithm were examined.
Performance was measured by the error rate of a speech recognition system which included this algorithm as part of the front-end
processing. System parameters which were varied included weight
(connection strength) quantization, adap tation quantization, distance measures and circuit approximations which include device
characteristics and process variability. Experiments using the TI
isolated word database for 16 speakers demonstrated degradation in
performance when weight quantization fell below 8 bits. The competitive nature of the algorithm rela..xes constraints on uniformity
and linearity which makes it an excellent candidate for a fully analog circuit implementation. Prototype circuits have been fabricated
and characterized following the constraints established through the
simulation efforts.
1
Introduction
The self-organizing feature map algorithm developed by Kohonen [Kohonen, 1988]
readily lends itself to the task of vector quantization for use in such areas as speech
recognition. However, in considering practical imp lementations, it is necessary to
The Effects of Circuit Integration on a Feature Map Vector Quantizer
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Figure 1: Recognition performance of the Euclidean and dot product activity
calculators plotted as a function of weight precision .
understand the limitations imposed by circuitry on algorithm performance. In order
to test the effects of these constraints on overall performance a simulation was
written which permits ready variation of critical system parameters.
The feature map algorithm was placed in the frontend of a discrete hidden Ylarkov
model (H111'I) speech recognition program as the vector quantizer (VQ) in order to
track the effects of feature map algorithm modifications by monitoring overall word
recognition accuracy. The system was tested on TI's 20 isolated word database
consisting of 16 speakers . Each speaker had 1 training session consisting of 10 repetitions of each word in the vocabulary and 8 test sessions consisting of 2 repetitions
of each word.
The key parameters tested include; quantization of both the weight coefficients and
learning rule, and several different activation computations, the dot product and
the mean squared error (i.e. squared Euclidean distance), as well as the circuit
approximations to these calculators.
2
Results
A unique dependency between weight quantization and distance measure emerged
from the simulations and is illustrated in the graph presented in Figure 1. The
network equipped with the mean squared error activity calculator shows a "knee"
in the word error rate at 6 bits of precision in the weight representation. The overall
performance dropped only slightly between the essentially ideal floating point case,
at 1.45% error rate, and the 6 bit case, at 2.99% error rate. At 4 bits, the error rate
climbs to 7.62%. This still corresponds to a recognition accuracy of better than
92% but does show a marked degradation in performance.
227
228
Mann
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-./-V
?
...L
I--
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Xo
Figure 2: .-\ circuit approximation to the dot product calculator.
The dot product does not degrade as gracefully with reduced precision in the weight
representation as the mean squared error activity calculation. This is due to the
normalization required on the input, and subsequently the weight vectors, which
compresses the space onto the unit hypersphere. This step is necessary because of
the inherent sensitivity of this metric to vector magnitude in making decisions of
relative distance. Here the" knee" in the error curve occurs at 8 bits. Below 8 bits,
performance drops off dramatically, reaching 40.6% error rate at 6 bits. The double
precision floating point case starts off at 1.68% and is 3.44% at 8 bits .
Circuit approximations to these activity calculators were also included in the simulations. An approximation to the dot product operation can be implemented with
single transistors operating in the ohmic region at each connection as illustrated in
Figure 2.
These area. related considerations can often overshadow the performance penalties
associated with their implementation. The simulation results from this circuit approximation match the performance of the digital calculation of the dot product
almost exactly as seen in Figure 3. This indicates that the performance of the
system depends more on the monotonicity of the product operation performed at
each connection then its linearity.
Effects of process variations on transistor thresholds were also examined. There
appears to be a gradual decrease in system performance with increasing variability
in transistor thresholds as seen in Figure 4. The cause of this phenomena remains
to be investigated .
A weight adjustment rule which simplifies circuitry consists of quantizing the learning rate gain term. An integer step is added to or subtracted from the weight
depending on the magnitude of the difference between it and the input. In the
The Effects of Circuit Integration on a Feature Map Vector Quantizer
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CIRCUIT
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Figure 3: Similarity between the transistor circuit simulation and the digital calculation of the dot product
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90 100
STD. DEV. (mV)
Figure 4: The effects of transistor threshold variation on recognition performance.
(8 bit weight; Gaussian distributed, mean(Vth) = 0.75 volts).
229
230
Mann
2.0
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MAX. WEIGHT ADJUST (+I-)
DBl
PRECISION
Figure 5: 'Nord recognition error rate as a function of learning rate gain quantization.
simplest case . a fixed increment or decrement operation is performed based only
upon the sign of the difference between the two terms. Even in this simplest case
no degradation in performance was noted while using an 8 bit weight representation as demonstrated in the graph shown in Figure 5. In fact, performance was
often improved over the original learning rule. The error rates using an increment/decrement learning rule with 8 weight bits was O.9i% and 2.0% for the mean
squared error and the dot product, respectively.
An additional learning rule is being tested, targeted at a floating gate implementation which uses a "flash" EPROM memory structure at each synapse. Weight
changes are restricted to positive adjustments locally while all negative adjustments
are made globally to all weights. This corresponds to a forgetting term, or constant
weight decay, in the learning rule. This rule was chosen to be compatible with one
technique in non-volatile charge storage which allows selective write but only block
erase.
3
Hardware
A prototype synaptic array and weight adaptation circuit have been designed and
fabricated [Mann, 1989]. A single transistor synapse computes its contribution to
the dot product activity calculation. The weight is stored dynamically as charge on
the gate of the synapse transistor. The input is represented as a voltage on the drain
of the transistor. The current through the transistor is proportional to the product
of the gate voltage (i.e. the weight) and the drain voltage (i.e. the input strength)
with the source connected to a virtual ground (see Figure 2). The sources of several
of these synapse connected together form the accumulation needed to realize the dot
product. Circuitry for accessing stored weight information has also been included.
The Effects of Circuit Integration on a Feature Map Vector Quantizer
The synapse array works as expected except for circuitry used to read the weight
contents. This circuit requires very high on-chip voltages causing other circuits to
latch-up when the clocks are turned on .
The weight adaptation circuit performs the simple increment/decrement operation
based on the comparison between the input and weight magnitudes. Both quantities are first converted to a digital representation by a flash A/D converter before
comparison. This circuit also performs the required refresh operation on weight.
contents, much like that required for dynamic RAM's but requiring analog charge
storage. This insures that weight drift is constrained to lie within boundaries defined
by the precision of the weight representation determined by the A/D con version process. This circuit was functional in the refresh and increment modes, but would not
decrement correctly.
Further tests are being conducted to establish the causes of the circuit problems
detected thus far. Additional work is proceeding on a non-volatile charge storage
version of this device. Some test structures have been fabricated and are currently
being characterized for compatibility with this task.
This work was supported by the Department of the Air Force.
References
T. Kohonen. (1988) Self-Organization and Associative Memory, Berlin: SpringerVerlag.
J. Mann & S. Gilbert. (1989) An Analog Self-Organizing Neural Network Chip .
In D. S. Touretzky (ed.), Advances in Neural Information Processing Systems 1,
739-747. San Mateo, CA: Morgan Kaufmann.
231
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1,628 | 2,480 | From Algorithmic to Subjective Randomness
Thomas L. Griffiths & Joshua B. Tenenbaum
{gruffydd,jbt}@mit.edu
Massachusetts Institute of Technology
Cambridge, MA 02139
Abstract
We explore the phenomena of subjective randomness as a case study in
understanding how people discover structure embedded in noise. We
present a rational account of randomness perception based on the statistical problem of model selection: given a stimulus, inferring whether the
process that generated it was random or regular. Inspired by the mathematical definition of randomness given by Kolmogorov complexity, we
characterize regularity in terms of a hierarchy of automata that augment
a finite controller with different forms of memory. We find that the regularities detected in binary sequences depend upon presentation format,
and that the kinds of automata that can identify these regularities are informative about the cognitive processes engaged by different formats.
1
Introduction
People are extremely good at finding structure embedded in noise. This sensitivity to patterns and regularities is at the heart of many of the inductive leaps characteristic of human
cognition, such as identifying the words in a stream of sounds, or discovering the presence
of a common cause underlying a set of events. These acts of everyday induction are quite
different from the kind of inferences normally considered in machine learning and statistics: human cognition usually involves reaching strong conclusions on the basis of limited
data, while many statistical analyses focus on the asymptotics of large samples.
The ability to detect structure embedded in noise has a paradoxical character: while it is
an excellent example of the kind of inference at which people excel but machines fail, it
also seems to be the source of errors in tasks at which machines regularly succeed. For
example, a common demonstration conducted in introductory psychology classes involves
presenting students with two binary sequences of the same length, such as HHTHTHTT and
HHHHHHHH, and asking them to judge which one seems more random. When students
select the former, they are told that their judgments are irrational: the two sequences are
equally random, since they have the same probability of being produced by a fair coin. In
the real world, the sense that some random sequences seem more structured than others
can lead people to a variety of erroneous inferences, whether in a casino or thinking about
patterns of births and deaths in a hospital [1].
Here we show how this paradox can be resolved through a proper understanding of what our
sense of randomness is designed to compute. We will argue that our sense of randomness is
actually extremely well-calibrated with a rational statistical computation ? just not the one
to which it is usually compared. While previous accounts criticize people?s randomness
judgments as poor estimates of the probability of an outcome, we claim that subjective
randomness, together with other everyday inductive leaps, can be understood in terms of the
statistical problem of model selection: given a set of data, evaluating hypotheses about the
process that generated it. Solving this model selection problem for small datasets requires
two ingredients: a set of hypotheses about the processes by which the data could have been
generated, and a rational statistical inference by which these hypotheses are evaluated.
We will model subjective randomness as an inference comparing the probability of a sequence under a random process, P (X|random), with the probability of that sequence
under a regular process, P (X|regular). In previous work we have shown that defining
P (X|regular) using a restricted form of Kolmogorov complexity, in which regularity is
characterized in terms of a simple computing machine, can provide a good account of human randomness judgments for binary sequences [2]. Here, we explore the consequences
of manipulating the conditions under which these sequences are presented. We will show
that the kinds of regularity to which people are sensitive depend upon whether the full sequence is presented simultaneously, or its elements are presented sequentially. By exploring how these regularities can be captured by different kinds of automata, we extend our
rational analysis of the inference involved in subjective randomness to a rational characterization of the processes underlying it: certain regularities can only be detected by automata
with a particular form of memory access, and identifying the conditions under which regularities are detectable provides insight into how characteristics of human memory interact
with rational statistical inference.
2
Kolmogorov complexity and randomness
A natural starting point for a formal account of subjective randomness is Kolmogorov complexity, which provides a mathematical definition of the randomness of a sequence in terms
of the length of the shortest computer program that would produce that sequence. The idea
of using a code based upon the length of computer programs was independently proposed
in [3], [4] and [5], although it has come to be associated with Kolmogorov. A sequence
X has Kolmogorov complexity K(X) equal to the length of the shortest program p for a
(prefix) universal Turing machine U that produces X and then halts,
K(X) =
min
p:U (p)=X
`(p),
(1)
where `(p) is the length of p in bits. Kolmogorov complexity identifies a sequence X
as random if `(X) ? K(X) is small: random sequences are those that are irreducibly
complex [4]. While not necessarily following the form of this definition, psychologists
have preserved its spirit in proposing that the perceived randomness of a sequence increases
with its complexity (eg. [6]). Kolmogorov complexity can also be used to define a variety
of probability distributions, assigning probability to events based upon their complexity.
One such distribution is algorithmic probability, in which the probability of X is
R(X) = 2?K(X) =
max
p:U (p)=X
2?`(p) .
(2)
There is no requirement that R(X) sum to one over all sequences; many probability distributions that correspond to codes are unnormalized, assigning the missing probability to an
undefined sequence.
There are three problems with using Kolmogorov complexity as the basis for a computational model of subjective randomness. Firstly, the Kolmogorov complexity of any particular sequence X is not computable [4], presenting a practical challenge for any modelling
effort. Secondly, while the universality of an encoding scheme based on Turing machines
is attractive, many of the interesting questions in cognition come from the details: issues of
representation and processing are lost in the asymptotic equivalence of coding schemes, but
play a key role in people?s judgments. Finally, Kolmogorov complexity is too permissive in
what it considers a regularity. The set of regularities identified by people are a strict subset
of those that might be expressed in short computer programs. For example, people are very
unlikely to be able to tell the difference between a binary sequence produced by a linear
congruential random number generator (a very short program) and a sequence produced by
flipping a coin, but these sequences should differ significantly in Kolmogorov complexity.
Restricting the set of regularities does not imply that people are worse than machines at
recognizing patterns: reducing the size of the set of hypotheses increases inductive bias,
making it possible to identify the presence of structure from smaller samples.
3
A statistical account of subjective randomness
While there are problems with using Kolmogorov complexity as the basis for a rational
theory of subjective randomness, it provides a clear definition of regularity. In this section
we will present a statistical account of subjective randomness in terms of a comparison between random and regular sources, where regularity is defined by analogues of Kolmogorov
complexity for simpler computing machines.
3.1
Subjective randomness as model selection
One of the most basic problems that arises in statistical inference is identifying the source
of a set of observations, based upon a set of hypotheses. This is the problem of model
selection. Model selection provides a natural basis for a statistical theory of subjective
randomness, viewing these judgments as the consequence of an inference to the process
that produced a set of observations. On seeing a stimulus X, we consider two hypotheses:
X was produced by a random process, or X was produced by a regular process. The
decision about the source of X can be formalized as a Bayesian inference,
P (random|X)
P (X|random) P (random)
=
,
P (regular|X)
P (X|regular) P (regular)
(3)
in which the posterior odds in favor of a random generating process are obtained from the
likelihood ratio and the prior odds. The only part of the right hand side of the equation
affected by X is the likelihood ratio, so we define the subjective randomness of X as
random(X) = log
P (X|random)
,
P (X|regular)
(4)
being the evidence that X provides towards the conclusion that it was produced by a random process.
3.2
The nature of regularity
In order to define random(X), we need to specify P (X|random) and P (X|regular). When
evaluating binary sequences, it is natural to set P (X|random) = ( 21 )`(X) . Taking the
logarithm in base 2, random(X) is ?`(X) ? log 2 P (X|regular), depending entirely on
P (X|regular). We obtain random(X) = K(X) ? `(X), the difference between the complexity of a sequence and its length, if we choose P (X|regular) = R(X), the algorithmic probability defined in Equation 2. This is identical to the mathematical definition of
randomness given by Kolmogorov complexity. However, the key point of this statistical
approach is that we are not restricted to using R(X): we have a measure of the randomness
of X for any choice of P (X|regular).
The choice of P (X|regular) will reflect the stimulus domain, and express the kinds of
regularity which people can detect in that domain. For binary sequences, a good candidate for specifying P (X|regular) is a hidden Markov model (HMM), a probabilistic finite
1
H
2
T
5
4
H
T
T
H
6
3
Figure 1: Finite state automaton used to define P (X|regular) to give random(X) ? DP .
Solid arrows are transitions consistent with repeating a motif, which are taken with probability ?. Dashed arrows are motif changes, using the prior determined by ?.
state automaton. In fact, specifying P (X|regular)in terms of a particular HMM results in
random(X) being equivalent to the ?Difficulty Predictor? (DP) [6] a measure of sequence
complexity that has been extremely successful in modelling subjective randomness judgments. DP measures the complexity of a sequence in terms of the number of repeating (eg.
HHHH) and alternating (eg. HTHT) subsequences it contains, adding one point for each
repeating subsequence and two points for each alternating subsequence. For example, the
sequence TTTHHHTHTH is a run of tails, a run of heads, and an alternating sub-sequence,
DP = 4. If there are several partitions into runs and alternations, DP is calculated on the
partition that results in the lowest score.
In [2], we showed that random(X) ? DP if P (X|regular) is specified by a particular HMM. This HMM produces sequences by motif repetition, using the transition graph
shown in Figure 1. The model emits sequences by choosing a motif, a sequence of symbols
of length k, with probability proportional to ?k , and emitting symbols consistent with that
motif with probability ?, switching to a new motif with probability 1 ? ?. In Figure 1,
state 1 repeats the motif H, state 2 repeats T, and the remaining states repeat the alternating motifs HT and TH. The randomness of a sequence under this definition of regularity
depends on ? and ?, but is generally affected by the number of repeating and alternating
subsequences. The equivalence to DP, in which a sequence scores a single point for each
repeating subsequence
and two points for each alternating subsequence, results from taking
?
? = 0.5 and ? = 3?1
2 , and choosing the the state sequence for the HMM that maximizes
the probability of the sequence.
Just as the algorithmic probability R(X) is a probability distribution defined by the length
of programs for a universal Turing machine, this choice of P (X|regular) can be seen as
specifying the length of ?programs? for a particular finite state automaton. The output of a
finite state automaton is determined by its state sequence, just as the output of a universal
Turing machine is determined by its program. However, since the state sequence is the
same length as the sequence itself, this alone does not provide a meaningful measure of
complexity. In our model, probability imposes a metric on state sequences, dictating a
greater cost for moves between certain states, which translates into a code length through
the logarithm. Since we find the state sequence most likely to have produced X, and thus
the shortest code length, we have an analogue of Kolmogorov complexity defined on a
finite state automaton.
3.3
Regularities and automata
Using a hidden Markov model to specify P (X|regular) provides a measure of complexity
defined in terms of a finite state automaton. However, the kinds of regularities people can
detect in binary sequences go beyond the capacity of a finite state automaton. Here, we
consider three additional regularities: symmetry (eg. THTHHTHT), symmetry in the com-
Finite state automaton
(motif repetition)
Queue automaton
(duplication)
Pushdown automaton
(symmetry)
Stack automaton
Turing machine
(all computable)
Figure 2: Hierarchy of automata used to define measures of complexity. Of the regularities
discussed in this paper, each automaton can identify all regularities identified by those
automata to its left as well as those stated in parentheses beneath its name.
plement (eg. TTTTHHHH), and the perfect duplication of subsequences (eg. HHHTHHHT
vs. HHHTHHHTH). These regularities identify formal languages that cannot be recognized
by a finite state automaton, suggesting that we might be able to develop better models of
subjective randomness by defining P (X|regular) in terms of more sophisticated automata.
The automata we will consider in this paper form a hierarchy, shown in Figure 2. This
hierarchy expresses the same content as Chomsky?s [7] hierarchy of computing machines
? the regularities identifiable by each machine are a strict superset of those identifiable
to the machine to the left ? although it features a different set of automata. The most
restricted set of regularities are those associated with the finite state automaton, and the
least restricted are those associated with the Turing machine. In between are the pushdown
automaton, which augments a finite controller with a stack memory, in which the last item
added is the first to be accessed; the queue automaton,1 in which the memory is a queue, in
which the first item added is the first to be accessed; and the stack automaton, in which the
memory is a stack but any item in the stack can be read by the controller [9, 10]. The key
difference between these kinds of automata is the memory available to the finite controller,
and exploring measures of complexity defined in terms of these automata thus involves
assessing the kind of memory required to identify regularities.
Each of the automata shown in Figure 2 can identify a different set of regularities. The
finite state automaton is only capable of identifying motif repetition, while the pushdown
automaton can identify both kinds of symmetry, and the queue automaton can identify
duplication. The stack automaton can identify all of these regularities, and the Turing
machine can identify all computable regularities. For each of the sub-Turing automata,
we can use these constraints to specify a probabilistic model for P (X|regular). For example, the probabilistic model corresponding to the pushdown automaton generates regular sequences by three methods: repetition, producing sequences with probabilities determined by the HMM introduced above; symmetry, where half of the sequence is produced
by the HMM and the second half is produced by reflection; and complement symmetry,
where the second half is produced by reflection and exchanging H and T. We then take
P (X|regular) = maxZ,M P (X, Z|M )P (M ), where M is the method of production and
Z is the state sequence for the HMM. Similar models can be defined for the queue and
stack automata, with the queue automaton allowing generation by repetition or duplication,
and the stack automaton allowing any of these four methods. Each regularity introduced
into the model requires a further parameter in specifying P (M ), so the hierarchy shown
in Figure 2 also expresses the statistical structure of this set of models: each model is a
special case of the model to its right, in which some regularities are eliminated by setting
P (M ) to zero. We can use this structure to perform model selection with likelihood ratio
tests, determining which model gives the best account of a particular dataset using just the
difference in the log-likelihoods. We apply this method in the next section.
1
An unrestricted queue automaton is equivalent to a Turing machine. We will use the phrase to
refer to an automaton in which the number of queue operations that can be performed for each input
symbol is limited, which is generally termed a quasi real time queue automaton [8].
4
Testing the models
The models introduced in the previous section differ in the memory systems with which
they augment the finite controller. The appropriateness of any one measure of complexity
to a particular task may thus depend upon the memory demands placed upon the participant. To explore this hypothesis, we conducted an experiment in which participants make
randomness judgments after either seeing a sequence in its entirety, or seeing each element
one after another. We then used model selection to determine which measure of complexity gave the best account of each condition, illustrating how the strategy of defining
more restricted forms of complexity can shed light into the cognitive processes underlying
regularity detection.
4.1
Experimental methods
There were two conditions in the experiment, corresponding to Simultaneous and Sequential presentation of stimuli. The stimuli were sequences of heads (H) and tails (T) presented
in 130 point fixed width sans-serif font on a 19? monitor at 1280 ? 1024 pixel resolution.
In the Simultaneous condition, all eight elements of the sequence appeared on the display
simultaneously. In the Sequential condition, the elements appeared one by one, being displayed for 300ms with a 300ms inter-stimulus interval.
The participants were 40 MIT undergraduates, randomly assigned to the two conditions.
Participants were instructed that they were about to see sequences which had either been
produced by a random process (flipping a fair coin) or by other processes in which the
choice of heads and tails was not random, and had to classify these sequences according
to their source. After a practice session, each participant classified all 128 sequences of
length 8, in random order, with each sequence randomly starting with either a head or a
tail. Participants took breaks at intervals of 32 sequences.
4.2
Results and Discussion
We analyzed the results by fitting the models corresponding to the four automata described above, using all motifs up to length 4 to specify the basic model. We computed
random(X) for each stimulus as in Eq. (4), with P (X|regular) specified by the probabilistic model corresponding to each of the automata. We then converted this log-likelihood
ratio into the posterior probability of a random generating process, using
P (random|X) =
1
1 + exp{?? random(X) ? ?}
where ? and ? are parameters weighting the contribution of the likelihoods and the priors respectively. We then optimized ?, ?, ?, ? and the parameters contributing to P (M )
for each model, maximizing the likelihood of the classifications of the sequences by the
20 participants in each of the 2 conditions. The results of the model-fitting are shown in
Figure 3(a) and (b), which indicate the relationship between the posterior probabilities predicted by the model and the proportion of participants who classified a sequence as random.
The correlation coefficients shown in the figure provide a relatively good indicator of the
fit of the models, and each sequence is labelled according to the regularity it expresses,
showing how accommodating particular regularities contributes to the fit.
The log-likelihood scores obtained from fitting the models can be used for model selection, testing whether any of the parameters involved in the models are unnecessary. Since
the models form a nested hierarchy, we can use likelihood ratio tests to evaluate whether
introducing a particular regularity (and the parameters associated with it) results in a statistically significant improvement in fit. Specifically, if model 1 has log-likelihood L1 and
df1 parameters, and model 2 has log-likelihood L2 and df2 > df1 parameters, 2(L2 ? L1 )
1
P(random|x)
Finite state
(a)
0.5
Pushdown
r=0.69
0
0
r=0.79
1
P(random|x)
0.5
0
Repetition
Symmetry
Complement
Duplication
Pushdown
r=0.70
r=0.70
0
Stack
r=0.83
0.5
1
Simultaneous data
Finite state
(b)
Queue
r=0.76
0.5
Sequential data
(c) 57.43 (1df, p < 0.0001)
Stack
r=0.76
r=0.77
1
Queue
Finite state
87.76 (2df, p < 0.0001)
Queue
75.41 (2df, p < 0.0001)
Stack
Pushdown
45.08 (1df, p < 0.0001)
(d) 33.24 (1df, p < 0.0001)
Queue
5.69 (2df, p = 0.0582)
Pushdown
31.42 (1df, p < 0.0001)
Finite state
1.82 (2df, p = 0.4025)
Stack
Figure 3: Experimental results for (a) the Simultaneous and (b) the Sequential condition,
showing the proportion of participants classifying a sequence as ?random? (horizontal axis)
and P (random|X) (vertical axis) as assessed by the four models. Points are labelled according to their parse under the Stack model. (c) and (d) show the model selection results
for the Simultaneous and Sequential conditions respectively, showing the four automata
with edges between them labelled with ?2 score (df, p-value) for improvement in fit.
should have a ?2 (df2 ? df1 ) distribution under the null hypothesis of no improvement in
fit. We evaluated the pairwise likelihood ratio tests for the four models in each condition,
with the results shown in Figure 3(c) and (d). Additional regularities always improved the
fit for the Simultaneous condition, while adding duplication, but not symmetry, resulted in
a statistically significant improvement in the Sequential condition.
The model selection results suggest that the best model for the Simultaneous condition
is the stack automaton, while the best model for the Sequential condition is the queue
automaton. These results indicate the importance of presentation format in determining
subjective randomness, as well as the benefits of exploring measures of complexity defined
in terms of a range of computing machines. The stack automaton can evaluate regularities
that require checking information in arbitrary positions in a sequence, something that is
facilitated by a display in which the entire sequence is available. In contrast, the queue
automaton can only access information in the order that it enters memory, and gives a
better match to the task in which working memory is required. This illustrates an important
fact about cognition ? that human working memory operates like a queue rather than a stack
? that is highlighted by this approach.
The final parameters of the best-fitting models provide some insight into the relative importance of the different kinds of regularities under different presentation conditions. For the
Simultaneous condition, ? = 0.66, ? = 0.12, ? = 0.26, ? = ?1.98 and motif repetition,
symmetry, symmetry in the complement, and duplication were given probabilities of 0.748,
0.208, 0.005, and 0.039 respectively. Symmetry is thus a far stronger characteristic of reg-
ularity than either symmetry in the complement or duplication, when entire sequences are
viewed simultaneously. For the Sequential condition, ? = 0.70, ? = 0.11, ? = 0.38, ? =
?1.24, and motif repetition was given a probability of 0.962 while duplication had a probability of 0.038, with both forms of symmetry being given zero probability since the queue
model provided the best fit. Values of ? > 0.5 for both models indicates that regular sequences tend to repeat motifs, rather than rapidly switching between them, and the low ?
values reflect a preference for short motifs.
5
Conclusion
We have outlined a framework for understanding the rational basis of the human ability to
find structure embedded in noise, viewing this inference in terms of the statistical problem of model selection. Solving this problem for small datasets requires two ingredients:
strong prior beliefs about the hypothetical mechanisms by which the data could have been
generated, and a rational statistical inference by which these hypotheses are evaluated.
When assessing the randomness of binary sequences, which involves comparing random
and regular sources, people?s beliefs about the nature of regularity can be expressed in
terms of probabilistic versions of simple computing machines. Different machines capture
regularity when sequences are presented simultaneously and when their elements are presented sequentially, and the differences between these machines provide insight into the
cognitive processes involved in the task. Analyses of the rational basis of human inference
typically either ignore questions about processing or introduce them as relatively arbitrary
constraints. Here, we are able to give a rational characterization of process as well as inference, evaluating a set of alternatives that all correspond to restrictions of Kolmogorov
complexity to simple general-purpose automata.
Acknowledgments. This work was supported by a Stanford Graduate Fellowship to the first author.
We thank Charles Kemp and Michael Lee for useful comments.
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1,629 | 2,481 | Warped Gaussian Processes
Edward Snelson?
Carl Edward Rasmussen?
Zoubin Ghahramani?
?
Gatsby Computational Neuroscience Unit
University College London
17 Queen Square, London WC1N 3AR, UK
{snelson,zoubin}@gatsby.ucl.ac.uk
?
Max Planck Institute for Biological Cybernetics
Spemann Stra?e 38, 72076 T?ubingen, Germany
[email protected]
Abstract
We generalise the Gaussian process (GP) framework for regression by
learning a nonlinear transformation of the GP outputs. This allows for
non-Gaussian processes and non-Gaussian noise. The learning algorithm chooses a nonlinear transformation such that transformed data is
well-modelled by a GP. This can be seen as including a preprocessing
transformation as an integral part of the probabilistic modelling problem,
rather than as an ad-hoc step. We demonstrate on several real regression
problems that learning the transformation can lead to significantly better
performance than using a regular GP, or a GP with a fixed transformation.
1
Introduction
A Gaussian process (GP) is an extremely concise and simple way of placing a prior on
functions. Once this is done, GPs can be used as the basis for nonlinear nonparametric
regression and classification, showing excellent performance on a wide variety of datasets
[1, 2, 3]. Importantly they allow full Bayesian predictive distributions to be obtained, rather
than merely point predictions.
However, in their simplest form GPs are limited by the nature of their simplicity: they
assume the target data to be distributed as a multivariate Gaussian, with Gaussian noise on
the individual points. This simplicity enables predictions to be made easily using matrix
manipulations, and of course the predictive distributions are Gaussian also.
Often it is unreasonable to assume that, in the form the data is obtained, the noise will be
Gaussian, and the data well modelled as a GP. For example, the observations may be positive quantities varying over many orders of magnitude, where it makes little sense to model
these quantities directly assuming homoscedastic Gaussian noise. In these situations it is
standard practice in the statistics literature to take the log of the data. Then modelling proceeds assuming that this transformed data has Gaussian noise and will be better modelled
by the GP. The log is just one particular transformation that could be done; there is a con-
tinuum of transformations that could be applied to the observation space to bring the data
into a form well modelled by a GP. Making such a transformation should really be a full
part of the probabilistic modelling; it seems strange to first make an ad-hoc transformation,
and then use a principled Bayesian probabilistic model.
In this paper we show how such a transformation or ?warping? of the observation space
can be made entirely automatically, fully encompassed into the probabilistic framework
of the GP. The warped GP makes a transformation from a latent space to the observation,
such that the data is best modelled by a GP in the latent space. It can also be viewed as a
generalisation of the GP, since in observation space it is a non-Gaussian process, with nonGaussian and asymmetric noise in general. It is not however just a GP with a non-Gaussian
noise model; see section 6 for further discussion.
For an excellent review of Gaussian processes for regression and classification see [4].
We follow the notation there throughout this paper and present a brief summary of GP
regression in section 2. We show in sections 4 and 5, with both toy and real data, that the
warped GP can significantly improve predictive performance over a variety of measures,
especially with regard to the whole predictive distribution, rather than just a single point
prediction such as the mean or median. The transformation found also gives insight into
the properties of the data.
2
Nonlinear regression with Gaussian processes
Suppose we are given a dataset D, consisting of N pairs of input vectors XN ? {x(n) }N
n=1
and real-valued targets tN ? {tn }N
n=1 . We wish to predict the value of an observation
tN +1 given a new input vector x(N +1) , or rather the distribution P (tN +1 |x(N +1) , D). We
assume there is an underlying function y(x) which we are trying to model, and that the
observations lie noisily around this. A GP places a prior directly on the space of functions
by assuming that any finite selection of points XN gives rise to a multivariate Gaussian distribution over the corresponding function values yN . The covariance between the function
value of y at two points x and x0 is modelled with a covariance function C(x, x0 ), which
is usually assumed to have some simple parametric form. If the noise model is taken to be
Gaussian, then the distribution over observations tN is also Gaussian with the entries of
the covariance matrix C given by
Cmn = C(x(m) , x(n) ; ?) + ?mn g(x(n) ; ?) ,
(1)
where ? parameterises the covariance function, g is the noise model, and ?mn is the Kronecker delta function.
Often the noise model is taken to be input-independent, and the covariance function is taken
to be a Gaussian function of the difference in the input vectors (a stationary covariance
function), although many other possibilities exist, see e.g. [5] for GPs with input dependent
noise. In this paper we consider only this popular choice, in which case the entries in the
covariance matrix are given by
?
! ?
D
(m)
(n) 2
X
x
?
x
1
d
d
? + v0 ?mn .
Cmn = v1 exp ??
(2)
2
rd
d=1
Here rd is a width parameter expressing the scale over which typical functions vary in the
dth dimension, v1 is a size parameter expressing the typical size of the overall process in
y-space, v0 is the noise variance of the observations, and ? = {v0 , v1 , r1 , . . . , rD }.
It is simple to show that the predictive distribution for a new point given the observed
data, P (tN +1 |tN , XN +1 ), is Gaussian. The calculation of the mean and variance of this
distribution involves doing a matrix inversion of the covariance matrix CN of the training
inputs, which using standard exact methods incurs a computational cost of order N 3 .
Learning, or ?training?, in a GP is usually achieved by finding a local maximum in the
likelihood using conjugate gradient methods with respect to the hyperparameters ? of the
covariance matrix. The negative log likelihood is given by
1
1
N
C?1 tN +
log 2? .
(3)
L = ? log P (tN |XN , ?) = log det CN + t>
2
2 N N
2
Once again, the evaluation of L, and its gradients with respect to ?, involve computing the
inverse covariance matrix, incurring an order N 3 cost. Rather than finding a ML estimate
?ML , a prior over ? can be included to find a MAP estimate ?MAP , or even better ? can be
numerically integrated out when computing P (tN +1 |x(N +1) , D) using for example hybrid
Monte Carlo methods [2, 6].
3
Warping the observation space
In this section we present a method of warping the observation space through a nonlinear
monotonic function to a latent space, whilst retaining the full probabilistic framework to
enable learning and prediction to take place consistently. Let us consider a vector of latent
targets zN and suppose that this vector is modelled by a GP,
1
N
1
?1
? log P (zN |XN , ?) = log det CN + z>
log 2? .
(4)
N CN z N +
2
2
2
Now we make a transformation from the true observation space to the latent space by
mapping each observation through the same monotonic function f ,
zn = f (tn ; ?)
?n ,
(5)
where ? parameterises the transformation. We require f to be monotonic and mapping on
to the whole of the real line; otherwise probability measure will not be conserved in the
transformation, and we will not induce a valid distribution over the targets tN . Including
the Jacobian term that takes the transformation into account, the negative log likelihood,
? log P (tN |XN , ?, ?), now becomes:
N
X
1
?f (t)
1
N
> ?1
L = log det CN + f (tN ) CN f (tN ) ?
log
+ 2 log 2? . (6)
2
2
?t
tn
n=1
3.1
Training the warped GP
Learning in this extended model is achieved by simply taking derivatives of the negative
log likelihood function (6) with respect to both ? and ? parameter vectors, and using a
conjugate gradient method to compute ML parameter values. In this way the form of both
the covariance matrix and the nonlinear transformation are learnt simultaneously under the
same probabilistic framework. Since the computational limiter to a GP is inverting the
covariance matrix, adding a few extra parameters into the likelihood is not really costing us
anything. All we require is that the derivatives of f are easy to compute (both with respect
to t and ?), and that we don?t introduce so many extra parameters that we have problems
with over-fitting. Of course a prior over both ? and ? may be included to compute a MAP
estimate, or in fact the parameters integrated out using a hybrid Monte Carlo method.
3.2
Predictions with the warped GP
For a particular setting of the covariance function hyperparameters ? (for example ? ML
or ?MAP ), in latent variable space the predictive distribution at a new point is just as for a
regular GP: a Gaussian whose mean and variance are calculated as mentioned in section 2;
2
P (zN +1 |x(N +1) , D, ?) = N z?N +1 (?), ?N
(7)
+1 (?) .
To find the distribution in the observation space we pass that Gaussian through the nonlinear
warping function, giving
"
2 #
1 f (tN +1 ) ? z?N +1
f 0 (tN +1 )
(N +1)
exp ?
P (tN +1 |x
, D, ?, ?) = q
. (8)
2
?N +1
2
2??N
+1
The shape of this distribution depends on the form of the warping function f , but in general
it may be asymmetric and multimodal.
If we require a point prediction to be made, rather than the whole distribution over t N +1 ,
then the value we will predict depends on our loss function. If our loss function is absolute
error, then the median of the distribution should be predicted, whereas if our loss function is
squared error, then it is the mean of the distribution. For a standard GP where the predictive
distribution is Gaussian, the median and mean lie at the same point. For the warped GP in
general they are at different points. The median is particularly easy to calculate:
?1
tmed
(?
zN +1 ) .
N +1 = f
(9)
Notice we need to compute the inverse warping function. In general we are unlikely to have
an analytical form for f ?1 , because we have parameterised the function in the opposite
direction. However since we have access to derivatives of f , a few iterations of NewtonRaphson with a good enough starting point is enough.
It is often useful to give an indication of the shape and range of the distribution by giving
the positions of various ?percentiles?. For example we may want to know the positions of
?2?? either side of the median so that we can say that approximately 95% of the density
lies between these bounds. These points in observation space are calculated in exactly the
same way as the median - simply pass the values through the inverse function:
tmed?2?
= f ?1 (?
zN +1 ? 2?N +1 ) .
N +1
(10)
To calculate the mean, we need to integrate tN +1 over the density (8). Rewriting this
integral back in latent space we get
Z
2
?1
E(tN +1 ) = dzf ?1 (z)Nz (?
zN +1 , ?N
).
(11)
+1 ) = E(f
This is a simple one dimensional integral under a Gaussian density, so Gauss-Hermite
quadrature may be used to accurately compute it with a weighted sum of a small number
of evaluations of the inverse function f ?1 at appropriate places.
3.3
Choosing a monotonic warping function
We wish to design a warping function that will allow for complex transformations, but we
must constrain the function to be monotonic. There are various ways to do this, an obvious
one being a neural-net style sum of tanh functions,
f (t; ?) =
I
X
ai tanh (bi (t + ci ))
ai , bi ? 0 ?i ,
(12)
i=1
where ? = {a, b, c}. This produces a series of smooth steps, with a controlling the size
of the steps, b controlling their steepness, and c their position. Of course the number of
(a)
2
1.5
3
1
P(t | x=??/4)
2.5
0.5
t
(b)
3.5
0
2
1.5
?0.5
1
?1
0.5
?1.5
?2
?pi
?pi/2
?pi/4
0
pi/2
pi
0
?2
?1
x
0
1
t
Figure 1: A 1D regression task. The dotted lines show the true generating distribution, the
dashed lines show a GP?s predictions, and the solid lines show the warped GP?s predictions.
(a) The triplets of lines represent the median, and 2? percentiles in each case. (b) Predictive
probability densities at x = ??/4; i.e. a cross section through (a) at the solid grey line
steps I needs to be set, and that will depend on how complex a function one wants. The
derivatives of this function with respect to either t, or the warping parameters ?, are easy
to compute. In the same spirit, sums of error functions, or sums of logistic functions, would
produce a similar series of steps, and so these could be used instead.
The problem with using (12) as it stands is that it is bounded; the inverse function f ?1 (z)
does not exist for values of z outside the range of these bounds. As explained earlier, this
will not lead to a proper density in t space, because the density in z space is Gaussian,
which covers the whole of the real line. We can fix this up by using instead:
f (t; ?) = t +
I
X
ai tanh (bi (t + ci ))
a i , bi ? 0
?i .
(13)
i=1
which has linear trends away from the tanh steps. In doing so, we have restricted ourselves
to only making warping functions with f 0 ? 1, but because the size of the covariance
function v1 is free to vary, the effective gradient can be made arbitrarily small by simply
making the range of the data in the latent space arbitrarily big.
A more flexible system of linear trends may be made by including,
in addition to the neural
net style function (12), some functions of the form ?1 log e?m1 (t?d) + e?m2 (t?d) , where
m1 , m2 ? 0. This function effectively splices two straight lines of gradients m1 and
m2 smoothly together with a ?curvature? parameter ?, and at position d. The sign of ?
determines whether the join is convex or concave.
4
A simple 1D regression task
A simple 1D regression task was created to show a situation where the warped GP should,
and does, perform significantly better than the standard GP. 101 points, regularly spaced
from ?? to ? on the x axis, were generated with Gaussian noise about a sine function.
These points were then warped through the function t = z 1/3 , to arrive at the dataset t
which is shown as the dots in Figure 1(a).
(a) sine
z
(c) abalone
(b) creep
z
z
t
t
(d) ailerons
z
t
t
Figure 2: Warping functions learnt for the four regression tasks carried out in this paper.
Each plot is made over the range of the observation data, from tmin to tmax .
A GP and a warped GP were trained independently on this dataset using a conjugate gradient minimisation procedure and randomly initialised parameters, to obtain maximum likelihood parameters. For the warped GP, the warping function (13) was used with just two
tanh functions. For both models the covariance matrix (2) was used. Hybrid Monte Carlo
was also implemented to integrate over all the parameters, or just the warping parameters
(much faster since no matrix inversion is required with each step), but with this dataset (and
the real datasets of section 5) no significant differences were found from ML.
Predictions from the GP and warped GP were made, using the ML parameters, for 401
points regularly spaced over the range of x. The predictions made were the median and
2? percentiles in each case, and these are plotted as triplets of lines on Figure 1(a). The
predictions from the warped GP are found to be much closer to the true generating distribution than the standard GP, especially with regard to the 2? lines. The mean line was also
computed, and found to lie close, but slightly skewed, from the median line.
Figure 1(b) emphasises the point that the warped GP finds the shape of the whole predictive
distribution much better, not just the median or mean. In this plot, one particular point on
the x axis is chosen, x = ??/4, and the predictive densities from the GP and warped GP
are plotted alongside the true density (which can be written down analytically). Note that
the standard GP must necessarily predict a symmetrical Gaussian density, even when the
density from which the points are generated is highly asymmetrical, as in this case.
Figure 2(a) shows the warping function learnt for this regression task. The tanh functions
have adjusted themselves so that they mimic a t3 nonlinearity over the range of the observation space, thus inverting the z 1/3 transformation imposed when generating the data.
5
Results for some real datasets
It is not surprising that the method works well on the toy dataset of section 4 since it was
generated from a known nonlinear warping of a smooth function with Gaussian noise. To
demonstrate that nonlinear transformations also help on real data sets we have run the
warped GP comparing its predictions to an ordinary GP on three regression problems.
These datasets are summarised in the following table which shows the range of the targets
(tmin , tmax ), the number of input dimensions (D), and the size of the training and test sets
(Ntrain , Ntest ) that we used.
Dataset
creep
abalone
ailerons
D
30
8
40
tmin
18 MPa
1 yr
?3.0 ? 10?3
tmax
530 MPa
29 yrs
?3.5 ? 10?4
Ntrain
800
1000
1000
Ntest
1266
3177
6154
Dataset
creep
abalone
ailerons
Model
GP
GP + log
warped GP
GP
GP + log
warped GP
GP
warped GP
Absolute error
16.4
15.6
15.0
1.53
1.48
1.47
1.23 ? 10?4
1.18 ? 10?4
Squared error
654
587
554
4.79
4.62
4.63
3.05 ? 10?8
2.72 ? 10?8
? log P (t)
4.46
4.24
4.19
2.19
2.01
1.96
-7.31
-7.45
Table 1: Results of testing the GP, warped GP, and GP with log transform, on three real
datasets. The units for absolute error and squared error are as for the original data.
The dataset creep is a materials science set, with the objective to predict creep rupture stress (in MPa) for steel given chemical composition and other inputs [7, 8]. With
abalone the aim is to predict the the age of abalone from various physical inputs [9].
ailerons is a simulated control problem, with the aim to predict the control action on
the ailerons of an F16 aircraft [10, 11].
For datasets creep and abalone, which consist of positive observations only, standard
practice may be to model the log of the data with a GP. So for these datasets we have
compared three models: a GP directly on the data, a GP on the fixed log-transformed data,
and the warped GP directly on the data. The predictive points and densities were always
compared in the original data space, accounting for the Jacobian of both the log and the
warped transforms. The models were run as in the 1D task: ML parameter estimates only,
covariance matrix (2), and warping function (13) with three tanh functions.
The results we obtain for the three datasets are shown in Table 1. We show three measures
of performance over independent test sets: mean absolute error, mean squared error, and
the mean negative log predictive density evaluated at the test points. This final measure
was included to give some idea of how well the model predicts the entire density, not just
point predictions.
On these three sets, the warped GP always performs significantly better than the standard
GP. For creep and abalone, the fixed log transform clearly works well too, but particularly in the case of creep, the warped GP learns a better transformation. Figure 2 shows
the warping functions learnt, and indeed 2(b) and 2(c) are clearly log-like in character. On
the other hand 2(d), for the ailerons set, is exponential-like. This shows the warped GP
is able to flexibly handle these different types of datasets. The shapes of the learnt warping functions were also found to be very robust to random initialisation of the parameters.
Finally, the warped GP also makes a better job of predicting the distributions, as shown by
the difference in values of the negative log density.
6
Conclusions, extensions, and related work
We have shown that the warped GP is a useful extension to the standard GP for regression,
capable of finding extra structure in the data through the transformations it learns. From
another viewpoint, it allows standard preprocessing transforms, such as log, to be discovered automatically and improved on, rather than be applied in an ad-hoc manner. We have
demonstrated an improvement in performance over the regular GP on several datasets.
Of course some datasets are well modelled by a GP already, and applying the warped GP
model simply results in a linear ?warping? function. It has also been found that datasets that
have been censored, i.e. many observations at the edge of the range lie on a single point,
cause the warped GP problems. The warping function attempts to model the censoring
by pushing those points far away from the rest of the data, and it suffers in performance
especially for ML learning. To deal with this properly a censorship model is required.
As a further extension, one might consider warping the input space in some nonlinear fashion. In the context of geostatistics this has actually been dealt with by O?Hagan [12],
where a transformation is made from an input space which can have non-stationary and
non-isotropic covariance structure, to a latent space in which the usual conditions of stationarity and isotropy hold.
Gaussian process classifiers can also be thought of as warping the outputs of a GP, through a
mapping onto the (0, 1) probability interval. However, the observations in classification are
discrete, not points in this warped continuous space. Therefore the likelihood is different.
Diggle et al. [13] consider various other fixed nonlinear transformations of GP outputs.
It should be emphasised that the presented method can be beneficial in situations where the
noise variance depends on the output value. Gaussian processes where the noise variance
depends on the inputs have been examined by e.g. [5]. Forms of non-Gaussianity which
do not directly depend on the output values (such as heavy tailed noise) are also not captured by the method proposed here. We propose that the current method should be used in
conjunction with methods targeted directly at these other issues. The force of the method
it that it is powerful, yet very easy and computationally cheap to apply.
Acknowledgements. Many thanks to David MacKay for useful discussions, suggestions
of warping functions and datasets to try. CER was supported by the German Research
Council (DFG) through grant RA 1030/1.
References
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1,630 | 2,482 | Salient Boundary Detection using Ratio Contour
Song Wang, Toshiro Kubota
Dept. Computer Science & Engineering
University of South Carolina
Columbia, SC 29208
{songwang|kubota}@cse.sc.edu
Jeffrey Mark Siskind
School Electrical & Comput. Engr.
Purdue University
West Lafayette, IN 47906
[email protected]
Abstract
This paper presents a novel graph-theoretic approach, named ratio contour, to extract perceptually salient boundaries from a set of noisy boundary fragments detected in real images. The boundary saliency is defined
using the Gestalt laws of closure, proximity, and continuity. This paper first constructs an undirected graph with two different sets of edges:
solid edges and dashed edges. The weights of solid and dashed edges
measure the local saliency in and between boundary fragments, respectively. Then the most salient boundary is detected by searching for an
optimal cycle in this graph with minimum average weight. The proposed
approach guarantees the global optimality without introducing any biases
related to region area or boundary length. We collect a variety of images
for testing the proposed approach with encouraging results.
1
Introduction
Human vision and neural systems possess very strong capabilities of identifying salient
structures from various images. Implementing such capabilities on a computer is an important but extremely challenging problem for artificial intelligence, computer vision, and
machine learning. The main challenges come from two closely related aspects: (a) the definition of the structural saliency, and (b) the design of efficient algorithms for finding the
salient structures. On one hand, we expect very comprehensive and advanced definitions
of the saliency so that it models accurately the human perceptual and visual process. On
the other hand, we expect simple definitions of saliency so that the global optimum can be
found in polynomial time.
Previous methods for salient-structure detection can be grouped into two classes. The first
class of methods aims to directly group or segment all the image pixels into some disjoint regions, which are expected to coincide with the underlying salient structures. Earlier
efforts include the region-merging/splitting methods, watershed methods, and the activecontour-like methods. Those methods usually have difficulties in finding the globally optimal boundaries in terms of the selected saliency definitions. Recently we have witnessed
some advanced methods, like ratio region [5], minimum cut[17], normalized cut [14], globally optimal region/cycle [9], and ratio cut [15], which aim to produce globally optimal
boundaries. However, those pixel-grouping based methods usually have difficulties in effectively incorporating perceptual rules, such as boundary smoothness, into their saliency
definitions.
Instead of operating directly on the image pixels, another class of methods is designed
based on some pre-extracted boundary fragments (or for brevity, fragments) 1 , which can
be obtained using some standard edge-detection methods like Canny detectors. As shown
in Fig. 1(a), although those fragments are disconnected and contain serious noise, they provide abundant information on boundary length, tangent directions, and curvatures, which
can greatly facilitate the incorporation of advanced perceptual rules like boundary smoothness. Shashua and Ullman [13] presents a parallel network model for detecting salient
boundary based on fragment proximity, boundary length, and boundary smoothness. Recent development in this class includes Alter and Basri [2], Jacobs [8], Sarkar and Boyer
[12], Guy and Medioni [7], Williams and Thornber [16, 11], and Amir and Lindenbaum
[3]. However, many of them still have difficulty in finding the closed boundaries in a sense
of global optimality with respect to the given boundary-saliency measure. Elder and Zucker
[6] use the shortest-path algorithm to connect fragments to form salient closed boundaries.
However, the results have a bias to produce boundaries with shorter length.
This paper presents a new graph based approach to extract salient closed boundaries from
a set of fragments detected from real images. This approach seeks a good balance between
the complexity of the saliency definition and the complexity of the optimization algorithm.
The boundary saliency is based on the well-known Gestalt laws of closure, proximity, and
continuity. To avoid the various biases as in Elder and Zucker [6], this paper defines the
boundary saliency as the average saliency along the whole boundary. We finally formulate
the salient-boundary detection problem into a problem for finding an optimal cycle in an
undirected graph. We show this problem is of polynomial time-complexity and give an
algorithm to solve it. The proposed algorithm is then tested on a variety of real images.
2
Problem Formulation
(a)
(b)
(c)
(d)
Figure 1: An illustration of detecting salient boundaries from some fragments. (a) Boundary fragments, (b) salient boundary by connecting some fragments with dashed curves, (c)
a solid-dashed graph, and (d) an alternate cycle in (c).
The basic primitives in the ratio-contour approach are a set of noisy (boundary) fragments
extracted by edge detection. For simplicity, here we assume each detected fragment is a
continuous open curve segment with two endpoints, as shown in Fig. 1(a). Our goal is
to identify and connect a subset of fragments to form the most salient structural boundary as shown in Fig. 1(b). In this paper, we measure the boundary saliency using simple Gestalt laws of closure, proximity, and continuity. The closure means that the salient
boundary must be a closed contour. The proximity implies that we desire relatively small
gaps (dashed curves in Fig. 1(b)) in connecting the fragments. The continuity indicates that
the resulting contour should be continuous and sufficiently smooth.
Let the parametric form of a boundary B be v(t), 0 ? t ? 1. We have v(0) = v(1) as the
boundary is closed. Considering the boundary proximity and the continuity, we define its
1
Most literatures use the terminology edge instead of fragment. However, in this paper edge has
other specified meaning in a graph model.
cost, which is negatively related to the boundary saliency, as
R
[?(t) + ? ? ?2 (t)]dt
T (B)
R
= B
R(B) ,
,
L(B)
dt
B
(1)
where ?(t) = 1 if v(t) is in the gap and ?(t) = 0, otherwise. ?(t) is the curvature at v(t).
We can see that the un-normalized cost T (B) combines the total gap-length and curvature
along the boundary B and has bias to produce a short boundary. The issue is addressed in
(1) through normalizing T (B) by the boundary length L(B). The most salient boundary
B is then the one with the minimum cost R(B). The parameter ? > 0 is set to balance the
weight between proximity and continuity.
We can formulate the above cost into an undirected graph G = (V, E) with vertices V =
{v1 , v2 , ? ? ? , vn } and edges E = {e1 , e2 , ? ? ? , em }. A unique vertex is constructed from
each fragment endpoint. Two different kinds of edges, solid edges and dashed edges, are
constructed between vertices. (a) If vi and vj correspond to the two endpoints of the same
fragment, we construct a solid edge between vi and vj to model this fragment. (b) Between
each possible vertex pair vi and vj , we construct a dashed edge to model the gap or a virtual
fragment (dashed curves in Fig. 1(b)). An example is shown in Fig. 1(c), which is made up
of 3 solid edges for three fragments and all 15 possible dashed edges. For clarity, sometimes
we call the boundary fragment a real fragment when both real and virtual fragments are
involved.
The constructed graph always has even number of vertices, as each real fragment has two
endpoints. More interestingly, no two solid edges are incident from the same vertex and
each vertex has exactly one incident solid edge. We name such a graph an (undirected)
solid-dashed graph. We further define an alternate cycle in a solid-dashed graph as a simple
cycle that traverses the solid edges and dashed edges alternately. Examples of a soliddashed graph and an alternate cycle are given in Fig. 1(c) and (d), respectively. Since
a boundary always traverses real fragments and virtual fragments alternately, it can be
described by an alternate cycle. Note that not all the cycles in a solid-dashed graph are
alternate cycles, because two adjacent dashed edges can appear sequentially in the same
cycle.
According to the cost function (1), we define a weight function w(e) and a length function
l(e) for each edge e. For convenience, we define B(e) as a function that gives the (real
corresponding to an edge e. Then the weight w(e) , T (B(e)) =
Ror virtual) fragment
[?(t) + ? ? ?2 (t)]dt is the un-normalized cost on B(e). The edge length l(e) is defined
B(e)
as the length of B(e). We can see that the most salient boundary with minimum cost (1)
corresponds to an alternate cycle C with minimum cycle ratio
P
w(e)
CR(C) = Pe?C
.
e?C l(e)
Fragments extracted from real images usually contain noise, intersections, and even some
closed curves, which cause difficulties in estimating the curve length, curvature, and therefore, the edge weight and length. We will describe a spline-based method to address this
problem in Section 4. In the following, we first present a polynomial-time algorithm to
identify the alternate cycle with the minimum cycle ratio CR(C).
3
Ratio-Contour Algorithm
For simplicity, we denote the alternate cycle with minimum cycle ratio as MRA (Minimum
Ratio Alternate) cycle. In this section, we introduce a graph algorithm for finding the MRA
cycle in polynomial time. This algorithm consists of three reductions. (a) Both the weight
and edge length of the solid edges can be set to zero by merging them into the weight
and length of their adjacent dashed edges, without changing the underlying MRA. (b) The
problem of finding an MRA cycle can be reduced to a problem of detecting a negativeweight alternate (NWA) cycle in the same graph. (c) Finding NWA cycles in a solid-dashed
graph with zero solid-edge weights and zero solid-edge lengths can be reduced to finding a
minimum-weight perfect matching (MWPM) in the same graph. Finding MWPM has been
shown to be of polynomial-time complexity with various efficient algorithms available.
3.1
Setting Zero-Weight and Zero-Length to Solid Edges
As illustrated in Fig. 2(a) and (b), each solid edge e can only be adjacent to a set of dashed
edges, say {e1 , e2 , ? ? ? , eK }, in a solid-dashed graph, and no two solid edges are adjacent
to each other. Therefore, we can directly merge the solid-edge weight and length to its
adjacent dashed edges by
(
w(ek ) ? w(ek ) + w(e)
Nk
l(ek ) ? l(ek ) + l(e)
k = 1, 2, ? ? ? K,
Nk ,
where Nk = 2 if ek shares one vertex with e as in Fig. 2(a) and Nk = 1 if ek shares both
vertices with e as in Fig. 2(b). Then we reset the weight and length of this solid edge to
zero, i.e., w(e) = 0, l(e) = 0. This merging process is performed on all solid edges. While
solid and dashed edges are traversed alternately in an alternate cycle, it is not difficult to
achieve the following conclusion.
Lemma 3.1 The above processing of edge weights and edge-lengths does not change the
cycle ratio of any alternate cycles.
e1
e1
e
e
ek
(a)
(b)
(c)
(d)
Figure 2: An illustration of reductions in ratio-contour algorithm. (a) Merging the weight
and length of a solid edge to its adjacent dashed edges. (b) A special case for weight
merging. (c) A perfect matching in a solid-dashed graph. (d) Derived cycle from the
perfect matching shown in (c).
3.2
Reducing to Negative-Alternate-Cycle Detection
The following lemma claims that MRA cycles are invariant to some more general linear
edge-weight transforms.
Lemma 3.2 The MRA cycle in a solid-dashed graph G = (V, E) is invariant to the following linear transform on the edge weight
w(e) ? w(e) ? b ? l(e), ?e ? E.
(2)
The proof for this lemma is similar to the one we gave for general ratio-cycle detection
problem [15]. Notice that all the edge lengths are non-negative. There always exists an
optimal b = b? so that after weight transform (2), the MRA cycle has the cycle ratio of
zero. In this case, the MRA cycle is the same as the cycle with total edge weight of zero.
The detection of the optimal b? and the MRA cycle can then be reduced into a problem
of finding the NWA cycle (negative weight alternate cycle). Basically, if we can detect an
NWA cycle after the edge weight transform (2), we know b > b? . Otherwise, we know that
b ? b? . With an NWA cycle detection algorithm, we can use binary or sequential search to
locate the optimal b? and the desired MRA cycle. This search process is polynomial if all
the edge weight are integers [15]. In addition, with the first reduction mentioned in Section
3.1, it is easy to see that the linear transform (2) always preserves zero weight and zero
length for all solid edges in this search process.
3.3
Reducing to Minimum Weight Perfect Matching
The problem of detecting an NWA cycle in a solid-dashed graph can be reduced to a problem of finding a minimum weight perfect matching (MWPM) in the same graph. A perfect
matching in G denotes a subgraph that contains all the vertices in G while each vertex only
has one incident edge. An example is shown in Fig. 2(c), where three thick edges together
with their vertices form a perfect matching. The MWPM is the perfect matching with minimum total edge weight. As all the solid edges form a trivial perfect matching with total
weight zero, the MWPM in our solid-dashed graph should have non-positive total weight.
We can construct a set of cycles from a perfect matching P by (a) removing from P all
the solid edges and their endpoints, and (b) adding to P any solid edges in the solid-dashed
graph G whose two endpoints are still in P after the removal in (a). The remaining subgraph
must consist of a set of cycles because each remaining vertex has two incident edges: one
is solid and the other one is dashed. This also confirms that all the resulting cycles are
alternate cycles. An example of this reduction is shown in Fig. 2(d), which is constructed
from (c). As all the solid edges have zero weight and zero length, it is not difficult to see
that the total weight of the perfect matching is the same as the total weight of the resulting
cycles. Therefore, the NWA detection problem is reduced into a problem of finding a
perfect matching with negative total weight. This is the same as the problem of finding the
MWPM, which is of polynomial-time complexity [1].
4
Edge-Weight and Edge-Length Functions
We need to estimate the curvature and length of both real and virtual fragments for defining
w(e) and l(e) of solid and dashed edges. To deal with the noise and aliasing in detected
fragments, we impose a pre-smoothing process on those fragments. In this paper, we approximate a fragment by a set of quadratic splines with the parametric form
2
xi (ti )
xi
Ai Bi
ti
=
+
,
yi (ti )
yi
Ci Di
ti
where 0 ? ti ? 1 is the parameter for the spline. We developed an iterative algorithm [10]
to estimate the optimal parameters xi , yi , Ai , Bi , Ci , and Di minimizing a comprehensive
cost function that measures smoothness, under the constraint of C 0 and C 1 continuities
across the fragment. An example is illustrated in Fig. 3 where solid curves in (a) and (b)
are fragments before and after smoothing. More discussion and analysis on this curvesmoothing method can be found in our previous work [10].
With the parametric form of quadratic splines, the total length and the curvature along a real
fragment can be computed by summing over each spline its length and its total curvature
as
Z 1p
li =
(2Ai t + Bi )2 + (2Ci t + Di )2 dt,
0
Z
1
0
?2i (t)dt
=
Z
1
0
4(Ai Di ? Bi Ci )2
dt,
[(2Ai t + Bi )2 + (2Ci t + Di )2 ]3
where li is the length and ?i (t) is the curvature function of the ith spline.
However, estimating these quantities for a virtual fragment is not trivial, as no information
is given on how the virtual fragment should look like. We take the following approach to
compute the dashed-edge weight. First, a pair of endpoints involved in forming a particular
dashed edge is connected with a straight line. Then a new curve segment is constructed
by connecting this straight line and adjacent fragments. The smoothing process described
above is applied to this new curve segment. In this smoothed curve segment, the virtual
fragment is then the part corresponding to the straight line before the smoothing. The
dashed curve in Fig. 3(b) shows a resulting virtual fragment used for estimating curvature,
length, and finally edge weight.
(a)
(b)
(c)
(d)
Figure 3: An illustration of the edge weight estimation process. (a) Two noisy fragments.
(b) Smoothed real fragments and an estimated virtual fragment. (c) Fragments obtained
by Canny detector. (d) Smoothed fragments after breaking undesired connections, corresponding to the portion of the box in (c). Crossings specify the endpoints and breaking
points.
In real implementation, another issue is that the detected fragments using edge detectors
may not be disjoint open curves as assumed in Section 2. It is common that some fragments are connected in the form of intersections, attachments, and even undesired closure,
as shown in Fig. 4. Therefore, we need to break those connections to construct the graph
model. First, we identify the intersection points and split them to get multiple open fragments. An example is shown in Fig. 4(a) and (d), where an intersection point is broken into
three endpoints. In the constructed graph, they (u1 , u2 , and u3 ) are connected by dashed
edges with zero weight and zero length. Attachment specifies the case where two fragments
are undesirably connected into a single fragment as shown in Fig. 4(b). This greatly hurts
the reliability of salient boundary detection as those attached fragments may exclude many
desired dashed edges from the graph. We alleviate this problem by splitting all the fragments at their high-curvature points, as illustrated in Figs. 4(b) and (e). Similarly, we can
break closed fragments into open fragments at high-curvature points, as shown in Fig. 4(c)
and (f). Note that the identification of high-curvature points requires the smoothing of the
noisy fragments. We apply the same smoothing technique described above to each fragment for this purpose. Figures 3(c) and (d) show an example of dealing with the above
special cases.
5
Experiments and Discussion
In this section, we test the proposed ratio-contour algorithm to extract the salient boundaries
from real images. For initial fragment detection, we use the standard Canny edge detector
in the Matlab software with its default threshold settings. We also adopt the Blossom4
implementation [4] of the minimum-weight perfect matching.
One problem in the implementation is the construction of dashed edges, which may be of
a very large number (O(n2 )) if we connect every two possible vertices. In this paper, we
constrain the proximity to reduce the number of dashed edges. In the implementation, for
each vertex, we only keep certain number of incident dashed edges with smallest length.
B(e 2)
B(e 2)
B(e 3)
B(e 3)
B(e 1)
B(e 1)
u2
e3
(d)
u2
e2
u3
e1
(c)
(b)
(a)
u1
B(e 1)
e2
u1
e3
e1
u1
e1
u2
(f)
(e)
Figure 4: An illustration of fragment identification and graph construction in some special
cases. (a), (b), and (c) show the detected fragments with intersections, attachments, and
closures. (d), (e), and (f) are the constructed graphs from (a), (b), and (c), respectively.
This number is uniformly set to 20 in all experiments. Meanwhile, we set the parameter
? = 50 in the edge-weight definition. Figure 5 shows salient boundaries detected from
seven real images, together with the initial fragments from Canny detector. It can be seen
that the proposed method integrates well the Gestalt laws of proximity, continuity, and
closure.
(d)
(a)
(e)
(f)
(b)
(g)
(c)
Figure 5: Salient boundaries detected from some real images using the proposed ratiocontour algorithm. Each subfigure from (a) to (g) contains three images, left: original
images, middle: Canny detection results, and right: the detected most salient boundaries.
6
Conclusions
We have presented a novel graph-theoretic approach, named ratio contour, for extracting
perceptually salient boundaries from a set of noisy boundary fragments detected in real
images. The approach guarantees the global optimality without introducing any biases re-
lated to region area or boundary length, and exhibits promising performance in extracting
salient objects from real cluttered images. One potential extension of this research is to extract multiple salient objects that are overlapped or share part of boundaries by performing
ratio-contour algorithm iteratively. We are currently investigating this extension and plan
on reporting the result in the future.
Acknowledgements
The authors would like to thank David Jacobs and anonymous reviewers for important
comments. This work was funded, in part, by National Science Foundation grant EIA0312861, and the USC SOM-COEIT research development fund.
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1,631 | 2,483 | Approximate Analytical Bootstrap Averages for
Support Vector Classifiers
D?orthe Malzahn1,2
Manfred Opper3
Informatics and Mathematical Modelling, Technical University of Denmark,
R.-Petersens-Plads, Building 321, Lyngby DK-2800, Denmark
2
Institute of Mathematical Stochastics, University of Karlsruhe,
Englerstr. 2, Karlsruhe D-76131, Germany
3
Neural Computing Research Group, School of Engineering and Applied Science,
Aston University, Birmingham B4 7ET, United Kingdom
[email protected]
[email protected]
1
Abstract
We compute approximate analytical bootstrap averages for support vector classification using a combination of the replica method of statistical
physics and the TAP approach for approximate inference. We test our
method on a few datasets and compare it with exact averages obtained
by extensive Monte-Carlo sampling.
1
Introduction
The bootstrap method [1, 2] is a widely applicable approach to assess the expected qualities
of statistical estimators and predictors. Say, for example, in a supervised learning problem,
we are interested in measuring the expected error of our favorite prediction method on test
points 1 which are not contained in the training set D0 . If we have no hold out data, we
can use the bootstrap approach to create artificial bootstrap data sets D by resampling with
replacement training data from the original set D0 . Each data point is taken with equal
probability, i.e., some of the examples will appear several times in the bootstrap sample
and others not at all. A proxy for the true average test error can be obtained by retraining
the model on each bootstrap training set D, calculating the test error only on those points
which are not contained in D and finally averaging over all possible sets D.
While in general bootstrap averages can be approximated to any desired accuracy by the
Monte-Carlo method, by generating a large enough number of random samples, it is useful to have also analytical approximations which avoid the time consuming retraining of
the model for each new sample. Existing analytical approximations (based on asymptotic techniques) such as the delta method and the saddle point method require usually
explicit analytical formulas for the estimators of the parameters for a trained model (see
e.g. [3]). These may not be easily obtained for more complex models in machine learning such as support vector machines (SVMs). Recently, we introduced a novel approach
for the approximate calculation of bootstrap averages [4] which avoids explicit formulas
for parameter estimates. Instead, we define statistical estimators and predictors implicitly
1
The average is over the unknown distribution of training data sets.
as expectations with suitably defined pseudo-posterior Gibbs distributions over model parameters. Within this formulation, it becomes possible to perform averages over bootstrap
samples analytically using the so-called ?replica trick? of statistical physics [5]. The latter
involves a specific analytic continuation of the original statistical model. After the average, we are left with a typically intractable inference problem for an effective Bayesian
probabilistic model. As a final step, we use techniques for approximate inference to treat
the probabilistic model. This combination of techniques allows us to obtain approximate
bootstrap averages by solving a set of nonlinear equations rather than by explicit sampling.
Our method has passed a first test successfully on the simple case of Gaussian process (GP)
regression, where explicit predictions are still cheaply computed. Also, since the original
model is a smooth probabilistic one, the success of approximate inference techniques may
be not too surprising. In this paper, we will address a more challenging problem, that of
the support vector machine. In this case, the connection to a probabilistic model (a type
of GP) can be only established by introducing a further parameter which must eventually
diverge to obtain the SVM predictor. In this limit, the probabilistic model is becoming
highly nonregular and approaches a deterministic model. Hence it is not clear a priori if
our framework would survive these delicate limiting manipulations and still be able to give
good approximate answers.
2
Hard Margin Support Vector Classifiers
The hard margin SVM is a classifier which predicts binary class labels y = sign[ f?D0 (x)] ?
{?1, 1} for inputs x ? IRd based on a set of training points D0 = (z1 , z2 , . . . , zN ), where
zi = (xi , yi ) (for details see [6]). The usually nonlinear activation function f?D0 (x) (which
PN
we will call ?internal field?) is expressed as f?D0 (x) = i=1 yi ?i K(x, xi ), where K(x, x0 )
is a positive definite kernel and the set of ?i ?s is computed from D0 by solving a certain
convex optimization problem.
For bootstrap problems, we fix the pool of training data D0 , and consider the statistics
of vectors ?f D = (f?D (x1 ), . . . , f?D (xN )) at all inputs xi ? D0 , when the predictor f? is
computed on randomly chosen subsets D of D0 . Unfortunately, we do not have an explicit
analytical expression for ?fD , but it is obtained implicitly as the vector f = (f1 , . . . , fN )
which solves the constraint optimization problem
Minimize f T K?1 f
with fi yi ? 1
for all i such that (xi , yi ) ? D (1)
K is the kernel matrix with elements K(xi , xj ).
3
Deriving Predictors from Gibbs Distributions
In this section, we will show how to obtain the SVM predictor f?D formally as the expectation over a certain type of Gibbs distribution over possible f ?s in the form
Z
?fD = hf i = df f P [f |D]
(2)
with respect to a density P [f |D] = Z1 ?[f ] P (D|f ) which is constructed from a suitable
prior distribution ?[f ], a certain type of ?likelihood? P (D|f ) and a normalizing partition
function
Z
Z = df ?[f ] P (D|f ) .
(3)
Our general notation suggests that this principle will apply to a variety of estimators and
predictors of the MAP type.
To represent the SVM in this framework, we use a well established relation between SVM?s
and Gaussian process (GP) models (see e.g. [7, 8]). We choose the GP prior
?
1
exp ? f T K?1 f .
?[f ] = p
(4)
2
(2?)N ? ?N det(K)
The pseudo-likelihood 2 is defined by
Y
P (D|f ) =
P (zj |fj ) =
j: zj ?D
Y
j: zj ?D
?(yj fj ? 1)
(5)
where ?(u) = 1 for u > 0 and 0 otherwise. In the limit ? ? ?, the measure P [f |D] ?
?[f ]P (D|f ) obviously concentrates at the vector ?f which solves Eq. (1).
4
Analytical Bootstrap Averages Using the Replica Trick
With the bootstrap method, we would like to compute average properties of the estimator
?fD , Eq. (2), when datasets D are random subsamples of D0 . An important class of such
averages are of the type of a generalization error ? which are expectations of loss functions
g(f?D (xi ); xi , yi ) over test points i, i.e., those examples which are in D0 but not contained
in the bootstrap training set D. Hence, we define
h
i
?D (xi ); xi , yi )
N E
?
g(
f
X
D
s
,0
i
. 1
(6)
?=
N i=1
ED [?si ,0 ]
where ED [? ? ? ] denotes the expectation over random bootstrap samples D which are created
from the original training set D0 . Each sample D is represented by a vector of ?occupation?
numbers s = (s1 , . . . , sN ) where si is the number of times example zi appears in the set
PN
D and i=1 si = S. The Kronecker symbol, defined by ?si ,0 = 1 for si = 0 and 0 else,
guarantees that only realizations of bootstrap training sets D contribute to Eq. (6) which
do not contain the test point. For fixed bootstrap sample size S, the distribution of s i ?s is
multinomial. It is simpler (and does not make a big difference when S is sufficiently large)
when we work with a Poisson distribution for the size of the set D with S as the mean
number of data points in the sample. Then we get the simpler, factorizing joint distribution
P (s) =
N
Y
( S )si e?S/N
N
i=1
(7)
si !
S
for the occupation numbers si . From Eq. (7) we get ED [?si ,0 ] = e? N .
Since we can represent general loss functions g by their Taylor expansions in powers of
f?D (or polynomial approximations in case of non-smooth losses) it is sufficient to consider
only monomials g(f?D (x); x, y) = (f?D (x))r for arbitrary r in the following and regain the
general case at the end by resumming the series. Using the definition of the estimator ?fD ,
Eq. (2), the bootstrap expectation Eq. (6) can be rewritten as
o
r n
R Q
QN
?r
a
a
a
a sj
df ?[f ] fi
N ED ?si ,0 Z
j=1 (P (zj |fj ))
1 X
a=1
?(S) =
.
(8)
N i=1
ED [?si ,0 ]
which involves r copies3 , i.e. replicas f 1 , . . . , f r of the parameter vector f . If the partition
functions Z in the numerator of Eq. (8) were raised to positive powers rather than negative
2
3
It does not allow a full probabilistic interpretation [8].
The superscripts should NOT be confused with powers of the variables.
ones, one could perform the bootstrap average over the distribution Eq. (7) analytically. To
enable such an analytical average over the vector s (which is the ?quenched disorder? in
the language of statistical physics) one introduces the following ?trick? extensively used in
statistical physics of amorphous systems [5]. We introduce the auxiliary quantity
??
?
?
Z Y
r ?
N
N
?
Y
X
1
(P (zj |fja ))sj ? (9)
ED ??si ,0 Z n?r
df a ?[f a ] fia
?n (S) =
S
?
?
e? N N
a=1
i=1
j=1
for arbitrary real n, which allows to write
?(S) = lim ?n (S).
(10)
n?0
The advantage of this definition is that for integers n ? r, ?n (S) can be represented in
terms of n replicas f 1 , f 2 , . . . , f n of the original variable f for which an explicit average
over si ?s is possible. At the end of all calculations an analytical continuation to arbitrary
real n and the limit n ? 0 must be performed. For integer n ? r, we use the definition of
the partition function Eq. (3), exchange the expectation over datasets with the expectation
over f ?s and use the explicit form of the distribution Eq. (7) to perform the average over
bootstrap sets. The resulting expressions can be rewritten as 4
** r
++
\i N
Y
?n X
?n (S) =
fia
,
(11)
N i=1
a=1
\i
where hh? ? ? ii\i denotes an average with respect to the so called cavity distribution P\i for
replicated variables f~i = (fi1 , . . . , fin ) defined by
Z Y
N
1
df~j P (f~1 , . . . , f~N ) .
(12)
P\i (f~i ) ?
Li (f~i ) j=1,j6=i
Qn
QN
~
The joint distribution of replica variables P (f~1 , . . . , f~N ) ? a=1 ?[f a ]
j=1 Lj (fj ) is
defined by the new likelihoods
!#
"
n
Y
S
1?
P (zj |fja )
.
(13)
Lj (f~j ) = exp ?
N
a=1
5
TAP Approximation
We have mapped the original bootstrap problem to an inference problem for an effective
Bayesian probabilistic model (the hidden variables have the dimensionality N ? n) for
which we have to find a tractable approximation which allows analytical continuation of
n ? 0 and ? ? ?. We use the adaptive TAP approach of Opper and Winther [9] which
is often found to give more accurate results than, e.g., a simple mean field or a variational
Gaussian approximation. The ADATAP approach replaces the analytically intractable cavity distribution Eq. (12) by a Gaussian distribution. In our case this can be written as
T ~
1 ~T
~
P\i (f~i ) ? e? 2 f ?c (i)f +?c (i) f ,
(14)
where the parameters ?c and ?c are computed selfconsistently from the dataset D0 by
solving a set of coupled nonlinear equations. Details are given in the appendix.
The form Eq. (14) allows a simple way of dealing with the parameters n and ?. We utilize the exchangeability of variables fi1 , . . . , fin and assume replica symmetry and further
4
\i
\i
P\i (f~i ), Eq. (12), has the normalizing partition function ?n where ?n ? 1 for n ? 0.
introduce an explicit scaling of all parameters with ?. This scaling was found to make all
final expressions finite in the limit ? ? ?. We set
2
?ab
c (i) = ?c (i) = ? ?c (i)
?aa
c (i)
=
?0c (i)
=
? 2 ?0c (i)
for a 6= b
(15)
?ca (i)
and
= ??c (i) for all a = 1, . . . , n .
.
We also assume that ??c (i) = ? ?1 (?0c (i) ? ?c (i)) remains finite for ? ? ?. The ansatz
Eq. (15) keeps the number of adjustable parameters independent of n and allows to perform
the ?replica limit? n ? 0 and the ?SVM-limit? ? ? ? in all equations analytically before
we start the final numerical parameter optimization.
Computing the expectation Eq. (11) with Eq. (14) and (15) and resumming the power series
over r yields the final theoretical expression for Eq. (6)
!
p
N Z
?c (i) + u ??c (i)
1 X
dG(u) g
; xi , yi
?(S) =
(16)
N i=1
??c (i)
1
u2
where dG(u) = du(2?)? 2 e? 2 and g is an arbitrary loss function.
g(f?D (xi ); xi , yi ) = ?(?yi f?D (xi )) we obtain the bootstrapped classification error
!
N
yi ?c (i)
1 X
? ?p
?(S) =
N i=1
??c (i)
where ?(x) =
Rx
??
With
(17)
dG(u).
Besides the computation of generalization errors, we can use our method to quantify the
uncertainty of the SVM prediction at test points. This can be obtained by computing the
bootstrap distribution of the ?internal fields? f?D (xi ) at a test input xi . This is obtained
from Eq. (16) by inserting g(f?D (xi ); xi , yi ) = ?(f?D (xi ) ? h) using the Dirac ?-function
??c (i)
(h??c (i) ? ?c (i))2
?i (h) = p
exp ?
,
(18)
2(??c (i))
?2??c (i)
?c (i)
?c (i)
i.e., mci = ??
and Viic = ? (??
2 are the predicted mean and variance of the internal
c (i)
c (i))
field. (The predicted posterior variance of the internal field is (???c (i))?1 and goes to zero
as ? ? ? indicating the transition to a deterministic model.) It is possible to extend the
result Eq. (18) to ?real? test inputs x ?
/ D0 , which is of greater importance to applications.
This replaces ??c (i), ?c (i), ?c (i) by
??c (x)
?c (x)
=
=
K(x, x) ?
??c (x)
N
X
N
X
K(x, xi )??(i)Ti (x)
i=1
!?1
(19)
Ti (x)?(i)
i=1
?c (x)
=
(??c (x))2
N
X
(Ti (x))2 ?(i)
i=1
PN
with Ti (x) = j=1 K(x, xj )(I + diag(??)K)?1
ji . The parameters ??(i), ?(i), ?(i) are
determined from D0 according to Eq. (22), (23).
0.3
Wisconsin, N=683
1.5
0
200
400
600
Pima, N=532
0.2
1.0
200
400
Bootstrap sample size S
0.8
0.6
0.4
0.2
0.0
S: 0.376
T: 0.405
Crabs, N=200
0.0
0
Simulation: p(-1|x)
0 0.2 0.4 0.6 0.8 1
0.5
Sonar, N=208
0.1
1.0
Theory: p(-1|x)
0.4
2.0
0.14
0.12
0.10
0.08
0.06
0.04
Density
Bootstrapped classification error
0.5
600
0.0
-2
-1.5
-1
-0.5
0
0.5
1
Bootstrapped local field at a test input x
Figure 1: Left: Average bootstrapped generalization error for hard margin support vector
classification on different data sets (simulation: symbols, theory: lines). Right: Bootstrapped distribution of the internal field for Sonar data at a test input x ?
/ D0 . Most
distributions are Gaussian-like and in good agreement with the theory Eq. (18). We show
an atypical case (simulation: histogram, theory line) which nevertheless predicts the relative weights for both class labels fairly well. The inset shows true versus estimated values
of the probability p(?1|x) for predicting label y = ?1 .
6
Results for Bootstrap of Hard Margin Support Vector Classifiers
We determined the set of theoretical parameters by solving Eq. (21)-(23) for four benchmark data sets D0 [10] and different sample sizes S using a RBF kernel K(x, x0 ) =
Pd
exp(? 21 k=1 vk (xk ? x0k )2 )) with individually customized hyperparameters vk . The left
panel of Fig.1 compares our theoretical results for the bootstrapped learning curves obtained by Eq. (17) (lines) with results from Monte-Carlo simulations (symbols). The Gaussian approximation of the cavity distribution is based on the assumption that the model
prediction at a training input is influenced by a sufficiently large number of neighboring
inputs. We expect it to work well for sufficiently broad kernel functions. This was the case
for the Crabs and Wisconsin data sets where our theory is very accurate. It predicts correctly the interesting non-monotonous learning curve for the Wisconsin data (inset Fig.1,
left). In comparison, the Sonar and Pima data sets were learnt with narrow RBF kernels.
Here, we see that the quality of the TAP approximation becomes less good. However, our
results provide still a reasonable estimate for the bootstrapped generalization error at sample size S = N . While for practical applications of estimating the ?true? generalization
error using Efron?s 0.632 bootstrap estimator the case S = N is of main importance, it
is also interesting to discuss the limit of extreme oversampling S ? ?. Since the hard
margin SVM gains no additional information from the multiple presentation of the same
data point, in this limit all bootstrap sets D supply exactly the same information as the data
set D0 and the data average ED [. . . ] becomes trivial. Variances with respect to ED [. . . ]
go to zero. With Eq. (21)-(23), we can write the average prediction mi at input xi ? D0 as
PN
??(j)??c (j)
mi = j=1 yj ?j K(xi , xj ) with weights ?j = ??(j)+??
(yj mj ? yj mcj ) and recover
c (j)
for S ? ? the Kuhn-Tucker conditions ?i ? 0 and ?i ?(yi mi ? 1) = 0. The bootstrapped
generalization error Eq. (17) is found to converge to the approximate leave-one-out error
of Opper and Winther [8]
N
SV
X
1 X
?i
lim ?(S) =
? (?yi mci ) =
?
?
1
(20)
S??
N i=1
[K?1
SV ]ii
i
where the weights ?i are given by the SVM algorithm on D0 and KSV is the kernel matrix
on the set of SV?s. While the leave-one-out estimate is a non-smooth function of model
parameters, Efron?s 0.632 ?(N ) bootstrap estimate [2] of the generalization error approximated within our theory results in a differentiable expression Eq. (17) which may be used
for kernel hyperparameter estimation. Preliminary results are promising.
The right panel of Fig. 1 shows results for the bootstrapped distribution of the internal field
on test inputs x ?
/ D0 . The data set D0 contained N = 188 Sonar data and the bootstrap
is at sample size S = N . We find that the true distribution is often very Gaussian-like and
well described by the theory Eq. (18). Figure 1 (right) shows a rare case where a bi-modal
distribution (histogram) is found. Nevertheless, the Gaussian (line) predicted by our theory
estimates the probability p(?1|x) of a negative output quite accurately in comparison to
the probability obtained from the simulation.
Both SVM training and the computation of our approximate SVM bootstrap requires the
running of iterative algorithms. We compared the time ttrain for training a single SVM
on each of the four benchmark data sets D0 with the time ttheo needed to solve our theory
for SVM bootstrap estimates on these data for S = N . For sufficiently broad kernels we
find ttrain ? ttheo and our theory is reliable. The exception are extremely narrow kernels.
For the latter (Pima example in Fig.1 (left)) we find ttheo > ttrain where our theory is still
faster to compute but less reliable than a good Monte-Carlo estimate of the bootstrap.
7
Outlook
Our experiments on SVMs show that the approximate replica bootstrap approach appears
to be highly robust to apply to models which only fit into our framework after some delicate
limiting process. The SVM is also an important application because the prediction for each
dataset requires the solution of a costly optimization problem. Experiments on benchmark
data showed that our theory is appreciably faster to compute than a good Monte-Carlo
estimate of the bootstrap and yields reliable results for kernels which are sufficiently broad.
It will be interesting to apply our approach to other kernel methods such as kernel PCA.
Since our method is based on a fairly general framework, we will also investigate if it can
be applied to models where the bootstrapped parameters have a more complicated structure
like, e.g., trees or hidden Markov models.
Acknowledgments
DM gratefully acknowledges financial support by the Copenhagen Image and Signal Processing Graduate School and by the Postgraduate Programme ?Natural Disasters? at the
University of Karlsruhe.
Appendix: TAP Equations
The ADATAP approach computes the set of parameters ?c (i), ?c (i) by constructing an
? j (f~) = e? 21 f~T ?(j)f~+?(j)T f~ defining an auxiliary
alternative set of tractable likelihoods L
Qn
QN ? ~
Gaussian joint distribution PG (f~1 , . . . , f~N ) ? a=1 ?(f a ) j=1 L
j (fj ). We use replica
symmetry and a specific scaling of the parameters with ?: ? a (j) = ??(j), ?aa (j) =
?0 (j) = ? 2 ?0 (j) for all a, ?ab (j) = ?(j) = ? 2 ?(j) for a 6= b and ??(j) = ? ?1 (?0 (j)?
?(j)). All unknown parameters are found by moment matching: We assume that the first
two marginal moments mi = lim hhfia ii, Vii = lim hhfia fib ii?(mi )2 , ?ii = ? lim hhfia fia ?
n?0
n?0
n?0
fia fib ii of the variables f~i can be computed 1) by marginalizing PG and 2) by using the
relations between cavity distribution and marginal distributions P (f~i ) ? Li (f~i )P\i (f~i ) as
? i (f~i )P\i (f~i ) for all i = 1, . . . , N . This yields
well as PG (f~i ) ? L
S
(21)
?ii = ?cii 1 ? (1 ? e? N )?(?ci )
!
p
Vc 1 c 2
S
S
mi = mci 1 ? (1 ? e? N )?(?ci ) + yi (1 ? e? N ) ?(?ci ) + ? ii e? 2 (?i )
2?
S
Vii = Viic 1 ? (1 ? e? N )?(?ci ) + (1 ? yi mi )(yi mi ? yi mci )
where mci =
?c (i)
??c (i) ,
?c (i)
c
Viic = ? (??
2 , ?ii =
c (i))
?ii
mi
Vii
1
??c (i)
c
1?y m
and ?ci = ? i c i . Further
= (G)ii
= (G ?)i
= ? (G diag(?) G)ii
Vii
(22)
with the N ? N matrix G = (K?1 + diag(??))?1 and
1
(23)
?ii =
??(i) + ??c (i)
?(i) + ?c (i)
mi =
??(i) + ??c (i)
?(i) + ?c (i)
Vii = ?
(??(i) + ??c (i))2
We solve Eq. (21)-(23) by iteration using Eqs. (21) and (22) to evaluate the moments
{mi , Vii , ?ii } and Eq. (23) to update the sets of parameters {?c (i), ??c (i), ?c (i)} and
{?(i), ??(i), ?(i)}, respectively. Reasonable start values are ??(i) = ??, ?(i) = ???,
PN
?i ??
?(i) = yi ?? where ?? is obtained as the root of 0 = 1 ? N1 i=1 1+?
? (1 ? (1 ?
i ??
?S/N
c
c
e
)?(? )) with ? = ?0.5 and ?i are the eigenvalues of kernel matrix K.
References
[1] B. Efron. Ann. Statist., 7: 1-26, 1979.
[2] B. Efron, R. J. Tibshirani. An Introduction to the Bootstrap. Monographs on Statistics
and Applied Probability 57, Chapman & Hall, 1993.
[3] J. Shao, D. Tu, The Jackknife and Bootstrap, Springer Series in Statistics, Springer,
1995.
[4] D. Malzahn, M. Opper, A statistical mechanics approach to approximate analytical
Bootstrap averages, NIPS 15, S. Becker, S. Thrun, K. Obermayer eds., MIT Press,
2003.
[5] M. M?ezard, G. Parisi, M. A. Virasoro, Spin Glass Theory and Beyond, Lecture Notes
in Physics 9, World Scientific, 1987.
[6] B. Sch?olkopf, C. J. C. Burges, A. J. Smola (eds.), Advances in Kernel Methods: Support Vector Learning, MIT, Cambridge, MA, 1999.
[7] P. Sollich, Probabilistic interpretation and Bayesian methods for Support Vector Machines, In: ICANN99, pp.91-96, Springer 1999.
[8] M. Opper, O. Winther, Neural Computation, 12: 2655-2684, 2000.
[9] M. Opper, O. Winther, Phys. Rev. Lett. , 86: 3695, 2001.
[10] From http://www1.ics.uci.edu/?mlearn/MLSummary.html
and http://www.stats.ox.ac.uk/pub/PRNN/.
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1,632 | 2,484 | Insights from Machine Learning Applied to
Human Visual Classification
Arnulf B. A. Graf and Felix A. Wichmann
Max Planck Institute for Biological Cybernetics
Spemannstra?e 38
72076 T?ubingen, Germany
{arnulf.graf, felix.wichmann}@tuebingen.mpg.de
Abstract
We attempt to understand visual classification in humans using both psychophysical and machine learning techniques. Frontal views of human
faces were used for a gender classification task. Human subjects classified the faces and their gender judgment, reaction time and confidence
rating were recorded. Several hyperplane learning algorithms were used
on the same classification task using the Principal Components of the
texture and shape representation of the faces. The classification performance of the learning algorithms was estimated using the face database
with the true gender of the faces as labels, and also with the gender estimated by the subjects. We then correlated the human responses to the
distance of the stimuli to the separating hyperplane of the learning algorithms. Our results suggest that human classification can be modeled by
some hyperplane algorithms in the feature space we used. For classification, the brain needs more processing for stimuli close to that hyperplane
than for those further away.
1
Introduction
The last decade has seen tremendous technological advances in neuroscience from the microscopic to the macroscopic scale (e.g. from multi-unit recordings to functional magnetic
resonance imaging). On an algorithmic level, however, methods and understanding of brain
processes are still limited. Here we report on a study combining psychophysical and machine learning techniques in order to improve our understanding of human classification of
visual stimuli. What algorithms best describe the way the human brain classifies? Might
humans use something akin to hyperplanes for classification? If so, is the learning rule as
simple as in mean-of-class prototype learners or are more sophisticated algorithms better
candidates?
In our experiments, subjects and machines classified human faces according to gender. The
stimuli were presented and we collected the subjects? responses, which are the estimated
gender, reaction time and confidence rating (sec.2). For every subject two personal new
datasets were created: the original faces either with the true or with the subject?s labels
(true or estimated gender response). We then applied a Principal Component Analysis to
a texture and shape representation of the faces. Various algorithms such as Support Vec-
tor Machines, Relevance Vector Machines, Prototype and K-means Learners (sec.3) were
applied on this low-dimensional dataset with either the true or the subjects? labels. The
resulting classification performances were compared, the corresponding decision hyperplanes were computed and the distances of the faces to the hyperplanes were correlated
with the subjects? responses, the data being pooled among all subjects and stimuli or on a
stimulus-by-stimulus basis (sec.4).
2
Human Classification Behaviour
We used grey-scale frontal views of human faces taken from the MPI face database [1].
Because of technical inhomogeneities of the faces in the database we post-processed each
face such that all faces have same mean intensity, same pixel-surface area and are centred [2]. This processing stage is followed by a slight low-pass filtering of each face in
the database in order to eliminate, as much as possible, scanning artifacts. The database
is gender-balanced and contains 200 Caucasian faces (see Fig.1). Twenty-seven human
15
13
i
eigenvalue log(? )
14
12
11
10
9
8
7
0
20
40
60
80
100
120
140
160
180
200
index of component i
Figure 1: Female and male faces from the processed database (left). Eigenvalue spectrum
from the PCA of our texture-shape representation (see sec.4): ?min = 1.01 ? 103 (the last
eigenvalue being 0 is not plotted) and ?max = 2.47 ? 106 (right).
subjects were asked to classify the faces according to their gender and we recorded three
responses: estimated class (i.e. female/male), reaction time (RT) and, after each estimatedclass-response, a confidence rating (CR) on a scale from 1 (unsure) to 3 (sure). The stimuli
were presented sequentially to the subjects on a carefully calibrated display using a modified Hanning window (a raised cosine function with a raising time of ttransient = 500ms
and a plateau time of tsteady = 1000ms, for a total presentation time t = 2000ms per
face). Subjects were asked to answer as fast as possible to obtain perceptual, rather than
cognitive, judgements. Most of the time they responded well before the presentation of
the stimulus had ended (mean RT over all stimuli and subjects was approximately 900ms).
All subjects had normal or corrected-to-normal vision and were paid for their participation.
Most of them were students from the University of Tu? bingen and all of them were naive to
the purpose of the experiment.
Analysis of the classification performance of humans is based on signal detection theory [3] and we assume that, on the decision axis, the internal signal and noise distributions are Gaussian with same unit variance but different means. We define correct response probabilities for males (+) and females (?) as P+ = P (?
y = 1|y = 1) and
P? = P (?
y = ?1|y = ?1) where y? is the estimated class and y the true class of the stimulus. The discriminability of both stimuli can then be computed as: d0 = Z(P+ ) + Z(P? )
where Z = ??1 , and ? is the cumulative normal distribution with zero mean and unit
variance. Averaged across subjects we obtain d0 = 2.85 ? 0.73. This value indicates
that the classification task is comparatively easy for the subjects, although without being trivial (no ceiling effect). We observe a strong male bias (a large number of females
classified as males but very few males classified as females) and express this bias as:
? = Z 2 (P+ ) ? Z 2 (P? ) = 3.14 ? 2.61. The subplots of Fig.2 show the correlations
of (a) RT and classification error, (b) classification error and CR, and (c) RT and CR. First,
(a)
(b)
1
0.9
1.1
0.4
RT [s]
mean error
1.1
RT [s]
(c)
0.6
0.2
1
0.9
0.8
0.8
0
no error
error
1
2
CR
3
1
2
CR
3
Figure 2: Human classification behaviour: mutual dependencies of the subject?s responses.
RT?s are longer for incorrect answers than for correct ones (a). Second, a high CR is correlated with a low classification error (b) and thus subjects have veridical knowledge about
the difficulty of individual responses?this is certainly not the case in many low-level psychophysical settings. Third, the RT decreases as the CR increases (c), i.e. stimuli easy to
classify are also classified rapidly. It may thus be concluded that a high error (or equivalently a low CR) implies higher RT?s. This may suggest that patterns difficult to classify
need more computation, i.e. longer processing, by the brain than patterns easy to classify.
3
Machine Learning Classifiers
In the following, various hyperplane classification algorithms are expressed as weighted
dual space learners with different learning rules. Given a dataset {~xi , yi }pi=1 , we assume
classification is done in the input space, i.e. we consider linear kernels. Moreover, the input space is normalized since this has proved to be effective for some classifiers [4]. The
hyperplanes can be written using a weight (or normal) vector w
~ and an offset b in order to
yield a classification rule as y(~x) = sign(hw|~
~ xi + b) in the first three cases whereas in the
last one, the decision rule is a collection of hyperplanes. These classifiers are compared on
a two-dimensional toy dataset in Fig.3.
P
Support Vector Machine (SVM, [5]).
vector is given as: w
~ = P i ?i yi ~xi
P The 1weight
P
where ?
~ is obtained by maximising i ?i ? 2 ij yi yj ?i ?j h~xi |~xj i subject to i ?i yi =
0 and 0 ? ?i ? C where C is a regularisation parameter, determined using for instance
cross-validation. The offset is computed as: b = hyi ? hw|~
~ xi iii|0<?i <C .
Relevance Vector Machine
(RVM,
[6]).
The
weight
vector
(incorporating here the offP
set) is expressed as w
~ =
xi . A Bernoulli distribution describes P (~y |X, ?
~ ) where
i ?i ~
p
~
X = {~xi }i=1 . A hyperparameter ? is introduced in order to retrieve a sparse and smooth
~ Learning amounts to maximising
solution for ?
~ using a Gaussian distribution for P (~
?|?).
R
~
~
~ Since the latter is not integrable
P (~y |X, ?) = P (~y |X, ?
~ )P (~
?|?)d~
? with respect to ?.
analytically, the Laplace approximation (local approximation of the integrand by a Gaus~
sian) is used for resolution, yielding an iterative update scheme for ?.
Pp
~
xi (yi ?1)
Prototype Learner (Prot, [7]). Defining the prototypes p~? = Pi=1
=
p
i=1 (yi ?1)
P
xi as the centre of mass of each class, the weight vector is then expressed
i|yi =?1 ?i ~
P
2
P
hw|
~
xi i
p + k2
i ?i ~
as: w
~ = p~+ ? p~? = i ?i yi ~xi and the offset as: b = k~p? k ?k~
=
?
.
2
2
K-means Clustering with Nearest-neighbor Learner (Kmean, [8]). Once the K centres of the clusters for each class are computed using the K-means algorithm, one mean
~k? (~x) = P ?? (~x)~xi for each class is selected for a pattern ~x using the nearest-neighbour
i i
P
rule. The weight is then computed as: w(~
~ x) = ~k+ (~x) ? ~k? (~x) = i (?+
x) ? ??
x))~xi ,
i (~
i (~
~
2
~
2
k+ (~
x)k
the offset being given by: b(~x) = kk? (~x)k ?k
. Since the nearest-neighbour rule is
2
used for each pattern, the decision function is piecewise linear. The appropriate value of K
is determined for instance using cross-validation.
SVM
RVM
Prot
Kmean
Figure 3: Two-dimensional toy example illustrating classification for a SVM, RVM, Prot
and Kmean: the lines indicate the separating hyperplanes and the circles show the SVs,
RVs, prototypes or means respectively.
4
Human Classification Behaviour Revisited by Machine
Each face taken from the MPI database is represented by three vectors: an intensitystandardised texture map, and space-standardised x- and y-flowfields representing the
shape. The texture and shape vectors contain the information required to generate a specific
face from an ?average? reference face by putting each face of the database into correspondence. This format makes intensity and structural information about the faces explicit.
For the sake of numerical tractability, especially when using cross-validation methods, the
dimension of the image vectors has to be reduced to be usable by machine learning algorithms. We use Principal Component Analysis (PCA) to represent the concatenated textureand shape vectors of each face of size 3 ? 2562 in only 200 dimensions. In contrast to [9]
where PCA is applied only to the intensity (or pixel) information of standard images, the
use of PCA on the texture-shape representation forces learning machines to encode information about local structure and spatial correspondences.
It may be argued that the Principal Components of faces form a biologically-plausible basis for representation of faces [10], the so-called eigenfaces. Standard PCA on the images
themselves may thus be considered a biologically-plausible representation of faces. Given
that we use PCA on texture and shape, any claim of biological plausibility of our representation is somewhat tenuous, however.
The variant of PCA considered in this paper searches to express the eigenvectors as linear
combinations of the data vectors [10, 11]. It has the computational advantage over classic
PCA that it does not require the computation of a correlation between the dimensions of the
input but between the patterns of the input. For the stimuli considered here, the eigenvalue
spectrum as shown in Fig.1 is a monotonically decreasing function with no flat regions.
Thus PCA seems to be a sensible choice to represent the human face stimuli used in this
study (for a comparative study of PCA against Locally Linear Embedding, where PCA is
clearly superior for machine learning purposes, see [2]).
4.1 Classification Performance of Man and Machine
We compare the classification performance of man and machine in plot (a) of Fig.4. For
humans, the classification error on the true dataset is obtained by comparing the estimated
gender (class) to the true one. The classification error on the subject dataset, seen as a
measure for the mean consistency between subjects, is the mean over all subjects of the
(a)
0.35
0.3
(b)
classification error on true dataset
classification error on subject dataset
0.3
0.25
|?| to SH
error
0.25
0.2
0.15
0.2
0.15
0.1
0.1
0.05
0.05
0
man
SVM RVM
0
Prot Kmean
SVM
RVM
Prot
Kmean
no error
(d)
0.3
0.3
0.25
0.25
|?| to SH
|?| to SH
(c)
0.2
0.15
0.2
0.15
0.1
0.1
0.05
0.05
0
bin 1
bin 2
RT (binified)
error
bin 3
0
1
2
CR
3
Figure 4: Classification performance of man and machine on the true and subject datasets
(a) and correlation of behaviour of man (classification error, RT and CR) with machine (|?|)
for data pooled across subjects and stimuli (b-d).
mean classification error the other subjects made on the stimuli presented to each subject
by defining as an error when the other subjects responded differently than the considered
subject. For machines the classification error is obtained for the dataset with either the true
or the subject?s labels by using a single 10-fold cross-validation for RVM and Prot and a
double 10-fold cross-validation to determine C for SVMs and K for Kmeans. Since every
subject gets a different set of 148 randomly chosen faces from the 200 available, the mean
and standard error of the classification errors of man or machine for each dataset is plotted.
When classifying the dataset with the true labels, the combination of PCA with Kmean
yields a classification performance comparable to that of humans. The better classification
performance of Kmean compared to the simple prototype classifier may be explained by
the piecewise linear decision function. The prototype classifier, popular in neuroscience,
psychology and philosophy, performs on average worse than humans. Either humans do
not classify gender using prototypes in the linear PCA space, or they use prototypes but not
the PCA representation, or, of course, they use neither.
An intriguing fact is that SVMs and RVMs perform better than man, which is contrary to
what is reported in [5, 12] where human experts and machines are tested on digits from the
postal service database USPS. The context of the study presented here is different, however.
Our subjects were presented with human faces with some high-level features such as hair,
beards, or glasses removed. However, such features were likely used by the subjects to
create their representation of gender-space during their lifetime. Subjects are thus trained
on one type of data and tested on another. The machines on the other hand are trained and
tested on the same type of stimuli. This may explain the quite disappointing performance
of man in such a biologically-relevant task compared to machine.
However while humans learn the gender classification during their lifetime, it seems that
they solve the problem in a manner not as optimal from a statistical point of view as SVMs
or RVMs, but similarly to Kmeans and better than prototype learners.
The classification on the subject?s labels represents the ability of the classifier to learn what
we, based on the responses of the subjects, presume to be their internal representation of
face-space. The machines have more difficulty in learning the dataset with the subject?s
labels than the one with the true labels. Given our aim of re-creating the subjects? decision
boundaries using artificial classifiers?to compare human response patterns to machine
learning concepts?this makes SVM and RVM good, Kmean a mediocre, and the prototype
learner a rather poor candidate for this enterprise using the PCA representation.
4.2
Correlation of Behaviour of Man with Machine
Here we correlate the classification behaviours of man and machine. The results are summarized in plot (b-d) of Fig.4 and in Fig.5 where the parameters are averaged over the
subjects as before. This type of data analysis simply correlates the subject?s classification
~ xi i+b|
error, RT, and CR to the distance |?(~xi )| = |hw|~
of the face stimuli to the separatkwk
~
ing hyperplane (SH) obtained for the four types of hyperplane classifiers (in the case of
Kmean this distance is computed for each pattern with respect to the SH constructed using
its nearest mean of each class.). The hyperplanes are determined using cross-validation
(see above) on the dataset with the subject?s labels. The distance of a pattern ~x to the SH is
then calculated using the hyperplane computed using the training set corresponding to the
testing set ~x is belonging to. Notice that |?| reflects the construction rule of the classification hyperplane rather than the generalisation ability of the algorithm. SVMs maximise the
distance to the nearest point but not the average distance to all points, which may yield a
small value of |?|. Moreover the number of SVs, here ](SV ) = 74 ? 1 out of 148 patterns,
indicates that most patterns are close to the SH since classification is done in a space of
dimensionality 200. The number of RVs, ](RV ) = 9 ? 0, is comparatively small, this
sparsity being a well-known feature of RVMs.
Looking at Fig.4 (b-d) where the data is averaged across subjects and stimuli, we observe,
first, that the error of the subjects is high for |?| low, suggesting that elements near the SH
are more difficult to classify. Second |?| is low for high RT?s: the elements near the SH
seem to require more processing in the brain resulting in a higher RT. Third, the high CR
for high |?| indicates that the subjects are sure when stimuli are far from SH. Thus elements
far from the SH are classified more accurately, faster and with higher confidence than those
near to the SH. In order to compare the classifiers, we proceed as below.
Thus far we only considered data averaged across all face-stimuli. In the following we
assess the relation between the distance of each face representation to the SH and the mean
across all subjects of one of their responses (classification error, RT or CR) for that face.
We perform a non-parametric rank correlation analysis using the tied rank of the subject?s
response and of |?| across the set of 200 faces. Fig.5 presents the resulting scatter plots for
each classifier and for each type of response. Qualitatively, it seems that RVMs show most
and prototype learners least correlation between the subject?s response and |?|. In order to
compare these behaviours in a more quantitative manner, we indicate in fig.5 Spearman?s
rank correlation coefficients r (linear correlation between the tied rank of one variable and
the tied rank of the other) between the parameter of machine (distance of a face to the SH)
and the responses of man (classification error, RT and CR). Under
the null hypothesis of
?
no correlation between man and machine, the variable z = r N ? 1 follows a standard
normal distribution, N = 200 being the number of points in the scatter plots, and the
significance of the hypothesis test is computed as P = ?(z) where ? is the cumulative
normal distribution with zero mean and unit variance. We get for all cases P < 5 ? 10 ?4
which allows us to reject the null hypothesis with a high degree of confidence.
subject error
SVM r=?0.60 ? 0.02
RVM r=?0.65 ? 0.01
Prot r=?0.29 ? 0.02 Kmean r=?0.39 ? 0.02
200
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RT [s]
SVM r=?0.69 ? 0.01
RVM r=?0.71 ? 0.01
Prot r=?0.35 ? 0.02 Kmean r=?0.45 ? 0.02
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Kmean r=0.40 ? 0.02
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1
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Prot r=0.24 ? 0.02
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1
50 100 150 200
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1
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50 100 150 200
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1
1
50 100 150 200
|?| to SH
Figure 5: Scatter plots relating the subjects? responses (classification error, RT and CR)
to the distance |?| to the SH for each face in the database, the pooling being done across
subjects.
From these results it can be seen that RVMs correlate best all the subject?s responses with
the distances of the stimuli to the SH. The RT seems to be the performance measure where
most correlation between man and machine can be asserted although all performance measures are related as shown in sec.2. The prototype algorithm again behaves in the least
human-like manner of the four classifiers. The correlation between the classification behaviour of man and machine indicates for RVMs, and to some extent SVMs, that heads
far from the SH are more easily processed by humans. It may be concluded that the brain
needs to do more processing (higher RT) to classify stimuli close to the decision hyperplane, while stimuli far from it are classified more accurately (low error) and with higher
confidence (high CR). Human classification behaviour can thus be modeled by hyperplane
algorithms; a piecewise linear decision function as found in Kmean seems however to be
not biologically-plausible.
5
Conclusions
Our study compared classification of faces by man and machine. Psychophysically we
noted that a high classification error and a low CR for humans is accompanied by a longer
processing of information by the brain (a longer RT). Moreover, elements far from the SH
are classified more accurately, faster and with higher confidence than those near to the SH.
We also find three noteworthy results. First, SVMs and RVMs can learn to classify faces
using the subjects? labels but perform much better when using the true labels. Second, correlating the average response of humans (classification error, RT or CR) with the distance
to the SH on a face-by-face basis using Spearman?s rank correlation coefficients shows that
RVMs recreate human performance most closely in every respect. Third, the mean-of-class
prototype, its popularity in neuroscience notwithstanding, is the least human-like classifier
in all cases examined.
Obviously our results rely on a number of crucial assumptions: first, all measurements were
done in a linear space; second, the conclusions are only valid given the PCA representation
(pre-processing). Third, when rejecting the prototype learner as a plausible candidate for
human classification we assume the representativeness of our face space: we assume that
the mean face of our human subjects? is close to the sample mean of our database. Clearly,
a larger face database would be welcome, but is not trivial as we need texture maps and
the corresponding shapes. Finally, there is the different learning regime. Machines were
trained on the dataset proper, whereas humans were assumed to have extracted the relevant
information during their lifetime, and they were tested on faces with some cues removed.
However, the representation we used does allow the genders to be separated well, as shown
by the SVM classification performance on the true labels. As a first attempt to extend
the neuroscience community?s toolbox with machine learning methods we believe to have
shown the fruitfulness of this approach.
Acknowledgements
The authors would like to thank Volker Blanz for providing the face database and the flowfield algorithms. In addition we are grateful to Go? khan Bak?r, Heinrich B?ulthoff, Jez Hill,
Carl Rasmussen, Gunnar R?atsch, Bernhard Sch?olkopf and Vladimir Vapnik for helpful
comments and suggestions. AG was supported by a grant from the European Union (IST
2000-29375 COGVIS).
References
[1] V. Blanz and T. Vetter. A Morphable Model for the Synthesis of 3D Faces. Proc.
Siggraph99, pp. 187-194. Los Angeles: ACM Press, 1999.
[2] A. B. A. Graf and F. A. Wichmann. Gender Classification of Human Faces. Proceedings of the BMCV, Springer LNCS 2525, 491-501, 2002.
[3] T. D. Wickens. Elementary Signal Detection Theory. Oxford University Press, 2002.
[4] A. B. A. Graf, A. J. Smola, and S. Borer. Classification in a Normalized Feature
Space using Support Vector Machines. IEEE Transactions on Neural Networks 14(3),
597-605, 2003.
[5] V. N. Vapnik. The Nature of Statistical Learning Theory. Springer, 1995.
[6] M. E. Tipping. Sparse Bayesian learning and the relevance vector machine. Journal
of Machine Learning Research 1, 211-244, 2001.
[7] S. K. Reed. Pattern Recognition and Categorization. Cognitive Psychology 3, 382407, 1972.
[8] R. O. Duda, P.E. Hart, and D.G. Stork. Pattern Classification. John Wiley & Sons,
2001.
[9] L. Sirovich and M. Kirby. Low-Dimensional Procedure for the Characterization of
Human Faces. Journal of the Optical Society of America A, 4(3), 519-524, 1987.
[10] M. Turk and A. Pentland. Eigenfaces for Recognition. Journal of Cognitive Neuroscience, 3(1), 1991.
[11] B. Sch?olkopf, A. Smola, and K.-R. M?uller. Nonlinear Component Analysis as a Kernel Eigenvalue Problem. Neural Computation, 10, 1299-1319, 1998.
[12] J. Bromley and E. S?ackinger. Neural-network and K-nearest-neighbor Classifiers.
Technical Report 11359-910819-16TM, AT&T, 1991.
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1,633 | 2,485 | Approximate Planning in POMDPs with
Macro-Actions
Georgios Theocharous
MIT AI Lab
200 Technology Square
Cambridge, MA 02139
[email protected]
Leslie Pack Kaelbling
MIT AI Lab
200 Technology Square
Cambridge, MA 02139
[email protected]
Abstract
Recent research has demonstrated that useful POMDP solutions do not
require consideration of the entire belief space. We extend this idea with
the notion of temporal abstraction. We present and explore a new reinforcement learning algorithm over grid-points in belief space, which uses
macro-actions and Monte Carlo updates of the Q-values. We apply the
algorithm to a large scale robot navigation task and demonstrate that with
temporal abstraction we can consider an even smaller part of the belief
space, we can learn POMDP policies faster, and we can do information
gathering more efficiently.
1 Introduction
A popular approach to artificial intelligence is to model an agent and its interaction with
its environment through actions, perceptions, and rewards [10]. Intelligent agents should
choose actions after every perception, such that their long-term reward is maximized. A
well defined framework for this interaction is the partially observable Markov decision process (POMDP) model. Unfortunately solving POMDPs is an intractable problem mainly
due to the fact that exact solutions rely on computing a policy over the entire belief-space
[6, 3], which is a simplex of dimension equal to the number of states in the underlying
Markov decision process (MDP). Recently researchers have proposed algorithms that take
advantage of the fact that for most POMDP problems, a large proportion of the belief space
is not experienced [7, 9].
In this paper we explore the same idea, but in combination with the notion of temporally
extended actions (macro-actions). We propose and investigate a new model-based reinforcement learning algorithm over grid-points in belief space, which uses macro-actions
and Monte Carlo updates of the Q-values. We apply our algorithm to large scale robot navigation and demonstrate the various advantages of macro-actions in POMDPs. Our experimental results show that with macro-actions an agent experiences a significantly smaller
part of the belief space than with simple primitive actions. In addition, learning is faster because an agent can look further into the future and propagate values of belief points faster.
And finally, well designed macros, such as macros that can easily take an agent from a high
entropy belief state to a low entropy belief state (e.g., go down the corridor), enable agents
to perform information gathering.
2 POMDP Planning with Macros
We now describe our algorithm for finding an approximately optimal plan for a known
POMDP with macro actions. It works by using a dynamically-created finite-grid approximation to the belief space, and then using model-based reinforcement learning to compute
a value function at the grid points. Our algorithm takes as input a POMDP model, a resolution r, and a set of macro-actions (described as policies or finite state automata). The
output is a set of grid-points (in belief space) and their associated action-values, which via
interpolation specify an action-value function over the entire belief space, and therefore a
complete policy for the POMDP.
Dynamic Grid Approximation A standard method of finding approximate solutions to
POMDPs is to discretize the belief space by covering it with a uniformly-spaced grid (otherwise called regular grid as shown in Figure 1, then solve an MDP that takes those grid
points as states [1]. Unfortunately, the number of grid points required rises exponentially
in the number of dimensions in the belief space, which corresponds to the number of states
in the original space.
Recent studies have shown that in many cases, an agent actually travels through a very small
subpart of its entire belief space. Roy and Gordon [9] find a low-dimensional subspace of
the original belief space, then discretize that uniformly to get an MDP approximation to
the original POMDP. This is an effective strategy, but it might be that the final uniform
discretization is unnecessarily fine.
S2
(0,1,0)
S2
S2
(0.25,0.75,0)
(0.5,0.5,0)
(0,0,1)
S1 (1,0,0)
S3 S1
S3
RESOLUTION 1
RESOLUTION 2
S1
S3
RESOLUTION 4
Figure 1: The figure depicts various regular dicretizations of a 3 dimensional belief simplex. The belief-space is the surface of the triangle, while grid points are the intersection
of the lines drawn within the triangles. Using resolution of powers of 2 allows finer discretizations to include the points of coarser dicretizations.
In our work, we allocate grid points from a uniformly-spaced grid dynamically by simulating trajectories of the agent through the belief space. At each belief state experienced, we
find the grid point that is closest to that belief state and add it to the set of grid points that
we explicitly consider. In this way, we develop a set of grid points that is typically a very
small subset of the entire possible grid, which is adapted to the parts of the belief space
typically inhabited by the agent.
In particular, given a grid resolution r and a belief state b we can compute the coordinates (grid points gi ) of the belief simplex that contains b using an efficient method called
Freudenthal triangulation [2]. In addition to the vertices of a sub-simplex, Freundenthal
triangulation also produces barycentric coordinates ?i , with respect to gi , which enable
effective interpolation for the value of the belief state b from the values of the grid points
gi [1]. Using the barycentric coordinates we can also decide which is the closest grid-point
to be added in the state space.
Macro Actions The semi-Markov decision process (SMDP) model has become the preferred method for modeling temporally extended actions. An SMDP is defined as a fivetuple (S,A,P ,R,F ), where S is a finite set of states, A is the set of actions, P is the state
and action transition probability function, R is the reward function, and F is a function giving probability of transition times for each state-action pair. The transitions are at decision
epochs only. The SMDP represents snapshots of the system at decision points, whereas the
so-called natural process [8] describes the evolution of the system over all times. Discretetime SMDPs represent transition distributions as F (s0 , N |s, a), which specifies the expected number of steps N that action a will take before terminating in state s 0 starting in
state s. Q-learning generalizes nicely to discrete SMDPs. The Q-learning rule for discretetime discounted SMDPs is
Qt+1 (s, a) ? (1 ? ?)Qt (s, a) + ? R + ? k 0max 0 Qt (s0 , a0 ) ,
a ?A(s )
where ? ? (0, 1), and action a was initiated in state s, lasted for k steps, and terminated
in state s0 , while generating a total discounted sum of rewards of R. Several frameworks
for hierarchical reinforcement learning have been proposed, all of which are variants of
SMDPs, such as the ?options? framework [11].
Macro actions have been shown to be useful in a variety of MDP situations, but they have a
special utility in POMDPs. For example, in a robot navigation task modeled as a POMDP,
macro actions can consist of small state machines, such as a simple policy for driving down
a corridor without hitting the walls until the end is reached. Such actions may have the
useful property of reducing the entropy of the belief space, by helping a robot to localize
its position. In addition, they relieve us of the burden of having to choose another primitive
action based on the new belief state. Using macro actions tends to reduce the number of
belief states that are visited by the agent. If a robot navigates largely by using macro-actions
to move to important landmarks, it will never be necessary to model the belief states that
are concerned with where the robot is within a corridor, for example.
Algorithm Our algorithm works by building a grid-based approximation of the belief
space while executing a policy made up of macro actions. The policy is determined by
?solving? the finite MDP over the grid points. Computing a policy over grid points equally
spaced in the belief simplex, otherwise called regular discretization, is computationally
intractable since the number of grid-points grows exponentially with the resolution [2].
Nonetheless, the value of a belief point in a regular dicretization can be interpolated efficiently from the values of the neighboring grid-points [2]. On the other hand, in variable
resolution non-regular grids, interpolation can be computationally expensive [1]. A better
approach is variable resolution with regular dicretization which takes advantage of fast interpolation and increases resolution only in the necessary areas [12]. Our approach falls
in this last category with the addition of macro-actions, which exhibit various advantages
over approaches using primitive actions only. Specifically, we use a reinforcement-learning
algorithm (rather than dynamic programming) to compute a value function over the MDP
states. It works by generating trajectories through the belief space according to the current
policy, with some added exploration. Reinforcement learning using a model, otherwise
called real time dynamic programming (RTDP) is not only better suited for huge spaces
but in our case is also convenient in estimating the necessary models of our macro-actions
over the experienced grid points.
While Figure 2 gives a graphical explanation of the algorithm, below we sketch the entire
algorithm in detail:
1. Assume a current true state s. This is the physical true location of the agent, and
it should have support in the current belief state b (that is b(s) 6= 0).
2. Discretize the current belief state b ? gi , where gi is the closest grid-point
(with the maximum barycentric coordinate) in a regular discretization of the belief
space. If gi is missing add it to the table. If the resolution is 1 initialize its value
to zero otherwise interpolate its initial value from coarser resolutions.
b
g1
b?
g2
b??
g
g3
Figure 2: The agent finds itself at a belief state b. It maps b to the grid point g, which has
the largest barycentric coordinate among the sub-simplex coordinates that contain b. Now,
it needs to do a value backup for that grid point. It chooses a macro action and executes it
starting from the chosen grid-point, using the primitive actions and observations that it does
along the way to update its belief state. It needs to get a value estimate for the resulting
belief state b00 . It does so by using the barycentric coordinates from the grid to interpolate a
value from nearby grid points g1, g2, and g3. In case the nearest grid-point g i is missing, it
is interpolated from coarser resolutions and added to the representation. If the resolution is
1, the value of gi is initialized to zero. The agent executes the macro-action from the same
grid point g multiple times so that it can approximate the probability distribution over the
resulting belief-states b00 . Finally, it can update the estimated value of the grid point g and
execute the macro-action chosen from the true belief state b. The process repeats from the
next true belief state b0 .
3. Choose a random action % of the time. The rest of the time choose the best
macro-action ? by interpolating over the Q values of the vertices of the subP
simplex that contains b: ? = argmax??M |S|+1
i=1 ?i Q(gi , ?).
4. Estimate E [R(gi , ?) + ? t V (b0 )] by sampling:
(a) Sample a state s from the current grid-belief state gi (which like all belief
states represents a probability distribution over world states).
i. Set t = 0
ii. Choose the appropriate primitive action a according to macro-action ?.
iii. Sample the next state s0 from the transition model T (s, a, ?).
iv. Sample an observation z from observation model O(a, s0 , ?).
v. Store the reward R(gi , ?) := R(gi , ?) + ? t ? R(s, a). For faster learning
we use reward-shaping: R(gi , ?) := R(gi , ?) + ? t+1 V (s0 ) ? ? t V (s),
where V (s) are the values of the underlying MDP
P[5].
vi. Update the belief state: b0 (j) := ?1 O(a, j, z) i?S T (i, a, j), for all
states j, where ? is a normalizing factor.
vii. Set t = t+1, b = b0 , s = s0 and repeat from step 4(a)ii until ? terminates.
(b) Compute the value of the resulting belief state b0 by interpolating over the
P|S|+1
vertices in the resulting belief sub-simplex: V (b0 ) =
?i V (gi ). If
i
the closest grid-point (with the maximum barycentric coordinate) is missing,
interpolate it from coarser resolutions, and add it to the hash-table.
(c) Repeat steps 4a and 4b multiple times, and average the estimate
[R(gi , ?) + ? t V (b0 )].
5. Update the state action value: Q(gi , ?) = (1 ? ?)Q(gi , ?) + ? [R + ? t V (b0 )].
6. Update the state value: V (gi ) = argmax??M Q(gi , ?).
7. Execute the macro-action ? starting from belief state b until termination. During
execution, generate observations by sampling the POMDP model, starting from
the true state s. Set b = b0 and s = s0 and go to step 2.
8. Repeat this learning epoch multiple times starting from the same b.
3 Experimental Results
We tested this algorithm by applying it to the problem of robot navigation, which is a
classic sequential decision-making problem under uncertainty. We performed experiments
in a corridor environment, shown in Figure 3. Such a topological map can be compiled
into POMDPs, in which the discrete states stand for regions in the robot?s pose space (for
example 2 square meters in position and 90 degrees in orientation). In such a representation, the robot can move through the different environment states by taking actions such as
?go-forward?, ?turn-left?, and ?turn-right?. A macro-actions is implemented as a behavior
(could be a POMDP policy) that takes as inputs observations and outputs actions. In our
experiments we have a macro-action for going down the corridor until the end. In this navigation domain, our robot can only perceive sixteen possible observations, which indicate
the presence of a wall and opening on the four sides of the robot. The observations are
extracted from trained neural nets where the inputs are local occupancy grids constructed
from sonar sensors and outputs are probabilities of walls and openings [4]. The POMDP
model of the corridor environment has a reward function with value -1 in every state, except
for -100 for going forward into a wall and +100 for taking any action from the four-way
junction.
66
14
24
26
18
40
2
32
40
96
40
20
20
20
26
36
4
40
8
32
Figure 3: The figure on the left shows the floor plan of our experimental environment. The
figure on the right is a topological map representation of the floor, which compiles into a
POMDP with 1068 world states. The numbers next to the edges are the distances between
the nodes in meters.
We ran the algorithm starting with resolution 1. When the average number of training steps
stabilized we increased the resolution by multiplying it by 2. The maximum resolution we
considered was 4. Each training episode started from the uniform initial belief state and
was terminated when the four-way junction was reached or when more than 200 steps were
taken. We ran the algorithm with and without the macro-action go-down-the-corridor. We
compared the results with the QMDP heuristic which first solves the underlying MDP and
then given any belief state, chooses the action that maximizes the dot product of the belief
P|S|
and Q values of state action pairs: QM DPa = argmaxa s=1 b(s)Q(s, a).
With Reward Shaping The learning results in Figure 4 demonstrate that learning with
macro-actions requires fewer number of training steps, which means the agent is getting
to the goal faster. An exception is when the resolution is 1, where training with only
primitive actions requires a small number of steps too. Nonetheless as we increase the
resolution, training with primitive actions only does not scale well, because the number of
states increases dramatically.
In general, the number of grid points used with or without macro-actions is significantly
smaller than the total number of points allowed for regular dicretization. For example, for
a regular discretization the number of grid points can be computed by the formula given
10
in [2], (r+|S|?1)!
for r = 4 and |S| = 1068. Our algorithm with macro
r!(|S|?1)! , which is 5.4
actions uses only about about 3000 and with primitive actions only about 6500 grid points.
primitive
macro
Number of States
7000
Number of states
Average # of training steps
Training Steps
180
160
140
120
100
80
60
40
20
0
primitive
macro
6000
5000
4000
3000
2000
1000
0
0 100 200 300 400 500 600 700 800 900
0 100 200 300 400 500 600 700 800 900
Number of training episodes
Number of training episodes
Figure 4: The graph on the left shows the average number of training-steps per episode as a
function of the number of episodes. The graph on the right shows the number of grid-points
added during learning. The sharp changes in the graph are due to the resolution increase.
We tested the policies that resulted from each algorithm by starting from a uniform initial
belief state and a uniformly randomly chosen world state and simulating the greedy policy
derived by interpolating the grid value function. We tested our plans over 200 different
sampling sequences and report the results in Figure 5. A run was considered a success if
the robot was able to reach the goal in fewer than 200 steps.
Success
qmdp
primitive
macro
Success %
80
60
40
20
Testing Steps
140
primitive
macro
120
Steps to goal
100
100
80
60
40
20
0
0
0
1
2
3
4
Resolution
0
1
2
3
4
Resolution
Figure 5: The figure on the left shows the success percentage for the different methods
during testing. The results are reported after training for each resolution. The graph on the
right shows the number of steps during testing. For the primitive-actions only algorithm
we report the result for resolution 1 only, since it was as successful as the macro-action
algorithm.
From Figure 5 we can conclude that the QMDP approach can never be 100% successful, while the primitive-actions algorithm can perform quite well with resolution 1 in this
environment. It is also evident from Figure 5 that as we increase the resolution, the macroaction algorithm maintains its robustness while the primitive-action algorithm performs
considerably worse, mainly due to the fact that it requires more grid-points. In addition,
when we compared the average number of testing steps for resolution 1 the macro-action
algorithm seems to have learned a better policy. The macro-action policy policy seems to
get worse for resolution 4 due to the increasing number of grid-points added in the repre-
sentation. This means that more training is required.
Without Reward Shaping We also performed experiments to investigate the effect of
reward-shaping. Figure 6 shows that with primitive actions only, the algorithm fails completely. However, with macro-actions the algorithm still converges and is more successful
than the QMDP-heuristic.
Success
100
primitive
macro
primitive
macro
80
Success %
Average # of training steps
Training Steps
200
190
180
170
160
150
140
130
120
110
60
40
20
0
0 100 200 300 400 500 600 700 800 900
Number of training episodes
0
1
Resolution
Figure 6: The The graph on the left shows the average number of training-steps (without
reward shaping). The figure on the right shows the success percentage
Information Gathering Apart from simulated experiments we also wanted to compare
the performance of QMDP with the macro-action algorithm on a platform more closely
related to a real robot. We used the Nomad 200 simulator and describe a test in Figure 7 to
demonstrate how our algorithm is able to perform information gathering, as compared to
QMDP.
4 Conclusions
In this paper we have presented an approximate planning algorithm for POMDPs that uses
macro-actions. Our algorithm is able to solve a difficult planning problem, namely the
task of navigating to a goal in a huge space POMDP starting from a uniform initial belief,
which is more difficult than many of the tasks that similar algorithms are tested on. In
addition, we have presented an effective reward-shaping approach to POMDPs that results
in faster training (even without macro-actions).
In general macro-actions in POMDPs allow us to experience a smaller part of the state
space, backup values faster, and do information gathering. As a result we can afford to
allow for higher grid resolution which results in better performance. We cannot do this
with only primitive actions (unless we use reward shaping) and it is completely out of the
question for exact solution over the entire regular grid. In our current research we are
investigating methods for dynamic discovery of ?good? macro-actions given a POMDP.
References
[1] M. Hauskrecht. Value-function approximations for partially observable Markov decision processes. Journal of Artificial Intelligence Research, 13:33?94, 2000.
[2] W. S. Lovejoy. Computationally feasible bounds for partially observed Markov decision processes. Operations Research, 39(1):162?175, January-February 1991.
[3] O. Madani, S. Hanks, and A. Gordon. On the undecidability of probabilistic planning and
infinite-horizon partially observable Markov decision processes. In Proceedings of the Sixteenth
National Conference in Artificial Intelligence, pages 409?416, 1999.
J2
J3
START
J1
J5
J4
GOAL
Figure 7: The figure shows the actual floor as it was designed in the Nomad 200 simulator.
For the QMDP approach the robot starts from START with uniform initial belief. After
reaching J2 the belief becomes bi-modal concentrating on J1 and J2. The robot then keeps
turning left and right. On the other hand, with our planning algorithm, the robot again starts
from START and a uniform initial belief. Upon reaching J2 the belief becomes bimodal
over J1 and J2. The agent resolves its uncertainty by deciding that the best action to take is
the go-down-the-corridor macro, at which point it encounters J3 and localizes. The robot
then is able to reach its goal by traveling from J3, to J2 , J1, J4, and J5.
[4] S. Mahadevan, G. Theocharous, and N. Khaleeli. Fast concept learning for mobile robots.
Machine Learning and Autonomous Robots Journal (joint issue), 31/5:239?251, 1998.
[5] A. Y. Ng, D. Harada, and S. Russell. Theory and application to reward shaping. In Proceedings
of the Sixteenth International Conference on Machine Learning, 1999.
[6] C. Papadimitriou and J. Tsitsiklis. The complexity of Markov decision processes. Mathematics
of Operation Research, 12(3), 1987.
[7] J. Pineau, G. Gordon, and S. Thrun. Point-based value iteration: An anytime algorithm for
POMDPs. In International Joint Conference on Artificial Intelligence, 2003.
[8] M. Puterman. Markov Decision Processes: Discrete Dynamic Stochastic Programming. John
Wiley, 1994.
[9] N. Roy and G. Gordon. Exponential family PCA for belief compression in POMDPs. In
Advances in Neural Information Processing Systems, 2003.
[10] S. J. Russell and P. Norvig. Artificial Intelligence: A Modern Approach. Prentice Hall, 2nd
edition, 2003.
[11] R. S. Sutton, D. Precup, and S. Singh. Between MDPs and semi-MDPs: A framework for temporal abstraction in reinforcement learning. Artificial Intelligence, pages 112:181?211, 1999.
[12] R. Zhou and E. A. Hansen. An improved grid-based approximation algorithm for POMDPs. In
Proceedings of the Seventeenth International Conference in Artificial intelligence (IJCAI-01),
Seattle, WA, August 2001.
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1,634 | 2,486 | Feature Selection in Clustering Problems
Volker Roth and Tilman Lange
ETH Zurich, Institut f. Computational Science
Hirschengraben 84, CH-8092 Zurich
Tel: +41 1 6323179
{vroth, tilman.lange}@inf.ethz.ch
Abstract
A novel approach to combining clustering and feature selection is presented. It implements a wrapper strategy for feature selection, in the
sense that the features are directly selected by optimizing the discriminative power of the used partitioning algorithm. On the technical side, we
present an efficient optimization algorithm with guaranteed local convergence property. The only free parameter of this method is selected
by a resampling-based stability analysis. Experiments with real-world
datasets demonstrate that our method is able to infer both meaningful
partitions and meaningful subsets of features.
1
Introduction
The task of selecting relevant features in classification problems can be viewed as one of
the most fundamental problems in the field of machine learning. A major motivation for
selecting a subset of features from which a learning rule is constructed is the interest in
sparse and interpretable rules, emphasizing only a few relevant variables. In supervised
learning scenarios, feature selection has been studied widely in the literature. The methods
used can be subdivided in filter methods and wrapper methods. The main difference is
that a wrapper method makes use of the classifier, while a filter method does not. From a
conceptual viewpoint, wrapper approaches are clearly advantageous, since the features are
selected by optimizing the discriminative power of the finally used classifier.
Selecting features in unsupervised learning scenarios is a much harder problem, due to the
absence of class labels that would guide the search for relevant information. Problems of
this kind have been rarely studied in the literature, for exceptions see e.g. [1, 9, 15]. The
common strategy of most approaches is the use of an iterated stepwise procedure: in the
first step a set of hypothetical partitions is extracted (the clustering step), and in the second step features are scored for relevance (the relevance determination step). A possible
shortcoming is the way of combining these two steps in an ?ad hoc? manner: usually the
relevance determination mechanism implements a filter approach and does not take into
account the properties of the clustering method used. Usual scoring methods make an implicit independence assumption, while ignoring feature correlations. It is thus of particular
interest to combine wrapper selection strategies and clustering methods. The approach
presented in this paper can be viewed as a method of this kind. It combines a Gaussian
mixture model with a Bayesian feature selection principle. The usual combinatorial problems involved with wrapper approaches are overcome by using a Bayesian marginalization
mechanism. We present an efficient optimization algorithm for our model with guaranteed
convergence to a local optimum.
The only free model parameter is selected by a resampling-based stability analysis. The
problem of many ambiguous and equally high-scoring splitting hypotheses, which seems to
be a an inherent shortcoming of many other approaches, is successfully overcome. A comparison with ground-truth labels in control experiments indicates that the selected models
induce sample clusters and feature subsets which both provide a clear interpretation.
Our approach to combining clustering and feature selection is based on a Gaussian mixture model, which is optimized by way of the classical expectation-maximization (EM)
algorithm. In order to incorporate the feature selection mechanism, the M-step is first reformulated as a linear discriminant analysis (LDA) problem, which makes use of the ?fuzzy
labels? estimated in the preceding E-step. We then use the well-known identity of LDA and
linear regression to restate the M-step in a form which easily allows us to regularize the estimation problem by specifying a prior distribution over the regression coefficients. This
distribution has the functional form of an Automatic Relevance Determination (ARD) prior.
For each regression coefficient, the ARD prior contains a free hyperparameter, which encodes the ?relevance? of the corresponding variable in the linear regression. In a Bayesian
marginalization step, these hyperparameters are then integrated out. We finally arrive at an
M-step with integrated feature selection mechanism.
2
Clustering and Bayesian relevance determination
Gaussian mixtures and LDA. The dataset is given as a collection of N samples x i ? Rd .
For the purpose of finding clusters, consider now a Gaussian mixture model with 2 mixture
components which share an identical covariance matrix ?. Under this model, the loglikelihood for the dataset reads
P
PN
2
lmix = i=1 log
(1)
?=1 ?? ?(xi ; ?? , ?) ,
where the mixing proportions ?? sum to one, and ? denotes a Gaussian density. The
classical EM-algorithm, [2], provides a convenient method for maximizing l mix :
?? ?(xi ; ?? , ?)
E-step: set p?i = Prob(xi ? class ?) = P2
.
?=1 ?? ?(xi ; ?? , ?)
PN
N
2
1 XX
i=1 p?i xi
p?i (xi ? ?? ) (xi ? ?? )> .
M-step: set ?? = P
, ?=
N
N
p
i=1 ?i
?=1 i=1
The likelihood equations in the M-step can be viewed as weighted mean and covariance
maximum likelihood estimates in a weighted and augmented problem: one replicates the
N observations 2 times, with the ?-th such replication having observation weights p ?i . In
[5] it is proven that the M-step can be carried out via a weighted and augmented linear discriminant analysis (LDA). Following [6], any LDA problem can be restated as an optimal
scoring problem. Let the class-memberships of the N data vectors be coded as a matrix Z,
the i, ?-th entry of which equals one if the i-th observation belongs to class ?. The point
of optimal scoring is to turn categorical variables into quantitative ones: the score vector
? assigns the real number ?? to the entries in the ?-th column of Z. The simultaneous
estimation of scores and regression coefficients ? constitutes the optimal scoring problem:
minimize
M (?, ?) = kZ? ? X?k22
(2)
1
2
under the constraint N kZ?k2 = 1. The notion k ? k22 stands for the squared `2 ?norm, and
X denotes the (centered) data matrix of dimension N ? d. In [6] an algorithm for carrying
out this optimization has been proposed, whose main ingredient is a linear regression of the
data matrix X against the scored indicator matrix Z?.
Returning from a standard LDA-problem to the above weighted and augmented problem, it
turns out that it is not necessary to explicitly replicate the observations: the optimal scoring
version of LDA allows an implicit solution of the augmented problem that still uses only N
observations. Instead of using a response indicator matrix Z, a blurred response matrix Z?
is employed, whose rows consist of the current class probabilities for each observation. At
each M-step this Z? enters in the linear regression, see [5]. After iterated application of the
E- and M-step, an observation xi is finally assigned to the class ? with highest probability
of membership p?i . Note that the EM-iterations converge to a local maximum.
LDA and Automatic Relevance Determination. We now focus on incorporating the automatic feature selection mechanism into the EM-algorithm. According to [6], the 2-class
LDA problem in the M-step can be solved by the following algorithm:
? 0 = 1, and
1. Choose an initial N -vector of scores ? 0 which satisfies N ?1 ? T0 Z? T Z?
?
?
is orthogonal to a k-vector of ones, 1k . Set ? = Z? 0 ;
c? = X(X T X)?1 X T ? ? ? X?.
2. Run a linear regression of X on ? ? : ?
The feature selection mechanism can now be incorporated in the M-step by imposing a
certain constraint on the linear regression. In [6, 4] it has been proposed to use a ridgetype penalized regression. Taking a Bayesian perspective, such a ridge-type penalty can
be interpreted as introducing a spherical Gaussian prior over the coefficients: p(?) =
N (0, ??1 I). The main idea of incorporating an automatic feature selection mechanism
consists of replacing the Gaussian prior with an automatic relevance determination (ARD)
prior1 of the form
P
Q
2
(3)
p(?| ?) = i N (0, ??1
i ) ? exp[?
i ?i ?i ].
In this case, each coefficient ?i has its own prior variance ??1
i . Note that in the above
ARD framework only the functional form of the prior (3) is fixed, whereas the parameters
?i , which encode the ?relevance? of each variable, are estimated from the data. In [3] the
following Bayesian inference procedure for the prior parameters has been introduced: given
i
exponential hyperpriors (the variances ??1
must be nonnegative), p(?i ) = ?2 exp{? ??
i
2 },
one can analytically integrate out the hyperparameters from the prior distribution over the
coefficients ?i :
R?
?
(4)
p(?i ) = 0 p(?i |?i )p(?i ) d?i = ?2 exp{? ?|?i |}.
Switching to the maximum a posteriori (MAP) solution in log-space, this marginalization
directly leads us to the following penalized functional:
? k?k1 ,
? ? X?k2 + ?
M (?, ?) = kZ?
(5)
2
?
? ? ? has the role of a Lagrange parameter in the `1 ?constrained problem: minwhere ?
? ? X?k2 subject to k?k1 < ?. In the statistical literature, this model is known
imize kZ?
2
as the Least Absolute Shrinkage and Selection Operator (LASSO) model, [14].
Returning to equation (3), we are now able to interpret the LASSO estimate as a Bayesian
feature selection principle: for the purpose of feature selection, we would like to estimate
the value of a binary selection variable S for each feature: Si equals one, if the i-th feature
is considered relevant for the given task, and zero otherwise. Taking into account feature correlations, estimation of Si necessarily involves searching the space of all possible
subsets of features containing the i-th one. In the Bayesian ARD formalism, this combinatorial explosion of the search space is overcome by relaxing the binary selection variable to
a positive real-valued variance of a Gaussian prior over each component of the coefficient
vector. Following the Bayesian inference principle, we introduce hyperpriors and integrate
out these variances, and we finally arrive at the `1 ?constrained LASSO problem.
Optimizing the final model. Since space here precludes a detailed discussion of ` 1 ?
constrained regression problems, the reader is referred to [12], where a highly efficient
algorithm with guaranteed global convergence has been proposed. Given this global convergence in the M-step, for the EM-model we can guarantee convergence to a local maximum of the constrained likelihood. Consider two cases: (i) the unconstrained solution
is feasible. In this case our algorithm simply reduces to the standard EM procedure, for
which is it known that in every iteration the likelihood monotonically increases; (ii) the
`1 ?constraint is active. Then, in every iteration the LASSO algorithm maximizes the likelihood within the feasible region of ?-values defined by k?k1 < ?. The likelihood cannot
be decreased in further stages of the iteration, since any solution ? found in a preceding
iteration is also a valid solution for the actual problem (note that ? is fixed!). In this case,
the algorithm has converged to a local maximum of the likelihood within the constraint
region.
1
For an introduction to the ARD principle the reader is referred to [10].
3
Model selection
Our model has only one free parameter, namely the value of the `1 ?constraint ?. In the
following we describe a method for selecting ? by observing the stability of data partitions.
For each of the partitions which we have identified as ?stable?, we then examine the fluctuations involved in the feature selection process. It should be noticed that the concept of
measuring the stability of solutions as a means of model selection has been successfully
applied to several unsupervised learning problems, see e.g. [8, 11].
We will usually find many potential splits of a dataset, depending on how many features are
selected: if we select only one feature, it is likely to find many competing hypotheses for
splits. The problem is that most of the feature vectors usually vote for a different partition.
If, on the other hand, we select too many features, we face the usual problems of finding
structure in high-dimensional datasets: our functional which we want to optimize will have
many local minima, and with high probability, the EM-algorithm will find suboptimal solutions. Between these two extremes, we can hope to find relatively stable splits, which are
robust against noise and also against inherent instabilities of the optimization method.
To obtain a quantitative measure of stability, we propose the following procedure: run the
class discovery method once, corrupt the data vectors by a small amount of noise, repeat
the grouping procedure, and calculate the Hamming distance between the two partitions as
a measure of (in-)stability. For computing Hamming distances, the partitions are viewed
as vectors containing the cluster labels. Simply taking the average stability over many
such two-sample comparisons, however, would not allow an adequate handling of situations where there are two equally likely stable solutions, of which the clustering algorithm
randomly selects one. In such situations, the averaged stability will be very low, despite the
fact that there exist two stable splitting hypotheses. This problem can be overcome by looking for compact clusters of highly similar partitions, leading to the following algorithm:
Algorithm for identifying stable partitions: for different values of the `1 ?constraint ? do
(i) compute m noisy replications of the data
(ii) run the class discovery algorithm for each of these datasets
(iii) compute the m ? m matrix of pairwise Hamming distances between all partitions
(iv) cluster the partitions into compact groups and score the groups by their frequency
(v) select dominant groups of partitions and choose representative partitions
In step (i) a ?suitable? noise level must be chosen a priori. In our experiments we make
use of the fact that we have normalized the input data to have zero mean and unit variance.
Given this normalization, we then add Gaussian noise with 5% of the total variance in the
dataset, i.e. ? 2 = 0.05. In step (iii) we use Hamming distances as a dissimilarity measure
between partitions. In order to make Hamming distances suitable for this purpose, we have
to consider the inherent permutation symmetry of the clustering process: a cluster called
?1? in the first partition can be called ?2? in the second one. When computing the pairwise Hamming distances, we thus have to minimize over the two possible permutations of
cluster labels. Steps (iv) and (v) need some further explanation: the problem of identifying
compact groups in datasets which are represented by pairwise distances can by solved by
optimizing the pairwise clustering cost function, [7]. We iteratively increase the number of
clusters (which is a free parameter in the pairwise clustering functional) until the average
dissimilarity in each group does not exceed a predefined threshold. Reasonable problemspecific thresholds can be defined by considering the following null-model: given N samples, the average Hamming distance between two randomly drawn 2?partitions P 1 and P2
is roughly dHamming (P1 , P2 ) ? N/2. It may thus be reasonable to consider only clusters
which are several times more homogeneous than the expected null-model homogeneity (in
the experiments we have set this threshold to 10 times the null-model homogeneity).
For the clusters which are considered homogeneous, we observe their populations, and
out of all models investigated we choose the one leading to the partition cluster of largest
size. For this dominating cluster, we then select a prototypical partition. For selecting such
prototypical partitions in pairwise clustering problems, we refer the reader to [13], where
it is shown that the pairwise clustering problem can be equivalently restated as a k-means
problem in a suitably chosen embedding space. Each partition is represented as a vector
Partitions
8
0
Noisy resample 99
0
Dissimilarity matrix
Noisy resample 100
0.35
Cluster population
0.3
0.25
0.2
0.15
Hamming distance
0.1
0.05
0
2
4
6
8
10
Cluster index
Histogramming cluster populations
Hamming distances
Noisy resample 1
Sample n?1
Sample n
Sample 1
Sample 2
Sample 3
in this space. This property allows us to select those partitions as representants, which
are closest to the partition cluster centroids. The whole work-flow of model selection is
summarized schematically in figure 1.
Embedding & Clustering
Figure 1: Model selection: schematic work-flow for one fixed value of the `1 ?constraint ?.
4
Experiments
Clustering USPS digits. In a first experiment we test our method for the task of clustering digits from the USPS handwritten digits database. Sample images are shown in figure
2.
Figure 2: Sample images of digits ?6? and ?7? from the USPS database.
The 16 ? 16 gray-value images of the digits are treated as 256-dimensional vectors. For
this experiment, we extracted a subset of 200 images, consisting of randomly selected digits
?6? and ?7?. Based on this dataset, we first selected the most stable model according to the
model selection procedure described in section 3. We observed the stability of the solutions
for different constraint values ? on the interval [0.7, 1.8] with a step-size of 0.1.
? = 0.7 / #(features) = 2.3
? = 1.0 / #(features) = 18.7
? = 1.8 / #(features) = 34.6
0.8
0.8
0.8
0.7
0.7
0.7
0.6
0.6
0.5
0.5
0.4
0.4
0.4
0.3
0.3
0.3
0.2
0.2
0.2
0.1
0.1
0
0
Cluster population.
0.6
Average Hamming distance
within clusters.
0.5
5
10
15
20
Cluster index
0.1
1
2
3
4
Cluster index
0
0
5
10
15
20
25
Cluster index
Figure 3: Model selection: three different choices of the `1 -constraint ?. The histograms show
the relative population of partition clusters. The solid line indicates the average pairwise Hamming
distance between partitions (divided by 100).
Figure 3 exemplarily shows the outcomes of the stability analysis: in the left panel, the
solution is so highly constrained that on average only 2.3 features (pixels) are selected.
One can see that the solutions are rather instable. Subsets of only two features seem to be
too small for building a consistent splitting hypothesis. Even the most populated partition
cluster (index 3) contains only 30% of all partitions. If, on the other hand, the constraint is
relaxed too far, we also arrive at the very instable situation, depicted in the right panel: for
? = 1.8, on average 34.6 pixels are selected. Optimizing the model in this 35-dimensional
feature space seems to be difficult, probably because the EM-algorithm is often trapped
by suboptimal local optima. In-between these models, however, we find a highly stable
solution for ? = 1.0 in moderate dimensions (on average 18.7 features), see the middle
panel. In this case, the dominating partition cluster (cluster no. 1 in the histogram) contains
almost 75% of all partitions.
Having selected the optimal model parameter ? = 1.0, in a next step we select the representative partition (= the one nearest to the centroid) of the dominating partition cluster
(no. 1 in the middle panel of figure 3). This partition splits the dataset into two clusters,
which highly agree with the true labeling. In the upper part of figure 4, both the inferred
labels and the true labels are depicted by horizontal bar diagrams. Only three samples out
of 200 are mislabeled (the rightmost three samples). The lower panel of this figure shows
several rows, each of which represents one automatically selected feature. Each of the 200
grey-value coded pixel blocks in a row indicates the feature value for one sample. For a better visualization, the features (rows) are permuted according to either high values (black)
or low values (white) for one of the two clusters.
Figure 4: Optimal model: representative partition. Upper horizontal bar: true labels of the 200
samples (black = ?6?, grey = ?7?). Lower bar: inferred labels. Lower panel: each row consists of
grey-value coded values of the selected features for all samples (1 pixel block = 1 sample).
We are not only interested in the stability of splittings of the dataset, but also in the stability
of the feature selection process. In order to quantify this latter stability, we return to the
dominating partition cluster no. 1 in the middle panel of figure 3, and for each of the 73
partitions in this cluster, we count how often a particular feature has been selected. The 22
features (pixels) which are selected in at least one halve of the partitions, are plotted in the
second panel of figure 5. The selection stability is grey-value coded (black = 100% stable).
To the left and to the right we have again plotted two typical sample images of both classes
from the database. A comparison with the selected features leads us to the conclusion,
that we were not only able to find reasonable clusters, but we also have exactly selected
those discriminative features which we would have expected in this control experiment. In
the rightmost panel, we have also plotted one of the three mislabeled ?7?s which has been
assigned to the ?6? cluster.
Figure 5: From left to right: First: a typical ?6?. Second: automatically extracted features. Third:
a typical ?7?. Fourth: one of the three mislabeled ?7?s.
Clustering faces. In a second experiment we applied our method to the problem of
clustering face images. From the Stirling Faces database (http://pics.psych.stir.ac.uk/cgibin/PICS/New/pics.cgi) we selected all 68 grey-valued front views of faces and all 105
profile views. The images are rather inhomogeneous, since they show different persons
with different facial expressions. Some sample images are depicted in figure 6. For a complete overview over the whole image set, we refer the reader to our supplementary web
page http://www.cs.uni-bonn.de/?roth/FACES/split.html, where all images can be viewed in
higher quality.
Figure 6: Example images form the Stirling Faces database.
Since it appears to be infeasible to work directly on the set of pixels of the high-resolution
images, in a first step we extracted the 10 leading eigenfaces of the total dataset (eigenfaces
are simply the eigenvectors v i of the images treated as pixel-wise vectorial objects). These
eigenfaces are depicted in figure 7. We then applied our method to these image objects,
which are represented as 10-dimensional vectors. Note that the original images I j can be
P10
(partially) reconstructed from this truncated eigenvector expansion as Ij0 = i=1 v i v >
i Ij
(assuming the image vectors Ij to be centered).
1.0
0.46
0.28
0.17
0.16
0.15
0.13
0.1
0.09
0.08
Figure 7: First 10 leading eigenfaces and their relative eigenvalues.
We again start our analysis with selecting an optimal model. Figure 8 depicts the outcome of the model selection procedure. The left panel shows both the number of extracted
features and the relative population of the largest partition cluster for different values of ?.
The most stable model is obtained for ? = 1.0. On average, 3.04 features (eigenfaces) have
been selected. A detailed analysis of the selected features within the dominating partition
cluster (no. 5 in the right panel) shows that the eigenfaces no. 2, 3 and 7 are all selected
with a stability of more than 98%. It is interesting to notice that the leading eigenface no. 1
with the distinctly largest eigenvalue has not been selected.
0.35
5
Cluster population
0.3
4
#(features)
0.25
0.2
3
0.15
Hamming distance
2
Relative population of dominating cluster (x10)
1
0.1
0.05
0
0.7
0.8
0.9
1
1.1
1.2
constraint value ?
1.3
0
2
4
6
8
10
Cluster index
Figure 8: Model selection. Left: average number of selected features and relative population of the
dominating partition cluster vs. ?. Right: partition clusters for optimal model with ? = 1.
In every M-step of our algorithm, a linear discriminant analysis is performed, in which
a weight vector ? for all features is computed (due to the incorporated feature selection
mechanism, most weights will be exactly zero). For a given partition of the objects, the
linear combination of the eigenface-features induced by this weight vector is known as
the Fisherface. Our method can, thus, be interpreted as a clustering method that finds
a partition and simultaneously produces a ?sparse? Fisherface which consists of a linear
combination of the most discriminative eigenfaces. Figure 9 shows the derived Fisherface,
reconstructed from the weight vector of the representative partition (no. 5 in the right panel
of figure 8). Note that there are only 3 nonzero weights ?2 = 0.8, ?3 = 0.05 and ?7 = 0.2.
0.8 *
+ 0.2 *
+ 0.05 *
eigenface 2
eigenface 3
=
eigenface 7
Fisherface
Figure 9: The inferred Fisherface as a linear combination of 3 eigenfaces.
The representative partition of the dominating cluster (no. 5 in the right panel of figure 8)
splits the images in two groups, which again highly coincide with the original groups of
frontal and profile faces. Only 7 out of all 173 images are mislabeled w.r.t. this ?groundtruth? labeling. The success of the clustering method can be understood by reconstructing
the original images from the inferred Fisherface (which is nothing but a weighted and
truncated eigenvector reconstruction of the original images). Figure 10 shows the same
images as in figure 6, this time, however, reconstructed from the Fisherface. For better
visualization, all images are rescaled to the full range of 255 grey values. One can see the
clear distinction between frontal and profile faces, which mainly results from different signs
of the projections of the images on the Fisherface. Again, the whole set of reconstructed
images can be viewed on our supplementary material web page in higher quality.
Figure 10: Images from figure 6, reconstructed from the Fisherface.
5
Conclusions
The problem tackled in this paper consists of simultaneously clustering objects and automatically extracting subsets of features which are most discriminative for this object partition. Some approaches have been proposed in the literature, most of which, however,
bear several inherent shortcomings, such as an unclear probabilistic model, the simplifying
assumption of features as being uncorrelated, or the absence of a plausible model selection
strategy. The latter issue is of particular importance, since many approaches seem to suffer
from ambiguities caused by contradictory splitting hypotheses. In this work we have presented a new approach which has the potential to overcome these shortcomings. It has a
clear interpretation in terms of a constrained Gaussian mixture model, which combines a
clustering method with a Bayesian inference mechanism for automatically selecting relevant features. We further present an optimization algorithm with guaranteed convergence
to a local optimum. The model has only one free parameter, ?, for which we propose a
stability-based model selection procedure. Experiments demonstrate that this method is
able to correctly infer partitions and meaningful feature sets.
Our method currently only implements partitions of the object set into two clusters. For
finding multiple clusters, we propose to iteratively split the dataset. Such iterative splits
have been successfully applied to the problem of simultaneously clustering gene expression datasets and selecting relevant genes. Details on these biological applications of our
method will appear elsewhere.
Acknowledgments. The authors would like to thank Joachim M. Buhmann for helpful
discussions and suggestions.
References
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[15] A. v.Heydebreck, W. Huber, A. Poustka, and M. Vingron. Identifying splits with clear separation: a new class discovery method for gene expression data. Bioinformatics, 17, 2001.
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1,635 | 2,487 | Application of SVMs for Colour Classification
and Collision Detection with AIBO Robots
Michael J. Quinlan, Stephan K. Chalup and Richard H. Middleton?
School of Electrical Engineering & Computer Science
The University of Newcastle, Callaghan 2308, Australia
{mquinlan,chalup,rick}@eecs.newcastle.edu.au
Abstract
This article addresses the issues of colour classification and collision detection as they occur in the legged league robot soccer environment of
RoboCup. We show how the method of one-class classification with support vector machines (SVMs) can be applied to solve these tasks satisfactorily using the limited hardware capacity of the prescribed Sony AIBO
quadruped robots. The experimental evaluation shows an improvement
over our previous methods of ellipse fitting for colour classification and
the statistical approach used for collision detection.
1 Introduction
Autonomous agents offer a wide range of possibilities to apply and test machine learning
algorithms, for example in vision, locomotion, and localisation. However, training-time
requirements of sophisticated machine learning algorithms can overstrain the hardware of
real world robots. Consequently, in most cases, ad hoc methods, hard coding of expert
knowledge, and hand-tuning of parameters, or similar approaches were preferred over the
use of learning algorithms on the robot. Application of the latter was often restricted to
simulations which sometimes could support training or tuning of the real world robot parameters. However, often the gap between simulation and the real world was too wide
so that a transfer of training results from the simulated to the real robot turned out to be
useless.
A few years ago it may have been regarded as infeasible to consider the use of support
vector machines [1, 2, 3] on real world robots with restricted processing capabilities. During the first years after their invention support vector machines had the reputation to be
more a theoretical concept than a method which could be efficiently applied in real world
situations. One of the main reasons for this was complexity of the quadratic programming
part. In recent years it has become possible to speed up optimisations for SVMs in various
ways [4]. SVMs have since been successfully applied on many tasks but primarily in the
areas of data mining and pattern classification.
With the present study we explore the feasibility and usefulness of one-class SVM classification [5] for tasks faced by AIBO robots within the legged league environment of
RoboCup [6]. We focus on two particularly critical issues: detection of objects based on
?
http://www.robots.newcastle.edu.au
correct colour classification and detection of robot-to-robot collisions. Both issues seemed
not to be sufficiently solved and implemented by the teams of RoboCup2002 and caused
significant deterioration in the quality of play even in the world-best teams of that league.
The article has five more sections addressing the environment and tasks, the methods, followed by the experiments and applications for colour classification and collision detection,
respectively. The article concludes with a summary.
2 Environment and tasks
The restricted real world environment and the uniformly prescribed hardware of the legged
league [6] of RoboCup provide a good compromise for testing machine learning algorithms
on autonomous agents with a view towards possible applications in more general real world
environments.
A soccer team in the legged league consists of four robots, including one goal keeper. Each
team is identified by robots wearing either a red or blue coloured ?uniform?. The soccer
matches take place on a green enclosed carpeted field with white boundaries. Two goals,
a blue and a yellow, are positioned on opposite ends of the field. To aid localisation six
beacons are placed regularly around the field, each uniquely identifiable by a specific colour
pattern. The ball used is orange plastic and of a suitable size to be easily moved around
by the robots. The games consist of two ten minute halves under strict rules imposed by
independent referees.
The legged league of RoboCup 2003 prescribed the use of Sony AIBO entertainment
robots, models ERS-210 or the newer ERS-210A. Both have an internal 64-bit RISC
processor with clock speeds of 192MHz and 384MHz, respectively. The robots are programmed in a C++ software environment using the Sony?s OPEN-R software development
kit [7]. They have 16MB of memory accessible by user programs. The dimensions of the
robot (width ? height ? length) are 154 mm ? 266 mm ? 274 mm (not including the
tail and ears) and the mass is approximately 1.4 kg. The AIBO has 20 degrees of freedom
(DOF): neck 3DOF (pan, tilt, and roll), ear 1DOF x 2, chin 1DOF, legs 3DOF (abductor,
rotator, knee) x 4 and tail 2DOF (up-down, left-right).
Among other sensors the AIBO has a 1/6 inch colour CMOS camera capable of 25 frames
per seconds. The images are gathered at a resolution of 352(H) ? 288(V) but middleware
restricts the available resolution to a maximum of 176(H) ? 144(V). The lens has an aperture of 2.0 and a focal length of 2.18 mm. Additionally, the camera has a field of vision
of 23.9? up and down and 28.8? left and right. To help achieve results in different lighting
conditions the camera allows the modification of parameters: White balance, Shutter Speed
and Gain.
2.1 Colour classification task
The vision system for most teams consists of four main tasks, Colour Classification, Run
Length Encoding, Blob Formation and Object Recognition (Figure 1).
The classification process takes the image from the camera in a YUV bitmap format [8].
Each pixel in the image is assigned a colour label (i.e. ball orange, beacon pink etc.)
based on its YUV values. A lookup table (LUT) is used to determine which YUV values
correspond to which colour labels. The critical point is the initial generation of the LUT.
Since the robot is extremely reliant on colour for object detection a new LUT has to be
generated with any change in lighting conditions. Currently this is a manual task which
requires a human to take hundreds of images and assign a colour label on a pixel-by-pixel
basis. Using this method each LUT can take hours to create, yet it will still contain holes
and classification errors.
176x144 Pixels
(3x8bit YUV)
Image
CMOS Camera
176x144 Pixels
(Enum)
Colour Lookup Table
3x6bit YUV Pixels
Run Length Encoding
& Blob Formation
List of Blobs
List of Objects
Object Recognition
Figure 1: Vision System of the NUbots Legged League Team [9]
2.2 Collision detection task
The goal is to detect collisions using the limited sensors provided by the AIBO robot. The
camera and infrared distance sensor on the AIBO don?t provide enough support in avoiding
obstacles unless the speed of the robot is dramatically decreased. For these reasons we have
chosen to use information obtained from the joint sensors (i.e. the angle of the joint) as the
input to our collision detection system [10].
3 One-class SVM classification method
An approach to one-class SVM classification was proposed by Sch?olkopf et al. [5]. Their
strategy is to map data into the feature space corresponding to the kernel function and to
separate them from the origin with maximum margin. This implies the construction of a
hyperplane such that w ? ?(xi ) ? p ? 0. The result is a function f that returns the value +1
in the region containing most of the data points and -1 elsewhere. Assuming the use of an
RBF kernel and i, j ? 1, ..., `, we are presented with the dual problem:
X
1X
1
?i ?j k(xi , xj ) subject to 0 ? ?i ?
min
,
?i = 1
(1)
? 2
?`
ij
i
p can be found by the fact that for any such ?i , a corresponding pattern xi satisfies:
X
p=
?j k(xj , xi )
j
The resulting decision function f (the support of the distribution) is:
X
f (x) = sign(
?i k(xi , x) ? p)
i
An implementation of this approach is available in the LIBSVM library [11]. It solves a
scaled version of (1):
X
1X
?i ?j k(xi , xj ) subject to 0 ? ?i ? 1 ,
?i = ?`
min
? 2
ij
i
For our applications we use a RBF kernel with parameter ? in the form k(x, y) =
2
e??kx?yk . The parameter ? approximates the fraction of outliers and support vectors [5].
3.1 Method for colour classification
The classification
functions we seek ?
take data that has been manually clustered to produce
?
sets X k = xki ? R3 ; i = 1, ..., Nk of colour space data for each object colour k. Each
X k corresponds to sets of colour values in the YUV space corresponding to one of the
known colour labels.
An individual one-class SVM is created for each colour, with X k being used as the training
data (each element in the set is scaled between -1 and 1). By training with an extremely
low ? and a large ? the boundary formed by the decision function approximates the region
that contains the majority (1-?) of the points in X k . In addition the SVM has the advantage
of simultaneously removing the outliers that occur during manual classification.
The new colour set is constructed by attempting to classify every point in the YUV space
(643 elements). All points that return a value of +1 are inside the region and therefore
deemed to be of colour k.
One-class SVM was chosen because it allows us to optimally treat each individual colour.
To avoid misclassification each point in YUV space that does not strongly correspond to
one of the known colours must remain classified as unknown. In addition the colours were
originally selected because they are located in different areas of the YUV space. Because
of this we can choose to treat each colour without regard to the location and shape of the
other colours. For these reasons we are not interested in using a multi-class technique to
form a hyperplane that provides an optimal separation between the colours.
3.2 Method for collision detection
For collision detection the one-class SVM is employed as a novelty detection mechanism.
In our implementation each training point is a vector containing thirteen elements. These
include five walk parameters, stepFrequency, backStrideLength, turn, strafe and timeParameter along with a sensor reading from the abductor and rotator joints on each of the
four legs. Upon training the SVMs decision function will return +1 for all values that relate
to a ?normal? step, and -1 for all steps that contain a fault.
Speed is of the greatest importance in the Robocup domain. For this reason a collision
detection system must attempt to minimise the generation of false-positives (detecting a
collision that we deemed not to have happened) while still finding a high percentage of
actual collisions. Low false-positives are achieved by keeping the kernel parameter ? high
but this has the side effect of lowering the generalisation to the data set, which results in
the need for an increased number of training points. In a real world robotic system the
need for more training points greatly increases the training time and in-turn the wear on the
machinery.
4 Experiments and application to colour classification
The SVM can be used in two situations during the colour classification procedure. Firstly
during the construction of a new LUT where it can be applied to increase the speed of
classification.
By lowering ? while the number of training points is low, a rough estimation of the final
shape can be obtained. By continuing the manual classification and increasing ? a closer
approximation to the area containing the training data is obtained. In this manner a continually improving LUT can be constructed until it is deemed adequate.
An extreme example of this application is during the set-up phase at a competition. In the
past when we arrived at a new venue all system testing was delayed until the generation
of a LUT. Of critical importance is testing the locomotion engine on the new carpet and in
particular ball chasing. The task of ball chasing relies on the classification of ball orange.
Thus a method of quickly but roughly classifying orange is valuable. By manually classifying a few images of the ball and then training the SVM with ? < 0, a sphere containing
all possible values for the ball is generated.
The second situation in which we use the one-class SVM is on a completed LUT. Either all
colours in the table can be trained (i.e. updating of an old table) or an individual colour is
trained due to an initial classification error. This procedure can be performed either on the
robot or a remote computer.
Empirical tests have indicated that ? = 0.025 and ? = 250 provide excellent results on a
previously constructed LUT. The initial table contained 3329 entries while after training
the table contained 6989 entries. The most evident change can be seen in the classification
of colour white, see Figure 2.
The LUTs were compared over 60 images, which equates to 1,520,640 individual pixel
comparisons. The initial table generated 144,098 classification errors. The new LUT produced 117,652 errors, this equates to an 18% reduction in errors.
Figure 2: Image Comparison: The left image is classified with the original LUT and the
image on the right is the using the updated LUT. Black pixels indicate an unknown colour.
4.1 Comparison with ellipsoid fitting
The previous method involved converting the existing LUT values from YUV to the HSI
colour space [8] and fitting an ellipsoid, E, which can be represented by the quadratic form:
n
o
T
E (x0 , Q) = x ? R3 : (x ? x0 ) Q?1 (x ? x0 ) ? 1
(2)
where x0 is the centre of the ellipsoid, and the size, orientation and shape of the ellipsoid
are contained in the positive definite symmetric matrix Q = QT > 0 ? R3?3 .
Note that this definition of the shape can be alternatively represented by the linear matrix
inequality (LMI):
?
?
Q
(xi ? x0 )
xi ? E =
?0
(3)
T
(xi ? x0 )
1
The LMI (3) is linear in the unknowns Q and x0 and this therefore leads to the convex
optimisation:
(Q, x0 ) =
argmin
{tr(Q)}
Q = QT > 0, x0 :
(3) is true for i = 1..Nk
Note that minimising the trace of Q (tr(Q)) is the same as minimising the sum of the
diagonal elements of Q which is the same as minimising the sum of the squares of the
lengths of the principal axes of the ellipsoid. The ellipsoidal shape defined in (2) has the
disadvantage of restricting the shape of possible regions in the colour space. However, it
does have the advantage of having a simple representation and a convex shape.
Before the ellipsoid can be fitted, potential outliers and duplicate points were identified and
removed. The removal of outliers is important in avoiding too large a region. Duplicate
points were removed, since these increase computations without adding any information.
For the comparison we use the initial LUT from the above example. Figure 3 shows the
effects of each method on the colour white. To make the comparison with ellipsoids, the
initial LUT and the generated LUT from the SVM procedure are shown in the HSI colour
space.
Figure 3: Colour classification in HSI colour space: A) Points manually classified at
white. B) Ellipsoid fitted to these white points. C) Result of the one-class SVM technique,
?=0.025 and ?=10. D) Result of the one-class SVM technique, ?=0.025 and ?=250.
It is evident that the manual classification of white is rather incomplete and contains many
holes that should be classified as white. The negative results of these holes can be seen as
noise in the left image of Figure 2.
Using the ellipsoid fitting method these holes are filled but with the potential drawback
of over classification. From image B in Figure 3 it is evident that the top section and the
bottom left of the ellipsoid contain no white entries and therefore it is highly questionable
that this area should be classified as white.
Images C and D in the figure show the results of our one-class SVM method. It is clear
from image D that the area now classified as white is a region that tightly fits the original
training set.
5 Experiments and application to collision detection
The collision detection system is designed with the aim that the entire system can be run on
the robot. This means adhering to the memory and processing capabilities of the device. On
the AIBO we have a maximum of 8MB memory available for collision detection, a total of
20,000 training points. This is the equivalent of 1000 steps which equates to approximately
10 minutes of training time. The training set is generated by having the robot behave
normally on the field but with the stipulation that all collisions are avoided.
The trained classifier analyses the on-line stream of joint data measurements in samples
of ten consecutive data points. If more than 2 points in one sample are classified as -1 a
collision is declared to be detected.
Initial parameters of ? = 0.05 and ? = 5 were chosen, this was based on the assumption
that a collision point would lie considerably outside the training set. The results from these
parameters were less then satisfying, only the largest of collisions (i.e. physically holding
multiple legs) were detected. The solution to this problem could involve increasing ? due
to the possibility that the initial training set contained many outliers and/or increasing ? to
improve the tightness of the classification.
By a series of tests, all of which tended to lead to either an over classification or an under
classification, parameters of ? = 0.05 and ? = 100 were settled on. In our system these
parameters appear to give the best balance between minimising false-positives and maximising correct detection of collisions.
5.1 Comparison with the previous statistical method
The previous method, described in [10], for collision detection involves observing a joint
position substantially differing from its expected value. In our case an empirical study
found two standard deviations to be a practical measure, see Figure 4. Initially we would
have considered a collision to have occurred if a single error is found, but further investigation has shown that finding multiple errors (in most cases three) in quick succession is
necessary to warrant a warning that can be acted upon by the robot?s behaviour system.
Figure 4: Rear Rotators for a forwards walking boundary collision on both front legs, front
right leg hitting first. The bold line shows the path of a collided motion. The dotted line
represents the mean ?normal? path of the joint (that is, during unobstructed motion), with
the error bars indicating two standard deviations above and below.
One drawback of this method is that it relied on domain knowledge to arrive at two standard
deviations. In addition it required considerable storage space to hold the table of means and
standard deviations for each parameter combination.
The previous statistical method had the advantage of extremely low computational expense,
in fact it was a table look up. The trade-off is increased space, this method required the
allocation of approximately 6MB of memory during both the training and detection stages.
Conversely the SVM approach requires only about 1MB of memory during the detection
phase, but this comes at the side effect of increased computation. Since the SVM approach
was capable of running without reducing the frame rate, the extra memory could now be
used for other applications.
With respect to accuracy the SVM approach slightly outperformed the original statistical
method for particular types of steps, these include the common steps associated with chasing the ball. Other step types, such as an aggressive turn did not show the same improvement. This is due to the movement of the joints in some motions being more inconsistent,
thus making accurate classification harder.
A possible solution may involve using multiple SVMs associated with different combinations of walk parameters, allowing the tuning of parameters on a specific basis. This
solution would have the downside of requiring more memory.
6 Summary
The method of one-class classification with SVMs was successfully applied to the tasks
of colour classification and collision detection using the restricted memory and processing
power of the AIBO hardware. It was possible to run the SVM algorithm implemented
in the C++ libraries of LIBSVM off and on the robot. In a comparison with previously
used methods the SVM based methods generated better results, and in the case of colour
classification the SVM approach was more efficient and convenient.
Acknowledgments
We would like to thank William McMahan and Jared Bunting for their work on the previous vision classification method and Craig Murch for his extensive contributions to both
the vision and locomotion systems. Michael J. Quinlan was supported by a University of
Newcastle Postgraduate Research Scholarship.
References
[1] B. E. Boser, I. M. Guyon, and V. N. Vapnik. A training algorithm for optimal margin classifiers. In D. Haussler, editor, Proceedings of the 5th Annual ACM Workshop on Computational
Learning Theory, pages 144?152, Pittsburgh, PA, July 1992. ACM Press.
[2] C. Cortes and V. Vapnik. Support vector networks. Machine Learning, 20:273 ? 297, 1995.
[3] V. Vapnik. The Nature of Statistical Learning Theory. Springer Verlag, New York, 1995.
[4] Bernhard Sch?olkopf and Alexander J. Smola. Learning with Kernels, Support Vector Machines,
Regularization, Optimization and Beyond. The MIT Press, 2002.
[5] B. Sch?olkopf, J. C. Platt, J. Shawe-Taylor, A. J. Smola, and R. C. Williamson. Estimating the
support of a high-dimensional distribution. Neural Computation, 13:1443?1471, 2001.
[6] RoboCup Legged League web site. http://www.openr.org/robocup/index.html.
[7] OPEN-R SDK. http://openr.aibo.com.
[8] Linda G. Shapiro and George C. Stockman. Computer Vision. Prentice Hall, 2001.
[9] J. Bunting, S. Chalup, M. Freeston, W. McMahan, R. Middleton, C. Murch, M. Quinlan,
C. Seysener, and G. Shanks. Return of the NUbots! The 2003 NUbots Team Report, 2003.
http://robots.newcastle.edu.au/publications/NUbotFinalReport2003.pdf.
[10] Michael J. Quinlan, Craig L. Murch, Richard H. Middleton, and Stephan K. Chalup. Traction
monitoring for collision detection with legged robots. In RoboCup 2003 Symposium, 2003.
[11] Chih-Chung Chang and Chih-Jen Lin. LIBSVM: a library for support vector machines, 2001.
Software available at http://www.csie.ntu.edu.tw/?cjlin/libsvm.
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1,636 | 2,488 | Sample Propagation
Mark A. Paskin
Computer Science Division
University of California, Berkeley
Berkeley, CA 94720
[email protected]
Abstract
Rao?Blackwellization is an approximation technique for probabilistic inference that flexibly combines exact inference with sampling. It is useful
in models where conditioning on some of the variables leaves a simpler inference problem that can be solved tractably. This paper presents
Sample Propagation, an efficient implementation of Rao?Blackwellized
approximate inference for a large class of models. Sample Propagation
tightly integrates sampling with message passing in a junction tree, and
is named for its simple, appealing structure: it walks the clusters of a
junction tree, sampling some of the current cluster?s variables and then
passing a message to one of its neighbors. We discuss the application of
Sample Propagation to conditional Gaussian inference problems such as
switching linear dynamical systems.
1
Introduction
Message passing on junction trees is an efficient means of solving many probabilistic inference problems [1, 2]. However, as these are exact methods, their computational costs
must scale with the complexity of the inference problem, making them inapplicable to very
demanding inference tasks. This happens when the messages become too expensive to
compute, as in discrete models of large treewidth or conditional Gaussian models [3].
In these settings it is natural to investigate whether junction tree techniques can be combined with sampling to yield fast, accurate approximate inference algorithms. One way to
do this is to use sampling to approximate the messages, as in HUGS [4, 5]. This strategy
has two disadvantages: first, the samples must be stored, which limits the sample size by
space constraints (rather than time constraints); and second, variables are sampled using
only local information, leading to samples that may not be likely under the entire model.
Another way to integrate sampling and message passing is via Rao?Blackwellization,
where we repeatedly sample a subset of the model?s variables and then compute all of the
messages exactly, conditioned on these sample values. This technique, suggested in [6] and
studied in [7], yields a powerful and flexible approximate inference algorithm; however, it
can be expensive because the junction tree algorithm must be run for every sample.
In this paper, we present a simple implementation of Rao?Blackwellized approximate inference that avoids running the entire junction tree algorithm for every sample. We develop
a new message passing algorithm for junction trees that supports fast retraction of evidence,
and we tightly integrate it with a blocking Gibbs sampler so that only one message must
be recomputed per sample. The resulting algorithm, Sample Propagation, has an appealing
structure: it walks the clusters of a junction tree, resampling some of the current cluster?s
variables and then passing a message to the next cluster in the walk.
2
Rao?Blackwellized approximation using junction tree inference
We start by presenting our notation and assumptions on the probability model. Then we
summarize the three basic ingredients of our approach: message passing in a junction tree,
Rao?Blackwellized approximation, and sampling via Markov chain Monte Carlo.
2.1
The probability model
Let X = (Xi : i ? I) be a vector of random variables indexed by the finite, ordered set
I, and for each index i let Xi be the range of Xi . We will use the symbols A, B, C, D and
E to denote subsets of the index set I. For each subset A, let XA ? (Xi : i ? A) be the
corresponding subvector of random variables and let XA be its range.
It greatly simplifies the exposition to develop a simple notation for assignments of values to
subsets of variables. An assignment to a subset A is a set of pairs {(i, xi ) : i ? A}, one per
index i ? A, where xi ? Xi . We use the symbols u, v, and w to represent assignments, and
we use XA to denote the set of assignments to the subset A (with the shorthand X ? XI ).
We use two operations to generate new assignments from old assignments. Given assignments u and v to disjoint subsets A and B, respectively, their union u ? v is an assignment
to A ? B. If u is an assignment to A then the restriction of u to another subset B is
uB ? {(i, xi ) ? u : i ? B}, an assignment to A ? B. We also let functions act on
assignments in the natural way: if u = {(i, xi ) : i ? D} is an assignment to D and f is a
function whose domain is XD , then we use f (u) to denote f (xi : i ? D).
We consider probability densities of the form
Y
p(u) ?
?C (uC ),
u?X
(1)
C?C
where C is a set of subsets of I and each ?C is a potential function over C (i.e., a nonnegative function of XC ). This class includes directed graphical models (i.e., Bayesian
networks) and undirected graphical models such as Markov random fields. Observed variables are reflected in the model by evidence potentials. We use pA (?) to denote the marginal
density of XA and pA|B (? | ?) to denote the conditional density of XA given XB . Finally,
we use the notation of finite measure spaces for simplicity, but our approach extends to the
continuous case.
2.2
Junction tree inference
Given a density of the form (1), we view the problem ofPprobabilistic inference as that of
computing the expectation of a function f : E [f (X)] = u?X p(u)f (u). This sum can be
expensive to compute when X is a large space. When the desired expectation is ?local? in
that f depends only upon some subset of the variables XD , we can compute the expectation
more cheaply using a marginal density as
X
E [f (XD )] =
pC (u)f (uD )
(2)
u?XC
where pC is the marginal density of XC and C ? D ?covers? the input of the function. If
this sum is tractable, then we have reduced the problem to that of computing pC .
We can compute this marginal via message passing on a junction tree [1, 2]. A junction
tree for C is a singly-connected, undirected graph (C, E) with the junction tree property:
for each pair of nodes (or clusters) A, B ? C that contain some i ? I, every cluster on
the unique path between A and B also contains i. In what follows we assume we have a
junction tree for C with a cluster that covers D, the input of f . (Such a junction tree can
always be found, but we may have to enlarge the subsets in C.)
Whereas the HUGIN message passing algorithm [2] may be more familiar, Sample Propagation is most easily described by extending the Shafer?Shenoy algorithm [1]. In this
algorithm, we define for each edge B ? C of the junction tree a potential over B ? C:
Y
X
?B (u ? v)
?AB (uA ? vA ),
u ? XB?C
(3)
?BC (u) ?
v?XB\C
(A,B)?E
A6=C
?BC is called the message from B to C. Note that this definition is recursive?messages
can depend on each other?with the base case being messages from leaf clusters of the
junction tree. For each cluster C we define a potential ?C over C by
Y
?C (u) ? ?C (u)
?BC (uB ),
u ? XC
(4)
(B,C)?E
?C is called the cluster belief of C, and it follows that ?C ? pC , i.e., that the cluster beliefs
are the marginals over their respective variables (up to renormalization). Thus we can use
the (normalized) cluster beliefs ?C for some C ? D to compute the expectation (2).
In what follows we will also be interested in computing conditional cluster densities given
an evidence assignment w to a subset of the variables XE . Because pI\E|E (u | w) ?
p(u ? w), we can ?enter in? this evidence by instantiating w in every cluster potential ?C .
The cluster beliefs (4) will then be proportional to the conditional density pC\E|E (? | w).
Junction tree inference is often the most efficient means of computing exact solutions to
inference problems of the sort described above. However, the sums required by the messages (3) or the function expectations (2) are often prohibitively expensive to compute. If
the variables are all finite-valued, this happens when the clusters of the junction tree are
too large; if the model is conditional-Gaussian, this happens when the messages, which are
mixtures of Gaussians, have too many mixture components [3].
2.3
Rao?Blackwellized approximate inference
In cases where the expectation is intractable to compute exactly, it can be approximated by
a Monte Carlo estimate:
N
1 X
n
f (vD
)
E [f (XD )] ?
N n=1
(5)
where {vn : 1 ? n ? N } are a set of samples of X. However, obtaining a good estimate
will require many samples if f (XD ) has high variance.
Many models have the property that while computing exact expectations is intractable,
there exists a subset of random variables XE such that the conditional expectation
E [f (XD ) | XE = xE ] can be computed efficiently. This leads to the Rao?Blackwellized
estimate, where we use a set of samples {wn : 1 ? n ? N } of XE to approximate
E [f (XD )] = E [E [f (XD ) | XE ]] ?
N
1 X
E [f (XD ) | wn ]
N n=1
(6)
The first advantage of this scheme over standard Monte Carlo integration is that the Rao?
Blackwell theorem guarantees that the expected squared error of the estimate (6) is upper
bounded by that of (5), and strictly so when f (XD ) depends on XD\E . A second advantage
is that (6) requires samples from a smaller (and perhaps better-behaved) probability space.
Algorithm 1 Rao?Blackwell estimation on a junction tree
Input: A set of samples {wn : 1 ? n ? N } of XE , a function f of XD , and a cluster C ? D
Output: An estimate f? ? E [f (XD )]
1: Initialize the estimator f? = 0.
2: for n = 1 to N do
3:
Enter the assignment wn as evidence into the junction tree.
4:
Use message passing to compute the beliefs ?C ? pC\E|E (? | wn ) via (3) and (4).
5:
Compute the expectation E [f (XD ) | wn ] via (7).
6:
Set f? = f? + E [f (XD ) | wn ].
7: Set f? = f?/N .
However, the Rao?Blackwellized estimate (6) is more expensive to compute than (5) because we must compute conditional expectations. In many cases, message passing in a
junction tree can be used to implement these computations (see Algorithm 1). We can enter each sample assignment wn as evidence into the junction tree and use message passing
to compute the conditional density pC\E|E (? | wn ) for some cluster C that covers D. We
then compute the conditional expectation as
X
n
pC\E|E (u | wn )f (uD ? wD
)
(7)
E [f (XD ) | wn ] =
u?XC\E
2.4
Markov chain Monte Carlo
We now turn to the problem of obtaining the samples {wn } of XE . Markov chain Monte
Carlo (MCMC) is a powerful technique for generating samples from a complex distribution
p; we design a Markov chain whose stationary distribution is p, and simulate the chain
to obtain samples [8]. One simple MCMC algorithm is the Gibbs sampler, where each
successive state of the Markov chain is chosen by resampling one variable conditioned on
the current values of the remaining variables. A more advanced technique is ?blocking?
Gibbs sampling, where we resample a subset of variables in each step; this technique can
yield Markov chains that mix more quickly [9].
To obtain the benefits of sampling in a smaller space, we would like to sample directly from
the marginal pE ; however, this requires us to sum out the nuisance variables XI\E from the
joint density p. Blocking Gibbs sampling is particularly attractive in this setting because
message passing can be used to implement the required marginalizations.1 Assume that the
current state of the Markov chain over XE is wn . To generate the next state of the chain
wn+1 we choose a cluster C (randomly, or according to a schedule) and resample XC?E
n
given wE\C
; i.e., we resample the E variables within C given the E variables outside C.
n
The transition density can be computed by entering the evidence wE\C
into the junction
tree, computing the cluster belief at C, and marginalizing down to a conditional density
over XC?E . The complete Gibbs sampler is given as Algorithm 2.2
3
Sample Propagation
Algorithms 1 and 2 represent two of the three key ideas behind our proposal: both Gibbs
sampling and Rao?Blackwellized estimation can be implemented efficiently using message
passing on a junction tree. The third idea is that these two uses of message passing can be
interleaved so that each sample requires only one message to be computed.
1
Interestingly, the blocking Gibbs proposal [9] makes a different use of junction tree inference
than we do here: they use message passing within a block of variables to efficiently generate a sample.
2
n
In cases where the transition density pC?E|E\C (? | wE\C
) is too large to represent or too difficult
to sample from, we can use the Metropolis-Hastings algorithm, where we instead sample from a
simpler proposal distribution qC?E and then accept or reject the proposal [8].
Algorithm 2 Blocking Gibbs sampler on a junction tree
Input: A subset of variables XE to sample and a sample size N
Output: A set of samples {wn : 1 ? n ? N } of XE
1: Choose an initial assignment w0 ? XE .
2: for n = 1 to N do
3:
Choose a cluster C ? C.
n?1
4:
Enter the evidence wE\C
into the junction tree.
n?1
5:
Use message passing to compute the beliefs ?C ? pC|E\C (? | wE\C
) via (3) and (4).
n?1
6:
Marginalize over XC\E to obtain the transition density pC?E|E\C (? | wE\C
).
n?1
n?1
n
n
7:
Sample wC?E ? pC?E|E\C (? | wE\C ) and set wE\C = wE\C .
3.1
Lazy updating of the Rao?Blackwellized estimates
Algorithms 1 and 2 both process the samples sequentially, so the first advantage of merging
them is that the sample set need not be stored. The second advantage is that, by being
selective about when the Rao?Blackwellized estimator is updated, we can compute the
messages once, not twice, per sample.
When the Gibbs sampler chooses to resample a cluster C that covers D (the input of f ),
we can update the Rao?Blackwellized estimator for free. In particular, the Gibbs sampler
n?1
computes the cluster belief ?C ? pC|E\C (? | wE\C
) in order to compute the transition
n?1
n
density pC?E|E\C (? | wE\C ). Once it samples wC?E from this density, we can instantiate
the sample in the belief ?C to obtain the conditional density pC\E|E (? | wn ) needed by the
n?1
n
Rao?Blackwellized estimator. (This follows from the fact that wE\C
= wE\C
.) In fact,
when it is tractable to do so, we can simply use the cluster belief ?C to update the estimator
in (7); because it treats more variables exactly, it can yield a lower-variance estimate.
Therefore, if we are willing to update the Rao?Blackwellized estimator only when the
Gibbs sampler chooses a cluster that covers the function?s inputs, we can focus on reducing
the computational requirements of the Gibbs sampler. In this scheme the estimate will be
based on fewer samples, but the samples that are used will be less correlated because they
are more distant from each other in the Markov chain. In parallel estimation problems
where every cluster is computing expectations, every sample will be used to update an
estimate, but not every estimate will be updated by every sample.
3.2
Optimizing the Gibbs sampler
We now turn to the Gibbs sampler. The Gibbs sampler computes the messages so that it can
compute the cluster belief ?C when it resamples within a cluster C. An important property
of the computation (4) is that it requires only those messages directed towards C; thus, we
have again reduced by half the number of messages required per sample.
The difficulty in further minimizing the number of messages computed by the Gibbs sampler is that the evidence on the junction tree is constantly changing. It will therefore be
useful to modify the message passing so that, rather than instantiating all the evidence and
then passing messages, the evidence is instantiated on the fly, on a per-cluster basis. For
each edge B ? C we define a potential ?BC|E by
X
Y
?BC|E (u, w) ?
?B (u ? v ? wB\C )
?AB|E ((u ? v ? wB\C )A , w) (8)
v?XB\(C?E)
A6=C
(A,B)?E
where u ? XB?C and w ? XE . This is the conditional message from B to C given
evidence w on XE . Figure 1 illustrates how the ranges of the assignment variables u, v,
and w cover the variables of B; the intuition is that when we send a message from B to C,
we instantiate all evidence variables that are in B but not those that are in C; this gives us
Algorithm 3 Sample Propagation
Input: A function f of XD , a cluster C ? D, a subset E to sample, and a sample size N
Output: An estimate f? ? E [f (XD )]
1: Choose an initial assignment w0 ? XE and compute the messages ?AB|E (?, w0 ) via (8).
2: Choose a cluster C1 ? C, initialize the estimator f? = 0, and set the sample count M = 0.
3: for n = 1 to N do
n?1
4:
Compute the conditional cluster belief ?Cn |E (?, wn?1 ) ? pCn |E\Cn (? | wE\C
) via (9).
n
Advance the Markov chain:
n?1
5:
Marginalize over XCn \E to obtain the transition density pCn ?E|E\Cn (? | wE\C
).
n
n?1
n?1
n
n
6:
Sample wCn ?E ? pCn ?E|E\Cn (? | wE\Cn ) and set wE\Cn = wE\Cn .
Update any estimates to be computed at Cn :
7:
if D ? Cn ? E then
n
8:
Instantiate wC
in ?C n |E and normalize to obtain pCn \E|E (? | wn ).
n
9:
Compute the expectation E [f (XD ) | wn ] via (7).
10:
Set f? = f? + E [f (XD ) | wn ] and increment the sample count M .
Take the next step of the walk:
11:
Choose a cluster Cn+1 that is a neighbor of Cn .
12:
Recompute the message ?Cn Cn+1 |E (?, wn ) via (8).
13: Set f? ? f?/M .
the freedom to later instantiate XC?E as we wish, or not at all. It is easy to verify that the
conditional belief ?C|E given by
Y
?C|E (u, w) ? ?C (u)
?BC|E (uB , w),
u ? XC , w ? XE
(9)
(B,C)?E
is proportional to the conditional density pC|E\C (u | wE\C ).3
Using these modified definitions, we can dramatically reduce the
number of messages computed per sample. In particular, the conditional messages have the following important property:
Proposition. Let w and w0 be two assignments to E such that
0
wE\D = wE\D
for some cluster D. Then for all edges B ? C
with C closer to D than B, ?BC|E (u, w) = ?BC|E (u, w0 ).
Figure 1: A Venn diagram showing how the
ranges of the assignment variables in (8)
cover the cluster B.
Proof. Assume by induction that the messages into B (except
the one from C) are equal given w or w0 . There are two cases
to consider. If (E ? D) has no overlap with (E ? B), then
0
wB\C = wB\C
and the equality follows from (8). Otherwise,
by the junction property we know that if i ? B and i ? D, then
0
i ? C, so again we get wB\C = wB\C
.
n?1
n
Thus, when we resample a cluster Cn , we have wE\C
= wE\C
n
n
and so only those messages directed away from Cn change. In addition, as argued above,
when we resample Cn+1 in iteration n + 1, we only require the messages directed towards
Cn+1 . Combining these two arguments, we find that only the messages on the directed
path from Cn to Cn+1 must be recomputed in iteration n. If we choose Cn+1 to be a
neighbor of Cn , we only have to recompute a single message in each iteration.4 Putting all
of these optimizations together, we obtain Algorithm 3, which is easily generalized to the
case where many function expectations are computed in parallel.
3
The modified message passing scheme we describe can be viewed as an implementation of fast
retraction for Shafer-Shenoy messages, analogous to the scheme described for HUGIN in [2, ?6.4.6].
4
A similar idea has recently been used to improve the efficiency of the Unified Propagation and
Scaling algorithm for maximum likelihood estimation [10].
3.3
Complexity of Sample Propagation
For simplicity of analysis we assume finite-value variables and tabular potentials. In the
Shafer?Shenoy algorithm, the space complexity of representing the exact message (3) is
O(|XB?C |), and the time complexity of computing it is O(|XB |) (since for each assignment to B ? C we must sum over assignments to B\C). In contrast, when computing the
conditional message (8), we only sum over assignments to B\(C ? E), since E ? (B\C) is
instantiated by the current sample. This makes the conditional message cheaper to compute
than the exact message: in the finite case the time complexity is O(|XB\(E?(B\C)) |). The
space complexity of representing the conditional message is O(|XB?C |)?the same as the
exact message, since it a potential over the same variables.
As we sample more variables, the conditional messages become cheaper to compute. However, note that the space complexity of representing the conditional message is independent
of the choice of sampled variables E; even if we sample a given variable, it remains a free
parameter of the conditional message. (If we instead fixed its value, the proposition above
would not hold.) Thus, the time complexity of computing conditional messages can be
reduced by sampling more variables, but only up to a point: the time complexity of computing the conditional message must be o(|XB?C |). This contrasts with the approach of
Bidyuk & Dechter [7], where the asymptotic time complexity of each iteration can be reduced arbitrarily by sampling more variables. However, to achieve this their algorithm runs
the entire junction tree algorithm in each iteration, and does not reuse messages between
iterations. In contrast, Sample Propagation reuses all but one of the messages between
iterations, leading to a greatly reduced ?constant factor?.
4
Application to conditional Gaussian models
A conditional Gaussian (CG) model is a probability distribution over a set of discrete variables {Xi : i ? ?} and continous variables {Xi : i ? ?} such that the conditional
distribution of X? given X? is multivariate Gaussian. Inference in CG models is harder
than in models that are totally discrete or totally Gaussian. For example, consider polytree
models: when all of the variables are discrete or all are Gaussian, exact inference is linear
in size of the model; but if the model is CG then approximate inference is NP-hard [11].
In traditional junction tree inference, our goal is to compute the marginal for each cluster.
However, when p is a CG model, each cluster marginal is a mixture of |X? | Gaussians,
and is intractable to represent. Instead, we can compute the weak marginals, i.e., for each
cluster we compute the best conditional Gaussian approximation of pC . Lauritzen?s algorithm [12] is an extension of the HUGIN algorithm that computes these weak marginals
exactly. Unfortunately, it is often intractable because it requires strongly rooted junction
trees, which can have clusters that contain most or all of the discrete variables [3].
The structure of CG models makes it possible to use Sample Propagation to approximate
the weak cluster marginals: we choose E = ?, since conditioning on the discrete variables
leaves a tractable Gaussian inference problem.5 The expectations we must compute are
of the sufficient statistics of the weak cluster marginals: for each cluster C, we need the
distribution of XC?? and the conditional means and covariances of XC?? given XC?? .
As an example, consider the model given in Figure 2(a) for tracking an object whose state
(position and velocity) at time t is Xt . At each time step, we obtain a vector measurement
Yt which is either a noisy measurement of the object?s position (if Zt = 0) or an outlier (if
Zt = 1). The Markov chain over Zt makes it likely that inliers and outliers come in bursts.
The task is to estimate the position of the object at all time steps (for T = 100).
5
We cannot choose E = ? because computing the conditional messages (8) may require summing
discrete variables out of CG potentials, which leads to representational difficulties [3]. In this case
one can instead use Bidyuk & Dechter?s algorithm, which does not require these operations.
Acknowledgements. I thank K. Murphy and S.
Russell for comments on a draft of this paper.
This research was supported by ONR N00014-001-0637 and an Intel Internship.
References
X1
X2
Y1
Y2
Z1
Z2
XT
...
YT
ZT
...
(a)
X1, X2,
X2, X3,
Z1, Z2
XT - 1, XT,
...
Z2, Z3
Z T - 1, Z T
(b)
Assumed Density Filtering
Sample Propagation
Gibbs Sampling
9
average position error
Lauritzen?s algorithm is intractable in this case because any strongly rooted junction tree for this network must have a cluster containing all of the discrete variables [3, Thm. 3.18]. Therefore, instead of comparing our approximate position estimates to the correct answer, we sampled a trajectory from the network and computed the average position error to the (unobserved) ground truth.
Both Gibbs sampling and Sample Propagation were
run with a forwards?backwards sampling schedule; Sample Propagation used the junction tree of
Figure 2(b).6 Both algorithms were started in the
same state and both were allowed to ?burn in? for
five forwards?backwards passes. We repeated this
10 times and averaged the results over trials. Figure 2(c) shows that Sample Propagation converged
much more quickly than Gibbs sampling. Also,
Sample Propagation found better answers than Assumed Density Filtering (a standard algorithm for
this problem), but at increased computational cost.
8
7
6
5
4
3 5
10
6
7
8
10
10
10
floating point operations
9
10
(c)
Figure 2: The TRACKING example.
[1] G. Shafer and P. Shenoy. Probability propagation. Annals of Mathematics and Artificial Intelligence, 2:327?352, 1990.
[2] R. Cowell, P. Dawid, S. Lauritzen, and D. Spiegelhalter. Probabilistic Networks and Expert
Systems. Springer, 1999.
[3] U. Lerner. Hybrid Bayesian Networks for Reasoning About Complex Systems. PhD thesis,
Stanford University, October 2002.
[4] A. Dawid, U. Kj?rulff, and S. Lauritzen. Hybrid propagation in junction trees. In Advances in
Intelligent Computing, volume 945 of Lecture Notes in Computer Science. Springer, 1995.
[5] U. Kj?rulff. HUGS: Combining exact inference and Gibbs sampling in junction trees. In Proc.
of the 11th Conf. on Uncertainty in Artificial Intelligence (UAI-95). Morgan Kaufmann, 1995.
[6] A. Doucet, N. de Freitas, K. Murphy, and S. Russell. Rao-Blackwellised particle filtering for
dynamic Bayesian networks. In Proc. of the 16th Conf. on Uncertainty in AI (UAI-00), 2000.
[7] B. Bidyuk and R. Dechter. An empirical study of w-cutset sampling for Bayesian networks. In
Proc. of the 19th Conf. on Uncertainty in AI (UAI-03). Morgan Kaufmann, 2003.
[8] R. Neal. Probabilistic inference using Markov chain Monte Carlo methods. Technical Report
CRG-TR-93-1, University of Toronto, 1993.
[9] C. S. Jensen, A. Kong, and U. Kj?rulff. Blocking Gibbs sampling in very large probabilistic
expert systems. International Journal of Human-Computer Studies, 42:647?666, 1995.
[10] Y. W. Teh and M. Welling. On improving the efficiency of the iterative proportional fitting
procedure. In Proc. of the 9th Int?l. Workshop on AI and Statistics (AISTATS-03), 2003.
[11] U. Lerner and R. Parr. Inference in hybrid networks: Theoretical limits and practical algorithms.
In Proc. of the 17th Conf. on Uncertainty in AI (UAI-01). Morgan Kaufmann, 2001.
[12] S. Lauritzen. Propagation of probabilities, means, and variances in mixed graphical association
models. Journal of the American Statistical Association, 87(420):1098?1108, 1992.
[13] C. Carter and R. Kohn. Markov chain Monte Carlo in conditionally Gaussian state space models. Biometrika, 83:589?601, 1996.
6
Carter & Kohn [13] describe a specialized algorithm for this model that is similar to a version of
Sample Propagation that does not resample the discrete variables on the backwards pass.
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1,637 | 2,489 | How to Combine Expert (or Novice) Advice
when Actions Impact the Environment
Daniela Pucci de Farias?
Department of Mechanical Engineering
Massachusetts Institute of Technology
Cambridge, MA 02139
[email protected]
Nimrod Megiddo
IBM Almaden Research Center
650 Harry Road, K53-B2
San Jose, CA 95120
[email protected]
Abstract
The so-called ?experts algorithms? constitute a methodology for choosing actions repeatedly, when the rewards depend both on the choice of
action and on the unknown current state of the environment. An experts
algorithm has access to a set of strategies (?experts?), each of which may
recommend which action to choose. The algorithm learns how to combine the recommendations of individual experts so that, in the long run,
for any fixed sequence of states of the environment, it does as well as the
best expert would have done relative to the same sequence. This methodology may not be suitable for situations where the evolution of states of
the environment depends on past chosen actions, as is usually the case,
for example, in a repeated non-zero-sum game.
A new experts algorithm is presented and analyzed in the context of repeated games. It is shown that asymptotically, under certain conditions,
it performs as well as the best available expert. This algorithm is quite
different from previously proposed experts algorithms. It represents a
shift from the paradigms of regret minimization and myopic optimization to consideration of the long-term effect of a player?s actions on the
opponent?s actions or the environment. The importance of this shift is
demonstrated by the fact that this algorithm is capable of inducing cooperation in the repeated Prisoner?s Dilemma game, whereas previous
experts algorithms converge to the suboptimal non-cooperative play.
1
Introduction
Experts algorithms. A well-known class of methods in machine learning are the socalled experts algorithms. The goal of these methods is to learn from experience how to
combine advice from multiple experts in order to make sequential decisions in an online
environment. The general idea can be described as follows. An agent has to choose repeatedly from a given set of actions. The reward in each stage is a function of the chosen action
and the choices of Nature or the environment (also referred to as the ?adversary? or the ?opponent?). A set of strategies {1, . . . , r} is available for the agent to choose from. We refer
?
Work done while at IBM Almaden Research Center, San Jose, California.
to each such strategy as an ?expert,? even though some of them might be simple enough to
be called a ?novice.? Each expert suggests a choice of an action based on the history of
the process and the expert?s own choice algorithm. After each stage, the agent observes his
own reward. An experts algorithm directs the agent with regard to which expert to follow
in the next stage, based on the past history of actions and rewards.
Minimum Regret. A popular criterion in decision processes is called Minimum Regret
(MR). Regret is defined as the difference between the reward that could have been achieved,
given the choices of Nature, and what was actually achieved. An expert selection rule is said
to minimize regret if it yields an average reward as large as that of any single expert, against
any fixed sequence of actions chosen by the opponent. Indeed, certain experts algorithms,
which at each stage choose an expert from a probability distribution that is related to the
reward accumulated by the expert prior to that stage, have been shown to minimize regret
[1, 2]. It is crucial to note though that, since the experts are compared on a sequence-bysequence basis, the MR criterion ignores the possibility that different experts may induce
different sequences of choices by the opponent. Thus, MR makes sense only under the
assumption that Nature?s choices are independent of the decision maker?s choices.
Repeated games. We consider a multi-agent interaction in the form of a repeated game.
In repeated games, the assumption that the opponent?s choices are independent of the
agent?s choices is not justified, because the opponent is likely to base his choices of actions on the past history of the game. This is evident in nonzero-sum games, where players
are faced with issues such as how to coordinate actions, establish trust or induce cooperation. These goals require that they take each other?s past actions into account when making
decisions. But even in the case of zero-sum games, the possibility that an opponent has
bounded rationality may lead a player to look for patterns to be exploited in the opponent?s
past actions.
We illustrate some of the aforementioned issues with an example involving the Prisoner?s
Dilemma game.
The Prisoner?s Dilemma. In the single-stage Prisoner?s Dilemma (PD) game, each
player can either cooperate (C) or defect (D). Defecting is better than cooperating regardless of what the opponent does, but it is better for both players if both cooperate than if both
defect. Consider the repeated PD. Suppose the row player consults with a set of experts,
including the ?defecting expert,? who recommends defection all the time. Let the strategy
of the column player in the repeated game be fixed. In particular, the column player may be
very patient and cooperative, willing to wait for the row player to become cooperative, but
eventually becoming non-cooperative if the row player does not seem to cooperate. Since
defection is a dominant strategy in the stage game, the defecting expert achieves in each
step a reward as high as any other expert against any sequence of choices of the column
player, so the row player learns with the experts algorithm to defect all the time. Obviously,
in retrospect, this seems to minimize regret, since for any fixed sequence of actions by the
column player, constant defection is the best response. Obviously, constant defection is
not the best response in the repeated game against many possible strategies of the column
player. For instance, the row player would regret very much using the experts algorithm if
he were told later that the column player had been playing a strategy such as Tit-for-Tat.1
In this paper, we propose and analyze a new experts algorithm, which follows experts judiciously, attempting to maximize the long-term average reward. Our algorithm differs from
previous approaches in at least two ways. First, each time an expert is selected, it is followed for multiple stages of the game rather than a single one. Second, our algorithm takes
1
The Tit-for-Tat strategy is to play C in the first stage, and later play in every stage whatever the
opponent played in the preceding stage.
into account only the rewards that were actually achieved by an expert in the stages it was
followed, rather than the reward that could have been obtained in any stage. Our algorithm
enjoys the appealing simplicity of the previous algorithms, yet it leads to a qualitatively
different behavior and improved average reward. We present two results:
1. A ?worst-case? guarantee that, in any play of the game, our algorithm achieves an
average reward that is asymptotically as large as that of the expert that did best
in the rounds of the game when it was played. The worst-case guarantee holds
without any assumptions on the opponent?s or experts? strategies.
2. Under certain conditions, our algorithm achieves an average reward that is asymptotically as large as the average reward that could have been achieved by the best
expert, had it been followed exclusively. The conditions are required in order to
facilitate learning and for the notion of a ?best expert? to be well-defined.
The effectiveness of the algorithm is demonstrated by its performance in the repeated PD
game, namely, it is capable of identifying the opponent?s willingness to cooperate and it
induces cooperative behavior.
The paper is organized as follows. The algorithm is described in section 2. A bound
based on actual expert performance is presented in section 3. In section 4, we introduce
and discuss an assumption about the opponent. This assumption gives rise to asymptotic
optimality, which is presented in section 5.
2
The algorithm
We consider an ?experts strategy? for the row player in a repeated two-person game in
normal form. At each stage of the game, the row and column player choose actions i ? I
and j ? J, respectively. The row player has a reward matrix R, with entries 0 ? Rij ? u.
The row player may consult at each stage with a set of experts {1, . . . , r}, before choosing
an action for the next stage. We denote by ?e the strategy proposed by expert e, i.e.,
?e = ?e (hs ) is the proposed probability distribution over actions in stage s, given the
history hs . We refer to the row player as the agent and to the column player as the opponent.
Usually, the form of experts algorithms found in the literature is as follows. Denote by
Me (s ? 1) the average reward achieved by expert e prior to stage s of the game2 . Then,
a reasonable rule is to follow expert e in stage s with a probability that is proportional to
some monotone function of Me (s ? 1). In particular, when this probability is proportional
to exp{?s Me (s?1)}, for a certain choice of ?s , this algorithm is known to minimize regret
[1, 2]. Specifically, by letting js (s = 1, 2, . . .) denote the observed actions of the opponent
up to stage s, and letting ?X denote the strategy induced by the experts algorithm, we have
s
s
1X
1X
E[R(i, js ) : i ? ?X (hs )] ? sup
E[R(i, js ) : i ? ?e (hs )] ? o(s).
s 0
e s 0
s =1
(1)
s =1
The main deficiency of the regret minimization approach is that it fails to consider the influence of chosen actions of a player on the future choices of the opponent ? the inequality
(1) holds for any fixed sequence (js ) of the opponent?s moves, but does not account for the
fact that different choices of actions by the agent may induce different sequences of the opponent. This subtlety is also missing in the experts algorithm we described above. At each
2
In different variants of the algorithm and depending on what information is available to the row
player, Me (s ? 1) could be either an estimate of the average reward based on reward achieved by
expert e in the stages it was played, or the reward it could have obtained, had it been played in all
stages against the same history of play of the opponent.
stage of the game, the selection of expert is based solely on how well various experts have,
or could have, done so far. There is no notion of learning how an expert?s actions affect
the opponent?s moves. For instance, in the repeated PD game described in the introduction,
assuming that the opponent is playing Tit-for-Tat, the algorithm is unable to establish the
connection between the opponent?s cooperative moves and his own.
Based on the previous observations, we propose a new experts algorithm, which takes into
account how the opponent reacts to each of the experts. The idea is simple: instead of
choosing a (potentially different) expert at each stage of the game, the number of stages an
expert is followed, each time it is selected, increases gradually. We refer to each such set
of stages as a ?phase? of the algorithm. Following is the statement of the Strategic Experts
Algorithm (SEA). The phase number is denoted by i. The number of phases during which
expert e has been followed is denoted by Ne . The average payoff from phases in which
expert e has been followed is denoted by Me .
Strategic Experts Algorithm (SEA):
1. For e = 1, . . . , r, set Me = Ne = 0. Set i = 1.
2. With probability 1/i perform an exploration phase, namely, choose an expert e
from the uniform distribution over {1, . . . , r}; otherwise, perform an exploitation
phase, namely, choose an expert e from the uniform distribution over the set of
experts e0 with maximum Me0 .
3. Set Ne = Ne +1. Follow expert e?s instructions for the next Ne stages. Denote by
? the average payoff accumulated during the current phase (i.e., these Ne stages),
R
and set
? ? Me ) .
Me = Me + Ne2+1 (R
4. Set i = i + 1 and go to step 2.
Throughout the paper, s will denote a stage number, and i will denote a phase number.
We denote by M1 (i), . . . , Mr (i) the values of the registers M1 , . . . , Mr , respectively, at
the end of phase i. Similarly, we denote by N1 (i), . . . , Nr (i) the values of the registers
N1 , . . . , Nr , respectively, at the end of phase i. Thus, Me (i) and Ne (i) are, respectively,
the average payoff accumulated by expert e and the total number of phases this expert was
followed on or before phase i. We will also let M (s) and M (i) denote, without confusion,
the average payoff accumulated by the algorithm in the first s stages or first i phases of the
game.
3
A bound based on actual expert performance
When the SEA is employed, the average reward Me (i) that was actually achieved by each
available expert e is being tracked. It is therefore interesting to compare the average reward
M (s) achieved by the SEA, with the averages achieved by the various experts. The following theorem states that, in the long run, the SEA obtains almost surely at least as much as
the actual average reward obtained by any available expert during the same play.
Theorem 3.1.
Pr lim inf M (s) ? max lim inf Me (i) = 1 .
s??
e
i??
(2)
Although the claim of Theorem 3.1 seems very close to regret minimization, there is an essential difference in that we compare the average reward of our algorithm with the average
reward actually achieved by each expert in the stages when it was played, as opposed to
the estimated average reward based on the whole history of play of the opponent.
Note that the bound (2) is merely a statement about the average reward of the SEA in
comparison to the average reward achieved by each expert, but nothing is claimed about
the limits themselves. Theorem 5.1 proposes an application of this bound in a case when
an additional assumption about the experts? and opponent?s strategies allows us to analyze
convergence of the average reward for each expert. Another interesting case occurs when
one of the experts plays a maximin strategy; in this case, bound (2) ensures that the SEA
achieves at least the maximin value of the game. The same holds if one of the experts is a
regret-minimizing experts algorithm, which is known to achieve at least the maximin value
of the game.
The remainder of this section consists of a sketch of the proof of Theorem 3.1.
Sketch of proof: Denote by V be the random variable maxe lim inf i?? Me (i), and denote
? the expert that achieves that maximum (if there is more than one, let E
? be the one
by E
with the least index). For any logical proposition L, let ?(L) = 1 if L is true; otherwise
?(L) = 0. The proof of Theorem 3.1 relies on establishing that, for all > 0 and any
expert e,
Ne (i) ? ?(Me (i) ? V ? )
Pr lim
=0 =1.
(3)
i??
i
In words, if the average reward of an expert falls below V by a non negligible amount, it
must have been followed only a small fraction of the total number of phases. There are
three possible situations for any expert e: (a) When lim inf i?? Me (i) > V ? , the inequality is satisfied trivially. (b) When lim supi?? Me (i) < V , there is a phase I such that
for all i ? I, Me (i) < ME? (i), so that expert e is played only on exploration phases, and a
large deviations argument establishes that (3) holds. (c) The most involved situation occurs
when lim inf i?? Me (i) ? V ? and lim supi?? Me (i) ? V . To show that (3) holds in
this case, we are going to focus on the trajectory of Me (i) each time it goes from above
V ?/2 to below V ?+?/2, for some 0 < ? < . We offer the two following observations:
1. Let Ik be the k th phase such that Me (i) ? V ? + ?/2, and let Ik0 be the first
phase before Ik such that Me (i) ? V ? /2. Then, between phases Ik0 and Ik ,
expert e is selected at least Ne (Ik0 )( ? ?)/(6u) times.
Denoting by Ikj , j = 1, . . . , Pk , the phases when expert e is selected, between Ik0
and Ik , we have
Me (Ikj ) ?
Me (Ikj?1 )(Ne (Ik0 ) + j ? 1)(Ne (Ik0 ) + j)
.
(Ne (Ik0 ) + j)(Ne (Ik0 ) + j + 1)
A simple induction argument shows that, in order to have
Me (Ik ) ? V ?
??
? Me (Ik0 ) ?
,
2
2
expert e must be selected a number of times Pk ? Ne (Ik0 )( ? ?)/(6u).
2. For all large enough k, the phases Ikj when expert e is selected are exclusively
exploration phases.
This follows trivially from the fact that, after a certain phase I, we have ME? (i) ?
V ? /2, for all i ? I, whereas Me (i) < V ? /2 for all i between Ik0 and Ik .
From the first observation, we have
Ne (Ik )
Ne (Ik0 ) + Pk
(1 + 6u)Pk
?
?
,
Ik
Ik ? Ik0
( ? ?)Ik ? Ik0
Since expert e is selected only during exploration phases between Ik0 and Ik , a large deviations argument allows us to conclude that the ratio of the number of times Pk expert e is
selected, to the total number of phases Ik ? Ik0 , converges to zero with probability one. We
conclude that (3) holds.
We now observe that
P
Ne (i)(Ne (i) + 1)Me (i)
M (i) = eP
.
(4)
e Ne (i)(Ne (i) + 1)
By a simple optimization argument, we can show that
X
Ne (i)(Ne (i) + 1) ? i(i/r + 1).
(5)
e
Using (3) and (5) to bound (4), we conclude that (2) holds for the subsequence of stages
s corresponding to the end of each phase of the SEA. It is easy to show that the average
reward M (s) in stages s in the middle of phase i becomes arbitrarily close to the average
reward at the end of that phase M (i), as i goes to infinity, and the theorem follows .
2
4
The flexible opponent
In general, it is impossible for an experts algorithm to guarantee, against an unknown opponent, a reward close to what the best available expert would have achieved if it had been
the only expert. It is easy to construct examples which prove this impossibility.
Example: Repeated Matching Pennies. In the Matching Pennies (MP) game, each of
the player and the adversary has to choose either H (?Heads?) or T (?Tails?). If the choices
match, the player loses 1; otherwise, he wins 1. A possible strategy for the adversary in the
repeated MP game is:
Adversary: Fix a positive integer s and a string ? s ? {H, T }s . In each of the first s
stages, play the 50:50 mixed strategy. In each of the stages s + 1, s + 2, . . . , if the sequence
of choices of the player during the first s stages coincided with the string ? s , then play T ;
otherwise, play the 50:50 mixed strategy.
Suppose each available expert e corresponds to a strategy of the form:
Expert: Fix a string ?e ? {H, T }s . During the first s stages play according to ?e . In each
of the stages s + 1, s + 2, . . . , play H.
Suppose an expert e? with ?e? = ? s is available. Then, in order for an experts algorithm
to achieve at least the reward of e? , it needs to follow the string ? s precisely during the
first s stages. Of course, without knowing what ? s is, the algorithm cannot play it with
probability one, nor can it learn anything about it during the play.
In view of the repeated MP example, some assumption about the opponent must be made
in order for the player to be able to learn how to play to against that opponent. The essence
of the difficulty with the above strategy of the opponent is that it is not flexible ? the
player has only one chance to guess who the best expert is and thus cannot recover from
a mistake. Here, we introduce the assumption of flexibility as a possible remedy to that
problem. Under the assumption of flexibility, the SEA achieves an average reward that is
asymptotically as high as what the best expert could be expected to achieve.
Definition 4.1 (Flexibility). (i) An opponent playing strategy ?(s) is said to be flexible
with respect to expert e (e = 1, . . . , r) if there exist constants ?e , ? > 0.25 and c such that
for every stage s0 , every possible history hs0 at stage s0 and any number of stages s,
h P
i
c
s0 +s
E 1s s=s
R(a
(s),
b(s))
?
?
:
a
(s)
?
?
(h
),
b(s)
?
?(h
)
? ?
e
e
e
e
s
s
+1
0
s
(ii) Flexibility with respect to a set of experts is defined as flexibility with respect to every
member of the set.
In words, the expected average reward during the s stages between stage s0 and stage s0 +s
converges (as s tends to infinity) to a limit that does not depend on the history of the play
prior to stage s0 .
Example 4.1 : Finite Automata. In the literature on ?bounded rationality?, players are
often modelled as finite automata. A probabilistic automaton strategy (PAS) is specified
by a tuple A = hM, O, A, ?, P i, where M = {1, . . . , m} is the finite set of internal
states of the automaton, A is the set of possible actions, O is the set of possible outcomes,
?i (a) is the probability of choosing action a while in state i (i = 1, . . . , m) and P o =
(Pijo ) (1 ? i, j ? m) is the matrix of state transition probabilities, given an outcome
o ? O. Thus, at any stage of the game, the automaton picks an action from a probability
distribution associated with its current state and transitions into a new state, according to
a probability distribution which depends on the outcome of the stage game. If both the
opponent and an expert play PASs, then a Markov chain is induced over the set of pairs
of the respective internal states. If this Markov chain has a single class of recurrent states,
then the flexibility assumption holds. Note that we do not limit the size of the automata; a
larger set of internal states implies a slower convergence of the average rewards, but does
not affect the asymptotic results for the SEA.
Example 4.2 : Bounded dependence on the history. The number of possible histories at
stage s grows exponentially with s. Thus, it is reasonable to assume that the choice of action
would be based not on the exact detail of the history but rather on the empirical distribution
of past actions or patterns of actions. If the opponent is believed not to be stationary,
then discounting previous observations by recency may be sensible. For instance, if the
frequency of play of action j by the opponent is relevant, the player might condition his
Ps
0
choice at stage s + 1 on the quantities ?j = s0 =1 ? s?s ?jjs where ? < 1 and ? is the
Kronecker delta. In this case, only actions js at stages s that are relatively recent have a
significant impact on ?j . Therefore strategies based on ?j should exhibit behavior similar
to that of bounded recall, and lead to flexibility in the same circumstances as the latter.
5
A bound based on expected expert performance
In this section we show that if the opponent is ?flexible? with respect to the available
experts, then the SEA achieves almost surely an average payoff that is asymptotically as
large as what the best expert could achieve against the same opponent.
Theorem 5.1. If an opponent ? is flexible with respect to the experts 1, . . . , r, then the
average payoff up to stage s, M (s), satisfies
Pr lim inf M (s) ? max ?e = 1 .
s??
e
Theorem 5.1 follows from Lemma 5.1, stated and proven below, and Theorem 3.1.
Flexibility comes into play as a way of ensuring that the value of following any given
expert is well-defined, and can eventually be estimated as long as the SEA follows that
expert sufficiently many times. In other words, flexibility ensures that there is a best expert
to be learned, and that learning can effectively occur because actions taken by other experts,
which could affect the behavior of the opponent, are eventually forgotten by the latter.
We now present Lemma 5.1, which shows that, under the flexibility assumption, the average
reward achieved by each expert is asymptotically almost surely the same as the reward that
would have been achieved by the same expert, had he been the only available expert.
Lemma 5.1. If the opponent is flexible with respect to expert e, then with probability one,
limi?? Me (i) = ?e .
Sketch of proof: Let e be any expert. By the Borel-Cantelli lemma, exploration occurs
infinitely many times, hence e is followed during infinitely many phases. Let Ij = Ij (e),
(j = 1, 2, . . .) be the phase numbers in which e is followed. By Markov?s inequality, for
every > 0,
Pr(|Me (Ij ) ? ?e | > ) ? ?4 E[(Me (Ij ) ? ?e )4 ] .
If we could show that
?
X
E[(Me (Ij ) ? ?e )4 ] < ? ,
(6)
j=1
then we could conclude, by the Borel-Cantelli lemma, that with probability one, the inequality |Me (Ij ) ? ?e | > holds only for finitely many values of j. This implies that, with
probability one, limi?? Me (i) = ?e . It follows that if the opponent is flexible with respect
to expert e, then for some ? > 0, as j tends to infinity, E[(Me (Ij ) ? ?e )4 ] = O(j ?1?? ),
which suffices for (6).
2
Example 5.1 : Repeated Prisoner?s Dilemma revisited. Consider playing the repeated
PD game against an opponent who plays Tit-for-Tat, and suppose there are only two experts: ?Always defect? (AD) and ?Always cooperate? (AC). Thus, AC induces cooperation
in every stage and yields a payoff higher than AD, which induces defection in every stage
of the game except the first one. It is easy to verify that Tit-for-Tat is flexible with respect
to the experts AC and AD. Therefore, Theorem 5.1 holds and the SEA achieves an average
payoff at least as much as that of AC. By contrast, as mentioned in the introduction, in
order to minimize regret, the standard experts algorithm must play D in almost every stage
of the game, and therefore achieves a lower payoff.
References
[1] Auer, P., Cesa-Bianchi, N., Freund, Y. & Schapire, R.E. (1995) Gambling in a rigged casino:
The adversarial multi-armed bandit problem. In Proc. 36th Annual IEEE Symp. on Foundations
of Computer Science, pp. 322?331, Los Alamitos, CA: IEEE Computer Society Press.
[2] Freund, Y. & Schapire, R.E. (1999) Adaptive game playing using multiplicative weights. Games
and Economic Behavior 29:79?103.
[3] Foster, D. & Vohra, R. (1999) Regret and the on-line decision problem. Games and Economic
Behavior 29:7?35.
[4] Fudenberg, D. & Levine, D.K. (1997) The Theory of Learning in Games. Cambridge, MA: The
MIT Press.
[5] Littlestone, N. & Warmuth, M.K. (1994) The weighted majority algorithm. Information and
Computation 108 (2):212?261.
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1,638 | 249 | 28
Lockery t Fang and Sejnowski
Neu.?al Network Analysis of
Distributed Representations of Dynamical
Sensory-Motor rrransformations in the Leech
Shawn R. LockerYt Van Fang t and Terrence J. Sejnowski
Computational Neurobiology Laboratory
Salk Institute for Biological Studies
Box 85800, San Diego, CA 92138
ABSTRACT
Interneurons in leech ganglia receive multiple sensory inputs and make
synaptic contacts with many motor neurons. These "hidden" units
coordinate several different behaviors. We used physiological and
anatomical constraints to construct a model of the local bending reflex.
Dynamical networks were trained on experimentally derived input-output
patterns using recurrent back-propagation. Units in the model were
modified to include electrical synapses and multiple synaptic time
constants. The properties of the hidden units that emerged in the
simulations matched those in the leech. The model and data support
distributed rather than localist representations in the local bending reflex.
These results also explain counterintuitive aspects of the local bending
circuitry.
INTRODUCTION
Neural network modeling techniques have recently been used to predict and analyze the
connectivity of biological neural circuits (Zipser and Andersen, 1988; Lehley and
Sejnowski, 1988; Anastasio and Robinson, 1989). Neurons are represented as simplified
processing units and arranged into model networks that are then trained to reproduce the
input-output function of the reflex or brain region of interest. After training, the
receptive and projective field of hidden units in the network often bear striking similarities
to actual neurons and can suggest functional roles of neurons with inputs and outputs that
are hard to grasp intuitively. We applied this approach to the local bending reflex of the
leech, a three-layered, feed-forward network comprising a small number of identifiable
Neural Network Analysis of Distributed Representations in the Leech
neurons whose connectivity and input-output function have been determined
physiologically. We found that model local bending networks trained using recurrent
back-propagation (pineda, 1987; Pearlmutter, 1989) to reproduce a physiologically
detennined input-output function contained hidden units whose connectivity and temporal
response properties closely resembled those of identified neurons in the biological
network. The similarity between model and actual neurons suggested that local bending
is produced by distributed representations of sensory and motor infonnation.
THE LOCAL BENDING REFLEX
In response to a mechanical stimulus, the leech withdraws from the site of contact (Fig.
la). This is accomplished by contraction of longitudinal muscles beneath the stimulus
and relaxation of longitudinal muscles on the opposite side of the body, resulting in a Ushaped local bend (Kristan, 1982). The fonn of the response is independent of the site of
stimulation: dorsal, ventral, and lateral stimuli produce an appropriately oriented
a
.~~\
b
Rutlng
/
"
.,
Left
Right
"",,-
Sensory
neurons
~~
Dorsal
,
Interneurons
,
0000-
Motor
neurons
~0
Dorsal
0
~ -local
bending
-
excitatory
-
inhib~ory
interneurons
Unidentifi8d
0000- local bending
interneurons
Figure 1: a. Local bending behavior. Partial view of a leech in the resting position
and in response to dorsal, ventral, and lateral stimuli. b. Local bending circuit. The main
input to the reflex is provided by the dorsal and ventral P cells (PD and PV). Control of
local bending is largely provided by motor neurons whose field of innervation is restricted
to single left-right, dorsal-ventral quadrants of the body; dorsal and ventral quadrants are
innervated by both excitatory (DE and VE) and inhibitory (DI and VI) motor neurons.
Motor neurons are connected by electrical and chemical synapses. Sensory input to motor
neurons is mediated by a layer of intemeurons. Intemeurons that were excited by PD and
which in tum excite DE have been identified (hatched); other types of intemeurons
remain to be identified (open).
29
30
Lockery, Fang and Sejnowski
withdrawal. Major input to the local bending reflex is provided by four pressure sensitive
mechanoreceptors called P cells, each with a receptive field confined to a single quadrant
of the body wall (Fig. Ib). Output to the muscles is provided by eight types of
longitudinal muscle motor neurons, one to four excitatory and inhibitory motor neurons
for each body wall quadrant (Stuart, 1970; Ort et al., 1974). Motor neurons are connected
by chemical and electrical synapses that introduce the possibility of feedback among the
motor neurons.
Dorsal. ventral. and lateral stimuli each produce a pattern of P cell activation that results
in a unique pattern of activation and inhibition of the motor neurons (Lockery and
Kristan, 1990a). Connections between sensory and motor neurons are mediated by a layer
of interneurons (Kristan, 1982). Nine types of local bending interneurons have been
identified (Lockery and Kristan, 1990b). These comprise the subset of the local bending
interneurons which contribute to dorsal local bending because they are excited by the
dorsal P cell and in turn excite the dorsal excitatory motor neuron. There appear to be no
functional connections between interneurons. Other interneurons remain to be identified,
such as those which inhibit the dorsal excitatory motor neurons.
Interneuron input connections were determined by recording the amplitude of the
postsynaptic potential in an interneuron while each of the P cells was stimulated with a
standard train of impulses (Lockery and Kristan, 1990b). Output connections were
detennined by recording the amplitude of the postsynaptic potential in each motor neuron
when an interneuron was stimulated with a standard current pulse. Interneuron input and
output connections are shown in Figure 2, where white squares are excitatory
connections, black squares are inhibitory connections, and the size of each square indicates
connection strength. Most interneurons received substantial input from three or four P
cells, indicating that the local bending network fonns a distributed representation of
sensory input.
dorsal
ventral
c
Figure 2: Input and output connections of the nine types of dorsal local bending
interneurons. Within each gray box, the upper panel shows input connections from
sensory neurons, the middle panel shows output connections to inhibitory motor
neurons, and the lower panel shows output connections to excitatory motor neurons.
Side-length of each box is proportional to the amplitude of the connection detennined
from intracellular recordings of interneurons or motor neurons. White boxes indicate
excitatory connections and black boxes indicated inhibitory connections. Blank spaces
denote conections whose strength has not been detennined for technical reasons.
Neural Network Analysis of Distributed Representations in the Leech
NEURAL NETWORK MODEL
Because sensory input is represented in a distributed fashion, most interneurons are active
in all forms of local bending. Thus, in addition to contributing to dorsal local bending,
most interneurons are also active during ventral and lateral bending when some or all of
their output effects are inappropriate to the observed behavioral response. This suggests
that the inappropriate effects of the dorsal bending interneurons must be offset by other as
yet unidentified interneurons and raises the possibility that local bending is the result of
simultaneous activation of a population of interneurons with multiple sensory inputs and
both appropriate and inappropriate effects on many motor neurons. It was not obvious,
however, that such a population was sufficient, given the well-known nonlinearities of
neural elements and constraints imposed by the input-output function and connections
known to exist in the network. The possibility remained that intemeurons specific for
each form of the behavior were required to produce each output pattern. To address this
issue, we used recurrent back-propagation (Pearl mutter, 1989) to train a dynamical
network of model neurons (Fig 3a). The network had four input units representing the
IJ
a
Left
Before
RIght
@@
Sensory
neurons
c
Interneuron.
After
Target
1-
y-
y-
2-
J>--
-"'"-
3-
f.-
"-
4-
y-
y-
5 ---
y-
y-
6-
J---
-"'"-
f.-
"-
0
~
:s
CD
?:
~
0
0
'-
Motor
neuron.
7I
~ ??clatory
-
InPllbaory
---
electrical
Stlm
8-
y-
y-
~
~
---=-110 mV
5 sac
Figure 3: a. The local bending network model. Four sensory neurons were connected to
eight motor neurons via a layer of 10 interneurons. Neurons were represented as single
electrical compartments whose voltage varied as a function of time (see text). Known
electrical and chemical connections among motor neurons were assigned fixed connection
strengths (g's and w's) determined from intracellular recordings. Interneuron input and
output connections were adjusted by recurrent back-propagation. Chemical synaptic
delays were implemented by inserting s-units between chemically connected pairs of
neurons. S-units with different time constants were inserted between sensory and
interneurons to account for fast and slow components of synaptic potentials recorded in
interneurons. b. Output of the model network in response to simultaneous activation of
both PDs (stirn). The response of each motor neuron (rows) is shown before and after
training. The desired response contained in the training set is shown on the right for
comparison (target).
31
32
Lockery, Fang and Sejnowski
four P cells, and eight output units representing the eight motor neuron types. Between
input and output units was a single layer of 10 hidden units representing the intemeurons.
Neurons were represented as single electrical compartments with an input resistance and
time constant. The membrane potential (Vj) of each neuron was given by
where Ti and Ri are the time constant and input resistance of the neuron and Ie and Ie are
the sum of the electrical and chemical synaptic currents from presynaptic neurons.
Current due to electrical synapses was given by
where gij is the coupling conductance between neuron i and j. To implement the delay
associated with chemical synapses, synapse units (s-units) were inserted between between
pairs of neurons connected by chemical synapses. The activation of each s-unit was given
by
where Tij is the synaptic time constant and f(Vj) was a physiologically determined
sigmoidal function (0 S f S 1) relating pre- and postsynaptic membrane potential at an
identified monosynaptic connection in the leech (Granzow et al., 1985). Current due to
chemical synapses was given by
where Wij is the strength of the chemical synapse between units i and j. Thus, synaptic
current is a graded function of presynaptic voltage, a common feature of neurons in the
leech (Friesen, 1985; Granzow et al., 1985; Thompson and Stent, 1976) and other
invertebrates (Katz and Miledi, 1967; Burrows and Siegler, 1978; Nagayama and Hisada.
1987).
Chemical and electrical synaptic strengths between motor neurons were determined by
recording from pairs of motor neurons and were not adjusted by the training algorithm.
Interneuron input and output connections were given small initial values that were
randomly assigned and subsequently adjusted during training. During training, input
connections were constrained to be positive to reflect the fact that only excitatory
interneuron input connections were seen (Fig. 2), but no constraints were placed on the
number of input or output connections. Synaptic time constants were assigned fixed values. These were adjusted by hand to fit the time course of motor neuron synaptic
potentials (Lockery and Kristan, 1990a), or determined from pairwise motor neuron
recordings (Granzow et al., 1985).
Neural Network Analysis or Distributed Representations in the Leech
a
left
right
\
I
dorsal
ventral -
b
Data
Model
Slow
Fast
~l-ttL-- -------Stirn
110mV
--=---1100 mV
Stirn
400ms
Figure 4:
Q. Input and output connections of model local bending intemeurons. Model
interneurons, like the actual interneurons, received substantial inputs from three or four
sensory neurons and had significant effects on most of the motor neurons. Symbols as in
figure 2. o. Actual (data) and simulated (model) synaptic potentials recorded from three
types of interneuron. Actual synaptic potentials were recorded in response to a train of P
cell impulses. Simulated synaptic potentials were recorded in response to a pulse of
current in the P cell which simulates a step change in P cell firing frequency.
RESULTS
Model networks were trained to produce the amplitude and time course of synaptic
potentials recorded in all eight motor neurons in response to trains of P cell impulses
33
34
Lockery t Fang and Sejnowski
(Lockery and Kristan. 1990a). The training set included the response of all eight motor
neurons when each P cell was stimulated alone and when P cells were stimulated in pairs.
After 6.000 - 10.000 training epochs. the output of the model closely matched the desired
output for all patterns in the training set (Fig. 3b). To compare intemeurons in the model
network to actual interneurons. simulated physiological experiments were performed.
Interneuron input connections were determined by recording the amplitude of the
postsynaptic potential in a model interneuron while each of the P cells was stimulated
with a standard current pulse. Output connections were detennined by recording the
amplitude of the postsynaptic potential in each motor neuron when an interneuron was
stimulated with a standard current pulse. Model interneurons. like those in the real
network. received three or four substantial connections from P cells and had significant
effects on most of the motor neurons (Fig. 4a). Most model interneurons were active
during each form of the behavior and the output connections of the interneurons were only
partially consistent with each fonn of the local bending response. Thus. the appropriate
motor neuron responses were produced by the summation of many appropriate and
inappropriate interneuron effects. This result explains the appropriate and inappropriate
effects of interneurons in the leech.
There was also agreement between the time course of the response of model and actual
interneurons to P cell stimulation (Fig. 4b). In the actual network. interneuron synaptic
potentials in response to trains of P cell impulses had a fast and slow component. Some
interneurons showed only the fast component. some only the slow. and some showed
both components (mixed). Although no constraints were placed on the temporal response
properties of interneurons. the same three types of interneuron were found in the model
network. The three different types of interneuron temporal response were due to different
relative connection strengths of fast and slow s-units impinging on a given interneuron
(Fig. 3a).
CONCLUSION
Our results show that the network modeling approach can be adapted to models with more
realistic neurons and synaptic connections. including electrical connections. which occur
in both invertebrates and vertebrates. The qualitative similarity between model and actual
interneurons demonstrates that a population of interneurons resembling the identified
dorsal local bending interneurons could mediate local bending in a distributed processing
system without additional interneurons specific for different forms of local bending.
Interneurons in the model also displayed the diversity in temporal responses seen in
interneurons in the leech. Clearly. the training algorithm did not produce exact matches
between model and actual intemeurons. but this was not surprising since the identified
local bending interneurons represent only a subset of the intemeurons in the reflex. More
exact matches could be obtained by using two pools of model interneurons. one to
represent identified neurons, the other to represent unidentified neurons. Model neurons in
the latter pool would constitute testable physiological predictions of the connectivity of
unidentified local bending intemeurons.
Acknowledgements
Supported by the Bank of America-Giannini Foundation. the Drown Foundation. and the
Mathers Foundation.
Neural Network Analysis of Distributed Representations in the Leech
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3S
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1,639 | 2,490 | .
Reasoning about Time and Knowledge In
Neural-Symbolic Learning Systems
Artur S. d' Avila Garcez" and Luis C. Lamb A
"Dept. of Computing, City University London
London, EC1V OHB, UK ([email protected])
ADept. of Computing Theory, PPGC-II-UFRGS
Porto Alegre, RS 91501-970, Brazil ([email protected])
Abstract
We show that temporal logic and combinations of temporal logics
and modal logics of knowledge can be effectively represented in artificial neural networks. We present a Translation Algorithm from
temporal rules to neural networks, and show that the networks
compute a fixed-point semantics of the rules. We also apply the
translation to the muddy children puzzle, which has been used as a
testbed for distributed multi-agent systems. We provide a complete
solution to the puzzle with the use of simple neural networks, capable of reasoning about time and of knowledge acquisition through
inductive learning.
1
Introduction
Hybrid neural-symbolic systems concern the use of problem-specific symbolic
knowledge within the neurocomputing paradigm (d'Avila Garcez et al., 2002a).
Typically, translation algorithms from a symbolic to a connectionist representation
and vice-versa are employed to provide either (i) a neural implementation of a logic,
(ii) a logical characterisation of a neural system, or (iii) a hybrid learning system
that brings together features from connectionism and symbolic artificial intelligence
(Holldobler, 1993).
Until recently, neural-symbolic systems were not able to fully represent, reason and
learn expressive languages other than propositional and fragments of first-order
logic (Cloete & Zurada, 2000). However, in (d'Avila Garcez et al., 2002b; d'Avila
Garcez et al., 2002c; d'Avila Garcez et al., 2003), a new approach to knowledge
representation and reasoning in neural-symbolic systems based on neural networks
ensembles has been introduced. This new approach shows that modal logics can be
effectively represented in artificial neural networks.
In this paper, following the approach introduced in (d'Avila Garcez et al., 2002b;
d'Avila Garcez et al., 2002c; d'Avila Garcez et al., 2003), we move one step further
and show that temporal logics can be effectively represented in artificial neural
oArtur Garcez is partly supported by the Nuffield Foundation. Luis Lamb is partly
supported by CNPq. The authors would like to thank the referees for their comments.
networks. This is done by providing a translation algorithm from temporal logic
theories to the initial architecture of a neural network. A theorem then shows
that the translation is correct by proving that the network computes a fixed-point
semantics of its corresponding temporal theory (van Emden & Kowalski, 1976) . The
result is a new learning system capable of reasoning about knowledge and time. We
have validated the Connectionist Temporal Logic (CTL) proposed here by applying
it to a distributed time and knowledge representation problem known as the muddy
children puzzle (Fagin et al., 1995).
CTL provides a combined (multi-modal) connectionist system of knowledge and
time, which allows the modelling of evolving situations such as changing environments or possible worlds. Although a number of multi-modal systems - e.g., combining knowledge and time (Halpern & Vardi, 1986; Halpern et al., 2003) and combining beliefs, desires and intentions (Rao & Georgeff, 1998) - have been proposed
for distributed knowledge representation, little attention has been paid to the integration of a learning component for knowledge acquisition. This work contributes
to bridge this gap by allowing the knowledge representation to be integrated in a
neural learning system. Purely from t he point of view of knowledge representation
in neural-symbolic systems, this work contributes to the long term aim of representing expressive and computationally well-behaved symbolic formalisms in neural
networks.
The remainder of this paper is organised as follows. We start , in Section 2, by
describing the muddy children puzzle, and use it to exemplify the main features
of CTL. In Section 3, we formally introduce CTL's Translation Algorithm, which
maps knowledge and time theories into artificial neural networks, and prove that
the t ranslation is correct. In Section 4, we conclude and discuss directions for future
work.
2
Connectionist Reasoning about Time and Knowledge
Temporal logic and its combination with other modalities such as knowledge and
belief operators have been the subject of intense investigation (Fagin et al., 1995). In
this section, we use the muddy children puzzle, a testbed for distributed knowledge
representation formalisms, t o exemplify how knowledge and t ime can be expressed
in a connectionist setting. We start by stating the puzzle (Fagin et al., 1995; Huth
& Ryan, 2000).
There is a number n of (truthful and intelligent) children playing in a garden. A
certain number of children k (k :S n) has mud on their faces . Each child can see if
the other are muddy, but not themselves. Now, consider the following situation: A
caret aker announces that at least one child is muddy (k 2': 1) and asks does any of
you know if you have mud on your own face? To help understanding the puzzle, let
us consider the cases in which k = 1, k = 2 and k = 3. If k = 1 (only one child is
muddy), the muddy child answers yes at the first instance since she cannot see any
other muddy child. All the other children answer no at the first instance. If k = 2,
suppose children 1 and 2 are muddy. At the first instance, all children can only
answer no. This allows 1 to reason as follows: if 2 had said yes the first time, she
would have been the only muddy child. Since 2 said no , she must be seeing someone
else muddy; and since I cannot see anyone else muddy apart from 2, I myself must
be muddy! Child 2 can reason analogously, and also answers yes the second time
round. If k = 3, suppose children 1, 2 and 3 are muddy. Every children can only
answer no the first two times round. Again, this allows 1 to reason as follows: if
2 or 3 had said yes the second time, they would have been the only two muddy
children. Thus, there must be a third p erson with mud. Since I can only see 2 and
3 with mud, this third person must be me! Children 2 and 3 can reason analogously
to conclude as well that yes, they are muddy.
The above cases clearly illustrate the need to distinguish between an agent's individual knowledge and common knowledge about the world in a particular situation.
For example, when k = 2, after everybody says no at the first round, it becomes
common knowledge that at least two children are muddy. Similarly, when k = 3,
after everybody says no twice, it becomes common knowledge that at least three
children are muddy, and so on. In other words, when it is common knowledge that
there are at least k -1 muddy children; after the announcement that nobody knows
if they are muddy or not , then it becomes common knowledge that there are at
least k muddy children, for if there were k - 1 muddy children all of them would
know that they had mud in their faces. I
In what follows, a modality K j is used to represent the knowledge of an agent j. In
addition, the term Pi is used to denote that proposition P is true for agent i. For
example, KjPi means that agent j knows that P is true for agent i. We use Pi to
say that child i is muddy, and qk to say that at least k children are muddy (k :s; n).
Let us consider the case in which three children are playing in the garden (n = 3).
Rule ri below states that when child 1 knows that at least one child is muddy and
that neither child 2 nor child 3 are muddy then child 1 knows that she herself is
muddy. Similarly, rule r~ states that if child 1 knows that there are at least two
muddy children and she knows that child 2 is not muddy then she must also be able
to know that she herself is muddy, and so on. The rules for children 2 and 3 are
interpreted analogously.
ri:
K Iql!\KI ""'P2!\KI ""'P3 ---+KIPI
rj: K Iq2!\K I ""'P3 ---+KIPI
d:
rl:
K Iq2!\KI ""'P2 ---+KIPI
K Iq3 ---+KIPI
Table 1: Snapshot rules for agent (child) 1
Each set of snapshot rules r~ (1 :s; I :s; n; mE N+) can be implemented in a single
hidden layer neural network Ni as follows. For each rule, a hidden neuron is created.
Each rule antecedent (e.g., KIql in ri) is associated with an input neuron. The rule
consequent (KIPI) is associated with an output neuron. Finally, the input neurons
are connected to the output neuron through the hidden neuron associated with
the rule (ri). In addition, weights and biases need to be set up to implement the
meaning of the rule. When a neuron is activated (i.e. has activation above a given
threshold), we say that its associated concept (e.g., KIql) is true. Conversely, when
a neuron is not activated, we say that its associated concept is false. As a result ,
each input vector of Ni can be associated with an interpretation (an assignment of
truth-values) to the set of rules. Weights and biases must be such that the output
neuron is activated if and only if the interpretation associated with the input vector
satisfies the rule antecedent. In the case of rule ri, the output neuron associated
with KIPI must be activated (true) if the input neuron associated with KIql, the
input neuron associated with K I ""'P2, and the input neuron associated with K I ""'P3
are all activated (true).
The Connectionist Inductive Learning and Logic Programming (C-ILP) System
(d'Avila Garcez et al., 2002a; d'Avila Garcez & Zaverucha, 1999) makes use of the
above kind of translation. C-ILP is a massively parallel computational model based
on an artificial neural network that integrates inductive learning from examples and
background knowledge with deductive learning through logic programming. FollowINotice that this reasoning process can only start once it is common knowledge that
at least one child is muddy, as announced by the caretaker.
ing (Holldobler & Kalinke , 1994) (see also (Holldobler et al. , 1999)) , a Translation
Algorithm maps any logic program P into a single hidden layer neural network N
such t hat N computes the least fixed point of P . This provides a massively parallel
model for computing the stable model semantics of P (Lloyd, 1987) . In addition,
N can b e t rained wit h examples using, e.g., Backpropagation, and using P as background knowledge (Pazzani & Kibler, 1992) . The knowledge acquired by training
can then be extracted (d'Avila Garcez et al. , 2001) , closing the learning cycle (as
in (Towell & Shavlik, 1994)).
For each agent (child) , a C-ILP network can be created. Each network can be
seen as representing a (learnable) possible world containing information about the
knowledge held by an agent in a distributed system . Figure 1 shows the implementation of rules ri to d. In addition, it contains output neurons PI 2 and Kql , Kq2
and Kq3 , all represented as facts. 3 This is highlighted in grey in Figure 1. Neurons
that appear on both the input and output layers of a C-ILP network (e.g., Kqd
are recurrently connected using weight one, as depicted in Figure 1. This allows the
network to iterate the computation of truth-values when chains occur in the set of
rules. For example, if a ---+ b and b ---+ C are rules of the theory, neuron b will appear
on both the input and output layers of the network, and if a is activated then c will
be activated through the activation of b.
Figure 1: The implementation of rules
{ri, ... , rn.
If child 1 is muddy, output neuron PI must be activat ed. Since, child 2 and 3 can
see child 1, they will know that PI is muddy. This can be represented as PI ---+ K 2PI
and PI ---+ K 3PI , and analogously for P2 and P3 . This means that the activation of
output neurons KI 'P2 and K I 'P3 in Figure 1 depends on the activation of neurons
that are not in this network (NI ), but in N2 and N 3 . We need, therefore, to model
how the networks in the ensemble interact with each other.
Figure 2 illustrat es the interaction between three C-ILP networks in the muddy
children puzzle. The arrows connecting the networks implement the fact that when
a child is muddy, the other children can see her . So if, e.g., neuron PI is activated
in N I , neuron KPI must be activat ed in N2 and N 3 . For the sake of clarit y, the
snapshot rules r;" shown in Figure 1 are omitted here, and this is indicat ed in Figure
2Note Pl means 'child 1 is muddy' while KPl means 'child 1 knows she is muddy'.
fact is normally represented as a rule with no antecedents . C-ILP represents fact s by
not connecting the rule's hidden neuron to any input neuron (in the case of fully-connected
n etworks, weights with initial value zero ar e used).
3A
2 by neurons highlighted in black. In addition, only positive information about the
problem is shown in Figure 2. Negative information such as -'PI, K-'PI, K-'P2 and
K -'P3 would be implemented analogously.
I
I
I
I
--------- - - -
Figure 2: Interaction between agents in t he muddy children puzzle.
Figure 2 illustrates well the idea behind this paper. By combining a number of
simple C-ILP networks , we are able to model individual and common knowledge .
E ach network represents a possible world or an agent 's current set of beliefs (d' Avila
Garcez et al. , 2002b). If we allow a number of ensembles like the one of Figure 2 to
be combined, we can represent the evolution in time of an agent's set of beliefs. This
is exactly what is required for a complet e solution of the muddy children puzzle, as
discussed below.
As we have seen, the solution to the muddy children puzzle illustrat ed in Figures 1
and 2 considers only snapshots of knowledge evolution along time rounds without
the addition of a time variable (Ruth & Ryan, 2000). A complete solution, however ,
requires the addition of a t emporal variable to allow reasoning about t he knowledge
acquired after each time round. The snapshot solution of Figures 1 and 2 should
then be seen as representing the knowledge held by the agents at an arbitrary time
t. The knowledge held by the agents at time t + 1 would then b e represented
by anot her set of C-ILP networks, appropriat ely connected t o the original set of
networks. Let us consider again the case where k = 3. There are alternative ways
of representing that , but one possible representation for child 1 would be as follows:
tl : -,KIPI /\ -,K 2P2 /\ -,K 3P3
t2 : -,KIPI /\ -,K2P2 /\ -,K3P3
---+
---+
O K I Q2
O K I Q3
Table 2: Temporal rules for agent(child) 1
Each t emporal rule is labelled by a time point ti in which the rule holds. In addition,
if a rule labelled t i makes use of the n ext time t emporal operator 0 then what ever
qualifies refers to the next time ti+l in a linear time flow. As a result , the first
t emporal rule above stat es that if, at tl, no child knows whether she is muddy or
not then, at t 2 , child 1 will know that at least two children are muddy. Similarly,
the second rule states that, at t2, if still no child knows whether she is muddy or
not then, at t3, child 1 will know that at least three children are muddy. As b efore,
analogous temporal rules exist for agents (children) 2 and 3. The temporal rules ,
together with the snapshot rules , provide a complete solution to the puzzle. This
is depicted in Figure 3 and discussed b elow. 4
o
In Figure 3, networks are replicat ed to represent an agent's knowledge evolution in
time. A network represents an agent 's knowledge today (or at tl), a network repre41t is worth noting that each network remains a simple, single hidden layer neura l
network that can be trained with the use of standard Backpropagation or other off-theshelf learning algorithm.
To Agents 2 and 3 (Kpl) at tl
To Agents 2 and 3 (Kp1) at t2
$"~;~~;'---:-\
?
CL)(). CLX I)
1 at t1~.
'"
-, ~K,
)
;'
,.'"
/
J,
)if~~;;;3) ~\o
.6~o:s;(t:).~~_~_ );::~AgrnU(Kp3)
' .!
\~ , 1 at t2
\ ....
"
'~K~'
~.~
--".. -- From Agent 2 (Kp2)
at t1
/
I
///
,.
\ : . '" ~//
",,"", __ ~ __~ _ l /~/
- ____ ~
,
-~~ /
~
___ - - From Agent 3 (p3) at t1
'",,' _____ ->~---.
From Agent 2 (p2) at t1
From Agent 3 (p3)
att2
From Agent 2 (p2)
at t2
Figure 3: Knowledge evolution of agent (child) 1 from time tl to time h
sents the same agent's knowledge tomorrow (t 2 ), and the appropriate connections
between networks model the relations between today and tomorrow according to
O. In the case of tl : ,KIPI 1\ ,K2 P2 1\ ,K3 P3 -+ OKl q2, for example, output
neuron KIPI of the network that represents agent 1 at t l , output neuron K 2P2 of
the network that represents agent 2 at tl, and output neuron K 3 P3 of the network
that represents agent 3 at tl need to be connected to output neuron K l q2 of the
network that represents agent 1 at t2 (the next time) such that K l q2 is activated
if KIPI, K 2P2 and K 3 P3 are not activated. In conclusion, in order to represent
time, in addition to knowledge, we need to use a two-dimensional C-ILP ensemble.
In one dimension we encode the knowledge interaction between agents at a given
time point, and in the other dimension we encode the agents' knowledge evolution
through time.
3
Temporal Translation Algorithm
In this section, we present an algorithm to translate temporal rules of the form
t : OKaLI' ... , OKbLk -+ OKcLk+I' where a, b, c ... are agents and 1 :s; t :s; n,5
into (two-dimensional) C-ILP network ensembles. Let P represent a number q of
ground 6 temporal rules. In such rules, we call Li (1 :s; i :s; k + 1) a literal, and
call KjL i (1 :s; j :s; m) an annotated literal. Each Li can be either a positive
literal (p) or a negative literal ('p). Similarly, KjL i can be preceded by , . We
use Amin to denote the minimum activation for a neuron to be considered active
(true), Amin E (0,1). We number the (annotated) literals 7 of P from 1 to v such
that, when a C-ILP network N is created, the input and output layers of N are
vectors of length v, where the i-th neuron represents the i-th (annotated) literal.
For convenience, we use a bipolar semi-linear activation function h(x) = l+e2- IlX -1,
and inputs in {-I, I}.
Let k z denote the number of (annotated) literals in the body of rule rl; f..L1, the
number of rules in P with the same (annotated) literal as consequent , for each
rule Tl; MAXrz (kl' f..L1), the greater element between kz and f..L1 for rule Tl; and
MAX p (kl' ... , kq, f..LI, ... , f..Lq), the greatest element among all kl's and f..Lz'S of P. We
5There may be n + 1 time points since, e.g., h : Kja, K k f3 -> OKj, means that if
agent j knows a and agent k knows f3 at time tl then agent j knows / at time t2.
6Variables such as ti are instantiated into the language's ground terms (tl, t2, t3 ... ).
7We use ' (annotated) literals' to refer to any literal, annotated or not annotated ones .
-----+
-----+
also use k as a shorthand for (k 1, ... , kq), and fJ, as a shorthand for (fJ,1, ... , fJ,q).
For example, for P = {r1 : b /\ c /\ ---,d ----+ a, r2 : e /\ f ----+ a, r3 : ----+ b}, k1 = 3,
k2 = 2, k3 = 0, fJ,1 = 2, fJ,2 = 2, fJ,3 = 1, MAXr 1 (k 1,fJ,1) = 3, MAX r2 (k 2,fJ,2) = 2,
-----+ -----+
M AXr 3 (k3, fJ,3) = 1 and M AXp( k , fJ, ) = 3.
CTL Translation Algorithm:
1. For each time point t in P do: For each agent j in P do: Create a C-ILP Neural
Network Nj,t.
In(l ?,~i n)-ln(l -Ami n)
2. Calculate W such that W 2': 2. .
(3
;
MAXp(k , M ).(Amin-1)+Amin+1
3. For each rule in P of the form t : OK1L 1, ... , OKm - 1L k ----+ OKm L k+1,8 do:
(a) Add a hidden neuron L O to N m ,t+1 and set h(x) as the activation function
of L O; (b) Connect each neuron OKjLi (1 ::; i ::; k) in Nj,t to LO. If L i is a
positive (annotated) literal then set the connection weight to W; otherwise, set the
connection weight to -W . Set the threshold eO
of L O to eO
= (1+ A min)(k l -1)W'
I
I
2
'
(c) Connect L O to KmLk+1 in N m ,t+ 1 and set the connection weight to W. Set the
threshold e;+l of KmLk+1 to e;+l = (1+ A mi;)(l- Md W ; (d) Add a hidden neuron L e
to Nm ,t and set h(x) as the activation function of L e ; (e) Connect neuron KmLk+1
in N m ,t+1 to Le and set the connection weight to W; Set the threshold ei of Le to
zero; (f) Connect L e to OKmLk+1 in Nm ,t and set the connection weight to W.
Set the threshold It of K m L k+1 to It = (1+A min2 )(l-Md W ?'
e
e
4. For each rule in P of the form t : OK1L 1, ... , OKm-1Lk ----+ KmLk+1 ' do:
(a) Add a hidden neuron L O to Nm, t and set h(x) as the activation function of
L O; (b) Connect each neuron OKjLi (1 ::; i ::; k) in Nj ,t to L O . If L i is a
positive (annotated) literal then set the connection weight to W; otherwise, set the
connection weight to -W . Set the threshold eO
of LO to eO
= (1+ A min)(k l -1)W'
I
I
2
'
(c) Connect LO to K m L k+1 in Nm ,t and set the connection weight to W . Set the
threshold ei+ 1 of K m L k+1 to e;+l = (1+ A mi;)(l- Md W;
5. If N ought to be fully-connected, set all other connections to zero.
In the above algorithm it is worth noting that, whenever a rule consequent is preceded by 0, a forward connection from t to t + 1 and a feedback connection from
t + 1 to t need to be added to the ensemble. For example, if t : a ----+ Ob is a
rule of P then not only must the activation of neuron a at t activate neuron b at
t + 1, but the activation of neuron b at t + 1 must also activate neuron Ob at t .
This is implemented in steps 3(d) to 3(1) of the algorithm. The remainder of the
algorithm is concerned with the implementation of snapshot rules (as in Figure 1).
The values of Wand come from C-ILP's Translation Algorithm (d'Avila Garcez
& Zaverucha, 1999), and are chosen so that the behaviour of the network matches
that of the temporal rules , as the following theorem shows.
e
Theorem 1 (Correctness of Translation Algorithm) For each set of ground temporal rules P, there exists a neural network ensemble N such that N computes the
fixed-point operator T p of P.
Proof. (sketch) This proof follows directly from the proof of the analogous theorem
for single C-ILP networks presented in (d 'Avila Garcez fj Zaverucha, 1999). This
is so because C-ILP's definition for Wand e values makes hidden neurons L O and
Le behave like and gates, while output neurons behave like or gates. D
8Note that 0 is not required to precede every rule antecedent. In the network, neurons
are labelled as OKILI or KILl to differentiate the two concepts.
4
Conclusions
In his seminal paper (Valiant, 1984), Valiant argues for the need of rich logic-based
knowledge representation mechanisms within learning systems. In this paper, we
have addressed such a need, yet complying with important principles of connectionism such as massive parallelism. In particular, a very important feature of the
system presented here (CTL) is the temporal dimension that can be combined with
an epistemic dimension. This paper provides the first account of how to integrate
such dimensions in a neural-symbolic learning system. The CTL framework opens
up several interesting research avenues in the domain of neural-symbolic integration, allowing for the representation and learning of expressive formalisms. In this
paper, we have illustrated this by providing a full solution to the muddy children
puzzle, where agents reason about their knowledge at different time points. In the
near future, we plan to also apply the system to a large, real world case study.
References
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The semantics of predicate logic as a programming
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1,640 | 2,491 | Decoding V1 Neuronal Activity using Particle
Filtering with Volterra Kernels
Ryan Kelly
Center for the Neural Basis of Cognition
Carnegie-Mellon University
Pittsburgh, PA 15213
[email protected]
Tai Sing Lee
Center for the Neural Basis of Cognition
Carnegie-Mellon University
Pittsburgh, PA 15213
[email protected]
Abstract
Decoding is a strategy that allows us to assess the amount of information
neurons can provide about certain aspects of the visual scene. In this
study, we develop a method based on Bayesian sequential updating and
the particle filtering algorithm to decode the activity of V1 neurons in
awake monkeys. A distinction in our method is the use of Volterra kernels to filter the particles, which live in a high dimensional space. This
parametric Bayesian decoding scheme is compared to the optimal linear
decoder and is shown to work consistently better than the linear optimal
decoder. Interestingly, our results suggest that for decoding in real time,
spike trains of as few as 10 independent but similar neurons would be
sufficient for decoding a critical scene variable in a particular class of
visual stimuli. The reconstructed variable can predict the neural activity
about as well as the actual signal with respect to the Volterra kernels.
1
Introduction
Cells in the primary visual cortex perform nonlinear operations on visual stimuli. This nonlinearity introduces ambiguity in the response of the neurons. Given a neuronal response,
an optimal linear decoder cannot accurately reconstruct the visual stimulus due to nonlinearities. Is there a strategy to resolve this ambiguity and recover the information that is
encoded in the response of these neurons?
Bayesian decoding schemes, which are nonlinear, might be useful in this context . Bayesian
sequential updating or belief propagation, implemented in the form of particle filtering, has
recently been used in estimating the hand trajectories of monkeys based on M1 neuron?s
responses [4] and the location of a rat based on the responses of the place cells in the
hippocampus[3]. However, linear methods have been shown to be quite adequate for decoding LGN, motor cortical, or hippocampal place cells? signals using population vectors
or the optimal linear decoder [10, 5, 8]. Bayesian methods, with proper probability model
assumptions, could work better than the linear methods, but they apparently are not critical to solving those problems. These methods may be more useful or important in the
decoding of nonlinear visual neuronal responses. Here, we implement an algorithm based
on Bayesian sequential updating in the form particle filtering to decode nonlinear visual
neurons in awake behaving monkeys. The strategy is similar to the one used by Brown et
al. [2] and Brockwell et al. [1] in their decoding of hippocampus place neurons or M1 neurons, except that we introduced the use of Volterra kernels [6, 7, 9] to filter the hypothesis
particle to generate feedback messages. The Volterra kernels integrate information from the
previous 200 ms. This window allows us to backtrack and update the hypotheses within a
200 ms window, so the hypothesis space does not grow beyond 200ms for lengthy signals.
We demonstrated that this method is feasible practically and indeed useful for decoding a
temporal variable in the stimulus input based on cells? responses and that it succeeds even
when the optimal linear decoder fails.
2
The Approach
Our objective is to infer the time-series of a scene variable based on the ongoing response
of one or a set of visual neurons. A hypothesis particle is then the entire history of the
scene variable of interest up to the present time t, i.e. (x1 , x2 , . . . , xt ) given the observed
neuronal activity (y1 , y2 , . . . , yt ). A key feature of our algorithm is the use of a decoded
or estimated hypothesis to predict the response of the neurons at the next time step. The
premise is that the scene variable we are inferring is sufficient to predict the activity of
the neuron. Since visual neurons have a temporal receptive field and typically integrate
information from the past 100-200 ms to produce a response, we cannot make the Markovian assumption made in other Bayesian decoding studies [1, 2, 3, 4]. Instead, we will use
the receptive field (kernel) to filter each hypothesis particle to generate a prediction of the
neural response. We propose to use the Volterra kernels, which have been used in previous
studies [6, 7, 9] to characterize the transfer function or receptive field of a neuron, to filter
the hypothesis (?
xt , . . . x
?1 ). The predicted response of the neuron according to the kernels
is based on the stimulus in the last 200 ms, optionally incorporating some lag which we
eliminated by shifting the response forward 40 ms in time to compensate for the 40 ms the
visual signal required to travel from the retina to V1.
Ongoing observation of the activity of
neurons is compared to the predicted response or proposal to yield a likelihood
measure.
The likelihood measure of
each hypothesis particle is proportional to
how close the hypothesis?s predicted response is to the actual observed neural response. As all the existing hypotheses
are weighted by their likelihood measures,
the posterior distribution of the hypothesis is effectively resampled. The hypotheses that tend to generate incorrect proposals will die off over time. Conversely, the
Figure 1: Two sample sinewave gratings.
hypotheses that give predicted responses
close to the actual response values will not only be kept alive, but also be allowed to give
birth to offspring particles in its vicinity in the hypothesis space, allowing the algorithm to
zoom in to the correct hypothesis more precisely.
After weighting, resampling and reproducing, the hypothesis particles are propagated forward according to the prior statistical distribution on how the scene variable tends to
progress. That is, p(xt |xt?1 ) yields a proposed hypothesis about the stimulus at time t + 1
based on the existing hypothesis which is defined at t and earlier times. These hypotheses
are then filtered though the Volterra kernels to predict the distribution p(yt |xt?200,...,t?1 ),
thus completing the loop. The entire flow-chart of our inference system is shown in Figure
4. Each step is described in detail below.
3
Neurophysiological Experiment
We applied the ideas above to the data obtained by the following experiment. This
experiment sought to understand the encoding and decoding of temporal visual information by V1 neurons. In each experimental session, a movie (2.2 seconds per
trial) of a sinewave grating stimulus was
presented while the monkey had to maintain fixation on a spot within a 0.8o ? 0.8o
window. The sinewave grating was constrained to move along one dimension in a
Figure 2: A sample time series of the scene direction perpendicular to the grating with
a step size in phase drawn from a random
variable, with a sample spike train below.
pink noise distribution which follow a 1/f
power spectrum in the Fourier domain, approximating the statistical correlational structures
in natural temporal stimuli. To ensure continuity of the input signals we took the cosine
of the phase, which is related to the image intensity value at a local area within the receptive field. In decoding the cos(phase), a hidden variable, was the scene variable inferred.
A sample stimulus is given in Figure 2. This scene variable, through the Volterra kernel
procedure, can predict the neural responses of this class of stimulus reasonably well.
400 trials of different sequences were presented. The known pair sequences of stimulus and
response in these trials were used to estimate the Volterra kernels by correlating the input
x with the neural response y. In addition, one particular stimulus sequence is repeated
60-80 trials to obtain a PSTH, which is smoothed with a 10 ms window to give an estimate
of the instantaneous firing rate. In our decoding work, we take the PSTH as input to our
algorithm; this is considered equivalent to assuming simultaneous access to a number of
identical, independent neurons. When the neurons are different, a kernel derivation for
each neuron is necessary.
4
Volterra Kernels
Volterra kernels have been used to characterize a cell?s transfer function. With Volterra
kernels with memory length L, the response yt can be predicted by convolution of the
kernels with the input xt ,
y(t) = yt = ho +
L
X
? =1
h? xt?? +
L
L X
X
?2
h?1 ,?2 xt??1 xt??2 ,
?1
where h0 corresponds to the mean firing rate, h? is the first order kernel and h?1 ,?2 the
second order kernel. We restrict all ? ?s to be positive, so we only consider causal filters.
This equation is easily expressed in matrix form as Y = XH, where time is now indexed
by matrix row in Y and X. H contains the concatenation of the terms
0
[h0 h1 ? ? ? hL h1,1 h1,2 ? ? ? hL,L ] ,
and row t of X is similarly
[1 xt?1 ? ? ? xt?L (xt?1 xt?1 ) ? ? ? (xt?L xt?L )]
?1
The standard solution for this regression problem is H = (X 0 X) X 0 Y . That is,
the parameters of the kernels are derived using the regression technique by correlating the input and the output, and are compensated by the covariance in the input, i.e.
PSTH
Predicted Response
0.2
Response (Prob of Spiking)
0.15
0.1
0.05
0
0
500
1000
1500
2000
2500
Time(ms)
Figure 3: The first and second order Volterra kernels of a V1 cell (left) and a typical prediction of the neuronal response compared to the actual response (right).
H = (X 0 X)?1 X 0 Y . Because of the correlations in the input signal xt , the matrix (X 0 X)
is ill conditioned. Instead of directly inverting this matrix, singular value decomposition
can be used, as U SU 0 = X 0 X where U S ?1 U 0 = (X 0 X)?1 and S is a diagonal matrix.
Only the first n largest dimensions as ranked by their eigenvalue are included, where n is
chosen to account for 99% of the variance in X [7].
Figure 3 depicts an example of the first and second order Volterra kernels and also shows
a typical example of their accuracy in predicting the response PSTH yt . For a majority of
these neurons, the Volterra kernels recovered are capable of predicting the neural response
to the input stimulus with a high level of accuracy. This observation forms the basis of
success for our scheme of particle filtering.
5
Decoding Scheme
We apply Bayesian decoding to the problem of determining the visual stimulus variable
xt for some time step t, given observations of the neural responses (y1 , y2 , . . . , yt ). The
global flow of the algorithm is shown in Figure 4.
5.1
Particle Prediction
At each time step of of decoding scheme, we can now filter a hypothesis particle
(?
x1 , x
?2 , . . . , x
?t ) by the Volterra kernels to generate a prediction of the response of the
neuron to test the validity of the hypothesis. (y1 , y2 , . . . , yt ) remains the observed neural
activity of a V1 neuron up to time t, and y?ti is the predicted neural activity at time t based on
hypothesis particle i. This gives us a set of predicted responses at time t, {?
yt1 , y?t2 , . . . , y?tN },
where the subscript is the particle index, and N is the number of particles.
5.2
Particle Resampling
The actual observed response of the neuron at time t is compared to each particle?s prediction as a means to evaluate the likelihood or fitness of the particle. If we assume yt is the
average of spike trains from a single neuron in independent trials or the average firing rate
of a population of independent neurons with identical tuning properties, then the resulting
error distribution can be assumed to be a Gaussian distribution, with ? representing the
uncertainty of the predicted response given the correct values of the stimulus variable. The
Figure 4: Flow chart of the PF decoding scheme. The effect of one resampling step is
shown in the two graphs. Each graph shows the particles? (n=100) values during a trial
over 200 ms. The thicknesses of the lines are proportional to the number of particles with
the corresponding values. Notice the change in the distribution of particles after sampling.
After the resampling there are many more particles concentrated around 1 instead of -1.
relative likelihood of an observation given each particle is then given by
i
p(yt |?
xi1 , . . . , x
?it )
2
2
e?(?yt ?yt ) /2?
=P
.
?(?
ytj ?yt )2 /2? 2
je
All the particles together provide a representation of the particle-conditional distribution,
p(yt |?
xt , x
?t?1 , . . . , x
?1 ).
This is used to resample the posterior distribution of the hypotheses based on all the observations up to time t ? 1,
p(?
xt |y1 , y2 , . . . yt ) ? p(yt |?
xt )p(?
xt |y1 , y2 , . . . yt?1 ),
to produce a current posterior distribution of the hypotheses.
5.3
Particle Propagation
The next step in the decoding scheme is to generate new value x
?t+1 and append it to the
hypothesis particle
Z
p(?
xt+1 |y1 , y2 , . . . yt ) = p(?
xt+1 |?
xt )p(?
xt |y1 , y2 , . . . yt )dxt ,
where p(?
xt+1 |?
xt ) is the state propagation model that provides the prior on how the stimulus
changes over time. For the state propagation model used in this study, all initial positions
for the stimulus are equally likely. The range of the stimulus (-1 to 1) is divided into 60
equally spaced intervals. A 60x60 probability table is constructed empirically from the
training data stimuli, corresponding to a discrete version approximating the conditional
prior above. Solving these priors analytically is difficult or even impossible. Besides, the
hypothesis space is enormous as there are 60 possible values at each time point, and information from a 200 ms window (20 time points at 10 ms intervals) is being integrated to
predict yt . The particle filtering algorithm is basically a way to approximate the distributions efficiently.
The algorithm consists of cycling through
the above steps, i.e. particle prediction,
particle resampling, and particle propagation. In summary,
1. Prediction step: Filter all particles by
the Volterra kernels to generate the prediction of neural responses.
2. Resampling step: Compare actual neural response with the predicted response of
each particle to assign a likelihood value
to each particle. Resample (with replacement) the posterior distribution of the particles based on their likelihood.
3. Propagation step: Sample from the state
model to randomly postulate a new stimulus value x
?t for each particle and add this
value to the end of the particle?s sequence
to obtain (?
x1 , x
?2 , . . . , x
?t ).
Figure 5: A scatter plot showing the least
squares regression line for the data.
In the propagation step, the state model will move the stimulus in ways that it has typically
been seen to move. In the prediction step, particles that predict a neural response close
to the actual observed response will be highly valued and will likely be duplicated in the
resampled set. Conversely, particles that predict a response which is not close to the actual
response will not be highly valued and thus will likely be removed from the resultant set.
6
Results and Discussion
Let xt , xt?1 , . . . , x1 be the inferred scene variable
(cos(phase)). sk (t) is the binary spike response of a
neuron during trial k. The instantaneous firing rate
of the neuron is given by
m
y(t) =
1 X
= 1sk (t)
m
k
where m is the number of trials. In general, for
cells that respond well to a stimulus, this first order
and the second order kernel can predict the response
well. Over all cells tested (n=33), the average error ratio ey in the energy of the actual response is
18.4%. Each of the cells was decoded using the
particle filtering algorithm with 1000 particles. The
average reconstruction error ex is 27.14%, and the
best cell has 10% error. A correlation exists between the encoding and decoding errors across trials as shown in Figure 5.
P
P
(?
xt ? xt )2
yt ? yt )2
t (?
P 2
, ex = Pt
,
ey =
2
t yt
t (xt + 1)
Figure 6: Reconstruction error when
input PSTH is constructed from
fewer trials. With 10 spike trains,
the PF has almost achieved the minimum error possible for this cell.
Figure 7: Particle filtering (PF) and optimal linear decoder (O.L.D.) reconstructions. The
top left is the best PF reconstruction, and the bottom right is the worst out of all the cells
tested.
? affects the rate at which the particle hypothesis space collapses around the correct solution. If ? is too large, all particles will become equally likely, while if ? is too small, only
a few particles will survive each time step. Ideally, the particles will converge on a value
for a number of time steps equal to the kernel?s length. The optimal value for ? was found
empirically and was used in all reconstructions.
Figure 7 shows sample reconstructions of some
good and bad cells. Decoding accuracy is limited
by the performance of the Volterra kernel. When
the kernel is unable to predict the neuronal response, particularly for cells that have low firing
rates, any decoding scheme will suffer because of
insufficient information. Thus the amount of error
is correlated to the inability of the kernel in predicting neuronal responses. This idea is consistent
with the error correlation between the particle filter
and kernel in Figure 5. These cells do not provide
enough relevant information about the visual stimulus in their spiking activities.
Figure 6 shows that reconstruction based on the Figure 8: A scatter plot comparing
PSTH constructed from as few as 5-10 spike trains the two decoding methods.
can reach an accuracy not far from reconstruction
based on the PSTH of 80 trials. This suggests that as few as 10 independent but similar
cells recorded simultaneously might be sufficient for decoding this scene variable.
We find that the optimal linear decoder does not decode these cells well. The decoded
output tends to follow the signal somewhat, but at a low amplitude as shown in Figure 7.
The problem for the optimal linear decoder is that at any single moment in time it can only
propose a single hypothesis, but there exist multiple signals that can produce the response.
The optimal linear decoder tends to average in these cases. The particle filter keeps alive
many independent hypotheses and can thus choose the most likely candidate by integrating
information.
The success of the particle filter relies mainly on three factors. First, in the particle prediction step, the Volterra kernels allow the particles to make reasonably accurate proposals
based on the observed neural activities. This gives a good measure for evaluating the fitness
of each particle. Second, in the resampling step, the weight of each particle embodies all
the earlier observations, and because our particle filter keeps track of all proposals within
the last 200 ms, earlier hypotheses can continue to be reevaluated and refined. Finally, in
the propagation step, the particle filter utilizes prior knowledge about the manner in which
the stimulus moves. This helps further in pruning down the hypothesis space.
Acknowledgments
This research is supported by NSF CAREER 9984706, NIH Vision Research core grant
EY08098, and a NIH 2P41PR06009-11 for biomedical supercomputing. Thanks to Rick
Romero, Yuguo Yu, and Anthony Brockwell for helpful discussion and advice.
References
[1] A. E. Brockwell, A. L. Rojas, and R. E. Kass. Bayesian decoding of motor cortical
signals by particle filtering. Submitted to J. Neurophysiology, 2003.
[2] E. Brown, L. Frank, D. Tang, M. Quirk, and M. Wilson. A statistical paradigm for
neural spike train decoding applied to position prediction from ensemble firing patterns of rat hippocampal place cells. J. Neuroscience, 18(18):7411?7425, 1998.
[3] U.T. Eden, L.M. Frank, R. Barbieri, and E.N. Brown. Particle filtering algorithms
for neural decoding and adaptive estimation of receptive field plasticity. In Proc.
Computational Neuroscience Meeting, CNS ?02, Santa Barbara, 2002.
[4] Y. Gao, M. J. Black, E. Bienenstock, S. Shoham, and J. P. Donoghue. Probabilistic
Inference of Hand Motion from Neural Activity in Motor Cortex, pages 213?220. MIT
Press, Cambridge, MA, 2002.
[5] A. P. Georgopoulos, A. B. Schwartz, and R. E. Kettner. Neuronal population coding
of movement direction. Science, 243:234?236, 1989.
[6] F. Rieke, D. Warland, R. deRuytervanSteveninck, and W. Bialek. Spikes: Exploring
the Neural Code. MIT Press, Cambridge, MA, 1997.
[7] R. Romero, Y. Yu, P Afhsar, and T. S. Lee. Adaptation of the temporal receptive fields
of macaque v1 neurons. Neurocomputing, 52-54:135?140, 2002.
[8] G. Stanley, F. Li, and Y. Dan. Reconstruction of natural scenes from ensemble responses in the lateral geniculate nucleus. J. Neuroscience, 19(18):8036?8042, 1999.
[9] G. B. Stanley. Adaptive spatiotemporal receptive field estimation in the visual pathway. Neural Computation, 14:2925?2946, 2002.
[10] K. Zhang, I. Ginzburg, B.L. McNaughton, and T. J. Sejnowski. Interpreting neuronal population activity by reconstruction: Unified framework with application to
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1,641 | 2,492 | A Low-Power Analog VLSI Visual
Collision Detector
Reid R. Harrison
Department of Electrical and Computer Engineering
University of Utah
Salt Lake City, UT 84112
[email protected]
Abstract
We have designed and tested a single-chip analog VLSI sensor that
detects imminent collisions by measuring radially expansive optic
flow. The design of the chip is based on a model proposed to
explain leg-extension behavior in flies during landing approaches.
A new elementary motion detector (EMD) circuit was developed to
measure optic flow. This EMD circuit models the bandpass nature
of large monopolar cells (LMCs) immediately postsynaptic to
photoreceptors in the fly visual system. A 16 ? 16 array of 2-D
motion detectors was fabricated on a 2.24 mm ? 2.24 mm die in a
standard 0.5-?m CMOS process. The chip consumes 140 ?W of
power from a 5 V supply. With the addition of wide-angle optics,
the sensor is able to detect collisions around 500 ms before impact
in complex, real-world scenes.
1
Introduction
Many animals ? from flies to humans ? are capable of visually detecting imminent
collisions caused either by a rapidly approaching object or self-motion towards an
obstacle. Neurons dedicated to this task have been found in the locust [1] and the
pigeon [2]. Borst and Bahde have shown that flies use visual information to time
the extension of their legs on landing approaches [3].
While several models have been proposed to explain collision detection, the model
proposed in [3] is particularly amenable to hardware implementation. The model,
shown in Fig. 1, employs a radially-oriented array of motion detectors centered in
the direction of flight. As the animal approaches a static object, an expansive optic
flow field is produced on the retina. A wide angle field of view is useful since optic
flow in the direction of flight will be zero. The response of this radial array of
motion detectors is summed and then passed through a leaky integrator (a lowpass
filter). If this response exceeds a fixed threshold, an imminent collision is detected
and the animal can take evasive action or prepare for a landing. This expansive
optic flow model has recently been used to explain landing and collision avoidance
responses in the fruit fly [4]. A similar algorithm has been implemented in a
traditional CPU for autonomous robot navigation [5]. In this work, we present a
single-chip analog VLSI sensor developed to implement this model.
Field of View
radially-oriented
D
elementary
motion detectors
(EMDs)
spatial
summation
leaky
integrator
comparator
? = RC
threshold
collision
detect
Figure 1: Diagram of collision detection algorithm.
2 E l emen tary Mo ti on Dete ctors
Our collision detection algorithm uses an array of radially-oriented elementary
motion detectors (EMDs) to sense image expansion. Simulations by the author have
shown that the structure and properties of the EMDs strongly affect the accuracy of
this algorithm [6]. We use an enhanced version of the familiar delay-and-correlate
or ?Reichardt? EMD first proposed by Hassenstein and Reichardt in the 1950s to
explain the optomotor response of beetles [7]. Fig. 2 shows a diagram of the EMD
used in our collision sensor.
The first stage of the EMD is photoreception, where light intensity is transduced to
a signal vphoto. Since light intensity is a strictly positive value, the mean intensity of
the scene must be subtracted. Since we are interested in motion, it is also
advantageous to amplify transient signals.
Suppressing dc illumination and enhancing ac components of photoreceptor signals
is a common theme in many biological visual systems. In flies, large monopolar
cells (LMCs) directly postsynaptic to photoreceptors exhibit transient biphasic
impulse responses approximately 40-200 ms in duration [8], [9]. In the frequency
domain, this can be seen as a bandpass filtering operation that attenuates dc signals
while amplifying signals in the 2-40 Hz range [9], [10]. In the lateral geniculate
nucleus of cats, ?lagged? and ?non-lagged? cells exhibit transient biphasic impulse
responses 200-300 ms in duration and act as bandpass filters amplifying signals in
the 1-10 Hz range [11]. This filtering has recently been explained in terms of
temporal decorrelation, and can be seen as way of removing redundant information
from the photoreceptor signal before further processing [9], [12].
After this ?transient enhancement?, or temporal decorrelation, the signals are
delayed using the phase lag of a lowpass filter. While not a true time delay, the
lowpass filter matches data from animal experiments and makes the Reichardt EMD
equivalent to the oriented spatiotemporal energy filter proposed by Adelson and
Bergen [13]. Before correlating the adjacent delayed and non-delayed signals, we
apply a saturating static nonlinearity to each channel. Without such a nonlinearity,
the delay-and-correlate EMD exhibits a quadratic dependence on image contrast. In
fly tangential neurons, motion responses show a quadratic dependence only at very
low contrasts, then quickly become largely independent of image contrast for
contrasts above 30%. Egelhaaf and Borst proposed the presence of this nonlinearity
in the biological EMD to explain this contrast independence [14]. Functionally, it is
necessary to prevent high-contrast edges from dominating the summed output of the
EMD array.
image motion
Fig. 3
vphoto-R
vphoto-L
LMC
LMC
temporal decorrelation
(Large Monopolar Cells)
vLMC-R
vLMC-L
delay
Fig. 4
photoreceptors
delay
vdelay-L
delay
vdelay-R
saturating nonlinearity
correlation (multiplication)
iout-L
iout-R
opponent subtraction
iout
Figure 2: Elaborated delay-and-correlate elementary motion detector (EMD)
After correlation, opponent subtraction produces a strong directionally selective
signal that is taken as the output of the EMD. Unlike algorithms that find and track
features in an image, the delay-and-correlate EMD does not measure true image
velocity independent of the spatial structure of the image. However, recent work
has shown that for natural scenes, these Reichardt EMDs give reliable estimates of
image velocity [15]. This reliability is improved by the addition of LMC bandpass
filters and saturating nonlinearities. Experiments using earlier versions of silicon
EMDs have demonstrated the ability of delay-and-correlate motion detectors to
work at very low signal-to-noise ratios [16].
3 Integrated C ir cuit Imple me nt at ion
We adapted the EMD shown in Fig. 2 to a small, low-power CMOS integrated
circuit. Fig. 3 shows a schematic of the photoreceptor and LMC bandpass filter. A
35 ?m ? 35 ?m well-substrate photodiode with diode-connected pMOS load
converts the diode photocurrent into a voltage vphoto that is a logarithmic function of
light intensity. A pMOS source follower biased by ISF = 700 pA buffers this signal
so that the input capacitance of the LMC circuit does not load the photoreceptor.
The LMC bandpass filter consists of two operational transconductance amplifiers
(OTAs) and three capacitors. The OTAs in the circuit are implemented with pMOS
differential pairs using diode-connected transistors for source degeneration for
extended linear range (see inset, Fig. 3). The transfer function of the LMC circuit is
given by
? ?1 ?
??1 ? s ??
? ?
v LMC (s )
N? 0 s ? (1 ? ? 0 s )
AN
= ?A?
=?
? ?
vin (s )
? ? s+ 1 + 1
(? 1s )2 + ? 1s + 1
1
Q ?1s
Q
(1)
where
?0 =
C
gm
? = N (A + 1)(K + 1) ? N ? NAK if A, K >> 1
(2)
(3)
C
vgm
out
v+
IB
ISF
gm /N
gm = ?IB/2UT
AC
vin
gm
vphoto
vLMC
v+
VREF
vout
KC
Figure 3: Schematic of photoreceptor/LMC circuit. Detail of operational
transconductance amplifier (OTA) shown in inset.
? 1 = ?? 0
Q=
?
(K + N )
(4)
(5)
The output signal v LMC is centered around VREF, a dc voltage which was set to 1.0 V.
We sized the capacitors in our circuit to give A = 20 and K = 5 (with C = 70 fF).
The transconductance of the lower OTA was set by adjusting its bias current IB:
gm =
I
?
? B
(? + 1) 2U T
(6)
where ? is the weak inversion slope (typically between 0.6 and 0.9) and UT is the
thermal voltage kT/q (approximately 26 mV at room temperature). We set the bias
current in the upper OTA five times smaller to achieve N = 5.
As we see from (1), the LMC circuit acts as an ac-coupled bandpass filter centered
at f1 = 1/2??1, with a quality factor Q set to 2.5 by capacitor and current ratios. The
circuit also has a zero at ?f 1, but since ? = 25 in our circuit, the zero takes effect
outside that passband and thus has little practical effect on the filter. We used a bias
current of IB = 35 pA in the lower OTA and 7 pA in the upper OTA to center the
passband near 20 Hz, which was chosen because it lies in the range of LMC
response measured in the fly.
This LMC circuit represents a significant
improvement over a previous silicon EMD design, which used only a first-order
highpass filter to block dc illumination [16]. The LMC circuit presented here
allows the designer to adjust the center frequency and Q factor to selectively
amplify frequencies present in moving images.
The LMC circuits from each photoreceptor pass their signals to the the delay-andcorrelate circuit shown in Fig. 4. The delay is implemented as a first-order lowpass
filter. The OTAs in this circuit used two diode-connected transistors in series for
extended linear range. The time constant of this filter is given by
? LPF =
C LPF
g m ? LPF
(7)
vLMC-L vLMC-R
vdelay-L
gm-LPF
CLPF
CLPF
Imult
Imult
VREF
VW
vdelay-R
gm-LPF
VREF
VREF
VW
VW
iout-L-
VREF
VW
iout-Riout-R+
iout-L+
iout+
iout-
Figure 4: Schematic of delay-and-correlate circuit. OTA-based g m-C filters are used
as low-pass filters. Subthreshold CMOS Gilbert multipliers are used for correlation.
We used CLPF = 700 fF and set ?LPF to around 25 ms, which is in the range of
biological motion detectors. This required a bias current of 9 pA for each OTA.
We implemented the correlation function using a CMOS Gilbert multiplier
operating in subthreshold [17]. The output currents of the multipliers in Fig. 4 can
be expressed as:
ioutL + ? ioutL ? = I mult tanh
ioutR + ? ioutR ? = I mult tanh
(
? v delay ? L ? V REF
)
tanh
? (v LMC ? R ? V REF )
2U T
(8)
)
tanh
? (v LMC ? L ? V REF )
2U T
(9)
2U T
(
? v delay ? R ? V REF
2U T
For small differential input voltages, tanh(x) ? x and the circuit acts as a linear
multiplier. As the input signals grow larger, the tanh nonlinearity dominates and the
circuit acts more like a digital exclusive-or gate. We use this inherent circuit
nonlinearity as the desired saturating nonlinearity in our EMD model (see Fig. 1).
The previous LMC circuit provides sufficient gain to ensure that we are usually
operating well outside the linear range of the multipliers.
Traditional CMOS Gilbert multipliers require that the dc level of the upper
differential input be shifted relative to the dc level of the lower differential input.
This is required to keep the transistors in saturation. To avoid the cost in chip area,
power consumption, and mismatch associated with level shifters, we introduce a
novel circuit modification that allows both the upper and lower differential inputs to
operate at the same dc level. We lower the well potential of the lower pMOS
transistors from VDD to a dc voltage VW (see Fig. 4). This lowered well voltage
causes the sources of these transistors to operate at a lower potential, which keeps
the upper transistors in saturation. We use VW = 2.5 V in our circuit. (Care must be
taken not to make VW too low, as parasitic source-well-substrate pnp transistors can
be activated.)
52??
74??
Figure 5: EMD pattern on chip. Ultra-wide-angle optics gave the chip a field of
view ranging from ?52? to ?74?.
The output of the Gilbert multiplier is a differential current. The signals from the
left and right correlators are easily subtracted by summing their currents
appropriately. Similarly, current summation on two global wires is used to sum the
motion signals over the entire EMD array.
4 Experimental Results
We fabricated a 16 ? 16 EMD array in a 0.5-?m 2-poly, 3-metal standard CMOS
process. The 2.24 mm ? 2.24 mm die contained a 17 ? 17 array of ?pixels,? each
measuring 100 ?m ? 100 ?m. Each pixel contained a photoreceptor, LMC circuit,
lowpass ?delay? filter, and four correlators. These correlators were used to
implement two independent EMDs: a vertical motion detector connected to the pixel
below and a horizontal motion detector connected to the pixel to the right. The
output signals from a subset of the EMDs representing radial outward motion were
connected to two global wires, giving a differential current signal that was taken off
chip on two pins.
Fig. 5 shows the EMDs that were summed to produce the global radial motion
signal. Diagonally-oriented EMDs were derived from the sum of a horizontal and a
vertical EMD. The center 4 ? 4 pixels were ignored, as motion near the center of
the field of view is typically very small in collision situations. We used custombuilt ultra-wide-angle optics to give the chip a field of view ranging from ?52? at
the sides to ?74? at the corners. Simulations revealed that a field of view of around
?60? was necessary for reasonable performance using this algorithm [6].
Before testing the array, we characterized an individual LMC circuit configured to
have a voltage input vphoto provided from off chip using a function generator. We
provided a 1.4 Hz, 100 mVpp square wave and observed the LMC circuit output.
As shown in Fig. 6a, the LMC circuit exhibits a transient oscillatory step response
similar to its biological counterpart. Using a spectrum analyzer, we measured the
transfer function of the circuit (see Fig. 6b). The LMC circuit acts as a bandpass
filter centered at 19 Hz, with a measured Q of 2.3.
Figure 6: Measurement of LMC circuit performance. (a) Step response of LMC
circuit. (b) Frequency tuning of LMC circuit.
The entire chip consumed 140 ?W of power. Most of this was consumed by
peripheral biasing circuits; the 17 ? 17 pixel array used only 5.2 ?W (18 nW per
pixel). To test the complete collision detection chip, we implemented the leaky
integrator (?leak = 50 ms) and comparator from Fig. 1 using off-chip components. In
future implementations, these circuits could be built on chip using little power.
We tested the chip by mounting it on a small motorized vehicle facing forward with
the lens centered 11 cm above the floor. The vehicle traveled in a straight path at 28
cm/s. Fig. 7 shows the output from the leaky integrator as the chip moves across the
floor and collides with the center of a 38 cm ? 38 cm trash can in our lab. The peak
response of the chip occurs approximately 500 ms before contact, which
corresponds to a distance of 14 cm. At this point, the edges of the trash can subtend
an angle of 54?. After this point, the edges of the can move beyond the chip?s field
of view, and the response decays rapidly. The rebound in response observed in the
last 100 ms may be due to the chip seeing the expanding shadow cast by its own
lens on the side of the can just before contact.
5
Conclusions
The response of our chip, which peaks and then collapses before impact, is similar
to activity patterns observed in the LGMD neuron in locusts [1] and ? neurons in
pigeons [2] during simulated collisions. While more complex models positing the
measurement of true image velocity and object size have been used to explain this
peculiar time course [1], we observe that a simple model integrating the output of a
radial EMD array gives qualitatively similar responses.
We have demonstrated that this model of collision detection can be implemented in
a small, low-power, single-chip sensor. Further testing of the chip on mobile
platforms should better characterize its performance.
A c k n o w l e d g me n t s
This work was partially supported by a contract from the Naval Air Warfare Center,
China Lake, CA.
References
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Figure 7: Measured output of collision detection chip.
[2] H. Sun and B.J. Frost, ?Computation of different optical variables of looming objects in
pigeon nucleus rotundus neurons,? Nature Neurosci. 1:296-303, 1998.
[3] A. Borst and S. Bahde, ?Visual information processing in the fly?s landing system,? J.
Comp. Physiol. A 163:167-173, 1988.
[4] L.T. Tammero and M.H. Dickinson, ?Collision-avoidance and landing responses are
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[12] D.W. Dong and J.J. Atick, ?Temporal decorrelation: a theory of lagged and nonlagged
responses in the lateral geniculate nucleus,? Network 6:159-178, 1995.
[13] E.H. Adelson and J.R. Bergen, ?Spatiotemporal energy models for the perception of
motion,? J. Opt. Soc. Am. A 2:284-299, 1985.
[14] M. Egelhaaf and A. Borst, ?Transient and steady-state response properties of movement
detectors,? J. Opt. Soc. Am. A 6:116-127, 1989.
[15] R.O. Dror, D.C. O?Carroll, and S.B. Laughlin, ?Accuracy of velocity estimation by
Reichardt correlators,? J. Opt. Soc. Am. A 18:241-252, 2001.
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Integrated Circuits and Signal Processing 24:213-229, 2000.
[17] C. Mead, Analog VLSI and Neural Systems, Reading, MA: Addison-Wesley, 1989.
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1,642 | 2,493 | Factorization with uncertainty and
missing data: exploiting temporal
coherence
Amit Gruber and Yair Weiss
School of Computer Science and Engineering
The Hebrew University of Jerusalem
91904 Jerusalem, Israel
{amitg,yweiss}@cs.huji.ac.il
Abstract
The problem of ?Structure From Motion? is a central problem in
vision: given the 2D locations of certain points we wish to recover
the camera motion and the 3D coordinates of the points. Under simplified camera models, the problem reduces to factorizing
a measurement matrix into the product of two low rank matrices.
Each element of the measurement matrix contains the position of
a point in a particular image. When all elements are observed, the
problem can be solved trivially using SVD, but in any realistic situation many elements of the matrix are missing and the ones that
are observed have a different directional uncertainty. Under these
conditions, most existing factorization algorithms fail while human
perception is relatively unchanged.
In this paper we use the well known EM algorithm for factor analysis to perform factorization. This allows us to easily handle missing
data and measurement uncertainty and more importantly allows us
to place a prior on the temporal trajectory of the latent variables
(the camera position). We show that incorporating this prior gives
a significant improvement in performance in challenging image sequences.
1
Introduction
Figure 1 illustrates the classical structure from motion (SFM) displays introduced by
Ullman [13]. A transparent cylinder with painted dots rotates around its elongated
axis. Even though no structure is apparent in any single frame, humans obtain a
vivid percept of a cylinder1 .
SFM has been dealt with extensively in the computer vision literature. Typically a
small number of feature points are tracked and a measurement matrix is formed in
1
An
online
animation
of
this
famous
stimulus
aris.ss.uci.edu/cogsci/personnel/hoffman/cylinderapplet.html
is
available
at:
Figure 1: The classical structure from motion stimulus introduced by Ullman [13].
Humans continue to perceive the correct structure even when each dot appears only
for a small number of frames, but most existing factorization algorithm fail in this
case. Replotted from [1]
which each element corresponds to the image coordinates of a tracked point. The
goal is to recover the camera motion and the 3D location of these points. Under
simplified camera models it can be shown that this problem reduces to a problem of
matrix factorization. We wish to describe the measurement matrix as a product of
two low rank matrices. Thus if all features are reliably tracked in all the frames, the
problem can be solved trivially using SVD [11]. In particular, performing an SVD
on the measurement matrix of the rotating cylinder stimulus recovers the correct
structure even if the measurement matrix is contaminated with significant amounts
of noise and if the number of frames is relatively small.
But in any realistic situation, the measurement matrix will have missing entries.
This is either because certain feature points are occluded in some of the frames and
hence their positions are unknown, or due to a failure in the tracking algorithm.
This has lead to the development of a number of algorithms for factorization with
missing data [11, 6, 9, 2].
Factorization with missing data turns out to be much more difficult than the full
data case. To illustrate the difficulty, consider the cylinder stimulus in figure 1.
Humans still obtain a vivid percept of a cylinder even when each dot has a short
?dot life?. That is, each dot appears at a random starting frame, continues to appear
for a small number of frames, and then disappears [12]. We applied the algorithms
in [11, 6, 9, 2] to a sequence of 20 frames of a rotating cylinder in which the dot life
was 10 frames. Thus the matrix was half full (or half empty). Surprisingly, none of
the algorithms could recover the cylinder structure. They either failed to find any
structure or they gave a structure that was drastically different from a cylinder.
Presumably, humans are using additional prior knowledge that the algorithms are
not.
In this paper we point out a source of information in image sequences that is usually
neglected by factorization algorithms: temporal coherence. In a video sequence, the
camera location at time t + 1 will probably be similar to its location at time t. In
other words, if we randomly permute the temporal order of the frames, we will get
a very unlikely image sequence. Yet nearly all existing factorization algorithms will
be invariant to this random permutation of the frames: they only seek a low rank
approximation to a matrix and permuting the rows of the matrix will not change
the approximation.
In order to enable the use of temporal coherence, we formulate factorization in
terms of maximum likelihood for a factor analysis model, where the latent variable corresponds to camera position. We use the familiar EM algorithm for factor
analysis to perform factorization with missing data and uncertainty. We show how
to add a temporal coherence prior to the model and derive the EM updates. We
show that incorporating this prior gives a significant improvement in performance
in challenging image sequences.
2
Model
A set of P feature points in F images are tracked along an image sequence. Let
(uf p , vf p ) denote image coordinates of feature point p in frame f . Let U = (uf p ),
V = (vf p ) and W = (wij ) where w2i?1,j = uij and w2i,j = vij for 1 ? i ? F , i.e.
W is an interleaving of the rows of U and V .
In the orthographic camera model, points in the 3D world are projected in parallel
onto the image plane. For example, if the camera?s optical center is in the origin
(w.r.t 3D coordinate system), and its x, y axes coincide with X, Y axes in the 3D
world, then taking a picture
? ? is a simple projection (in homogeneous coordinates):
?
? X
1 0 0 0 ?Y ?
(x, y) =
. The depth, Z, has no influence on the image. In this
0 1 0 0 ?Z?
1
model, a camera can undergo rotation, translation, or a combination of the two.
Under orthography, and in the absence of noise,
[W ]2F ?P = [M ]2F ?4 [S]4?P
(1)
?
?
?
?
X1 ? ? ? X P
M1
?
Y
? ? ? YP ?
.
?
?
..
where M =
and S = ? 1
. M describes camera
Z1 ? ? ? Z P ?
MF 2F ?4
1 ???
1 4?P
? T
?
mi d i
motion (rotation and translation, [Mi ]2?4 =
). mi and ni are 3 ? 1
nTi
ei
vectors that describe the rotation of the camera; di and ei are scalars describing
camera translation, 2 and S describes points location in 3D.
For noisy observations, the model becomes:
[W ]2F ?P = [M ]2F ?4 [S]4?P + [?]2F ?P
where ? is Gaussian noise.
(2)
If the elements of the noise matrix ? are uncorrelated and of equal variance then we
seek a factorization that minimizes the mean squared error between W and M S.
This can be solved trivially using the SVD of W . Missing data can be modeled
using equation 2 by assuming some elements of the noise matrix ? have infinite
variance. Obviously the SVD is not the solution once we allow different elements
of ? to have different variances.
2.1
Factorization as factor analysis
It is well known that the SVD calculation can be formulated as a limiting case of
maximum likelihood factor analysis [8]. In standard factor analysis we have a set
2
We do not subtract the mean of each row from it, since in case of missing data the
centroids of points do not coincide.
of observations {y(t)} that are linear combinations of a latent variable x(t):
y(t) = Ax(t) + ?(t)
(3)
N (0, ?x2 I)
2
with x(t) ?
and ?(t) ? N (0, ?t ). If ?t is a diagonal matrix with constant
elements ?t = ? I then in the limit ?/?x ? 0 the ML estimate for A will give the
same answer as the SVD. We now show how to rewrite the SFM problem in this
form.
In equation 1 the horizontal and vertical coordinates of the same point appear in
different rows. It can be rewritten as:
?
?
S 0
[U V ]F ?2P = [M N ]F ?8
+ [?
? ]F ?2P
(4)
0 S 8?2P
Let y(t) be the vector of noisy observations (noisy image locations) at time t,
i.e. y(t) = [u(t) v(t)], that is y(t) = [u1 (t), ? ? ? uP (t) v1 (t), ? ? ? vP (t)]T . Let x(t)
be a vector of length 8 that denotes
position at time t x(t) =
? T the camera
?
S
0
T
T
T
[m(t) d(t) n(t) e(t)] and let A =
. Identifying y(t) with the tth row
0 ST
of the matrix [U V ] and x(t) with the tth row of [m n], then equation 4 is equivalent
to equation 3.
We can now use the standard EM algorithm for factor analysis to find the ML
estimate for S.
E step:
E(x(t)|y(t))
V (x(t)|y(t))
=
=
?
?
?x?2 I + AT ??1
t A
?x?2 I + AT ??1
t A
< x(t) > = E(x(t)|y(t))
??1
AT ??1
t y(t)
??1
(5)
(6)
(7)
T
< x(t)x(t) > = V (x(t)|y(t))? < x(t) >< x(t) >
T
(8)
M step: In the M step we solve the normal equations for the structure S. The exact
form depends on the structure of ?t . Denote by sp a vector of length 3 that denotes
the 3D coordinates of point p then for a diagonal noise covariance matrix ?t the M
step is:
sp
= Bp Cp?1
(9)
where
Bp
=
X?
T
??1
t (p, p)(utp ? < dt >) < m(t) >
(10)
t
Cp
=
T
+ ??1
t (p + P, p + P )(vtp ? < et >) < n(t) >
X?
T
??1
t (p, p) < m(t)m(t) >
?
t
T
+ ??1
t (p + P, p + P ) < n(t)n(t) >
?
where the expectation required in the M step are the appropriate subvectors and
submatrices of < x(t) > and < x(t)x(t)T >.
?1
If we set ??1
t (p, p) = ?t (p + P, p + P ) = 0 if point p is missing in frame t then we
obtain an EM algorithm for factorization with missing data. Note that the form of
the updates means we can put any value we wish in the missing elements of y and
they will be ignored by the algorithm.
x(1)
y(1)
x(2)
x(3)
y(2)
y(3)
x(1)
x(2)
y(1)
x(3)
y(2)
a
y(3)
b
Figure 2: a. The graphical model assumed by most factorization algorithms for
SFM. The camera location x(t) is assumed to be independent of the camera location at any other time step. b. The graphical model assumed by our approach.
We model temporal coherence by assuming a Markovian structure on the camera
location.
A more realistic noise model for real images is that ?t is not diagonal but rather that
the noise in the horizontal and vertical coordinates of the same point are correlated
with an arbitrary 2 ? 2 inverse covariance matrix. This problem is usually called
factorization with uncertainty [5, 7]. It is easy to derive the M step in this case as
well. It is similar to equation 9 except that cross terms involving ??1
t (p, p + P ) are
also involved:
sp
= (Bp + Bp0 )(Cp + Cp0 )?1
(11)
where
Bp0
=
X?
T
??1
t (p, p +P )(vtp ?< et>) < m(t) >
(12)
t
Cp0
=
T
+ ??1
t (p + P, p)(utp ? < dt >) < n(t) >
X?
T
??1
t (p, p + P ) < n(t)m(t) >
?
t
T
+ ??1
t (p + P, p) < m(t)n(t) >
?
Regardless of uncertainty and missing data the complexity of the EM algorithm
grows linearly with the number of feature points and the number of frames. At
every iteration, the most computationally intensive step is an inversion of an 8 ? 8
matrix.
2.2
Adding temporal coherence
The factor analysis algorithm for factorization assumes that the latent variables
x(t) are independent. In SFM this assumption means that the camera location in
different frames is independent and hence permuting the order of the frames makes
no difference for the factorization. As mentioned in the introduction, in almost any
video sequence this assumption is wrong. Typically camera location varies smoothly
as a function of time.
Figure 2a shows the graphical model corresponding to most factorization algorithms:
the independence of the camera location is represented by the fact that every time
step is isolated from the other time steps in the graph. But it is easy to fix this
assumption by adding edges between the latent variables as shown in figure 2b.
Specifically, we use a second order approximation to the motion of the camera:
1
(13)
x(t) = x(t ? 1) + v(t ? 1) + a(t ? 1) + ?1
2
Truth
Structure:
factor analysis
Jacobs
5
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Figure 3: Comparison of factor analysis and Jacobs? algorithm on synthetic sequences. All other existing algorithms performed worse than Jacobs. They all fail
when there is noise and missing data while factor analysis with temporal coherence
succeeds. Structure and motion are shown from a top view.
v(t) = v(t ? 1) + a(t ? 1) + ?2
a(t) = a(t ? 1) + ?3
y(t) = Ax(t) + ?(t)
(14)
(15)
(16)
Note that we do not assume that the 2D trajectory of each point is smooth. Rather
we assume the 3D trajectory of the camera is smooth.
It is straightforward to derive the EM iterations for a ML estimate of S using the
model in equation 16. The M step is unchanged from the classical factor analysis
and is given by equation 9. The only change in the E step is that E(x(t)|y) and
V (x(t)|y) need to be calculated using a Kalman smoother. We use a standard RTS
smoother [4]. Note that the computation of the E step is still linear in the number
of frames and datapoints.
Kalman filtering has been used extensively in a more perspective SFM setting(e.g. [10]). However, in perspective projections the problem is no longer one
of factorization. Thus even for Gaussian noise, the Extended Kalman filter needs
to be used, smoothing is not performed and no guarantee of increase in likelihood
is obtained. Within the factorization framework, we can use the classical Kalman
filter and obtain a simple algorithm that provably increases the likelihood at every
iteration.
3
Experiments
In this section we describe the experimental performance of EM with time coherence
compared to ground truth and to previous algorithms for structure from motion with
missing data [11, 6, 9, 2]. For [11, 6, 9] we used the Matlab implementation made
public by D. Jacobs.
The first input sequence is the sequence of the cylinder shown in figure 1. 100
points uniformly drawn from the cylinder surface are tracked over 20 frames. Each
of the points appears for 10 frames, starting at a random time, and then disappears.
The observed image locations were added a Gaussian noise with standard deviation
? = 0.1.
We checked the performance of the different algorithms in the cases of: (1) full noise
free observation matrix , (2) noisy full observation matrix, (3) noiseless observations
Error as function of noise
Error as function of missing data
4
10
8<
EM with Temporal Coherence
EM
Jacobs
7
x 10
9
reconstruction square error
reconstruction square error
8
6
5
4
EM with Temporal Coherence
EM
Jacobs
7
6
5
4
3
2
3
1
2
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
0
0
0.5
0.1
0.2
0.3
0.4
0.5
noise level (sigma)
percentage of missing data
Figure 4: Graphs depict influence of noise and percentage of missing data on reconstruction results of factor analysis and [6].
200
150
100
50
0
?50
?100
100
50
0
?50
?100
?150
Figure 5: Results of scene reconstruction from a real sequence: A binder and is
placed on a rotating surface filmed with a static camera. Our algorithm succeeded
in (approximately) obtaining the right structure and all other algorithms failed.
Results are shown in top view.
with missing data and (4) noisy observations with missing data.
All algorithms performed well and gave similar results for the full matrix noiseless
sequence.
In the fully observed noisy case, factor analysis without temporal coherence gave
comparable performance to Tomasi-Kanade, which minimize kM S ? W k2F . When
temporal coherence was added, the reconstruction results were improved. The results of Shum?s algorithm were similar to Tomasi-Kanade. The algorithms of Jacobs
and Brand turned to be noise sensitive.
In the case of noiseless missing data (figure 3 top), our algorithm and Jacobs?
algorithm reconstruct the correct motion and structure. Tomasi-Kanade?s algorithm
and Shum?s algorithm could not handle this pattern of missing data and failed to
give any structure.
Once we add even very mild amounts of noise (figure 3 middle) all existing algorithms fail. While factor analysis with temporal coherence continues to extract the
correct structure even for significant noise values.
Figure 5 shows result on a real sequence.
4
Discussion
Despite progress in algorithms for factorization with uncertainty the best existing
algorithms still fall far short of human performance, even for seemingly simple
stimuli. Presumably, humans are using additional prior information. In this paper
we have focused on one particular prior: the temporal smoothness of the camera
motion. We showed how to formulate SFM as a factor analysis problem and how to
add temporal coherence to the EM algorithm. Our experimental results show that
this simple prior can give a significant improvement in performance in challenging
sequences.
Temporal coherence is just one of many possible priors. It has been suggested that
humans also use a smoothness prior on the 3D surface they are perceiving [12]. It
would be interesting to extend our framework in this direction.
The most drastic simplification our model makes is the assumption of Gaussian
noise. It would be interesting to extend the algorithm to non Gaussian settings.
This may require approximate inference algorithms in the E step as used in [3].
References
[1] R.A. Andersen and D.C Bradley. Perception of three-dimensional structure
from motion. In Trends in Cognitive Sciences, 2, pages 222?228, 1998.
[2] M.E. Brand. Incremental singular value decomposition of uncertain data with
missing values. In ECCV, pages 707?720, May 2002.
[3] F. Dellaert, S. M. Seitz, C. E. Thorpe, and S. Thrun. Structure from motion
without correspondence. In ICCV, pages 696?702, January 1999.
[4] Arthur Gelb, editor. Applied Optimal Estimation. MIT Press, 1974.
[5] M. Irani and P. Anandan. Factorization with uncertainty. In ECCV, pages
959?966, January 2000.
[6] D. Jacobs. Linear fitting with missing data: Applications to structure-frommotion and to characterizing intensity images. In CVPR, pages 206?212, 1997.
[7] D. D. Morris and T. Kanade. A unified factorization algorithm for points, line
segments and planes with uncertain models. In ICCV, pages 696?702, January
1999.
[8] S. Roweis. Em algorithms for pca and spca. In NIPS, pages 431?437, 1997.
[9] H. Y. Shum, K. Ikeuchi, and R. Reddy. Principal component analysis with
missing data and its application to polyhedral object modeling. pages 854?
867, September 1995.
[10] S. Soatto and P. Perona. Reducing structure from motion: a general framework
for dynamic vision. IEEE Trans. on Pattern Analysis and Machine Intelligence,
pages 943?960, 1999.
[11] C. Tomasi and T. Kanade. Shape and motion from image streams under orthography: A factorization method. Int. J. of Computer Vision, 9(2):137?154,
November 1992.
[12] S. Treue, M. Husain, and R. Andersen. Human perception of structure from
motion. Vision Research, 31:59?75, 1991.
[13] S. Ullman. The interpertation of visual motion. MIT Press, 1979.
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1,643 | 2,494 | Subject-Independent Magnetoencephalographic
Source Localization by a Multilayer Perceptron
Sung C. Jun
Biological and Quantum Physics Group
MS-D454, Los Alamos National Laboratory
Los Alamos, NM 87545, USA
[email protected]
Barak A. Pearlmutter
Hamilton Institute
NUI Maynooth
Maynooth, Co. Kildare, Ireland
[email protected]
Abstract
We describe a system that localizes a single dipole to reasonable accuracy from noisy magnetoencephalographic (MEG) measurements in real
time. At its core is a multilayer perceptron (MLP) trained to map sensor signals and head position to dipole location. Including head position
overcomes the previous need to retrain the MLP for each subject and session. The training dataset was generated by mapping randomly chosen
dipoles and head positions through an analytic model and adding noise
from real MEG recordings. After training, a localization took 0.7 ms with
an average error of 0.90 cm. A few iterations of a Levenberg-Marquardt
routine using the MLP?s output as its initial guess took 15 ms and improved the accuracy to 0.53 cm, only slightly above the statistical limits
on accuracy imposed by the noise. We applied these methods to localize
single dipole sources from MEG components isolated by blind source
separation and compared the estimated locations to those generated by
standard manually-assisted commercial software.
1
Introduction
The goal of MEG/EEG localization is to identify and measure the signals emitted by electrically active brain regions. A number of methods are in widespread use, most assuming
dipolar sources (H?am?al?ainen et al., 1993). Recently MLPs (Rumelhart et al., 1986) have
become popular for building fast dipole localizers (Abeyratne et al., 1991; Kinouchi et al.,
1996). Since it is easy to use a forward model to create synthetic data consisting of dipole
locations and corresponding sensor signals, one can train a MLP on the inverse problem.
Hoey et al. (2000) took EEG measurements for both spherical and realistic head models
and trained MLPs on randomly generated noise-free datasets. Integrated approaches to the
EEG/MEG dipole source localization, in which the trained MLPs are used as initializers
for iterative methods, have also been studied (Jun et al., 2002) along with distributed output
representations (Jun et al., 2003). Interestingly, all work to date trained with a fixed head
model. However, for MEG, head movement relative to the fixed sensor array is very difficult to avoid, and even with heroic measures (bite bars) the position of the head relative
to the sensor array varies from subject to subject and session to session. This either results
in significant localization error (Kwon et al., 2002), or requires laborious retraining and
revalidation of the system.
We propose an augmented system which takes head position into account, yet remains
able to localize a single dipole to reasonable accuracy within a fraction of a millisecond
on a standard PC, even when the signals are contaminated by considerable noise. The
system uses a MLP trained on random dipoles and random head positions, which takes
as inputs both the coordinates of the center of a sphere fitted to the head and the sensor
measurements, uses two hidden layers, and generates the source location (in Cartesian
coordinates) as its output. Adding head position as an extra input overcomes the primary
practical limitation of previous MLP-based MEG localization systems: the need to retrain
the network for each new head position.
We use an analytical model of quasi-static
electromagnetic propagation through a
spherical head to map randomly chosen dipoles and head positions to superconducting quantum interference device
(SQUID) sensor activities according to the
sensor geometry of a 4D Neuroimaging
Neuromag-122 MEG system, and trained
a MLP to invert this mapping in the presence of real brain noise. To improve the localization accuracy we use a hybrid MLPstart-LM method, in which the MLP?s
output provides the starting point for a
Levenberg-Marquardt (LM) optimization
(Press et al., 1988). We use the MLP and
MLP-start-LM methods to localize singledipole sources from actual MEG signal
components isolated by a blind source separation (BSS) algorithm (Vig?ario et al.,
2000; Tang et al., 2002) and compare the
results with the output of standard interactive commercial localization software.
Sensor surface
11.343 cm
10.851 cm
13.594 cm
10.851cm
3.605 cm
z
z
x
y
Saggital View
Coronal View
Head Model B
10.5 cm
7.5 cm
Head Model A
7.5 cm
A
B
3 cm
Training Region
4 cm
Training Region and various Head Models
Figure 1: Sensor surface and training region.
The center of the spherical head model was
varied within the given region. Diamonds denote sensors.
Section 2 describes our synthetic data, the
forward model, the noise used to additively contaminate the training data, and the MLP
structure. Section 3 presents the localization performance of both the MLP and MLP-startLM, and compares them with various conventional LM methods. In Section 3.2, comparative localization results for our proposed methods and standard Neuromag commercial
software on actual BSS-separated MEG signals are presented.
2
Data and MLP structure
We constructed noisy data using the procedure of Jun et al. (2002), except that an additional
input was associated with each exemplar, namely the (x, y, z) coordinates of the center of
a sphere fitted to the head, and the forward model was modified to account for this offset.
Each exemplar thus consisted of the (x, y, z) coordinates of the center of a sphere fitted to
the head, sensor activations generated by a forward model, and the target dipole location.1
We made two datasets: one for training and another for testing. Centers of spherical head
1
Given the sensor activations and a dipole location, the minimum error dipole moment can be
calculated analytically (H?am?al?ainen et al., 1993). Therefore, although the dipoles used in generating
the dataset had both location and moment, the moments were not included in the datasets used for
training or testing.
models in the training set were drawn from a ball of radius 3 cm centered 4 cm above the
bottom of the training region,2 as shown in Figure 1. The dipoles in the training set were
drawn uniformly from a spherical region centered at the corresponding center, with a radius
of 7.5 cm, and truncated at the bottom. Their moments were drawn uniformly from vectors
of strength ?200 nAm. The corresponding sensor activations were calculated by adding
the results of a forward model and a noise model. To check the performance of the network
during training, a test set was generated in the same fashion as the training set. We used the
sensor geometry of a 4D Neuroimaging Neuromag-122 whole-head gradiometer (Ahonen
et al., 1993) and a standard analytic model of quasistatic electromagnetic propagation in a
spherical head (Jun et al., 2002).
This work could be easily extended to a more realistic head model. In that case the integral
equations are solved by the boundary element method (BEM) or the finite element method
(FEM) numerically (H?am?al?ainen et al., 1993). The human skull phantom study in Leahy
et al. (1998) shows that the fitted spherical head model for MEG localization is slightly
inferior in accuracy to the realistic head model numerically calculated by BEM. In forward
calculation, a spherical head model has some advantages: it is more easily implemented and
is much faster. Despite its inferiority in terms of localization accuracy, we use a spherical
head model in this work.
In order to properly compare the performance of various localizers, we need a dataset for
which we know the ground truth, but which contains the sorts of noise encountered in
actual MEG recordings. To this end, we measured real brain noise and used it to additively
contaminate synthetic sensor readings (Jun et al., 2002). This noise was taken, unaveraged,
from MEG recordings during periods in which the brain region of interest in the experiment
was quiescent, and therefore included all sources of noise present in actual data: brain
noise, external noise, sensor noise, etc. This had a RMS (square root of mean square)
magnitude of roughly P n = 50?200 fT/cm, where we measure the SNR of a dataset using
the ratios of the powers in the signal and noise, SNR (in dB) = 20 log10 P s /P n , where P s
and P n are the RMS sensor readings from the dipole and noise, respectively. The datasets
used for training and testing were made by adding the noise to synthetic sensor activations
generated by the forward model, and exemplars whose resulting SNR was below ?4 dB
were rejected.
The MLP charged with approximating the inverse mapping had an input layer of 125 units
consisting of the three Cartesian coordinate of the center of the sphere fitted to the head, and
the 122 sensor activations. It had two hidden layers with 320 and 30 units respectively, and
an output layer of three units representing the Cartesian coordinates of the fitted dipole. The
output units had linear activation functions, while the hidden unit had hyperbolic tangent
activation functions. Adjacent layers were fully connected, with no cut-through connections. The 122 sensor activation inputs were scaled to an RMS value of 0.5, and the target
outputs were scaled into [?1, +1]. The network weights were initialized with uniformly
distributed random values between ?0.1, and online stochastic gradient decent with no momentum and an empirically chosen constant of proportionality was used for optimization.
2
Fitted spheres from twelve subjects performing various tasks on a 4D Neuroimaging Neuromag122 MEG system were collected, and this distribution of head positions was chosen to include all
twelve cases. Just as the position of the center of the head varies from session to session and subject to
subject, so does head orientation and radius. Because a sphere is rotationally symmetric, our forward
model is insensitive to orientation, and similarly the external magnetic field caused by a dipole in a
homogeneous sphere is invariant to the sphere?s radius. On the other hand, the noise process would
not be invariant to orientation or radius, so we might expect a slight increase in performance if the
network had orientation and radius available as inputs, rather than just the position of the center.
14
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Figure 2: Mean localization errors of the trained MLP as a function of correct dipole location, binned into regions. All units are in cm. Left: Coronal cross section. Right: Sagittal
cross section.
3.1
Results and discussion
Training and localization results
Datasets of 100,000 (training) and 25,000
(testing) patterns, all contaminated by real
brain noise, were constructed. As is typical, the incremental gains per epoch decrease exponentially with training. From
the training curves (not shown) it is
evident that additional training would
have further decreased the error, but we
nonetheless stopped after 1000 epochs,
which took about three days on 2.8 GHz
Intel Xeon CPU.
2.5
Mean Localization Error (cm)
3
fixed?4?start?LM
MLP
MLP?start?LM
optimal?start?LM
2
1.5
1
0.5
0
0
5
10
15
S/N (dB)
We investigated localization error distributions over various regions of interest. We Figure 3: Mean localization error vs. SNR.
considered two cross sections (coronal and MLP, MLP-start-LM, and optimal-start-LM
sagittal views) with width of 2 cm, and were tested on signals from 25,000 random
each of these was divided into 19 regions, dipoles, contaminated by real brain noise.
as shown in Figure 2. We extracted the noisy signals and the corresponding dipoles from
testing datasets. For each region 49?500 patterns were collected. A dipole localization was
performed using the trained MLP, and the average localization error for each region was
calculated. Figure 2 shows the localization error distribution over two cross sections. In
general, dipoles closer to the sensor surface were better localized.
We compared various automatic localization methods, most of which consist of LM used
in different ways:
? MLP-start-LM
LM was started with the trained MLP?s output.
? fixed-4-start-LM
LM was tuned for good performance using restarts at the four fixed initial points
(0, 0, 6), (?5, 2, ?1), (5, 2, ?1), and (0, ?5, ?1), in units of cm relative to the
center of the spherical head model. The best result among four results was chosen.
Table 1: Comparison of performance on real brain noise test set of Levenberg-Marquardt
source localizers with three LM restarts strategies, the trained MLP, and a hybrid system.
Each number is an average over 25,000 localizations, so the error bars are negligible.
Computation Localization
Algorithm
time (ms)
error (cm)
fixed-4-start-LM
random-20-start-LM
optimal-start-LM
MLP
MLP-start-LM
120
663
14
0.7
15
0.83
0.54
0.49
0.90
0.53
? random-n-start-LM
LM was restarted with n random (uniformly distributed) points within the spherical head model. We checked how many restarts were needed to match the accuracy
of the MLP-start-LM, yielding n = 20, which is the same as in Jun et al. (2002).
? optimal-start-LM
LM was started with the known exact dipole source location.
Figure 3 shows the localization performance as a function of SNR for fixed-4-start-LM,
optimal-start-LM, the trained MLP, and MLP-start-LM. Optimal-start-LM shows the best
localization performance across the whole range of SNRs, but the hybrid system shows
almost the same performance as optimal-start-LM except at very high SNRs, while the
trained MLP is more robust to noise than fixed-4-start-LM. In this experiment, most of the
sources with very high SNR were superficial, located around the upper neck or back of the
head. These sorts of sources are often very hard to localize well, as it is easy to become
trapped in a local minimum (Jun et al., 2002). It is expected that, under these conditions,
a better initial guess than the MLP output (which are 0.7 cm on average from the exact
source) would be required to obtain near-optimal performance from LM.
A grand summary, averaged across various SNR conditions, is shown in Table 1. The
trained MLP is fastest, and its hybrid system is about 40? faster than random-20-start-LM,
while the hybrid system is about 9? faster, yet more accurate than, fixed-4-start-LM. This
means that MLP-start-LM was about two times faster than might be naively expected.
3.2
Localization on real MEG signals and comparison with commercial software
The sensors in MEG systems have poor signal-to-noise ratios (SNRs) for single-trial data,
since MEG data is strongly contaminated by various noises. Blind source separation of
MEG data segregates noise from signal (Vig?ario et al., 2000; Tang et al., 2000a; Sander
et al., 2002), raising the SNR sufficiently to allow single-trial analysis (Tang et al., 2000b).
Even though the sensor attenuation vectors of the BSS-separated components can be well
localized to equivalent current dipoles (Vig?ario et al., 2000; Tang et al., 2002), the recovered field maps can be quite noisy. We applied the MLP and MLP-start-LM to localize
single dipolar sources from various actual BSS-separated MEG signals.3 The xfit program
3
Continuous 300 Hz MEG data for four right-handed subjects was collected using a cognitive
protocol developed by Michael P. Weisend, band-pass filtered at 0.03?100 Hz, separated using second order blind identification algorithm (SOBI), and scanned for neuronal sources of interest. The
following four visual reaction time tasks were performed by each subject: stimulus pre-exposure
task, trump card task, elemental discrimination task, and transverse patterning task. For each subject,
all four experiments were performed on the same day, but each in a separate session. Subjects were
permitted to move their heads between experiments.
15
20
MLP?start?LM
MLP
xfit
10
20
MLP?start?LM
MLP
xfit
15
5
MLP?start?LM
MLP
xfit
15
10
10
PV
0
PV
5
5
SV
SV
SV
PV
?5
0
0
?10
?5
?5
?15
?15
?10
?5
0
5
10
15
?10
?15
?10
?5
0
5
10
15
?10
?15
?10
?5
0
5
10
15
Figure 4: Dipole source localization results of Neuromag software (xfit), our MLP, MLPstart-LM for four BSS-separated primary visual and four secondary visual MEG signal
components of S01, over four sorts of tasks. PV and SVdenote primary visual source and
secondary visual source, respectively. Left: Axial view. Center: Coronal view. Right:
Sagittal view. The outer surface denotes the sensor surface, and diamonds on this surface
denote sensors. The inner surface denotes a spherical head model fit to the subject.
(standard commercial software bundled with the 4D Neuroimaging Neuromag-122 MEG
system) is compared with the methods developed here.
A field map of each component was scaled to an RMS of 0.5 and inputed to the trained
MLP. Their MLP?s outputs were scaled back to their dipole location vectors and were used
for initializing LM. Figure 4 shows the dipole locations estimated by the MLP, MLP-startLM, and Neuromag?s xfit software, for two sorts of sensory sources: primary visual sources
and secondary visual sources, respectively, over four tasks in subject S01. In Figure 5, the
estimated dipole locations are shown for somatosensory sources over three different subjects. Each figure consists of three viewpoints: axial (x-y plane), coronal (x-z plane), and
sagittal (y-z plane). The center of a fitted spherical head model (S01: trump card task) is
(0.335, 0.698, 3.157). All units are in cm. All dipole locations estimated by the MLP and
MLP-start-LM are clustered within about 3 cm, and about 0.7 cm, of xfit?s results, respectively. We see that the primary visual sources are more consistently localized, across all
four tasks, than the secondary visual sources. The secondary sources also had more variable stimulus-locked average time courses (Tang and Pearlmutter, 2003). It is noticeable
that somatosensory sources on the right hemisphere are localized poorly by the MLP, but
well localized by the hybrid method. Even though the auditory sources are the weakest (not
shown here), i.e. have the lowest SNRs, they are reasonably well localized.
While the MLP-estimated location is about 1.16 cm (|dx| ? 0.90, |dy| ? 0.57, |dz| ?
0.46) on average (N = 14) from those of xfit, the hybrid method?s result is about 0.35 cm
(|dx| ? 0.20, |dy| ? 0.22, |dz| ? 0.10) from xfit?s estimated location. Considering that
xfit had extra information, namely the identity of a subset of the sensors to use, this hybrid
method result is believed to be almost as good as the xfit result. The trained MLP and the
hybrid method are applicable to actual MEG signals, and seem to offer comparable and
perhaps superior localization relative to xfit, with clear advantages in both speed and in the
lack of required human interaction or subjective human input.
SOBI was performed on continuous 122-channel data collected during the entire period of the
experiment. It generated 122 components, each a one-dimensional time series with an associated field
map. Event triggered averages were calculated from their continuous single-trial time series for all
122 separated components. A dipole fitting method was applied to the identified neural components.
The input to the dipole fitting algorithm of xfit was the field map and the output was the location of
ECDs. From all separated components for four subjects and four sorts of tasks taken as in Tang et al.
(2002). only fourteen components were localized and compared. For further experimental details and
a detailed SOBI algorithm, see Tang et al. (2002).
15
20
MLP?start?LM
MLP
xfit
10
20
MLP?start?LM
MLP
15
5
10
10
0
5
5
?5
0
0
?10
?5
y
?15
?15
?10
?5
0
5
10
?5
z
x
MLP?start?LM
MLP
xfit
15
xfit
z
x
15
?10
?15
?10
?5
0
5
10
y
15
?10
?15
?10
?5
0
5
10
15
Figure 5: Dipole source localization results of Neuromag software (xfit), our MLP, MLPstart-LM for three real BSS-separated somatosensory MEG signal components from the
transverse patterning task over three different subjects (S01, S02, S03). Even the center of
a fitted spherical head model is varied over three subjects, the only fitted sphere of subject
S01 transverse patterning task, centered at (0.373, 0.642, 3.205), is depicted. Left: Axial
view. Center: Coronal view. Right: Sagittal view. The outer surface denotes the sensor
surface, and diamonds on this surface denote sensors. The inner surface denotes a spherical
head model fit to the subject.
4
Conclusion
We propose the inclusion of a head position input for MLP-based MEG dipole localizers.
This overcomes the limitation of previous MLP-based MEG localization systems, namely
the need to retrain the network for each session or subject. Experiments showed that the
trained MLP was far faster, albeit slightly less accurate, than fixed-4-start-LM. This motivated us to construct a hybrid system, MLP-start-LM, which improves the localization
accuracy while reducing the computational burden to less than one ninth than that of fixed4-start-LM. This hybrid method was comparable in accuracy to random-20-start-LM, at
1/40-th the computation burden, which is about two times faster than might be naively
expected. Over the whole range of SNRs, the hybrid system showed almost as good performance in accuracy and computation time as the hypothetical optimal-start-LM.
We applied the MLP and MLP-start-LM to localize single dipolar sources from actual BSSseparated MEG signals, and compared these with the results of the commercial Neuromag
program xfit. The MLP yielded dipole locations close to those of xfit, and MLP-start-LM
gave locations that were even closer to those of xfit.
In conclusion, our MLP can itself serve as a reasonably accurate real-time MEG dipole
localizer, even when the head position changes regularly. This MLP also constitutes an
excellent dipole guessor for LM. Because this MLP receives a head position input, the
need to retrain for various subjects or sessions has been eliminated without sacrificing the
many advantages of the universal approximator direct inverse approach to localization.
Acknowledgements
This work was supported by NSF CAREER award 97-02-311, the Mental Illness and Neuroscience Discovery Institute, a gift from the NEC Research Institute, NIH grant 2 R01
EB000310-05, and Science Foundation Ireland grant 00/PI.1/C067. We would like to thank
Guido Nolte for help with the forward model, Michael Weisend for allowing us to use his
data, and Michael Weisend, Akaysha Tang, and Natalie Malaszenko for providing experimental details.
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1,644 | 2,495 | Algorithms for Interdependent Security Games
Michael Kearns
Luis E. Ortiz
Department of Computer and Information Science
University of Pennsylvania
1 Introduction
Inspired by events ranging from 9/11 to the collapse of the accounting firm Arthur Andersen, economists Kunreuther and Heal [5] recently introduced an interesting game-theoretic
model for problems of interdependent security (IDS), in which a large number of players
must make individual investment decisions related to security ? whether physical, financial, medical, or some other type ? but in which the ultimate safety of each participant
may depend in a complex way on the actions of the entire population. A simple example is
the choice of whether to install a fire sprinkler system in an individual condominium in a
large building. While such a system might greatly reduce the chances of the owner?s property being destroyed by a fire originating within their own unit, it might do little or nothing
to reduce the chances of damage caused by fires originating in other units (since sprinklers
can usually only douse small fires early). If ?enough? other unit owners have not made the
investment in sprinklers, it may be not cost-effective for any individual to do so.
Kunreuther and Heal [5] observe that a great variety of natural problems share this basic interdependent structure, including investment decisions in airline baggage security (in which
investments in new screening procedures may reduce the risk of directly checking suspicious cargo, but nearly all airlines accept transferred bags with no additional screening 1 );
risk management in corporations (in which individual business units have an incentive to
avoid high-risk or illegal activities only if enough other units are similarly well-behaved);
vaccination against infectious disease (where the fraction of the population choosing vaccination determines the need for or effectiveness of vaccination); certain problems in computer network security; and many others. All these problems share the following important
properties:
There is a ?bad event? (condominium fire, airline explosion, corporate bankruptcy,
infection, etc.) to be avoided, and the opportunity to reduce the risk of it via some
kind of investment.
The cost-effectiveness of the security investment for the individual is a function
of the investment decisions made by the others in the population.
The original work by Kunreuther and Heal [5] proposed a parametric game-theoretic model
for such problems, but left the interesting question of computing the equilibria of model
largely untouched. In this paper we examine such computational issues.
1
El Al airlines is the exception to this.
2 Definitions
In an IDS game, each player i must decide whether or not to invest in some abstract security
mechanism or procedure that can reduce their risk of experiencing some abstract bad event.
The cost of the investment to i is Ci , while the cost of experiencing the bad event is Li ;
the interesting case is when Li >> Ci . Thus, player i has two choices for his action ai :
ai
means the player makes the investment, while ai
means he does not. It turns
out that the important parameter is the ratio of the two costs, so we define Ri Ci =Li .
=1
=0
=
For each player i, there is a parameter pi , which is the probability that player i will experience the bad event due to internal contamination if ai
? for example, this is the
probability of the condominium owner?s unit burning down due to a fire originating in his
own unit. We can also think of pi as a measure of the direct risk to player i ? as we shall
see, it is that portion of his risk under his direct control.
=0
=
To model sources of indirect risk, for each pair of players i; j; i 6 j , let qji be the probability that player i experiences the bad event as a result of a transfer from player j ? for
example, this is the probability that the condominium of player i burns P
down due to a fire
originating in the unit of player j . Note the implicit constraint that pi
q < .
j 6=i ji
+
1
An IDS game is thus given by the parameters pi , qji , Li , Ci for each player i, and the
expected cost to player i under the model is defined to be
( ) = a C +(1
Mi ~a
i
i
)
ai pi Li
+(1 (1
2
a )p ) 41
i
i
Y
n
6
(1 (1
)
aj qji
3
)5 L (1)
i
j =1;j =i
Let us take a moment to parse and motivate this definition, which is the sum of three terms.
The first term represents the amount invested in security by player i, and is either 0 (if
ai
) or Ci (if ai
). The second term is the expected cost to i due to internal or direct
risk of the bad event, and is either pi Li (which is the expected cost of internally generated
bad events in the case ai
), or is 0 (in the case of investment, ai
). Thus, there is a
natural tension between the first two terms: players can either invest in security, which costs
money but reduces risk, or gamble by not investing. Note that here we have assumed that
security investment perfectly eradicates direct risk (but not indirect risk); generalizations
are obviously possible, but have no qualitative effect on the model.
=0
=1
=0
=1
It is the third term of Equation (1) that expresses the interdependent nature of the problem.
This term encodes the assumption that there are n sources of risk to player i ? his own
internal risk, and a specific transfer risk from each of the other n
players ? and that
all these sources are statistically independent. The prefactor
ai pi is simply the
probability that player i does not experience the bad event due to direct risk. The bracketed
expression is the probability that player i experiences a bad event due to transferred risk:
each factor
aj qji in the product is the probability that a bad event does not
befall player i due to player j (and the product expresses the assumption that all of these
possible transfer events are independent). Thus 1 minus this product is the probability of
transferred contamination, and of course the product of the various risk probabilities is also
multiplied by the cost Li of the bad event.
(1 (1
(1 (1
1
) )
) )
The model parameters and Equation (1) define a compact representation for a multiplayer
game in which each player?s goal is to minimize their cost. Our interest is in the efficient
computation of Nash equilibria (NE) of such games 2 .
2
See (for example) [4] for definitions of Nash and approximate Nash equilibria.
3 Algorithms
We begin with the observation that it is in fact computationally straightforward to find a
single pure NE of any IDS game. To see this, it is easily verified that if there are any conditions under which player i prefers investing (ai
) to not investing (ai
) according
to the expected costs given by Equation (1), then it is certainly the case that i will prefer to
invest when all the other n
players are doing so. Similarly, the most favorable conditions for not investing occur when no other players are investing. Thus, to find a pure NE,
we can first check whether either all players investing, or no players investing, forms a NE.
If so, we are finished. If neither of these extremes are a NE, then there are some players for
whom investing or not investing is a dominant strategy (a best response independent of the
behavior of others). If we then ?clamp? such players to their dominant strategies, we obtain
a new IDS game with fewer players (only those without dominant strategies in the original
game), and can again see if this modified game has any players with dominant strategies.
At each stage of this iterative process we maintain the invariant that clamped players are
playing a best response to any possible setting of the unclamped players.
=1
=0
1
( )
Theorem 1 A pure NE for any n-player IDS game can be computed in time O n2 .
In a sense, the argument above demonstrates the fact that in most ?interesting? IDS games
(those in which each player is a true participant, and can have their behavior swayed by
that of the overall population), there are two trivial pure NE (all invest and none invest).
However, we are also interested in finding NE in which some players are choosing to invest
and others not to (even though no player has a dominant strategy). A primary motivation
for finding such NE is the appearance of such behavior in ?real world? IDS settings, where
individual parties do truly seem to make differing security investment choices (such as with
sprinkler systems in large apartment buildings). Conceptually, the most straightforward
way to discover such NE would be to compute all NE of the IDS game. As we shall
eventually see, for computational efficiency such a demand requires restrictions on the
parameters of the game, one natural example of which we now investigate.
3.1 Uniform Transfer IDS Games
A uniform transfer IDS game is one in which the transfer risks emanating from a given
player are independent of the transfer destination. Thus, for any player j , we have that
for all i 6 j , qji
?j for some value ?j . Note that the risk level ?j presented to the
population by different players j may still vary with j ? but each player spreads their risk
indiscriminately across the rest of the population. An example would be the assumption
that each airline transferred bags with equal probability to all other airlines.
=
=
In this section, we describe two different approaches for computing NE in uniform transfer IDS games. The first approach views a uniform transfer IDS game as a special type
of summarization game, a class recently investigated by Kearns and Mansour [4]. In an
n-player summarization game, the payoff of each player i is a function of the actions ~a i
of all the other players, but only through the value of a global and common real-valued
summarization function S ~a . The main result of [4] gives an algorithm for computing
approximate NE of summarization games, in which the quality of the approximation depends on the influence of the summarization function S . A well-known notion in discrete
functional analysis, the influence of S is the maximum change in S that any input (player)
can unilaterally cause. (See [4] for detailed definitions.)
()
It can be shown (details omitted) that any uniform transfer IDS game is in fact a summa-
rization game under the choice
S (~a) =
Y
n
(1 (1
) )
aj ?j
(2)
j =1
and that the influence of this function is bounded by the largest ?j . We note that in many
natural uniform transfer IDS settings, we expect this influence to diminish like =n with
the number of players n. (This would be the case if the risk transfer comes about through
physical objects like airline baggage, where each transfer event can have only a single
destination.) Combined with the results of [4], the above discussion can be shown to yield
the following result.
1
Theorem 2 There is an algorithm that takes as input any uniform transfer IDS game, and
any > , and computes an O -NE, where
pj =
?j g and
jf
=, and .
j f?j g. The running time of the algorithm is polynomial in n,
= max
0
( + )
= max (1
1
) (1
)
We note that in typical IDS settings we expect both the pj and ?j to be small (the bad event
is relatively rare, regardless of its source), in which case may be viewed as a constant.
Furthermore, it can be verified that this algorithm will in fact be able to compute approximate NE in which some players choose to invest and others not to, even in the absence of
any dominant strategies.
While viewing uniform transfer IDS games as bounded influence summarization games
relates them to a standard class and yields a natural approximation algorithm, an improved
approach is possible. We now present an algorithm (Algorithm UniformTransferIDSNash
in Figure 3.1) that efficiently computes all NE for uniform transfer IDS games. The algorithm (indeed, even the representation of certain NE) requires the ability to compute mth
roots.
0
: =0
0
We may assume without loss of generality that for all players i, ?i > , and pi > .
For a joint mixed strategy vector ~x 2 ; n , denote the set of (fully) investing players as
I fi xi
g; the set of (fully) non-investing players as N fi xi g; and the set
of partially investing players as P fi
< xi < g:
[0 1]
:0
: =1
1
The correctness of algorithm UniformTransferIDSNash follows immediately from two
lemmas that we now state without proof due to space considerations. The first lemma is a
generalization of Proposition 2 of [2], and essentially establishes that the values Ri =pi and
?i Ri =pi determine a two-level ordering of the players? willingness to invest. This double ordering generates the outer and inner loops of algorithm UniformTransferIDSNash.
Note that a player with small Ri =pi has a combination of relatively low cost of investing
compared to the loss of a bad event (recall Ri Ci =Li ), and relatively high direct risk pi ,
and thus intuitively should be more willing to invest than players with large Ri =pi . The
lemma makes this intuition precise.
(1 )
=
=
Lemma 3 (Ordering Lemma) Let ~x be a NE for a uniform transfer IDS game G
~ ~
n; R;
p; ~
? . Then for any i 2 I (an investing player), any j 2 N (a partially investing
player), and any k 2 P (a non-investing player), the following conditions hold:
(
)
Ri =pi
(1
)
Ri =pi
?j Rj =pj
<
Rj =pj
(1
<
)
(1 ? ) R =p
?k Rk =pk < Rk =pk
k
k
k
The second lemma establishes that if a NE contains some partially investing players, the
values for their mixed strategies is in fact uniquely determined. The equations for these
mixed strategies is exploited in the subroutine TestNash.
Algorithm UniformTransferIDSNash
p, transfer risk
Input: An n-player uniform transfer IDS game with direct risk parameters ~
~ , where Ri = Ci =Li .
parameters ~?, and cost parameters R
Output: A set S of all exact connected sets of NE for .
1. Initialize a partition of the players into three sets I; N; P (the investing, not investing,
and partially investing players, respectively) and test if everybody investing is a NE:
I
1; : : : ; n ; N
;P
;S
TestNash( ; I; N; P; S )
G
G
f
g
;
;
G
2. Let (i1 ; i2 ; :::; in ) be an ordering of the n players satisfying Ri1 =pi1
Rin =pin . Call this the outer ordering.
3. for k = 1; : : : ; n
Sf g
:::
(a) Move the next player in the outer ordering from the investing to the partiallyP
ik ; I
I
ik
investing sets: P
(b) Let (j1 ; :::; jk ) be an ordering of the players in P satisfying (1 ?j1 ) Rj1 =pj1
:::
(1
?jk ) Rjk =pjk . Call this the inner ordering.
(c) Consider a strategy with no not-investing players: N
;S
TestNash( ; I; N; P; S )
(d) for m = 1; : : : ; k
i. Move the next player in the inner ordering from the partially-investing to
non-investing sets, and test if there is a NE consistent with the partition:
N
N
jm ; P
P
jm ; S
TestNash( ; I; N; P; S )
f g
;
G
Sf g
f g
G
Subroutine TestNash
Inputs: An n-player uniform transfer IDS game ; a partition of the players I; N; P (as above);
S , the current discovered set of connected sets of NE for
Output: S with possibly one additional connected set of NE of consistent with I; N , and P
(assuming unit-time computation of m-roots of rational numbers)
G
G
G
1. Set pure strategies for not-investing and investing players, respectively:
N; xk
0, i
I; xi
1.
j j
82
T
2. if P = 1 (Lemma 4, part (a) applies)
8k 2
(a) Let P = j , U as in Equation 3 and U 0 = U
(0; 1)
(b) if Rj = pj k2N (1 ?k ) (i.e., player j is indifferent) and U 0 = , then return
S
f gQ
S ff~y : yj 2 U 0; ~y
6 ;
j = ~x j gg
3. else (Lemma 4, part (b) applies)
8 2
(a) Compute mixed strategies j P; xj as in Equation 4
(b) if j P; xj 0 or xj 1, return S
(c) if ~x is a NE for then return S
~
x
9 2
G
S ff gg
4. return S
Figure 1: Algorithm UniformTransferIDSNash
= [l; u] is an interval of < with endpoints l and u, and a; b 2 < then we define
+ b [al + b; au + b].
Lemma 4 (Partial Investment Lemma) Let ~x 2 [0; 1] be a mixed strategy for a uniform
~ p
~; ~
? ), and let P be the set of partially investing players in ~
x.
transfer IDS game G = (n; R;
Then (a) if jP j = 1, then letting P = fj g, V = [max 2 R =p ; min 2 (1 ? ) R =p ] ;
and
U = ((p =R ) V
(1 ? )) = ?
(3)
Q
it holds that ~x is a NE if and only if R = p
2 1 ? (i.e., player j is indifferent) and
player j mixed strategy satisfies x 2 U ; else, (b) if jP j > 1, and ~x is a NE, then for all
If I
aI
n
i
j
j
j
j
I
i
j
j
k
N
i
j
k
k
N
k
k
k
j
2 P,
where E
=
= ((p =R )E (1
(R =p )
(1
?
)
2
xj
Q
j
j
.Q
2
P
j
j
j
k
?j
j j
1=(
)) = ?
P
k
N
(4)
j
1)
:
The next theorem summarizes our second algorithmic result on uniform transfer IDS
games. The omitted proof follows from Lemmas 3 and 4.
Theorem 5 Algorithm UniformTransferIDSNash computes all exact (connected sets of)
NE for uniform transfer IDS games in time polynomial in the size of the model.
We note that it follows immediately from the description and correctness of the algorithm
that any n-player uniform transfer IDS game has at most n n
=
connected sets
of NE. In addition, each connected set of NE in a uniform transfer IDS game is either
a singleton or a simple interval where n
of the players play pure strategies and the
remaining player has a simple interval in ; of probability values from which to choose
its strategy. At most n of the connected sets of NE in a uniform transfer IDS game are
simple intervals.
( + 3) 2 + 1
1
[0 1]
3.2 Hardness of General IDS Games
In light of the results of the preceding section, it is of course natural to consider the computational difficulty of unrestricted IDS. We now show that even a slight generalization of
uniform transfer IDS games, in which we allow the ?j to assume two fixed values instead
of one, leads to the intractabilty of computing at least some of the NE.
A graphical uniform transfer IDS game, so named because it can be viewed as a marriage
between uniform transfer IDS games and the graphical games introduced in [3], is an IDS
game with the restriction that for all players j , qji 2 f ; ?j g, for some ?j > . Let
N j fi qji > g be the set of players that can be directly affected by player j ?s
behavior. In other words, the transfer risk parameter qji of player j with respect to player i
is either zero, in which case the player j has no direct effect on player i?s behavior; or it is
constant, in which case, the public safety eji
xj ?j of player j with respect
to player i 2 N j is the same as for any other player in N j .
()
:
0
0
= (1 (1
()
0
) )
()
The pure Nash extension problem for an n-player game with binary actions takes as input
a description of the game and a partial assignment ~a 2 f ; ; gn. The output may be any
complete assignment (joint action) ~b 2 f ; gn that agrees with ~a on all its 0 and 1 settings,
and is a (pure) NE for the game; or ?none? if no such NE exists. Clearly the problem of
computing all the NE is at least as difficult as the pure Nash extension problem.
01
01
Theorem 6 The pure Nash extension problem for graphical uniform transfer IDS games is
NP-complete, even if jN j j for all j , and ?j is some fixed value ? for all j .
() 3
The reduction (omitted) is from Monotone One-in-Three SAT [1].
4 Experimental Study: Airline Baggage Security
As an empirical demonstration of IDS games, we constructed and conducted experiments
on an IDS game for airline security that is based on real industry data. We have access
to a data set consisting of 35,362 records of actual civilian commercial flight reservations,
both domestic and international, made on August 26, 2002. Since these records contain
complete flight itineraries, they include passenger transfers between the 122 represented
commercial air carriers. As described below, we used this data set to construct an IDS
game in which the players are the 122 carriers, the ?bad event? corresponds to a bomb
exploding in a bag being transported in a carrier?s airplane, and the transfer event is the
physical transfer of a bag from one carrier to another.
( )
For each carrier pair i; j , the transfer parameter qji was set to be proportional to the
count of transfers from carrier j to carrier i in the data set. We are thus using the rate of
passenger transfers as a proxy for the rate of baggage transfers. The resulting parameters
(details omitted) are, as expected, quite asymmetric, as there are highly structured patterns of transfers resulting from differing geographic coverages, alliances between carriers,
etc. The model is thus far from being a uniform transfer IDS game, and thus algorithm
UniformTransferIDSNash cannot be applied; we instead used a simple gradient learning
approach.
The data set provides no guidance on reasonable values for the Ri and pi , which quantify
relative costs of a hypothetical new screening procedure and the direct risks of checking
contaminated luggage, respectively; presumably Ri depends on the specific economics of
the carrier, and pi on some notion of the risk presented by the carrier?s clientele, which
might depend on the geographic area served. Thus, for illustrative purposes, an arbitrary
value of pi
:
was chosen for all i 3 , and a common value for Ri of 0.009 (so an
explosion is roughly 110 times more costly to a carrier than full investment in security).
= 0 01
Since the asymmetries of the qji preclude the use of algorithm UniformTransferIDSNash,
we instead used a learning approach in which each player begins with a random initial
investment strategy xi 2 ; , and adjusts its degree of investment up or down based on
the gradient dynamics xi
xi i , where i is determined by computing the derivative
of Equation (1) and
: was used in the experiments to be discussed.
[0 1]
= 0 05
49
48
47
46
45
44
43
49
48
47
46
45
44
43
42
41
40
39
38
37
36
42
41
40
39
38
37
36
35
34
33
32
31
30
29
35
34
33
32
31
30
29
28
27
26
25
24
23
22
28
27
26
25
24
23
22
21
20
19
18
17
16
15
21
20
19
18
17
16
15
14
13
12
11
10
9
8
14
13
12
11
10
9
8
7
6
5
4
3
7
6
5
4
3
2
1
2
1
(a)
(b)
Figure 2: (a) Simulation of the evolution of security investment strategies for the 49 busiest carrier
using gradient dynamics under the IDS model. Above each plot is an index indicating the rank of the
carrier in terms of overall volume in the data set. Each plot shows the investment level xi (initialized
randomly in [0; 1]) for carrier i over 500 simulation steps. (b) Tipping phenomena. Simulation of
the evolution of security investment strategies for the 49 busiest carriers, but with the three largest
carriers (indices 1, 2 and 3) in the data set clamped (subsidized) at full investment. The plots are
ordered as in (a), and again show 500 simulation steps under gradient dynamics.
Figure 2(a) shows the evolution, over 500 steps of simulation time, of the investment level
xi for the 49 busiest carriers 4 . We have ordered the 49 plots with the least busy carrier
3
This is (hopefully) an unrealistically large value for the real world; however, it is the relationship
between the parameters and not their absolute magnitudes that is important in the model.
4
According to the total volume of flights per carrier in the data set.
(index 49) plotted in the upper left corner, and the busiest (index 1) in the lower right
corner. The horizontal axes measure the 500 time steps, while the vertical axes go from 0
to 1. The axes are unlabeled for legibility.
The most striking feature of the figure is the change in the evolution of the investment
strategy as we move from less busy to more busy carriers. Broadly speaking, there is a large
population of lower-volume carriers (indices 49 down to 34) that quickly converge to full
investment (xi
) regardless of initial conditions. The smallest carriers, not shown (ranks
122 down to 50), also all rapidly converge to full investment. There is then a set of mediumvolume carriers whose limiting strategy is approached more slowly, and may eventually
converge to either full or no investment (roughly indices 33 down to 14). Finally, the largest
carriers (indices 13 and lower) again converge quickly, but to no investment (xi
),
because they have a high probability of having bags transferred from other carriers (even if
they protect themselves against dangerous bags being loaded directly on their planes).
=1
=0
Note also that the dynamics can yield complex, nonlinear behavior that includes reversals of
strategy. The simulation eventually converges (within 2000 steps) to a (Nash) equilibrium
in which some carriers are at full investment, and the rest at no investment. This property
is extremely robust across initial conditions and model parameters,
The above simulation model enables one to examine how subsidizing several airlines to encourage it to invest in security can encourage others to do the same. This type of ?tipping?
behavior [6] can be the basis for developing strategies for inducing adoption of security
measures short of formal regulations or requirements. Figure2(b) shows the result of an
identical simulation to the one discussed above, except the three largest carriers (indices 1,
2 and 3) are now ?clamped? or forced to be at full investment during the entire simulation.
Independent of initial conditions, the remaining population now invariably converges to full
investment. Thus the model suggests that these three carriers form (one of perhaps many
different) tipping sets ? carriers whose decision to invest (due to subsidization or other
exogenous forces) will create the economic incentive for a large population of otherwise
skeptical carriers to follow. The dynamics also reveal a cascading effect ? for example,
carrier 5 moves towards full investment (after having settled comfortably at no investment)
only after a number of larger and smaller carriers have done so.
Acknowledgements: We give warm thanks to Howard Kunreuther, Geoffrey Heal and
Kilian Weinberger for many helpful discussions.
References
[1] Michael Garey and David Johnson. Computers and Intractability: A Guide to the
Theory of NP-completeness. Freeman, 1979.
[2] Geoffrey Heal and Howard Kunreuther. You only die once: Managing discrete interdependent risks. 2003. Working paper, Columbia Business School and Wharton Risk
Management and Decision Processes Center.
[3] M. Kearns, M. Littman, and S. Singh. Graphical models for game theory. In Proceedings of the Conference on Uncertainty in Artificial Intelligence, pages 253?260,
2001.
[4] M. Kearns and Y. Mansour. Efficient Nash computation in summarization games with
bounded influence. In Proceedings of the Conference on Uncertainty in Artificial Intelligence, 2002.
[5] Howard Kunreuther and Geoffrey Heal. Interdependent security. Journal of Risk and
Uncertainty (Special Issue on Terrorist Risks), 2003. In press.
[6] Thomas Schelling. Micromotives and Macrobehavior. Norton, 1978.
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1,645 | 2,496 | Semi-supervised protein classification using
cluster kernels
Jason Weston?
Max Planck Institute for Biological Cybernetics,
72076 T?ubingen, Germany
[email protected]
Christina Leslie
Department of Computer Science,
Columbia University
[email protected]
Dengyong Zhou, Andre Elisseeff
Max Planck Institute for Biological Cybernetics,
72076 T?ubingen, Germany
[email protected]
William Stafford Noble
Department of Genome Sciences
University of Washington
[email protected]
Abstract
A key issue in supervised protein classification is the representation of input sequences of amino acids. Recent work using string kernels for protein data has achieved state-of-the-art classification performance. However, such representations are based only on labeled data ? examples
with known 3D structures, organized into structural classes ? while
in practice, unlabeled data is far more plentiful. In this work, we develop simple and scalable cluster kernel techniques for incorporating unlabeled data into the representation of protein sequences. We show that
our methods greatly improve the classification performance of string kernels and outperform standard approaches for using unlabeled data, such
as adding close homologs of the positive examples to the training data.
We achieve equal or superior performance to previously presented cluster
kernel methods while achieving far greater computational efficiency.
1
Introduction
A central problem in computational biology is the classification of proteins into functional
and structural classes given their amino acid sequences. The 3D structure that a protein
assumes after folding largely determines its function in the cell. However, it is far easier
to determine experimentally the primary sequence of a protein than it is to solve the 3D
structure. Through evolution, structure is more conserved than sequence, so that detecting
even very subtle sequence similarities, or remote homology, is important for predicting
function.
The major methods for homology detection can be split into three basic groups: pairwise
sequence comparison algorithms [1, 2], generative models for protein families [3, 4], and
discriminative classifiers [5, 6, 7]. Popular sequence comparison methods such as BLAST
?
Supplemental information for the paper, including the data sets and Matlab source code can be
found on this author?s web page at http://www.kyb.tuebingen.mpg.de/bs/people/weston/semiprot
and Smith-Waterman are based on unsupervised alignment scores. Generative models such
as profile hidden Markov models (HMMs) model positive examples of a protein family,
but they can be trained iteratively using both positively labeled and unlabeled examples
by pulling in close homologs and adding them to the positive set. A compromise between
these methods is PSI-BLAST [8], which uses BLAST to iteratively build a probabilistic
profile of a query sequence and obtain a more sensitive sequence comparison score. Finally,
classifiers such as SVMs use both positive and negative examples and provide state-of-theart performance when used with appropriate kernels [5, 6, 7]. However, these classifiers still
require an auxiliary method (such as PSI-BLAST) to handle unlabeled data: one generally
adds predicted homologs of the positive training examples to the training set before training
the classifier.
In practice, relatively little labeled data is available ? approximately 30,000 proteins with
known 3D structure, some belonging to families and superfamilies with only a handful of
labeled members ? whereas there are close to one million sequenced proteins, providing
abundant unlabeled data. New semi-supervised learning techniques should be able to make
better use of this unlabeled data.
Recent work in semi-supervised learning has focused on changing the representation
given to a classifier by taking into account the structure described by the unlabeled data
[9, 10, 11]. These works can be viewed as cases of cluster kernels, which produce similarity metrics based on the cluster assumption: namely, two points in the same ?cluster?
or region of high density should have a small distance to each other. In this work, we
investigate the use of cluster kernels for protein classification by developing two simple
and scalable methods for modifying a base kernel. The neighborhood kernel uses averaging over a neighborhood of sequences defined by a local sequence similarity measure,
and the bagged kernel uses bagged clustering of the full sequence data set to modify the
base kernel. In both the semi-supervised and transductive settings, these techniques greatly
improve classification performance when used with mismatch string kernels, and the techniques achieve equal or superior results to all previously presented cluster kernel methods
that we tried. Moreover, the neighborhood and bagged kernel approaches are far more
computationally efficient than these competing methods.
2
Representations and kernels for protein sequences
Proteins can be represented as variable length sequences, typically several hundred characters long, from the alphabet of 20 amino acids. In order to use learning algorithms that
require vector inputs, we must first find a suitable feature vector representation, mapping
sequence x into a vector space by x 7? ?(x). If we use kernel methods such as SVMs,
which only need to compute inner products K(x, y) = h?(x), ?(y)i for training and testing, then we can accomplish the above mapping using a kernel for sequence data.
Biologically motivated sequence comparison scores, like Smith-Waterman or BLAST, provide an appealing representation of sequence data. The Smith-Waterman (SW) algorithm
[2] uses dynamic programming to compute the optimal local gapped alignment score between two sequences, while BLAST [1] approximates SW by computing a heuristic alignment score. Both methods return empirically estimated E-values indicating the confidence
of the score. These alignment-based scores do not define a positive definite kernel; however, one can use a feature representation based on the empirical kernel map
?(x) = hd(x1 , x), . . . , d(xm , x)i
where d(x, y) is the pairwise score (or E-value) between x and y and xi , i = 1 . . . m,
are the training sequences. Using SW E-values in this fashion gives strong classification
performance [7]. Note, however, that the method is slow, both because computing each SW
score is O(|x|2 ) and because computing each empirically mapped kernel value is O(m).
Another appealing idea is to derive the feature representation from a generative model for
a protein family. In the Fisher kernel method [5], one first builds a profile HMM for the
positive training sequences, defining a log likelihood function log P (x|?) for any protein
sequence x. Then the gradient vector ?? log P (x|?)|?=?0 , where ?0 is the maximum likelihood estimate for model parameters, defines an explicit vector of features, called Fisher
scores, for x. This representation gives excellent classification results, but the Fisher scores
must be computed by an O(|x|2 ) forward-backward algorithm, making the kernel tractable
but slow.
It is possible to construct useful kernels directly without explicitly depending on generative
models by using string kernels. For example, the mismatch kernel [6] is defined by a
histogram-like feature map that uses mismatches to capture inexact string matching. The
feature space is indexed by all possible k-length subsequences ? = a1 a2 . . . ak , where each
ai is a character in the alphabet A of amino acids. The feature map is defined on k-gram ?
by ?(?) = (?? (?))Ak where ?? (?) = 1 if ? is within m
Pmismatches of ?, 0 otherwise,
and is extended additively to longer sequences: ?(x) = k-grams?x ?(?). The mismatch
kernel can be computed efficiently using a trie data structure: the complexity of calculating
K(x, y) is O(cK (|x| + |y|)), where cK = k m+1 |A|m . For typical kernel parameters k = 5
and m = 1 [6], the mismatch kernel is fast, scalable and yields impressive performance.
Many other interesting models and examples of string kernels have recently been presented.
A survey of related string kernel work is given in the longer version of this paper.
String kernel methods with SVMs are a powerful approach to protein classification and
have consistently performed better than non-discriminative techniques [5, 7, 6]. However,
in a real-world setting, protein classifiers have access to unlabeled data. We now discuss
how to incorporate such data into the representation given to SVMs via the use of cluster
kernels.
3
Cluster kernels for protein sequences
In semi-supervised learning, one tries to improve a classifier trained on labeled data by
exploiting (a relatively large set of) unlabeled data. An extensive review of techniques
can be found in [12]. It has been shown experimentally that under certain conditions, the
decision function can be estimated more accurately in a semi-supervised setting, yielding
lower generalization error. The most common assumption one makes in this setting is
called the ?cluster assumption,? namely that the class does not change in regions of high
density.
Although classifiers implement the cluster assumption in various ways, we focus on classifiers that re-represent the given data to reflect structure revealed by unlabeled data. The
main idea is to change the distance metric so that the relative distance between two points
is smaller if the points are in the same cluster. If one is using kernels, rather than explicit
feature vectors, one can modify the kernel representation by constructing a cluster kernel.
In [10], a general framework is presented for producing cluster kernels by modifying the
eigenspectrum of the kernel matrix. Two of the main methods presented are the random
walk kernel and the spectral clustering kernel.
The random walk kernel is a normalized and symmetrized version of a transition matrix
corresponding to a t-step random walk. The random representation described in [11] interprets an RBF kernel as a transition matrix of a random walk on a graph with vertices
K
xi , P (xi ? xj ) = P Kijip . After t steps, the probability of going from a point xi to a
point xj should be high if the points are in the same cluster. This transition probability
can be calculated P
for the entire matrix as P t = (D?1 K)t , where D is a diagonal matrix
such that Dii =
p Kip . To obtain a kernel, one performs the following steps. Com-
pute L = D?1/2 KD?1/2 and its eigendecomposition L = U ?U > . let ?i ? ?ti , where
?D
? 1/2 , where D
? is a
? = U ?U
? > . Then the new kernel is K
? =D
? 1/2 L
?i = ?ii , and let L
?
diagonal matrix with Dii = 1/Lii .
The spectral clustering kernel is a simple use of the representation derived from spectral
clustering [13] using the first k eigenvectors. One computes the eigenvectors (v1 , . . . , vk )
1
1
of D? 2 KD? 2 , with D defined as before, giving the representation ?(xi )p = vpi . This
vector can also then be normalized to have length 1. This approach has been shown to
produce a well-clustered representation. While in spectral clustering, one then performs kmeans in this representation, here one simply gives the representation as input to a classifier.
A serious problem with these methods is that one must diagonalize a matrix the size of the
set of labeled and unlabeled data. Other methods of implementing the cluster assumption
such as transductive SVMs [14] also suffer from computational efficiency issues. A second
drawback is that these kernels are better suited to a transductive setting (where one is given
both the unlabeled and test points in advance) rather than a semi-supervising setting. In
order to estimate the kernel for a sequence not present during training, one is forced to
solve a difficult regression problem [10]. In the next two sections we will describe two
simple methods to implement the cluster assumption that do not suffer from these issues.
4
The neighborhood mismatch kernel
In most current learning applications for prediction of protein properties, such as prediction of three-state secondary structure, neural nets are trained on probabilistic profiles of
a sequence window ? a matrix of position-specific emission and gap probabilities ?
learned from a PSI-BLAST alignment rather than an encoding of the sequence itself. In this
way, each input sequence is represented probabilistically by its ?neighborhood? in a large
sequence database, where PSI-BLAST neighbors are sequences that are closely related
through evolution. We wish to transfer the notion of profiles to our mismatch representation of protein sequences.
We use a standard sequence similarity measure like BLAST or PSI-BLAST to define a
neighborhood Nbd(x) for each input sequence x as the set of sequences x0 with similarity
score to x below a fixed E-value threshold, together with x itself. Now given a fixed original
feature representation, we represent x by the average of the feature vectors for members of
P
1
its neighborhood: ?nbd (x) =
?orig (x0 ). The neighborhood kernel
0
|Nbd(x)| x ?Nbd(x)
is then defined by:
Knbd (x, y) =
1
|Nbd(x)||Nbd(y)|
X
Korig (x0 , y 0 ).
x0 ?Nbd(x),y 0 ?Nbd(y)
We will see in the experimental results that this simple neighborhood-averaging technique,
used in a semi-supervised setting with the mismatch kernel, dramatically improves classification performance.
To see how the neighborhood approach fits with the cluster assumption, consider a set of
points in feature space that form a ?cluster? or dense region of the data set, and consider
the region R formed by the union of the convex hulls of the neighborhood point sets. If the
dissimilarity measure is a true distance, the neighborhood averaged vector ?nbd (x) stays
inside the convex hull of the vectors in its neighborhood, all the neighborhood vectors stay
within region R. In general, the cluster contracts inside R under the averaging operation.
Thus, under the new representation, different clusters can become better separated from
each other.
5
The bagged mismatch kernel
There exist a number of clustering techniques that are much more efficient than the methods
mentioned in Section 3. For example, the classical k-means algorithm is O(rkmd), where
m is the number of data points, d is their dimensionality, and r is the number of iterations
required. Empirically, this running time grows sublinearly with k, m and d. In practice, it
is computationally efficient even to run k-means multiple times, which can be useful since
k-means can converge to local minima. We therefore consider the following method:
1. Run k-means n times, giving p = 1, . . . , n cluster assignments cp (xi ) for each i.
2. Build a bagged-clustering representation based upon the fraction of times that xi
and xj are in the same cluster:
P
p [cp (xi ) = cp (xj )]
Kbag (xi , xj ) =
.
(1)
n
3. Take the product between the original and bagged kernel:
K(xi , xj ) = Korig (xi , xj ) ? Kbag (xi , xj )
Because k-means gives different solutions on each run, step (1) will give different results;
for other clustering algorithms one could sub-sample the data instead. Step (2) is a valid
kernel because it is the inner product in an nk-dimensional space ?(xi ) = h[cp (xi ) = q] :
p = 1, . . . , n, q = 1, . . . , ki, and products of kernels as in step (3) are also valid kernels.
The intuition behind the approach is that the original kernel is rescaled by the ?probability?
that two points are in the same cluster, hence encoding the cluster assumption. To estimate
the kernel on a test sequence x in a semi-supervised setting, one can assign x to the nearest
cluster in each of the bagged runs to compute Kbag (x, xi ). We apply the bagged kernel
method with Korig as the mismatch kernel and Kbag built using PSI-BLAST.
6
Experiments
We measure the recognition performance of cluster kernels methods by testing their ability
to classify protein domains into superfamilies in the Structural Classification of Proteins
(SCOP) [15]. We use the same 54 target families and the same test and training set splits
as in the remote homology experiments in [7]. The sequences are 7329 SCOP domains
obtained from version 1.59 of the database after purging with astral.stanford.edu so that no
pair of sequences share more than 95% identity. Compared to [7], we reduce the number
of available labeled training patterns by roughly a third. Data set sequences that were
neither in the training nor test sets for experiments from [7] are included as unlabeled
data. All methods are evaluated using the receiver operating characteristic (ROC) score
and the ROC-50, which is the ROC score computed only up to the first 50 false positives.
More details concerning the experimental setup can be found at http://www1.cs.
columbia.edu/compbio/svm-pairwise.
In all experiments, we use an SVM classifier with a small soft margin parameter, set as
in [7] . The SVM computations are performed using the freely available Spider Matlab machine learning package available at http://www.kyb.tuebingen.mpg.de/
bs/people/spider. More information concerning the experiments, including data
and source code scripts, can be found at http://www.kyb.tuebingen.mpg.de/
bs/people/weston/semiprot.
Semi-supervised setting. Our first experiment shows that the neighborhood mismatch
kernel makes better use of unlabeled data than the baseline method of ?pulling in homologs? prior to training the SVM classifier, that is, simply finding close homologs of
1
0.8
50
Number of families
Mismatch(5,1)+homologs ROC?50
Using PSI?BLAST for homologs & neighborhoods
60
40
0.6
30
0.4
20
10
0
0
mismatch(5,1)
mismatch(5,1)+homologs
neighborhood mismatch(5,1)
0.2
0.4
0.6
ROC?50
0.8
0.2
1
0
0
0.2
0.4
0.6
0.8
1
Neighborhood Mismatch(5,1) ROC?50
Figure 1: Comparison of protein representations and classifiers using unlabeled data.
The mismatch kernel is used to represent proteins, with close homologs being pulled in
from the unlabeled set with PSI-BLAST. Building a neighborhood with the neighborhood
mismatch kernel improves over the baseline of pulling in homologs.
mismatch kernel
mismatch kernel + homologs
neighborhood mismatch kernel
BLAST
ROC-50
ROC
0.416
0.870
0.480
0.900
0.639
0.922
PSI-BLAST
ROC-50
ROC
0.416
0.870
0.550
0.910
0.699
0.923
Table 1: Mean ROC-50 and ROC scores over 54 target families for semi-supervised experiments, using BLAST and PSI-BLAST.
the positive training examples in the unlabeled set and adding them to the positive training
set for the SVM. Homologs come from the unlabeled set (not the test set), and ?neighbors? for the neighborhood kernel come from the training plus unlabeled data. We compare the methods using the mismatch kernel representation with k = 5 and m = 1, as
used in [6]. Homologs are chosen via PSI-BLAST as having a pairwise score (E-value)
with any of the positive training samples less than 0.05, the default parameter setting [1].
The neighborhood mismatch kernel uses the same threshold to choose neighborhoods.
For the neighborhood
kernel, we normalize before and after the averaging operation via
p
Kij ? Kij / Kii Kjj . The results are given in Figure 1 and Table 1. The former plots
the number of families achieving a given ROC-50 score, and a strongly performing method
thus produces a curve close to the top right of the plot. A signed rank test shows that the
neighborhood mismatch kernel yields significant improvement over adding homologs (pvalue 3.9e-05). Note that the PSI-BLAST scores in these experiments are built using the
whole database of 7329 sequences (that is, test sequences in a given experiment are also
available to the PSI-BLAST algorithm), so these results are slightly optimistic. However,
the comparison of methods in a truly inductive setting using BLAST shows the same improvement of the neighborhood mismatch kernel over adding homologs (p-value 8.4e-05).
Adding homologs to the (much larger) negative training set in addition to pulling in the positive homologs gives poorer performance than only adding the positive homologs (results
not shown).
Transductive setting. In the following experiments, we consider a transductive setting,
in which the test points are given to the methods in advance as unlabeled data, giving
slightly improved results over the last section. Although this setting is unrealistic for a
real protein classification system, it more easily enables comparison with random walk
and spectral clustering kernels, which do not easily work in another setting. In Figure 2
(left), we again show the mismatch kernel compared with pulling in homologs and the
neighborhood kernel. This time we also compare with the bagged mismatch kernel using
PSI?BLAST kernel, varying methods
Mismatch kernel, PSI?BLAST distance
50
50
Number of families
60
Number of families
60
40
40
30
20
10
0
0
30
20
mismatch(5,1)
mismatch(5,1)+homologs
neighborhood mismatch(5,1)
bagged mismatch(5,1) k=100
0.2
0.4
0.6
ROC?50
0.8
10
1
0
0
PSI?BLAST
+ close homologs
spectral cluster, k=100
random walk, t=2
0.2
0.4
0.6
ROC?50
0.8
1
Figure 2: Comparison of protein representations and classifiers using unlabeled data
in a transductive setting. Neighborhood and bagged mismatch kernels outperform pulling
in close homologs (left) and equal or outperform previous semi-supervised methods (right).
mismatch kernel
mismatch kernel + homologs
neighborhood mismatch kernel
bagged mismatch kernel (k = 100)
bagged mismatch kernel (k = 400)
ROC-50
0.416
0.625
0.704
0.719
0.671
ROC
0.875
0.924
0.917
0.943
0.935
PSI-BLAST kernel
PSI-BLAST+homologs kernel
spectral clustering kernel
random walk kernel
transductive SVM
ROC-50
0.533
0.585
0.581
0.691
0.637
ROC
0.866
0.873
0.861
0.915
0.874
Table 2: Mean ROC-50 and ROC scores over 54 target families for transductive experiments.
bagged k-means with k = 100 and n = 100 runs, which gave the best results. We found
the method quite insensitive to k. The result for k = 400 is also given in Table 2.
We then compare these methods to using random walk and spectral clustering kernels.
Both methods do not work well for the mismatch kernel (see online supplement), perhaps
because the feature vectors are so orthogonal. However, for a PSI-BLAST representation
via empirical kernel map, the random walk outperforms pulling in homologs. We take the
empirical map with ?(x) = hexp(??d(x1 , x)), . . . , exp(??(d(xm , x))i, where d(x, y)
1
are PSI-BLAST E-values and ? = 1000
, which improves over a linear map. We report
results for the best parameter choices, t = 2 for the random walk and k = 200 for spectral
clustering. We found the latter quite brittle with respect to the parameter choice; results
for other parameters can be found on the supplemental web site. For pulling in close
homologs, we take the empirical kernel map only for points in the training set and the
chosen close homologs. Finally, we also run transductive SVMs. The results are given
in Table 2 and Figure 2 (right). A signed rank test (with adjusted p-value cut-off of 0.05)
finds no significant difference between the neighborhood kernel, the bagged kernel (k =
100), and the random walk kernel in this transductive setting. Thus the new techniques are
comparable with random walk, but are feasible to calculate on full scale problems.
7
Discussion
Two of the most important issues in protein classication are representation of sequences
and handling unlabeled data. Two developments in recent kernel methods research, string
kernels and cluster kernels, address these issues separately. We have described two kernels
? the neighborhood mismatch kernel and the bagged mismatch kernel ? that combine
both approaches and yield state-of-the-art performance in protein classification. Practical
use of semi-supervised protein classification techniques requires computational efficiency.
Many cluster kernels require diagonalization of the full labeled plus unlabeled data kernel
matrix. The neighborhood and bagged kernel approaches, used with an efficient string kernel, are fast and scalable cluster kernels for sequence data. Moreover, these techniques can
be applied to any problem with a meaningful local similarity measure or distance function.
Future work will deal with additional challenges of protein classification: addressing the
full multi-class problem, which potentially involves thousands of classes; handling very
small classes with few homologs; and dealing with missing classes, for which no labeled
examples exist.
Acknowledgments
We would like to thank Eleazar Eskin for discussions that contributed to the neighborhood
kernel and Olivier Chapelle and Navin Lal for their help with this work.
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1,646 | 2,497 | Unsupervised Color Decomposition of
Histologically Stained Tissue Samples
A. Rabinovich
Department of Computer Science
University of California, San Diego
[email protected]
C. A. Laris
Q3DM, Inc.
[email protected]
S. Agarwal
Department of Computer Science
University of California, San Diego
[email protected]
J.H. Price
Department of Bioengineering
University of California, San Diego
[email protected]
S. Belongie
Department of Computer Science
University of California, San Diego
[email protected]
Abstract
Accurate spectral decomposition is essential for the analysis and
diagnosis of histologically stained tissue sections. In this paper we
present the first automated system for performing this decomposition. We compare the performance of our system with ground
truth data and report favorable results.
1
Introduction
Potentially cancerous tissue samples are analyzed by staining them with a combination of two or more dyes. We consider the problem of recovering the amount of
dye absorbed for each of the stains from a stack of hyperspectral images of the tissue sample. Since the exact spectral profile of the dyes varies from one experiment
to the next and is not available to the pathologist, the problem is an instance of
blind source separation. The problem is of special interest to clinical and research
pathologists as the amount of dye absorbed by the sample is used to determine a
quantitative estimate of the amount of cancerous cells present in the tissue.
The current state of the art solution requires an expert to hand click representative
points in the tissue image to indicate ?pure? dye spectra. This procedure requires
human intervention and hence is time consuming and error prone.
In this paper we present the first system capable of performing this color decomposition in a fully automated manner. We also describe a novel procedure for acquiring
the ground truth data and quantifying the performance of our system.
The organization of the paper is as follows. In section 2 we address the problem of
image alignment in hyperspectral stacks. Section 3 presents the problem of color
unmixing and proposes two unsupervised techniques as solutions. Data acquisition
and experiments are discussed in Section 4. Section 5 summarizes the study and
provides concluding remarks.
2
Multi-Spectral Alignment
Color unmixing is a challenging problem in itself, but it is complicated further by the
practicalities of multispectral imaging: the component spectral images are usually
misaligned, due to chromatic aberration and shifting of the stage. If the images
comprising the spectral stack are out of alignment by as little as half a pixel, the
estimated stain percentages at a given pixel can be altered drastically. This can
result in large inaccuracies in the resulting cancer diagnosis.
Empirically, we have observed that the misalignments between images in the spectral stack can be modeled as small affine transforms, i.e. global translation, stretching, and rotation. Letting I(x) and J(x) denote two images, where x = (x, y)> ,
this assumption is expressed as
J(Ax + d) = I(x)
where A is the 2 ? 2 matrix of affine coefficients
a11 a12
A=
a21 a22
and d is a 2D translation vector.
In the case of unimodal images, the iterative method of Shi and Tomasi [12] has
been very successful for the estimation of differential (subpixel) affine transforms,
e.g. from frame to frame in a video sequence. However, feeding cross-modal images
directly to this algorithm is ineffective since they violatie the brightness constancy
assumption [3]. We have observed, however, that the high spatial-frequency structures, e.g. edges and lines, tend to be consistent throughout the stack. This forms
the basis of our alignment technique. We use the Shi-Tomasi algorithm on a bandpass filtered version of the images in the stack. To perform the filtering we apply a
Laplacian of Gaussian (LoG) kernel [8], expressed as
h(x) = ?2 e?kxk
2
/2? 2
where ? controls the width of the filter, to each image. The LoG kernel acts as a
bandpass filter, suppressing constant regions and smooth shading, admitting edges
and lines, and suppressing high frequency noise. We empirically determined the
optimal parameters for the filtering to be ?=0.5 and a window size of 10 pixels. With
this step used as preprocessing, Shi and Tomasi?s algorithm is able to register this
pair of images. An example of a synthesized color image composed of a 3D spectral
stack is shown with and without this registration step in Figure 1; the blurring
caused by misalignment and the subsequent sharpening resulting from registration
is evident.
3
Color Unmixing
Once the registration problem is adequately addressed, we can proceed with the
determination of stain concentrations. The problem in its full generality is an
instance of the blind source separation problem. Given a spectral stack of ns images
Figure 1: Synthesized color image representation of the same tissue core from a 10
dimensional spectral stack (a) with and (b) without differential affine registration.
obtained from imaging a tissue sample stained with nd dyes, with ns > nd , we wish
to recover the staining due to each individual dye.
In an ideal world, the spectral profile of each dye would be exactly aligned with one
of the spectral bands, and the absorptions measured therein would directly yield
the stain concentrations. Realistically, however, the spectral profile of the dyes
overlap and extend over several spectral bands, and the goal of recovering the nd
components representing the dye percentages requires more careful analysis.
The problem of unmixing the dyes can be formulated as a matrix factorization
problem:
X = AS
(1)
Here X is an ns ? l column matrix, where l is the number of pixels and the entry
Xij is the brightness of the ith pixel in the image in to the j th spectral band. The
matrix A is an ns ? nd matrix where each column of the matrix corresponds to the
one of the dyes used in staining the tissue. S is a ns ? l matrix, with the entry Sij
indicating the contribution of the ith dye to the j th pixel.
The current state of the art solution for this problem in the field of automated
pathology is Color Deconvolution [11], which yields acceptable results, but requires
manual interaction in the form of mouse clicks on seed colors for the dyes. This is
an example of a supervised technique. However, given the data matrix X, there are
a number of ways in which Equation (1) can be solved in a completely automatic
manner without any human intervention. The three main classes of such methods are Principal Component Analysis (PCA), Non-negative Matrix Factorization
(NMF) and Independent Component Analysis (ICA).
In this work we assume that staining is an additive process. Once a part of a tissue
has been stained with a dye, addition of another stain can only increase the staining.
The additivity of the stains combined with the physical constraint that each dye
color will have a non-negative response in each frequency band implies that A and
B are forced to be restricted to the class of non-negative matrices.
Methods based on PCA work by enforcing orthogonality constraints on the columns
of A and are not well suited for recovering the factorization AS. PCA depends
heavily on cancellation effects, i.e. a balancing of positive and negative terms as
occurs with Gibbs? phenomenon in Fourier series. This will result in PCA returning
A and S with negative entries which have no physical basis. In the following we
shall investigate the use of algorithms based on NMF and ICA.
3.1
Non-negative Matrix Factorization
NMF is in principle well suited to the task of color unmixing, as it finds a factorization of X into A and S such that
[A, S] = argminkX ? ASk
(2)
A,S
subject to
Aij ? 0,
Sij ? 0
The above problem is underconstrained; it has a scale ambiguity. Given a solution
[A, S] of the above problem, [?A, S/?] for ? 6= 0 is also a solution to this problem.
We solve this problem by constraining each column of A to have unit norm. This
does not affect the final solution, since only the proportion of each stain is needed
in the final analysis; the exact intensity of the constituent stain is not important.
The choice of the norm k ? k decides the particular algorithm used for performing
the minimization. We have implemented an iterative algorithm for recovering the
non-negative factorization of a matrix due to Seung & Lee [7]. We use the L2 norm
as a measure of the error.
3.2
Independent Component Analysis
An alternate approach to matrix factorization is Independent Component Analysis
(ICA)[4]. While Non-negative Matrix Factorization is based on enforcing a nonnegativity constraint, it says nothing about the image formation process. ICA
is based on a generative view of the data, where the data is assumed to be a
result of superpositioning a number of stochastically independent processes. In the
case of histological staining, this corresponds to assuming that each dye stains the
tissue independently of all the other dyes. The rows of the matrix S represent the
individual stochastic processes and the columns of A code their interactions.
We implemented the Joint Approximate Diagonalization of Eigenmatrices (JADE)
algorithm to recover the independent components of X [2]. This algorithm calculates the ICA decomposition of X by calculating the eigenvalue decomposition of
the cumulant tensor of the data. The eigenvalues of cumulant tensor are vectors
corresponding to the independent components of the mixture.
4
4.1
Experimental Results
Sample Preparation and Data Acqusition
The histologically stained tissues used in this study were derived from human biopsies. The tissues were fixed in Bouin?s solution, and embedded in paraffin. Dewaxed
tissue sections were exposed to polyclonal antibodies (PAB) generated against synthetic peptides and confirmed to be specific for the proteins of interest. The sections
were stained using a diaminobenzidine (DAB)-based detection method employing
the Envision-Plus-Horseradish Peroxidase (HRP) system using an automated staining technique [5, 6]. The DAB immunohistochemistry stain used for the tissue
samples shown here covers the majority of the visible range of the color spectra
under the transmission of white light.
Great care must be taken in the acquisition of color images since the extraction of
spectral information is highly dependent on the quality of the raw data. Hyperspectral imaging has been shown to be the best means of doing so.
A spectral image stack can be acquired using a number of different approaches. We
use a setup based on a set of fixed bandpass filters. The filters are placed in the
optical path of the light in front of the light source or camera and transmit only
the desired wavelength bands.
In the following experiments the images were acquired on a scanning cytometer
[9, 10, 1] with a 20x, 0.5 NA Fluor Nikon objective lens using a set of 10 equally
spaced band pass filters ranging from 413 nm to 663 nm. The dynamic range of each
of the spectral bands was maximized by controlling the gain and the exposure of
the imaging system. This is required to ensure an accurate hyperspectral-to-RGB
reconstruction for result visualization. It is important to note that the gain and
exposure coefficients were inverted prior to the unmixing as they have no bearing
on the staining process.
In order to quantitatively evaluate the decomposition provided by NMF and ICA,
we prepared a set of ground truth data using the following procedure. Using a set of
four tissue samples, we first applied the DAB stain and captured the hyperspectral
image stack. We then added the hematoxylin stain and acquired a second image
stack. The second stack serves as the input to our algorithm and the resulting
decomposition, which estimates the DAB staining, is compared with the first stack,
which serves as the ground truth.
We now experimentally evaluate the use of NMF and ICA for the color decomposition problem. While reconstruction error represents a simple quantitative measure,
it does not provide a standard for judging how accurately the estimated components represent the dye concentrations. We quantify the performance by comparing
the ground truth single-stained image to the corresponding automatically extracted
component of the doubly-stained tissue sample. Figure 2 reports the performance
of the two algorithms. The error measure used is
P
(Ii ? I?i )2
(3)
error = 100 ? i P 2
i Ii
where the sum is over all pixels, and Ii and I?i denote the ground truth and the
estimate, respectively. Figure 3 shows the results of applying NMF and ICA to an
image patch.
set1
set2
set3
set4
overall
NMF
18.15
18.79
4.47
5.04
12.65
ICA
12.81
14.99
19.42
18.12
18.75
Figure 2: This table shows the percent error for the two unmixing algorithms across the four image sets. The four sets of images are available at
http://vision.ucsd.edu/.
5
Discussion
The above experiments indicate that both NMF and ICA are capable of performing
color decomposition of tissue samples stained with multiple histological dyes. However, there remain a number of sources of error, both during image acquisition as
well as in the decomposition stage. These include errors due to imperfect focussing
(a) DAB only
(b) DAB & Hematoxylin
(c) NMF
(d) ICA
Figure 3: Color unmixing using Non-negative Matrix factorization and Independent
Component Analysis. Figure (a) shows a segment of the tissue stained using DAB,
(b) shows the same tissue segment with DAB and Hematoxylin staining. The image
in figure (b) serves as input to the two unmixing algorithms, the output of which is
shown in (c) and (d). Figure (c) shows the DAB stain estimate produced by NMF
and (d) shows the DAB staining estimated by ICA
in the various spectral bands and distortion in the acquired images which cannot
be accounted for by optical flow based alignment methods such as Shi & Tomasi?s
algorithm. The principal source of discrepancy between the decomposition and the
ground truth images, however, is caused by the chemical interaction between the
various dyes used for staining. Measurement error due to dye interaction can be as
high as 15%[13]. In this light, both ICA and NMF provide good results, and we
expect that improvements in the image acquisition and registration procedure will
result in systems capable of delivering performance close to the theoretical optimum.
In conclusion, we have addressed the problem of image registration for the planes in
a hyperspectral stack for spectral information extraction and we proposed the use of
two unsupervised algorithms, Non-negative Matrix Factorization and Independent
Component Analysis, for extracting the contributions of various histological stains
to the overall spectral composition throughout the tissue sample. We demonstrate
the performance of these algorithms by comparing them with ground truth data.
We intend to address errors in the image acquisition and registration to further
reduce the decomposition error in future work.
References
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[4] A. Hyv?
arinen, J. Karhunen, and E. Oja. Independent Component Analysis. John
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Salvesen, D.E. Bredesen, R.E. Rosenthal, G. Fiskum, and J.C. Reed. Release of
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[8] David Marr. Vision: A Computational Investigation into the Human Representation
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[9] J. H. Price. Scanning cytometry for cell monolayers. PhD thesis, University of California, San Diego, 1990.
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[11] Arnout C. Ruifrok and Dennis A. Johnston. Quantification of histochemical staining
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1,647 | 2,498 | Approximability of Probability Distributions
Alina Beygelzimer?
IBM T. J. Watson Research Center
Hawthorne, NY 10532
[email protected]
Irina Rish
IBM T. J. Watson Research Center
Hawthorne, NY 10532
[email protected]
Abstract
We consider the question of how well a given distribution can be approximated with probabilistic graphical models. We introduce a new parameter, effective treewidth, that captures the degree of approximability as
a tradeoff between the accuracy and the complexity of approximation.
We present a simple approach to analyzing achievable tradeoffs that exploits the threshold behavior of monotone graph properties, and provide
experimental results that support the approach.
1
Introduction
One of the major concerns in probabilistic reasoning using graphical models, such as
Bayesian networks, is the computational complexity of inference. In general, probabilistic
inference is NP-hard and a typical approach to handling this complexity is to use an approximate inference algorithm that trades accuracy for efficiency. This leads to the following
question: How can we distinguish between distributions that are easy to approximate and
those that are hard? More generally, how can we characterize the inherent degree of distribution?s complexity, i.e. its approximability?
These questions also arise in the context of learning probabilistic graphical models from
data. Note that traditional model selection criteria, such as BIC/MDL, aim at fitting the
data well and minimizing the representation complexity of the learned model (i.e., the
total number of parameters). However, as demonstrated in [2], such criteria are unable to
capture the inference complexity: two models that have similar representation complexity
and fit data equally well may have quite different graph structures, making one model
exponentially slower for inference than the other. Thus, our goal is to develop learning
algorithms that can find good trade-offs between accuracy of a model and its inference
complexity.
Commonly used exact inference algorithms, such as the junction tree algorithm [12], or
closely related variable-elimination techniques [6], essentially triangulate the graph, and
their complexity is exponential in the size of largest clique induced during triangulation
(parameter known as treewidth). Generally, it can be shown that (in some precise sense)
any scheme for belief updating based on local calculations must contain a hidden triangulation [10]. Thus the treewidth arises as a natural measure of inference complexity in
graphical models.
?
The work was done while the author was at the Department of Computer Science, University of
Rochester.
Intuitively, a probability distribution is approximable, or easy, if it is close to a distribution
represented by an efficient, low-treewidth graphical model. We use the Kullback-Leibler
divergence dKL as a measure of closeness. 1 . The following example explains our intuition
behind approximable vs. nonapproximable distributions.
Motivating Example Consider the parity function on n binary random variables
{X1 , . . . , Xn }, and let our target distribution P be the uniform distribution on the values to
which it assigns 1 (i.e., on n-bit strings with an odd number of 1s). It is easy to see that any
approximation Q that decomposes over a network whose moralized graph misses at least
one edge, is precisely as inaccurate as the one that assumes all variables to be independent
(i.e., has no edges).
This follows from the fact that the probability
distribution induced on any proper subset of the
1
variables is uniform, and thus for any subset
{Xi1 , . . . , Xik } of k < n variables, P (Xi1 |
Xi2 , . . . , Xik ) = P (Xi1 ), uniform
on {0, 1}. It
P
is then readily seen that
P (x) log Q(x) =
x
P
Qn
2?(n?1) x:P (x)>0 log i=1 Q(xi | xi1 , . . . , xir ) =
treewidth
n?2
Qn
log i=1 Q(xi ) = log 2?n = ?n, 2 and dKL (P, Q) =
0
n?1
(empty graph)
(clique)
?H(P )+n = 1 since H(P ) = n?1. Thus, unless we
can afford the complexity of the complete graph, there
is absolutely no sense (i.e., absolutely no gain in accuracy and a potentially exponential loss
of efficiency) in using a model more complex than the empty graph (i.e., n isolated nodes
with no edges). This intuitively captures what we mean by a nonapproximable distribution.
dKL
On the other hand, one can easily construct a distribution with large weak dependencies
such that representing this distribution exactly requires a network with large treewidth;
however, if we are willing to sacrifice just a bit of accuracy, we get a very simple model.
For example, consider a distribution P ({X1 , . . . , Xn }) in which variables X1 , . . . , Xn?1
are independent and uniformly distributed; if all X1 , . . . , Xn?1 are true, Xn is true with
probability 1 (and false with probability 0); otherwise Xn is true with probability 1/2 (regardless of the values of X1 , . . . , Xn?1 ). The network yielding zero KL-divergence is the
n-node clique (after moralization). Tolerating KL-divergence 2?(n?1) (i.e., exponentially
vanishing with n) allows us to use an exponentially more efficient model for P (namely,
the empty graph).
The following questions naturally arise: If we tolerate a certain inaccuracy, what is the best
inference complexity we can hope to achieve? Or, what is the best achievable approximation accuracy given a constraint on the complexity (i.e., a bound on the treewidth)? The
tradeoff between the complexity and accuracy is monotonic; however, it may be far from
linear. The goal is to exploit these nonlinearities in choosing the best available tradeoff.
Our analysis of accuracy vs. complexity trade-offs is based on the results from random
graph theory which suggest that graph properties monotone in edge addition (e.g., such
as graph connectivity) appear rather suddenly: the transition from the property being very
unlikely to it being very likely occurs during a small change of the edge probability p
(density) in the random graph [7, 8].
This paper makes the following contributions. First, we show that both important properties of random graphical models, the property of ?being efficient? (i.e., having treewidth at
most some fixed integer k) and the property of ?being accurate? (i.e., being at distance at
most some ? from the target distribution), are monotone and demonstrate a threshold behavior, giving us two families of threshold curves parameterized by k and by ?, respectively.
Second, we introduce the notion of effective treewidth k(?), which denotes the smallest
1
Note that minimizing dKL from the empirical distribution (induced by a given set of samples)
also corresponds to maximizing the likelihood of observed data.
2
The second to last equality is due to the well-known fact that dKL (P, Q) is minimized by forcing
the conditional probabilities of Q to coincide with those computed from P .
achievable treewidth k given a constraint ? on KL-divergence (error) from the target (we
also introduce a notion of ?-achievable k(?) which requires at least ?-fraction of models
in a given set to achieve treewidth k and error ?). The effective treewidth captures the
approximability of the distribution, and is determined by relative position of the threshold curves, an inherent property of the target distribution. Finally, we provide an efficient
sampling-based approach that actually finds a model achieving k(?) with high probability. We estimate the threshold curves and, using their relative position, identify a class of
treewidth-bounded models such that the models in the class are still simple, yet this class
already contains (with high probability) a sufficiently good approximations to the target
distribution (otherwise, we suggest that the distribution is inherently hard to approximate).
2
Preliminaries and Related Work
Let P be a probability distribution on n discrete random variables X1 , X2 , . . . , Xn . A
Bayesian network exploits the independences among the Xi to provide a compact representation of P as a product of low-order conditional probability distributions. The independences are encoded by a directed acyclic graph (DAG) G with nodes corresponding to
X1 , X2 , . . . , Xn and edges representing direct dependencies. Each Xi is independent of
its non-descendants given its parents in the graph [12]. The dependencies are quantified
by associating each node Xi with a local conditional probability distribution PB (Xi | ?i ),
where ?i is the set of parents of Xi in G. TheQjoint probability distribution encoded by B
n
is given by the product PB (X1 , . . . , Xn ) = i=1 PB (Xi | ?i ). We say that a distribution P decomposes over a DAG G if there exist local conditional probability distributions
corresponding to G such that P can be written in such a form.
In general, exact probabilistic inference in Bayesian networks is NP-hard. For singlyconnected networks (i.e., networks with no undirected cycles), there is a linear time local
belief-propagation algorithm [12]. In order to use this algorithm in the presence of cycles, one typically constructs a junction tree of the network and runs the algorithm on this
tree [12]. Constructing a junction tree involves triangulating the graph, i.e., adding edges
so that every cycle of length greater than three has a chord (i.e., an edge between a pair
of non-adjacent nodes). Each triangulation corresponds to some order of eliminating variables when summing terms out during inference [6]. Exact inference can then be done in
time and space linear in the representation of clique marginals in the junction tree, which
is exponential in the size of the largest clique induced during triangulation. This number
(minus one) is known as the width of a given triangulation. The minimum width over all
possible triangulations is called the treewidth of the graph. The triangulation procedure
is defined for undirected graphs, so we must first make the network undirected while preserving the set of independence assumptions; this can be done by moralizing the network,
i.e., connecting (?marrying?) the parents of every node by a clique and then dropping the
direction of all edges.
Given a set of independent samples from P , the general goal is to learn a model (a Bayesian
network) of this distribution that involves dependencies only on limited subsets of the variables. Restricting the size of dependencies controls both overfitting and the complexity of
inference in the resulting model. The samples are in the form of tuples hx1 , . . . , xn i each
corresponding to a particular assignment hX1 = x1 , . . . , Xn = xn i. Given a target distribution P (X) and an approximation Q(X), the information divergence (or Kullback-Leibler
P
P (x)
, where x ranges
distance) between P and Q is defined as dKL (P, Q) = x P (x) log Q(x)
over all possible assignments to the variables in X (See [5].) Notice that dKL (P, Q) is not
necessarily symmetric.
A natural way of controlling the complexity of the learned model is to limit ourselves to a
class of treewidth-bounded networks. Let Dk denote the class of distributions decomposable on graphs with treewidth at most k (0 ? k < n), with D1 corresponding to the set of
tree-decomposable distributions. The distribution within Dk minimizing the information
divergence from the target distribution P is called the projection of P onto Dk . Again, if
P is the empirical distribution, then this is also the distribution within Dk maximizing the
likelihood of observing the data.
Learning bounded-treewidth models Chow and Liu [4] showed how to find a projection
onto the set of tree-decomposable distributions. For a fixed tree T , the projection of P
onto the set of T -decomposable distributions is uniquely given by the distribution in which
the conditional probabilities along the edges of T coincide with those computed from P .
Hence the tree yielding the closest projection is simply given by any maximum weight
spanning tree, where the edge weight is the mutual information between the corresponding
variables. Notice that candidate spanning trees can be compared without any knowledge of
P beyond that given by pairwise statistics. The tree can be efficiently found using any of
the well known algorithms. The additive decomposition of dKL used in the proof, can be
easily extended to ?wider? networks. Fix a network structure G, and let Q be a distribution
decomposable over G. Then
n
dKL (P, Q) =
X
x
P (x) log
X X
P (x)
=?
P (xi , ?i ) log Q(xi | ?i ) ? H(P ),
Q(x)
i=1 x ,?
i
i
where ?i ranges over all possible values of ?i . If P is the empirical distribution induced
by the given sample of size N (i.e., defined by frequencies of events in the sample), then
the first term can be shown to be ?LL(Q)/N .3 Thus minimizing dKL (P, Q) is equivalent
to maximizing the log likelihood LL(Q).
Standard arguments (see, for example, [12]) show that the first term is maximized by forcing all conditional probabilities Q(xi | ?i ) to coincide with those computed from P . If
P is the empirical distribution, this means forcing the parameters to be the corresponding
relative frequencies in the sample. Hence if G is fixed, the projection onto the set of Gdecomposable distributions is uniquely defined, and we will identify G with this projection
(ignoring some notational abuse). It remains, of course, to find G that is the closest to P
among all DAGs in some treewidth-bounded class Dk . As observed by H?offgen [9], the
problem readily reduces to the minimum-weight hypertree problem. The reverse reduction
is not known, so the NP-hardness of the hypertree problem does not imply the hardness of
the learning problem. Srebro [13] showed that a similar undirected decomposition holds
for bounded treewidth Markov networks (probabilistic models that use undirected graphs
to represent dependencies). He showed that the learning problem is equivalent to finding a minimum-weight undirected hypertree, and so is NP-hard. It is important to note
that Srebro [13] considered approximation in the context of density estimation rather than
model selection, thus the choice of k is directly driven by the size of the sample space;
the only rationale for limiting the class of hypothesis distributions is to prevent overfitting.
With an infinite amount of data, they would learn a clique, since adding edges would always decrease the divergence. Our goal, on the other hand, is to find the most appropriate
treewidth-bounded class onto which to project the distribution.
Threshold behavior of random graphs We use the model of random directed acyclic
graphs (DAGs) defined by Barak and Erd?os [1]. Consider the probability space G(n, p)
of random undirected graphs on n nodes with edge probability p (i.e., every pair of nodes
is connected with probability p, independently of every other pair). Let Gn,p stand for a
random graph from this probability space. We will also occasionally use Gn,m to denote
a graph
? ? chosen randomly from among all graphs with n nodes and m edges. When p =
m/ n2 , the two models are practically identical. A random DAG in the Barak-Erd?os model
is obtained from Gn,p by orienting the edges according to the ordering of vertices, i.e., all
edges are directed from higher to lower indexed vertices.
3
Since the true distribution P is given only by the sample, we let P also denote the empirical
distribution induced by the sample, ignoring some abuse of notation.
A graph property P is naturally associated with the set of graphs having P. A property
is monotone increasing if it is preserved under edge addition: If a graph G satisfies the
property, then every graph on the same set of nodes containing G as a subgraph must satisfy it as well. It is easy to see (and intuitively clear) that if P is a monotone increasing
property then the probability that Gn,p satisfies P is a non-decreasing function of p. A
monotone decreasing property is defined similarly. For example, the property of having
treewidth at most some fixed integer k is monotone decreasing: adding edges can only increase the treewidth. The theory of random graphs was initiated by Erd?os and R?enyi [7],
and one of the main observations they made was that many natural monotone properties
appear rather suddenly, i.e., as we increase p, there is a sharp transition from a property
being very unlikely to it being very likely. Friedgut [8] proved that every monotone graph
property of undirected graphs has such a threshold behavior. Random DAGs (corresponding to random partially ordered sets) have received less attention then random undirected
graphs, partially because of the additional structure that prevents the completely independent choice of edges. Nonetheless, many properties of random DAGs were also shown to
have threshold functions. (See, for example, [3] and references therein.) However, we are
not aware of any general result for random DAGs analogous to that of Friedgut [8].
3
Formalization
First we introduce two properties of networks essential for the rest of the paper.
Accuracy Recall that the information divergence of a given DAG G from the targetPdistribution
P is given by dKL (P, G) = W (G) ? H(P ), where W (G) =
n P
? i=1 xi ,?i P (xi , ?i ) log P (xi | ?i ). (In our case, P is the empirical distribution induced by the given sample S of size N . As mentioned before, W (G) = ?LL(G)/N ? 0.)
Fix a distance parameter ? > 0, and consider the property P? of n-node DAGs of having
W (G) ? ?. Notice that P? is monotone increasing: Adding edges to a graph can only
bring the graph closer to the target distribution, since any distribution decomposable on the
original graph is also decomposable on the augmented one. Thus if G is a subgraph of G0 ,
then W (G) ? ? only if W (G0 ) ? ?.
Complexity Fix an integer k, and consider the property of n-node DAGs of having
treewidth of their moralized graph at most k. Call this property Pk and note that it is
a structural property of a DAG, which does not depend on the target distribution and its
projection onto the DAG. It is also a monotone decreasing property, since if a graph has
treewidth at most k, then certainly any of its subgraphs does.
Recall that we identify each graph with the projection of the target distribution onto the
graph. We call a pair (k, ?) achievable for a distribution P , if there exists a distribution
Q decomposable on a graph with treewidth at most k such that dKL (P, Q) ? ?. The
effective treewidth of P , with respect to a given ?, is defined as the smallest k(?) such that
the pair (k, ?) is achievable, i.e., if all distributions at distance at most ? from P are not
decomposable on graphs with treewidth less than k(?). This formulation gives the level of
inevitable complexity (i.e., treewidth) k, given the desired level of accuracy ?. We will also
be interested in average-case analogs of these definitions. Fix ? > 0. We will say that a
pair (k, ?) is ?-achievable for P if at least an ?-fraction of all DAGs in Dk certify that (k, ?)
is achievable. Thus we not only care about the existence of an approximation with given ?
and k, but also in the number of such approximations.
4
Main Idea
Consider, for each treewidth bound k, the curve given by ?k (p) = Pr[width(Gn,p ) ? k],
and let pk be such that ?k (pk ) = 1/2 + ?, where 0 < ? < 21 is some fixed constant.
Similarly, for ? > 0, define the curve ?? (p) = Pr[W (Gn,p ) ? ?], and let p? be the critical
value of p given by ?? (p? ) = 1/2.
For reasons that will become clear in a moment, our goal will be to find, for each feasible
treewidth k, the value of ? such that p? = pk . To find each pk , the algorithm will simply
do a binary search on the interval (0, 1): For the current value of edge probability p, the
algorithm estimates ?k (Gn,p ) by random sampling and branches according to the estimate.
The search is continued until p gets sufficiently close to satisfying ?k (Gn,p ) = 1/2 +
?. To estimate ?k (Gn,p ) within an additive error ? with probability at least 1 ? ?, the
algorithm samples m = ln(2/?)
independent copies of Gn,p , and outputs the average value
2?2
of the 0/1 random variable indicating whether the treewidth of the sampled DAG is at
most k. The analysis is just a straightforward application of the Chernoff Bound. Note
that the values related to treewidth are independent of the target distribution and can be
precomputed offline. To find ? = ?(k) for a given value of k, the algorithm computes
the values of W (Gn,pk ) for the m sampled random DAGs in G(n, pk ), orders them and
chooses the median. Each pair (k, ?) gives a point on the threshold curve. We know
that at least a (1/2 + ?)-fraction of the DAGs in G(n, pk ) satisfy Pk . On the other hand,
at least half of them satisfy P? , and thus at least an ?-fraction satisfies both. Moreover,
there is a very simple probabilistic algorithm for finding a model realizing the tradeoff:
We just need to sample O(1/?) DAGs in G(n, pk ) and choose the closest one. Clearly
we are overcounting, since the same DAGs may contribute to both probabilities; however
not absurdly, since intuitively the graphs in G(n, pk ) with small treewidth will not fit the
distribution better than the ones with larger treewidth.
A small example should help make
the goals clear. A distribution is
called k-wise independent if any
subset of k variables is mutually independent (however, there may exist dependencies on larger subsets).
Figure 1 shows the curves for a 3wise independent distribution on 8
random variables. We can hardly
expect graphs with treewidth at most
2 to do well on this distribution,
since all triples are independent, and
their marginals do not reveal any
Figure 1: Threshold curves for a 3-wise indepen- higher-order structure; as we will
dent distribution on 8 random variables (using a see this is indeed the case. The xaxis in Figure 1 corresponds to the
construction from [11]).
number of edges m, the y-axis denotes the probability that Gn,m satisfies the property corresponding to a given curve. The
monotone decreasing curves correspond to the properties Pk for k = {1, . . . , 6} (from left
to right respectively). For k = 7, the curve is just ?m (Pk ) = 1. The monotone increasing curves correspond to the property of having dKL at most ?. The leftmost curve is for
? = 0.07, and it decreases by 0.01 as we go from left to right; the smaller ?, the higher
the quality of approximation, thus the smaller the probability of attaining it. The empty
graph (treewidth 0) had divergence 0.073. As m increases, the probability of having small
treewidth decreases, while the probability of getting close to the target increases. (Since n
is small, we computed the divergence exactly.) As the random graph evolves, we want to
capture the moment when the first probability is still high, while the second is already high.
As expected, graphs with treewidth at most 2 are as inaccurate as the empty graph since all
triples are independent. Given the desired level of closeness ?, we want to find the smallest
treewidth k such that the corresponding curves meet above some cut-off probability. For
example, to get within dKL at most 0.7, we may suggest, say, projecting onto graphs with
treewidth 4 (cutting at 0.4). The cut-off value determines the efficiency of finding a model
with such k and ? (see discussion above).
$1$
$0.8$
$0.6$
$0.4$
$0.2$
$0$
$5$
$10$
$15$
$20$
number of edges
$25$
$30$
Estimating dKL Fix a bounded-treewidth DAG G. Let the target distribution be the
empirical distribution P induced by a given sample. Recall that dKL (P, G) decomposes
into sum of conditional entropies induced by G (minus the entropy of P ). H?offgen [9]
showed how to estimate these conditional entropies with any fixed additive precision ?
using polynomially many samples. More precisely, he showed that a sample of size m =
k+1
m(?, ?) = O(( n? )2 log2 n? log n ? ) suffices to obtain good estimations of all induced
conditional entropies with probability at least 1 ? ?, which in turn suffices to estimate
dKL (P, G) with the additive precision ?.
Estimating Treewidth We, of course, will not attempt to compute the treewidth of the
randomly generated graphs exactly. The problem is NP-hard 4 . In practice, people often use
heuristics (based, for example, on eliminating vertices in the order of maximum cardinality,
minimum degree, or minimum separating vertex set). There are no theoretical guarantees in
general, but heuristics tend to perform reasonably well: used in combination with various
lower bound techniques, they can often pin down the treewidth to a small range, or even
identify it exactly 5 . We stress that the values related to treewidth are independent of the
target distribution and can be precomputed.
5
Experimental Results
We tested the approach presented in the paper on distributions ancestrally sampled from
real-life medical networks commonly used for benchmarking. The experiments support
the following conclusions: the properties capturing the complexity and accuracy of a model
indeed demonstrate a threshold behavior, which can be exploited in determining the best
tradeoff for the given distribution; the simple approach based on generating random graphs
and using them to approximate the thresholds is indeed capable of capturing the effective
width of a distribution. Due to page limit, we discuss an application of the method to a
single network known as A LARM (originating from anesthesia monitoring).
24
22
20
accuracy (W)
The network has 37 nodes, 46 directed edges,
19 additional undirected edges induced by
moralization; the treewidth is 4. A sample of
size N = 104 was generated using ancestral
sampling, inducing the empirical distribution
with support on 5570 unique variable assignments. The entropy of the empirical distribution P was 9.6 (maximum possible entropy
for a 5570-point distribution is 12.4). Figure 2 shows the curve illustrating the (estimated) tradeoffs available for P . For each
treewidth bound k, the curves gives an estimate of the best achievable value of W =
dKL ? H(P ). (Recall that LL = ?N ? W .)
18
16
14
12
10
0
5
10
15
20
25
complexity (treewidth)
30
35
Figure 2: Tradeoff curve for A LARM
The estimate is based on generating 400 random DAGs with 37 nodes and m edges, for
every possible m. Several points on the curve are worthy of note. The upper-left point
(0, 23.4) corresponds to the model that assumes all 37 variables to be independent. On the
other extreme, the lower-right point (36, 0) corresponds to the clique on 37 nodes, which of
course can model P perfectly, but with exponential complexity. The closer the area under
the curve to zero, the easier the distribution (in the sense discussed in this paper). Here we
see that the highest gain in accuracy from allowing the model to be more complex occurs
up to treewidth 4, less so 5 and 6; by further increasing the treewidth we do not gain much
in accuracy. We succeed in reconstructing the width in the sense that the distribution was
4
If k is fixed, the problem of determining whether a graph has treewidth k has a linear time
algorithm. As typical, the bound contains a large hidden constant with k in the exponent, making
the algorithm hardly applicable in practice. There is a number of constant-factor approximations
with an exponential dependence on k, and a polynomial-time O(log k)-factor approximation. No
polynomial-time constant-factor approximation is known.
5
Although one can construct graphs for which they produce solutions that are arbitrarily far from
optimal.
1
probability of satisfying the property
0.8
0.6
0.4
0.2
0
0
20
40
60
80
100
120
number of edges
Figure 3: Threshold curves for A LARM
simulated from a treewidth-4 model.6 Such tradeoff curves are similar to commonly used
ROC (Receiver Operating Characteristic) curves; the techniques for finding the cutoff value
in ROC curves can be used here as well. Instead of plotting the best achievable distance,
we can plot the best distance achievable by at least an ?-fraction of models in the class,
parameterizing the tradeoff curve by ?. Figure 3 shows the threshold curves. The axes
have the same meaning as in Figure 1. Varying sample size and the number of randomly
generated DAGs does not change the behavior of the curves in any meaningful way; not
surprisingly, increasing these parameters results in smoother curves.
References
[1] A. Barak and P. Erd?os. On the maximal number of strongly independent vertices in a random
acyclic directed graph. SIAM J. Algebraic and Discrete Methods, 5:508?514, 1984.
[2] A. Beygelzimer and I. Rish. Inference complexity as a model-selection criterion for learning
bayesian networks. In Proceedings of the Eighth International Conference on Principles of
Knowledge Representation and Reasoning (KR2002), Toulouse, France, 2002.
[3] B. Bollob?as and G. Brightwell. The structure of random graph orders. SIAM J. Discrete Mathematics, 10(2):318?335, 1997.
[4] C. Chow and C. Liu. Approximating discrete probability distributions with dependence trees.
IEEE Trans. on Inf. Theory, 14:462?467, 1968.
[5] T. Cover and J. Thomas. Elements of information theory. John Wiley & Sons Inc., New York,
1991. A Wiley-Interscience Publication.
[6] R. Dechter. Bucket elimination: A unifying framework for probabilistic reasoning. In M. I.
Jordan (Ed.), Learning in Graphical Models, Kluwer Academic Press, 1998.
[7] P. Erd?os and A. R?enyi. On the evolution of random graphs. Bull. Inst. Internat. Statist., 38:343?
347, 1961.
[8] E. Friedgut and G. Kalai. Every monotone graph property has a sharp threshold. Proceedings
of the American Mathematical Society, 124(10):2993?3002, 1996.
[9] K. H?offgen. Learning and robust learning of product distributions. In Proceedings of the 6th
Annual Workshop on Computational Learning Theory, pages 77?83, 1993.
[10] F. V. Jensen and F. Jensen. Optimal junction trees. In Proc. Tenth Conference on Uncertainty
and AI (UAI), 1994.
[11] J. Naor and M. Naor. Small-bias probability spaces: Efficient constructions and applications.
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[12] J. Pearl. Probabilistic Reasoning in Intelligent Systems: Networks of Plausible Inference. Morgan Kaufmann Publishers, 1988.
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17th Conference on Uncertainty in AI (UAI), pages 504?511, 2001.
6
Note, however, that it does not imply that the empirical distribution itself decomposes on a
treewidth-4 model. The simplest example of this is when the true distribution is uniform.
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1,648 | 2,499 | Attractive People: Assembling Loose-Limbed
Models using Non-parametric Belief Propagation
Leonid Sigal
Department of Computer Science
Brown University
Providence, RI 02912
[email protected]
Michael Isard
Microsoft Research Silicon Valley
Mountain View, CA 94043
[email protected]
Benjamin H. Sigelman
Department of Computer Science
Brown University
Providence, RI 02912
[email protected]
Michael J. Black
Department of Computer Science
Brown University
Providence, RI 02912
[email protected]
Abstract
The detection and pose estimation of people in images and video is made
challenging by the variability of human appearance, the complexity of
natural scenes, and the high dimensionality of articulated body models. To cope with these problems we represent the 3D human body as a
graphical model in which the relationships between the body parts are
represented by conditional probability distributions. We formulate the
pose estimation problem as one of probabilistic inference over a graphical model where the random variables correspond to the individual limb
parameters (position and orientation). Because the limbs are described
by 6-dimensional vectors encoding pose in 3-space, discretization is impractical and the random variables in our model must be continuousvalued. To approximate belief propagation in such a graph we exploit a
recently introduced generalization of the particle filter. This framework
facilitates the automatic initialization of the body-model from low level
cues and is robust to occlusion of body parts and scene clutter.
1
Introduction
Recent approaches to person detection and tracking exploit articulated body models
in which the body is viewed as a kinematic tree in 2D [14], 2.5D [16, 23], or 3D
[2, 5, 6, 19, 21] leading to a parametric state-space representation of roughly 25?35 dimensions. The high dimensionality of the resulting state-space has motivated the development of specialized stochastic search algorithms that either exploit the highly redundant
dynamics of typical human motions [19], or use hierarchical sampling schemes to exploit
the tree-structured nature of the model [5, 15]. These schemes have been effective for
tracking people wearing increasingly complex clothing in increasingly complex cluttered
backgrounds [21]. There are however a number of important shortcomings of these ap-
proaches. Hierarchical body models lead to ?top-down? search algorithms that make it
difficult to incorporate ?bottom-up? information about salient body parts available from
special-purpose detectors (e.g. face or limb detectors). As a result, few, if any, of the above
methods deal with the problem of automatic initialization of the body model. Furthermore,
the difficulty of incorporating bottom-up information means that the algorithms are brittle;
that is, when they lose track of the body, they have no way to recover. Finally, the fully
coupled kinematic model results in a computationally challenging search problem because
the search space cannot be naturally decomposed.
To address these problems, we propose a ?loose-limbed? body model in which the limbs
are not rigidly connected but are rather ?attracted? to each other (hence the title ?Attractive People?). Instead of representing the body as a single 33-dimensional kinematic tree,
each limb is treated quasi-independently with soft constraints between the position and
orientation of adjacent parts. The model resembles a Push Puppet toy which has elastic
connections between the limbs (Figure 1a).
This type of model is not new for finding or tracking articulated objects and dates back at
least to Fischler and Elschlager?s pictorial structures [9]. Variations on this type of model
have been recently applied by Burl et al. [1], Felzenszwalb and Huttenlocher [8], Coughlan
and Ferreira [3] and Ioffe and Forsyth [11, 17]. The main benefits are that it supports
inference algorithms where the computational cost is linear rather than exponential in the
number of body parts, it allows elegant treatment of occlusion, and it permits automatic
initialization based on individually unreliable low-level body-part detectors [25].
The work described here, like the previous work above, exploits this notion of flexible
?spring?-like constraints [8] defined over individually modeled body parts [11, 17, 23],
though we extend the approach to locate the parts in 3-space rather than the 2-dimensional
image plane. The body is treated as a graphical model [13], where each node in the graph
corresponds to an independently parameterized body part. The spatial constraints between
body parts are defined as directed edges in the graph. Each edge has an associated conditional distribution that models the probabilistic relationship between the parts. Each node
in the graph also has a corresponding image likelihood function that models the probability
of observing various image measurements conditioned on the position and orientation of
the part. Person detection (or tracking) then exploits belief propagation [24] to estimate
the belief distribution over the parameters which takes into account the constraints and the
observations.
This graphical inference problem is carried out using a recently proposed method that allows the parameters of the individual parts to be modeled using continuous-valued random
variables rather than the discrete variables used in previous approaches. This is vital in
our problem setting, since the discretization used in [8] is impractical once the body is
modeled in 3-space. Similar versions of the algorithm were independently introduced by
Sudderth et al. [22] under the name of Non-parametric Belief Propagation (NBP) and by
Isard [12] as the PAMPAS algorithm. We adopt the framework of Isard while making use
of the Gibbs sampler introduced by Sudderth et al. The algorithm extends the flexibility of
particle filters to the problem of belief propagation and, in our context, allows the model
to cope with general constraints between limbs, permits realistic appearance models, and
provides resilience to clutter.
We develop the loose-limbed model in detail, formulate the constraints between limbs using
mixture models, and outline the inference method. Using images from calibrated cameras
we illustrate the inference of 3D human pose with belief propagation. We simulate noisy,
bottom-up, feature detectors for the limbs and show how the inference method can resolve
ambiguities and cope with clutter. While our focus here is on static detection and pose
estimation, the body model can be extended in time to include temporal constraints on the
limb motion; we save tracking for future work.
9
7
5
0
8
6
1
3
?3,4
?1,2
2 ?2,1 ?4,3 4
(a)
(b)
(c)
Figure 1: (a) Toy Push Puppet with elastic joints. (b) Graphical model for a person. Nodes represent
limbs and arrows represent conditional dependencies between limbs. (c) Parameterization of part i.
2
A self-assembling body model
The body is represented by a graphical model in which each graph node corresponds to a
body part (upper leg, torso, etc.). Each part has an associated configuration vector defining
the part?s position and orientation in 3-space. Placing each part in a global coordinate frame
enables the part detectors to operate independently while the full body is assembled by inference over the graphical model. Edges in the graphical model correspond to spatial and
angular relationships between adjacent body parts, as illustrated in Figure 1b. As is standard for graphical models we assume the variables in a node are conditionally independent
of those in non-neighboring nodes given the values of the node?s neighbors1 .
Each part/limb is modeled by a tapered cylinder having 5 fixed (person specific) and 6 estimated parameters. The fixed parameters ?i = (li , wip , wid , opi , odi ) correspond respectively
to the part length, width at the proximal and distal ends and the offset of the proximal and
distal joints along the axis of the limb as shown in Figure 1c. The estimated parameters
XTi = (xTi , ?Ti ) represent the configuration of the part i in a global coordinate frame
where xi ? R3 and ?i ? SO(3) are the 3D position of the proximal joint and the angular
orientation of the part respectively. The rotations are represented by unit quaternions.
Each directed edge between parts i and j has an associated conditional distribution
?ij (Xi , Xj ) that encodes the compatibility between pairs of part configurations; that is,
it models the probability of configuration Xj of part j conditioned on the Xi of part i.
For notational convenience we define an ordering on body parts going from the torso out
towards the extremities and refer to conditionals that go along this ordering as ?forward?
conditionals. Conversely, the conditionals that go from the extremities towards the torso
are referred to as ?backward? conditionals. These intuitively correspond to kinematic and
inverse-kinematic constraints respectively.
Conditional distributions were constructed by hand to capture the physical constraints of
the joints and limbs of the human body. A typical range of motion information for the
various joints is approximated by the model. In general, these conditionals can, and should,
be learned from motion capture data.
Because we have chosen the local coordinate frame to be centered at the proximal joint of
1
Self-occlusion and self-intersection violate this assumption. These can be modeled by adding
additional edges in the graph between the possibly occluding or inter-penetrating parts. In the limit
this would lead to quadratic as opposed to linear computation time in the number of parts.
(a)
(b)
Figure 2: (a) For the forwards conditional the location of part i tightly constrains the proximal joint
of part j (light dots) while the position of the distal joint (dark dots) lies along an arc around the
principal axis of rotation, approximated by a Gaussian mixture. (b) For the backwards conditional
part i constrains the distal joint of part j (dark dots), so the proximal joint position (light dots) lies in
a non-Gaussian volume again approximated using a mixture distribution.
a part, the forward and backward conditionals are not symmetric. In both directions the
probability of Xj , conditioned on Xi , is non-Gaussian and it is approximated by a mixture
of Mij Gaussians (typically 5-7 in the experiments here): ?ij (Xi , Xj ) =
?0 N (Xj ; ?ij , ?ij ) + (1 ? ?0 )
PMij
m=1 ?ijm N (Xj ; Fijm (Xi ), Gijm (Xi ))
(1)
where ?0 is a fixed outlier probability, ?ij and ?ij are the mean and covariance of the
Gaussian outlier process, and Fijm (.) and Gijm (.) are functions computing the mean and
covariance matrix respectively of the m-th Gaussian mixture component. These functions
allow the mean and variance of the mixture components to be function of the limb pose X i .
PMij
?ijm is the relative weight of an individual component and m=1
?ijm = 1.
Figure 2a and b illustrate the forward and backward conditionals respectively. For the
forward case, we examine the distribution of calf configurations conditioned on the thigh.
To illustrate the conditional distribution we sample from it and plot the endpoints of the
sampled limb configurations. In the forward direction the conditional distribution over x j
(the position of the proximal joint of part j) is well approximated by a Gaussian so each
mixture component has the same mean and covariance for xj . This can be seen in the tight
clustering of the light dots which lie almost on top of each other. The probability of the
lower leg angle is restricted to a range of legal motions conditioned on the upper leg. This
distribution over rotations is modeled by giving each mixture component a different mean
rotation, ?j , spaced evenly around the principal axis of the joint. This angular uncertainty
is illustrated by the dark dots.
For the backward conditional we show the distribution over torso configurations conditioned on the thigh. In this direction the conditional predicting xj (e.g. torso position) is
more complicated. The location of xi restricts xj to lie near a hemisphere, and the orientation ?i and principal axis of rotation further restrict xj to a strip on that hemisphere
which can be seen in Figure 2b (light dots). Thus each mixture component in (1) is spaced
evenly in ?j and xj to represent this range of uncertainty. The combined uncertainty in
torso location and orientation can be seen in the distribution of the dark dots representing
the distal torso joint.
Image Likelihoods
The inference algorithm outlined in the next section combines the body model described
above with a probabilistic image likelihood model. In particular, we define ? i (Xi ) to be the
likelihood of observing the image measurements conditioned on the pose of limb i. Ideally
this model would be robust to partial occlusions, the variability of image statistics across
different input sequences, and variability among subjects. To that end, we combine a variety of cues including multi-scale edge and ridge filters as well as background subtraction
information. Following related work [18], the likelihoods are estimated independently for
each image view by projecting the 3D model of a limb into the corresponding image projection plane. These likelihoods are then combined across views, assuming independence,
and are weighted by the observability of the limb in a given view (more weight is given to
views in which the limb lies parallel to the image projection plane). For more information
on the formulation of the image likelihoods see [20].
3
Non-parametric Belief Propagation
Having defined the model it remains to specify an algorithm which will perform inference
and estimate a belief distribution for each of the body parts. If it were feasible to discretize
the Xi we could apply traditional belief propagation or a specialized inference algorithm
as set out in [8]. However, the 6-dimensional configuration vector compels the use of
continuous-valued random variables, and so we adopt the algorithm introduced in [12, 22]
for just such types of model. It is a generalization of particle filtering [7] which allows
inference over arbitrary graphs rather than just a chain. This generalization is achieved by
treating the particle set which is propagated in a standard particle filter as an approximation
to the ?message? used in the belief propagation algorithm, and replacing the conditional
distribution from the previous time step by a product of incoming message sets.
A message mij from node i ? j is written
mij (Xj ) =
Z
?ij (Xi , Xj )?i (Xi )
Y
mkj (Xi )dXi ,
(2)
k?Ai \j
where Ai is the set of neighbors of node i and ?i (Xi ) is the local likelihood associated
with node i. The message mij (Xj ) can be approximated by importance sampling N 0 =
(N ? 1)/Mij times from a proposal function f (Xi ), and then doing importance correction.
(See [22] for an alternative algorithm that uses more general potential functions than the
conditional distributions used here.) As discussed in [12] the samples may be stratified into
groups with different proposal functions f (?), so some samples come from the product of all
incoming messages Ai into the node, some from Ai \j (i.e. Ai excluding j) and some from
a static importance function Q(Xi ) ? we use a limb proposal distribution based on local
image measurements. For reasons of space we present only a simplified algorithm to update
message mij in Figure 3 which does not include the stratification but the full algorithm can
be found in [12]. We use the Gibbs sampler described in [22] to form message products of
D > 2 messages.
The algorithm must sample, evaluate, and take products over Gaussian distributions defined
? SO(3) and represented in terms of unit quaternions. We adopt the approximation given
in [4] for dealing with rotational distributions by treating the quaternions locally linearly in
R4 ? this approximation is only valid for kernels with small rotational covariance and can
in principle suffer from singularities if product distributions are widely distributed about
the sphere, but we have not encountered problems in practice.
1. Draw N 0 = (N ? 1)/Mij samples from the proposal function:
0
s?nij ? f (Xi ), n0 ? [1, N 0 ].
2. Compute importance corrections for n0 ? [1, N 0 ]:
0 Q
0
?i (?
snij ) k?Ai \j mki (?
snij )
n0
.
?ij =
0
f (?
snij )
3. Store normalized weights and mixture components for n0 ? [1, N 0 ], m ? [1, Mij ]:
(a) n = (n0 ? 1)Mij + m
0
(b) ?nij = Fijm (?
snij )
0
(c) ?nij = Gijm (?
snij )
0
? n ?ijm
n
(d) ?ij
= (1 ? ?0 ) PijN 0
k=1
k
?ij
.
N
0
N
0
4. Assign outlier component: ?ij
= ?0 , ?N
ij = ?ij , ?ij = ?ij
Figure 3: The simplified PAMPAS non-parametric belief propagation algorithm.
4
Experiments
We illustrate the approach by recovering 3D body pose given weak bottom-up information and clutter. The development of bottom-up part detectors is beyond the scope of this
paper. Here we exploit a realistic simulation of such detectors in which: 1) the limbs are
only detected 50% of the time ? the remaining samples are clutter; 2) the limb detectors
are non-specific in that they cannot distinguish the left and right sides of the body or the
upper from lower limbs (they do, however, distinguish between legs and arms) ? the result is that only a small fraction of bottom-up samples fall in the right place with the right
interpretation; 3) the detectors are noisy and do not detect the limb position and orientation accurately; 4) no correct initialization samples are generated for the torso, simulating
detector failure or occlusion.
Figure 4 shows results for two time instants in a video sequence taken from three calibrated
cameras. After 10 iterations of belief propagation, the algorithm has discarded the samples
which originated in clutter and has correctly assigned the limbs. The figure shows the
initialization and the final distribution over limb poses which is computed by sampling
from the belief distribution. Note that the torso is well localized even though there was no
bottom-up detector for it.
5
Conclusion
We present a new body model and inference method that supports the goals of automatically locating and tracking an articulated body in three dimensions. We show that a ?looselimbed? model with continuous-valued parameters can effectively represent a person?s location and pose, and that inference over such a model can be tractably performed using
belief propagation over particle sets. Moreover, we demonstrate robust location of the person starting from imperfect initialization using a simulated body-part detector. The detector
is assumed to generate both false positive initializations and false negatives; i.e. failures to
detect some body parts altogether.
It is straightforward to extend the graphical model across time to implement a person
(a)
(b)
Figure 4: Inferring attractive people: Two experiments are shown; (a) and (b) show results
for two different time instants in a walking cycle. Each experiment used three calibrated
camera views. Left: Initialization samples drawn from noisy simulated part detectors. Part
detectors are assumed to have high failure rate, generating 50% of the samples far away
from any true body part. They are also non-specific; e.g. the left thigh samples are equally
distributed over left and right thigh and calf. The torso is assumed to be undetectable.
Right: Belief after 10 iterations of PAMPAS. We use 100 particles to model the messages
between the nodes, and show 20 samples from the belief distribution, as well as the average
of the top 10 percent of the belief samples as the ?best? pose estimate. For brevity, (b) only
shows the best pose from a single view.
tracker. There are several advantages of this approach compared with traditional particle
filtering: the complexity of the search task is linear rather than exponential in the number
of body parts; bottom-up initialization information can be incorporated in every frame; and
forward-backward smoothing, either over a time-window or an entire sequence, is straightforward.
In future work we intend to build automatic body-part detectors. Constructing reliable
detectors using only low-level information (static appearance) is a challenging problem
but we have the advantage of being robust to imperfect detection as noted above. We
also intend to learn the conditional distributions between parts from a database of motion
capture data. Together these advances should allow reliable use of the presented body
model in the person tracking framework.
Acknowledgments. We thank Jianbo Shi for providing the image data. LS, BHS, and
MJB were supported in part by the DARPA HumanID Project (ONR N000140110886).
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1,649 | 25 | 422
COMPUTING MOTION USING RESISTIVE NETWORKS
Christof Koch, Jin Luo, Carver Mead
California Institute of Technology, 216-76, Pasadena, Ca. 91125
James Hutchinson
Jet Propulsion Laboratory, California Institute of Technology
Pasadena, Ca. 91125
INTRODUCTION
To us, and to other biological organisms, vision seems effortless. We open
our eyes and we "see" the world in all its color, brightness, and movement.
Yet, we have great difficulties when trying to endow our machines with similar
abilities. In this paper we shall describe recent developments in the theory of
early vision which lead from the formulation of the motion problem as an illposed one to its solution by minimizing certain "cost" functions. These cost
or energy functions can be mapped onto simple analog and digital resistive
networks. Thus, we shall see how the optical flow can be computed by injecting
currents into resistive networks and recording the resulting stationary voltage
distribution at each node. These networks can be implemented in cMOS VLSI
circuits and represent plausible candidates for biological vision systems.
APERTURE PROBLEM AND SMOOTHNESS ASSUMPTION
In this study, we use intensity-based schemes for recovering motion. Let us
derive an equation relating the change in image brightness to the motion of the
image (see l ). Let us assume that the brightness of the image is constant over
time: dI(~,y,t)/dt = o. On the basis of the chain rule of differentiation, this
transforms into
81 d~
8~ dt
81 dy
81
+ 8y dt + at = Izu + Iyv + It = 'V I? v + It =
0,
(1)
where we define the velocity v as (u,v) = (d:1)/dt,dy/dt). Because we assume
that we can compute these spatial and temporal image gradients, we are now
left with a single linear equation in two unknowns, u and v, the two components
of the velocity vector (aperture problem). Any measuring system with a finite
aperture, whether biological or artificial, can only sense the velocity component
perpendicular to the edge or along the spatial gradient (-It! 1'V I I). The
component of motion perpendicular to the gradient cannot, in principle, be
registered. The problem remains unchanged even if we measure these velocity
components at many points throughout the image.
How can this problem be made well-posed, that is, having a unique solution depending continuously on the data? One form of "regularizing" ill-posed
@ American Institute of Physics 1988
423
problems is to restrict the class of admissible solutions by imposing appropriate
constraints 2 ? Applying this method to motion, we shall argue that in general objects are smooth-except at isolated discontinuities-undergoing smooth
movements. Thus, in general, neighboring points in the world will have similar
velocities and the projected velocity field should reflect this fact. We therefore
impose on the velocity field the constraint that it should be the smoothest as well
as satisfying the data. As measure of smoothness we choose, the square of the
velocity field gradient. The final velocity field (u, v) is the one that minimizes
JJ [(::)' + (::)' + (~:)' + (:~)'] dz dy
A
+
+
+
.,
+
(2)
+
II:,
+
_I~
+
+
+
!al
(b)
Fig. 1. ( a) The location of the horizontal (lfj) and vertical (Iij) line processes
relative to the motion field nngrid. (b) The hybrid resistive network, computing
the optical flow in the presence of discontinuities. The conductances T c - ij connecting both grids depend on the brightness gradient, as do the conductances
gij and gij connecting each node with the battery. For clarity, only two such
elements are shown. The battery Eij depends on both the temporal and the
spatial gradient and is zero if no brightness change occurs. The ~ (resp. y) component of the velocity is given by the voltage in the top (resp. bottom) network.
Binary switches, which make or break the resistive connections between nodes,
424
implement motion discontinuities. These switches could be under the control of
distributed digital processors. Analog cMOS implementations are also feasible 3 ?
The first term implements the constraint that the final solution should follow
as closely as possible the measured data whereas the second term imposes the
smoothness constraint on the solution. The degree to which one or the other
terms are minimized is governed by the parameter).. If the data is very accurate, it should be "expensive" to violate the first term and), will be small.
If, conversely, the data is unreliable (low signal-to-noise), much more emphasis
will be placed on the smoothness term. Horn and Schunck1 first formulated this
variational approach to the motion problem.
The energy E( u, v) is quadratic in the unknown u and v. It then follows
from standard calculus of variation that the associated Euler-Lagrange equations
will be linear in u and v:
I~u
+ IzIyv I z I 1I u + I:v -
). \721.?
). \7 2 v
+ IzIt
+ Iylt
= 0
= O.
(3)
We now have two linear equations at every point and our problem is therefore
completely determined.
ANALOG RESISTIVE NETWORKS
Let us assume that we are formulating eqs. (2) and (3) on a discrete 2-D
grid, such as the one shown in fig. 1a. Equation (3) then transforms into
I~ijuij
+ IzijI1Iijvij - ). (UHlj + Uij+l Izijlyijuij + I:ijvij - ). (VHlj + Vij+l -
+ Ui-lj + Uij-l) + IZijltij
4Vij + Vi-lj + Vij-l) + Iyijltij
4Uij
= 0
= 0
(4)
where we replaced the Laplacian with its 5 point approximation on a rectangular
grid. We shall now show that this set of linear equations can be solved naturally
using a particular simple resistive network. Let us apply Kirchhoff's current law
to the nodne i, j in the top layer of the resistive network shown in fig. lb. We
then have the following update equation:
du??
C d;' = T (Ui+lj
+ Uij+l
+ gij (Eij -
- 4Uij
Uij)
+ Ui-lj + Uij-l)
+ Tc-ij( Vij
(5)
- Uij).
where Vij is the voltage at node i, j in the bottom network. Once dUij / dt = 0
and dVij/dt = 0, this equation is seen to be identical with eq. (4), if we identify
425
Tc-ij ~ -IzijIyij
+ IJlij)
gij ~ Iyii (Izii + Iyij)
gij
~ Izij (Izij
(6)
-It
Eij~----
Izii
+ Iyij
(a)
(b)
(c)
(d)
~
(e)
(f)
Fig. 2. Motion sequence using synthetic data. (a) and (b) Two images of
three high contrast squares on a homogeneous background. (c) The initial
velocity data. The inside of both squares contain no data. (d) The final state
426
of the network after 240 iterations, corresponding to the smooth optical flow
field. (e) Optical flow in the presence of motion discontinuities (indicated by
solid lines). (f) Discontinuities are strongly encouraged to form at the location
of intensity edges 4 ? Both (e) and (f) show the state of the hybrid network after
six analog-digital cycles.
Once we set the batteries and the conductances to the values indicated in
eq. (6), the network will settle-following Kirchhoff's laws-into the state of
least power dissipation. The associated stationary voltages correspond to the
sought solution: uii is equivalent to the :c component and Vii to the y component
of the optical flow field.
We simulated the behavior of these networks by solving the above circuit
equations on parallel computers of the Hypercube family. As boundary conditions we copied the initial velocity data at the edge of the image into the nodes
lying directly adjacent but outside the image.
The sequences in figs. 2 and 3 illustrate the resulting optical flow for synthetic and natural images. As discussed by Horn and Schunck 1 , the smoothness
constraint leads to a qualitatively correct estimate of the velocity field. Thus,
one undifferentiated blob appears to move to the lower right and one blob to
the upper left. However, at the occluding edge where both squares overlap, the
smoothness assumption results in a spatial average of the two opposing velocities, and the estimated velocity is very small or zero. In parts of the image
where the brightness gradient is zero and thus no initial velocity data exists (for
instance, the interiors of the two squares), the velocity estimates are simply the
spatial average of the neighboring velocity estimates. These empty areas will
eventually fill in from the boundary, similar to the How of heat for a uniform
flat plate with "hot" boundaries.
MOTION DISCONTINUITIES
The smoothness assumption of Horn and Schunck 1 regularizes the aperture
problem and leads to the qualitatively correct velocity field inside moving objects. However, this approach fails to detect the locations at which the velocity
changes abruptly or discontinuously. Thus, it smoothes over the figure-ground
discontinuity or completely fails to detect the boundary between two objects
with differing velocities because the algorithm combines velocity information
across motion boundaries.
A quite successful strategy for dealing with discontinuities was proposed by
Geman and Geman 5 ? We shall not rigorously develop their approach, which is
based on Bayesian estimation theory (for details see 5 ,6). Suffice it to say that
a priori knowledge, for instance, that the velocity field should in general be
smooth, can be formulated in terms of a Markov Random Field model of the
image. Given such an image model, and given noisy data, we then estimate
the "best" flow field by some likelihood criterion. The one we will use here
427
is the maximum a posteriori estimate, although other criteria are possible and
have certain advantages 6 ? This can be shown to be equivalent to minimizing an
expression such as eq. (2).
In order to reconstruct images consisting of piecewise constant segments,
Geman and Geman5 further introduced the powerful idea of a line process 1.
For our purposes, we will assume that a line process can be in either one of two
states: "on" (1 = 1) or "off" (1 = 0). They are located on a regular lattice set
between the original pixel lattice (see fig. 1a), such that each pixel i,j has a
horizontallfi and a verticallij line process associated with it. If the appropriate
line process is turned on, the smoothness term between the two adjacent pixels
will be set to zero. In order to prevent line processes from forming everywhere
and, furthermore, in order to incorporate additional knowledge regarding discontinuities into the line processes, we must include an additional term Vc(l)
into the new energy function:
E( 'IL, v, lh., IV) =
L (Iz'ILii + IyVii + I )2 +
t
i.i
). L
(1 -It) [('lLi+1i - 'lLii)2 + (Vi+li - Vii)2] +
(7)
i.i
). L
(1 -Iii) [('lLij+l - 'lLii)2 + (vii+1 - Vij)2] + Vc(l).
i.i
Vc contains a number of different terms, penalizing or encouraging specific
configurations of line processes:
i.;
i.i
plus the corresponding expression for the vertical line process Iii (obtained by interchanging i with j and Iii with Ifi). The first term penalizes each introduction
of a line process, since the cost C c has to be "payed" every time a line process
is turned on. The second term prevents the formation of parallel lines: if either
lfi+l or Ifi+2 is turned on, this term will tend to prevent
from turning on.
The third term, CIVI , embodies the fact that in general, motion discontinuities
occur along extended contours and rarely intersect (for more details see 7 ).
We obtain the optical flow by minimizing the cost function in eq. (7) with
respect to both the velocity v and the line processes Ih. and IV. To find an
optimal solution to this non-quadratic minimization problem, we follow Koch
et a1. 7 and use a purely deterministic algorithm, based on solving Kirchhoff's
equations for a mixed analogi digital network (see also 8). Our algorithm exploits
the fact that for a fixed distribution of line processes, the energy function (7)
is quadratic. Thus, we first initialize the analog resistive network (see fig. 2b)
according to eq. (6) and with no line processes on. The network then converges to
It
428
the smoothest solution. Subsequently, we update the line processes by deciding
at each site of the line process lattice whether the overall energy can be lowered
by setting or breaking the line proceSSj that is, lfi will be turned on if E( u, v, lfi =
1, IV) < E( u, v, Ifi = 0, IV); otherwise, Ifj = o. Line processes are switched on
by breaking the appropriate resistive connection between the two neighboring
nodes. After the completion of one such analog-digital cycle, we reiterate and
compute-for the newly updated distribution of line processes-the smoothest
state of the analog network. Although there is no guarantee that the system will
converge to the global minimum, since we are using a gradient descent rule, it
seems to find next-to-optimal solutions in about 10 to 15 analog-digital cycles.
(8)
(c)
(e)
Figure 3. Optical flow of a moving person. (a) and (b) Two 128 by 128
pixel images captured by a video camera. The person in the foreground is
moving toward the right while the person in the background is stationary. The
noise in the lower part of the image is a camera artifact. (c) Zero-crossings
superimposed on the initial velocity data. (d) The smooth optical flow after 1000
iterations. Note that the noise in the lower part of both images is completely
smoothed away. (e) The final piecewise smooth optical flow. The velocity
field is subsampled to improve visibility. The evolution of the hybrid network is
shown after the 1. (a), 3. (b), 5. (c), 7. (d), 10. (e), and 13. (f) analog-digital
cycle in the right part of the figure.
The synthetic motion sequence in fig. 2 demonstrates the effect of the line
429
processes. The optical flow outside the discontinuities approximately delineating
the boundaries of the moving squares is zero, as it should be (fig. 2e). However,
where the two squares overlap the velocity gradient is high and multiple intersecting discontinuities exist. To restrict further the location of discontinuities, we
adopt a technique used by Gamble and Poggio4 to locate depth discontinuities
by requiring that depth discontinuities coincide with the location of intensity
edges. Our rationale behind this additional constraint is that with very few
exceptions, the physical processes and the geometry of the 3-dimensional scene
giving rise to the motion discontinuity will also give rise to an intensity edge. As
edges we use the zero-crossings of a Laplacian of a Gaussian convolved with the
original image9 ? We now add a new term VZ-Cii to our energy function E, such
that Vz -Cii is zero if Iii is off or if Iii is on and a zero-crossing exists between
locations i and j. If Iii = 1 in the absence of a zero-crossing, V Z - Cii is set
to 1000. This strategy effectively prevents motion discontinuities from forming
at locations where no zero-crossings exist, unless the data strongly suggest it.
Conversely, however, zero-crossings by themselves will not induce the formation
of discontinuities in the absence of motion gradients (figs. 2f and 3).
ANALOG VLSI NETWORKS
Even with the approximations and optimizations described above, the computations involved in this and similar early vision tasks require minutes to hours
on computers. It is fortunate then that modern integrated circuit technology
gives us a medium in which extremely complex, analog real-time implementations of these computational metaphors can be realized3 ?
We can achieve a very compact implementation of a resistive network using
an ordinary cMOS process, provided the transistors are run in the sub-threshold
range where their characterstics are ideal for implementing low-current analog
functions. The effect of a resistor is achieved by a circuit configuration, such as
the one shown in fig. 4, rather than by using the resistance of a special layer in
the process. The value of the resulting resistance can be controlled over three
orders of magnitude by setting the bias voltages on the upper and lower current
source transistors. The current-voltage curve saturates above about 100 mVj a
feature that can be used to advantage in many applications. When the voltage
gradients are small, we can treat the circuit just as if it were a linear resistor.
Resistances with an effective negative resistance value can easily be realized.
In two dimensions, the ideal configuration for a network implementation is
shown in fig. 4. Each point on the hexagonal grid is coupled to six equivalent
neighbors. Each node includes the resistor apparatus, and a set of sample-andhold circuits for setting the confidence and signal the input and output voltages.
Both the sample-and-hold circuits and the output buffer are addressed by a
scanning mechanism, so the stored variables can be refreshed or updated, and
the map of node voltages read out in real time.
430
~
I,
I
v,
VI
(a)
v
(b)
Figure 4. Circuit design for a resistive network for interpolating and smoothing
noisy and sparsely sampled depth measurements. (a) Circuit-consisting of 8
transistors-implementing a variable nonlinear resistance. (b) If the voltage
gradient is below 100 mV its approximates a linear resistance. The voltage VT
controls the maximum current and thus the slope of the resistance, which can
vary between 1 MO and 1 GO 3. This cMOS circuit contains 20 by 20 grid
points on a hexagonal lattice. The individual resistive elements with a variable
slope controlled by VT correspond to the term governing the smoothness, A. At
those locations where a depth measurement dij is present, the battery is set to
this value (Vin = d ij ) and the value of the conductance G is set to some fixed
value. If no depth data is present at that node, G is set to zero. The voltage
at each node corresponds to the discrete values of the smoothed surface fitted
through the noisy and sparse measurements 7 ?
A 48 by 48 silicon retina has been constructed that uses the hexagonal
network of fig. 4 as a model for the horizontal cell layer in the vertebrate
retinal 0 ? In this application, the input potentials were the outputs of logarithmic photoreceptors-implemented via phototransistors-and the potential
difference across the conductance T formed an excellent approximation to the
Laplacian operator.
DISCUSSION
We have demonstrated in this study that the introduction of binary motion
431
discontinuities into the algorithm of Horn and Schunck1 leads to a dramatically
improved performance ~f their method, in particular for the optical flow in the
presence of a number of moving non-rigid objects. Moreover, we have shown
that the appropriate computations map onto simple resistive networks. We are
now implementing these resistive networks into VLSI circuits, using subtheshold
cMOS technology. This approach is of general interest, because a great number
of problems in early vision can be formulated in terms of similar non-convex
energy functions that need to be minimized, such as binocular stereo, edge
detection, surface interpolation, structure from motion, etc. 2 ,6,8.
These networks share several features with biological neural networks. Specifically, they do not require a system-wide clock, they rely on many connections
between simple computational nodes, they converge rapidly-within several time
constants-and they are quite robust to hardware errors. Another interesting
feature is that our networks only consume very moderate amounts of powerj the
entire retina chip requires about 100 J.L W 10
Acknowledgments: An early version of this model was developed and implemented in collaboration with A. L. Yuille8 ? M. Avalos and A. Hsu wrote the
code for the Imaging Technology system and E. Staats for the NCUBE. C.K.
is supported by an ONR Research Young Investigator Award and by the Sloan
and the Powell Foundations. C.M. is supported by ONR and by the System
Development Foundation. A portion of this research was carried out at the Jet
Propulsion Laboratory and was sponsored by NSF grant No. EET-8714710, and
by NASA.
REFERENCES
1. Horn, B. K. P. and Schunck, B. G. Artif. Intell. 17,185-203 (1981).
2. Poggio, T., Torre, V. and Koch, C. Nature 317,314-319 (1985).
3. Mead, C. Analog VLSI and Neural Systems. Addison-Wesley: Reading,
MA (1988).
4. Gamble, E. and Poggio, T. Artif. Intell. Lab. Memo. No. 970, MIT, Cambridge MA (1987).
5. Geman, S. and Geman, D. IEEE Trans. PAMI 6, 721-741 (1984).
6. Marroquin, J., Mitter, S. and Poggio, T. J. Am. Stat. Assoc. 82, 76-89
(1987).
7. Koch, C., Marroquin, J. and Yuille, A. Proc. Natl. Acad. Sci. USA 83,
4263-4267 (1986).
8. Yuille, A. L. Artif. Intell. Lab. Memo. No. 987, MIT, Cambridge, MA
(1987).
9. Marr, D. and Hildreth, E. C. Proc. R. Soc. Lond. B 207, 187-217 (1980).
10. Sivilotti, M. A., Mahowald, M. A. and Mead, C. A. In: 1987 Stanford VLSI
Conference, ed. P. Losleben, pp. 295-312 (1987).
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1,650 | 250 | 598
Le Cun, Denker and Solla
Optimal Brain Damage
Yann Le Cun, John S. Denker and Sara A. Sol1a
AT&T Bell Laboratories, Holmdel, N. J. 07733
ABSTRACT
We have used information-theoretic ideas to derive a class of practical and nearly optimal schemes for adapting the size of a neural
network. By removing unimportant weights from a network, several improvements can be expected: better generalization, fewer
training examples required, and improved speed of learning and/or
classification. The basic idea is to use second-derivative information to make a tradeoff between network complexity and training
set error. Experiments confirm the usefulness of the methods on a
real-world application.
1
INTRODUCTION
Most successful applications of neural network learning to real-world problems have
been achieved using highly structured networks of rather large size [for example
(Waibel, 1989; Le Cun et al., 1990a)]. As applications become more complex, the
networks will presumably become even larger and more structured. Design tools
and techniques for comparing different architectures and minimizing the network
size will be needed. More importantly, as the number of parameters in the systems
increases, overfitting problems may arise, with devastating effects on the generalization performance. We introduce a new technique called Optimal Brain Damage
(OBD) for reducing the size of a learning network by selectively deleting weights.
We show that OBD can be used both as an automatic network minimization procedure and as an interactive tool to suggest better architectures.
The basic idea of OBD is that it is possible to take a perfectly reasonable network,
delete half (or more) of the weights and wind up with a network that works just as
well, or better. It can be applied in situations where a complicated problem must
Optimal Brain Damage
be solved, and the system must make optimal use of a limited amount of training
data. It is known from theory (Denker et al., 1987; Baum and Haussler, 1989; Solla
et al., 1990) and experience (Le Cun, 1989) that, for a fixed amount of training
data, networks with too many weights do not generalize well. On the other hand.
networks with too few weights will not have enough power to represent the data
accurately. The best generalization is obtained by trading off the training error and
the network complexity.
One technique to reach this tradeoff is to minimize a cost function composed of two
terms: the ordinary training error, plus some measure of the network complexity.
Several such schemes have been proposed in the statistical inference literature [see
(Akaike, 1986; Rissanen, 1989; Vapnik, 1989) and references therein] as well as in
the NN literature (Rumelhart, 1988; Chauvin, 1989; Hanson and Pratt, 1989; Mozer
and Smolensky, 1989).
Various complexity measures have been proposed, including Vapnik-Chervonenkis
dimensionality (Vapnik and Chervonenkis, 1971) and description length (Rissanen,
1989) . A time-honored (albeit inexact) measure of complexity is simply the number
of non-zero free parameters, which is the measure we choose to use in this paper
[but see (Denker, Le Cun and Solla, 1990)]. Free parameters are used rather than
connections, since in constrained networks, several connections can be controlled by
a single parameter.
In most cases in the statistical inference literature, there is some a priori or heuristic
information that dictates the order in which parameters should be deleted; for
example, in a family of polynomials, a smoothness heuristic may require high-order
terms to be deleted first. In a neural network, however, it is not at all obvious in
which order the parameters should be deleted.
A simple strategy consists in deleting parameters with small "saliency", i.e. those
whose deletion will have the least effect on the training error. Other things being equal, small-magnitude parameters will have the least saliency, so a reasonable
initial strategy is to train the network and delete small-magnitude parameters in
order. After deletion, the network should be retrained. Of course this procedure
can be iterated; in the limit it reduces to continuous weight-decay during training
(using disproportionately rapid decay of small-magnitude parameters). In fact, several network minimization schemes have been implemented using non-proportional
weight decay (Rumelhart, 1988; Chauvin, 1989; Hanson and Pratt, 1989), or "gating coefficients" (Mozer and Smolensky, 1989). Generalization performance has
been reported to increase significantly on the somewhat small problems examined.
Two drawbacks of these techniques are that they require fine-tuning of the "pruning" coefficients to avoid catastrophic effects, and also that the learning process
is significantly slowed down. Such methods include the implicit hypothesis that
the appropriate measure of network complexity is the number of parameters (or
sometimes the number of units) in the network.
One of the main points of this paper is to move beyond the approximation that
"magnitude equals saliency" , and propose a theoretically justified saliency measure.
599
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Le Cun, Denker and Solla
Our technique uses the second derivative of the objective function with respect to
the parameters to compute the saliencies. The method was ,,-alidated using our
handwritten digit recognition network trained with backpropagation (Le Cun et aI.,
1990b).
2
OPTIMAL BRAIN DAMAGE
Objective functions playa central role in this field; therefore it is more than reasonable to define the saliency of a parameter to be the change in the objective
function caused by deleting that parameter. It would be prohibiti,-ely laborious to
evaluate the saliency directly from this definition, i.e. by temporarily deleting each
parameter and reevaluating the objective function.
Fortunately, it is possible to construct a local model of the error function and
analytically predict the effect of perturbing the parameter vector. "'e approximate
the objective function E by a Taylor series. A perturbation lL~ of the parameter
vector will change the objective function by
(1)
Here, the 6ui'S are the components of flJ, the gi's are the components of the gradient
G of E with respect to U, and the hi;'S are the elements of the Hessian matrix H
of E with respect to U:
8E
gi= -8
Ui
and
(2)
The goal is to find a set of parameters whose deletion will cause the least increase
of E . This problem is practically insoluble in the general case. One reason is
that the matrix H is enormous (6.5 x 10 6 terms for our 2600 parameter network),
and is very difficult to compute. Therefore we must introduce some simplifying
approximations. The "diagonal" approximation assumes that the 6E caused by
deleting several parameters is the sum of the 6E's caused by delet~ each parameter
individually; cross terms are neglected, so third term of the npt hand side of
equation 1 is discarded. The "extremal" approximation assumes that parameter
deletion will be performed after training has converged. The parameter vector is
then at a (local) minimum of E and the first term of the right hand side of equation 1
can be neglected. Furthermore, at a local minimum, all the hii's are non-negative,
so any perturbation of the parameters will cause E to increase or stay the same.
Thirdly, the "quadratic" approximation assumes that the cost fundion is nearly
quadratic 80 that the last term in the equation can be neglected. Equation 1 then
reduces to
6E=~~h"6u~
(3)
2L.i
" ?
i
Optimal Brain Damage
2.1
COMPUTING THE SECOND DERIVATIVES
Now we need an efficient way of computing the diagonal second derivatives hi i .
Such a procedure was derived in (Le Cun, 1987), and was the basis of a fast backpropagation method used extensively in \1lrious applications (Becker and Le Cun,
1989; Le Cun, 1989; Le Cun et al., 1990a). The procedure is very similar to the
back-propagation algorithm used for computing the first derivatives. We will only
outline the proced ure; details can be found in the references.
We assume the objective function is the usual mean-squared error (MSE); generalization to other additive error measures is straightforward. The following expressions apply to a single input pattern; afterward E and H must be averaged over
the training set. The network state is computed using the standard formulae
and
ai
=L
WijZj
( 4)
j
where Zi is the state of unit i, ai its total input (weighted sum), ! the squashing
function and Wij is the connection going from unit j to unit i. In a shared-weight
network like ours, a single parameter Uk can control one or more connections: Wij
Uk for all (i, j) E Vk, where Vk is a set of index pairs. By the chain rule, the diagonal
terms of H are given by
=
hu =
{)2E
L
(i,j)EV.
{)w~,
(5)
.,
The summand can be expanded (using the basic network equations 4) as:
{J2E
lP E
.
2
--=-z?
.,
{Jw~.
{Ja~'
(6)
The second derivatives are back-propagated from layer to layer:
(7)
We also need the boundary condition at the output layer, specifying the second
derivative of E with respect to the last-layer weighted BUms:
{J{J2
~ = 2!'(ai)2 -
ai
2(di - Zi)!"(ai)
(8)
for all units i in the output layer.
As can be seen, computing the diagonal Hessian is of the same order of complexity
as computing the gradient. In some cases, the second term of the right hand side of
the last two equations (involving the second derivative of I) can be neglected. This
corresponds to the well-known Levenberg-Marquardt approximation, and has the
interesting property of giving guaranteed positive estimates of the second derivative.
601
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Le Cun, Denker and Solla
2.2
THE RECIPE
The OBO procedure can be carried out as follows:
1. Choose a reasonable network architecture
2.
3.
4.
5.
6.
Train the network until a reasonable solution is obtained
Compute the second derivatives hu for each parameter
Compute the saliencies for each parameter: Sk = huu~/2
Sort the parameters by saliency and delete some low-saliency parameters
Iterate to step 2
Deleting a parameter is defined as setting it to 0 and freezing it there. Several
variants of the procedure can be devised, such as decreasing the ...41ues of the lowsaliency parameters instead of simply setting them to 0, or allowing the deleted
parameters to adapt again after they have been set to o.
2.3
EXPERIMENTS
The simulation results given in this section were obtained using back-propagation
applied to handwritten digit recognition. The initial network was highly constrained
and sparsely connected, having 10 5 connections controlled by 2578 free parameters.
It was trained on a database of segmented handwritten zip code digits and printed
digits containing approximately 9300 training examples and 3350 t.est examples.
More details can be obtained from the companion paper (Le Cun et al., 1990b).
16
16
<a>
14
1
10
pJ 8
Magnitude
~6
~6
b04
b04
-o
o
OBD
.9
o
~~--~--~---+--~----~
o
(b)
14
1
10
pJ 8
500
1000 1500 2000 2SOO
Parameters
-2~
o
__~__~__-+________~
SOO 1000 1500 laX) 2SOO
Parameters
Figure 1: (a) Objective function (in dB) versus number of paramet.ers for OBn
(lower curve) and magnitude-based parameter deletion (upper curve). (b) Predicted
and actual objective function versus number of parameters. The predicted value
(lower curve) is the sum of the saliencies of the deleted parameters.
Figure la shows how the objective function increases (from right to left) as the
number of remaining parameters decreases. It is clear that deletin~ parameters by
Optimal Brain Damage
order of saliency causes a significantly smaller increase of the objective function than
deleting them according to their magnitude. Random deletions were also tested for
the sake of comparison, but the performance was so bad that the curves cannot be
shown on the same scale.
Figure 1b shows how the objective function increases (from right to left) as the number of remaining parameters decreases, compared to the increase predicted by the
Quadratic-Extremum-Diagonal approximation. Good agrement is obtained for up
to approximately 800 deleted parameters (approximately 30% of the parameters).
Beyond that point, the curves begin to split, for several reasons: the off-diagonal
terms in equation 1 become disproportionately more important as the number of
deleted parameters increases, and higher-than-quadratic terms become more important when larger-valued parameters are deleted.
'
16
14
1
16
14
1
<a)
10
10
UJ 8
UJ 8
~
6
~
~4
~
-o
o
6
4.
~~
-2~--4----+----~--~--~
o
(b)
SOO
1000 1500 2000 2500
Parameters
-2~I--~,~
o
500
__
~, ~I
~I
+,____ __ ____
1000 1500 2000 2500
Parameters
Figure 2: Objective function (in dB) versus number of parameters, without retraining (upper curve), and after retraining (lower curve). Curves are given for the
training set (a) and the test set (b).
Figure 2 shows the log-MSE on the training set and the on the test set before and
after retraining. The performance on the training set and on the test set (after
retraining) stays almost the same when up to 1500 parameters (60% of the total)
are deleted.
We have also used OBn as an interactive tool for network design and analysis.
This contrasts with the usual view of weight deletion as a more-or-Iess automatic
procedure. Specifically, we prepared charts depicting the saliency of the 10,000
parameters in the digit recognition network reported last year (Le Cun et aI., 1990b).
To our surprise, several large groups of parameters were expendable. We were
able to excise the second-to-Iast layer, thereby reducing the number of parameters
by a factor of two. The training set MSE increased by a factor of 10, and the
generalization MSE increased by only 50%. The 10-category classification error
on the test set actually decreased (which indicates that MSE is not the optimal
603
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Le Cun, Denker and Solla
objective function for this task). OBD motivated other architectural changes, as
can be seen by comparing the 2600-parameter network in (Le Cun et aI., 1990a) to
the 1O,OOO-parameter network in (Le Cun et aI., 1990b).
3
CONCLUSIONS AND OUTLOOK
We have used Optimal Brain Damage interactively to reduce the number of parameters in a practical neural network by a factor of four. We obtained an additional
factor of more than two by using OBD to delete parameters automatically. The network's speed improved significantly, and its recognition accuracy increased slightly.
We emphasize that the starting point was a state-of-the-art network. It would be
too easy to start with a foolish network and make large improvements: a technique
that can help improve an already-good network is particularly valuable.
We believe that the techniques presented here only scratch the surface of the applications where second-derivative information can and should be used. In particular,
we have also been able to move beyond the approximation that "complexity equals
number of free parameters" by using second-derivative information. In (Denker, Le
Cun and Solla, 1990), we use it to to derive an improved measure of the network's
information content, or complexity. This allows us to compare network architectures on a given task, and makes contact with the notion of Minimum Description
Length (MDL) (Rissanen, 1989). The main idea is that a "simple" network whose
description needs a small number of bits is more likely to generalize correctly than
a more complex network, because it presumably has extracted the essence of the
data and removed the redundancy from it.
Acknowledgments
We thank the US Postal Service and its contractors for providing us with the
database. We also thank Rich Howard and Larry Jackel for their helpful comments
and encouragements. We especially thank David Rumelhart for sharing unpublished
ideas.
References
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Optimal Brain Damage
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1,651 | 2,500 | An iterative improvement procedure for
hierarchical clustering
David Kauchak
Department of Computer Science
University of California, San Diego
[email protected]
Sanjoy Dasgupta
Department of Computer Science
University of California, San Diego
[email protected]
Abstract
We describe a procedure which finds a hierarchical clustering by hillclimbing. The cost function we use is a hierarchical extension of the
k-means cost; our local moves are tree restructurings and node reorderings. We show these can be accomplished efficiently, by exploiting special properties of squared Euclidean distances and by using techniques
from scheduling algorithms.
1
Introduction
A hierarchical clustering of n data points is a recursive partitioning of the data into
2, 3, 4, . . . and finally n clusters. Each intermediate clustering is made more fine-grained
by splitting one of its clusters. It is natural to depict this process as a tree whose leaves are
the data points and whose interior nodes represent intermediate clusters. Such hierarchical
representations are very popular ? they depict a data set at multiple levels of granularity,
simultaneously; they require no prior specification of the number of the clusters; and there
are several simple heuristics for constructing them [2, 3].
Some of these heuristics ? such as average-linkage ? implicitly try to create clusters of
small ?radius? throughout the hierarchy. However, to the best of our knowledge, there
is so far no procedure which specifically hillclimbs the space of hierarchical clusterings
according to a precise objective function. Given the heuristic nature of existing algorithms,
it would be most helpful to be able to call an iterative improvement procedure on their
output. In particular, we seek an analogue of k-means for hierarchical clustering. Taken
literally this is possible only to a certain extent ? the basic object we are dealing with
is a tree rather than a partition ? but k-means has closely informed many aspects of our
procedure, and has determined our choice of objective function.
We use a canonical tree representation of a hierarchical clustering, in which the leaves are
data points, and the interior nodes are ordered; such a clustering is specified completely by
a tree structure and by an ordering of nodes. Our cost function is a hierarchical extension of
the k-means cost function, and is the same cost function which motivates average-linkage
schemes. Our iterative procedure alternates between two simple moves:
1. The ordering of nodes is kept fixed, and one subtree is relocated. This is the
natural generalization of a standard heuristic clustering move in which a data point
is transferred from one cluster to another.
2. The tree structure is kept fixed, and its interior nodes are reordered optimally.
We show that by exploiting properties of Euclidean distance (which underlies the k-means
cost function and therefore ours as well), these tasks can be performed efficiently. For instance, the second one can be transformed into a problem in VLSI design and job scheduling called minimum linear arrangement. In general this problem is NP-hard, but for our
particular case it is known [4] to be efficiently solvable, in O(n log n) time. After motivating and describing our model and our algorithm, we end with some experimental results.
2
2.1
The model
The space of trees
A hierarchical clustering of n points contains n different clusterings, nested within each
other. It is often depicted using a dendogram, such as the one below on the left (for a
data set of five points). We will use the term k-clustering, and the notation Ck , to denote
the grouping into k clusters. One of these clusters is divided in two to yield the (k + 1)clustering Ck+1 , and so on. Instead of a dendogram, it is convenient to use a rooted binary
tree (shown below on the right) in which the leaves are data points and internal nodes have
exactly two children, so there are 2n ? 1 nodes overall. Each internal node is annotated
with a unique ?split number? between 1 and n ? 1. These satisfy the property that the
split number of a parent is less than that of its children; so the root is numbered 1. The
k-clustering is produced by removing the internal nodes numbered 1, 2, 3, . . . , k ? 1; each
cluster consists of (the leaves in) one of the resulting connected components.
1
? 2-clustering:
{a, b, e}, {c, d}
4
2
? 3-clustering:
{a, b}, {e}, {c, d}
a
b
e
c
d
? 4-clustering:
{a}, {b}, {e}, {c, d}
e
3
a
c
d
b
Henceforth we will use ?node i? to mean ?the internal node with split number i?. The
maximal subtree rooted at this node is Ti ; the mean of its data points (leaves) is called ?i .
To summarize, a hierarchical clustering is specified by: a binary tree with the data points at
the leaves; and an ordering of the internal nodes.
2.2
Cost function
If the clusters of Ck are S1 , S2 , . . . , Sk , then the k-means cost function is
cost(Ck ) =
k X
X
kx ? ?(Sj )k2 ,
j=1 x?Sj
where ?(S) is the mean of set S. To evaluate a hierarchical clustering, we need to combine
the costs of all n intermediate clusterings, and we do so in the most obvious way, by a
linear combination. We take the overall cost of the hierarchical clustering to be
n
X
k=1
wk ? cost(Ck ),
where the wk are non-negative weights which add up to one. The default choice is to make
all wk = 1/n, but in general the specific application will dictate the choice of weights.
A decreasing schedule w1 > w2 > w3 > ? ? ? > wn places more emphasis upon coarser
clusterings (ie. small k); a setting wk = 1 singles out a particular intermediate clustering.
Although many features of our cost function are familiar from the simpler k-means setting,
there is one which is worth pointing out. Consider the set of six points shown here:
Under the k-means cost function, it is clear what the best 2-clustering is (three points in
each cluster). It is similarly clear what the best 3-clustering is, but this cannot be nested
within the best 2-clustering. In other words, the imposition of a hierarchical structure
forces certain tradeoffs between the intermediate clusterings. This particular feature is
fundamental to hierarchical clustering, and in our cost function it is laid bare. By adjusting
the weights wk , the user can bias this tradeoff according to his or her particular needs.
It is worth pointing out that cost(Ck ) decreases as k increases; as more clusters are allowed,
the data can be modeled with less error. This means that even when all the weights w k
are identical, the smaller values of k contribute more to the cost function, and therefore, a
procedure for minimizing this function must implicitly focus a little more on smaller k than
on larger k. This is the sort of bias we usually seek. If we wanted to further emphasize small
values of k, we could for instance use an exponentially decreasing schedule of weights, ie.
wk = c ? ?k , where ? < 1 and where c is a normalization constant.
Notice that any given subtree Tj can appear as an individual cluster in many of the clusterings Ck . If ?(j) denotes the parent of j, then Tj first appears as its own cluster in C?(j)+1 ,
and is part of all the successive clusterings up to and including Cj . At that point, it gets
split in two.
2.3
Relation to previous work
The most commonly used heuristics for hierarchical clustering are agglomerative. They
work bottom-up, starting with each data point in its own cluster, and then repeatedly merging the two ?closest? clusters until finally all the points are grouped together in one cluster.
The different schemes are distinguished by their measure of closeness between clusters.
1. Single linkage ? the distance between two clusters S and T is taken to be the
distance between their closest pair of points, ie. minx?S,y?T kx ? yk.
2. Complete linkage uses the distance between the farthest pair of points, ie.
maxx?S,y?T kx ? yk.
3. Average linkage seems to have now become a generic term encompassing at least
three different measures of distance between clusters.
(a) (Sokal-Michener) k?(S) ? ?(T )k2
P
1
2
(b) |S|?|T
x?S,y?T kx ? yk
|
(c) (Ward?s method)
|S|?|T |
|S|+|T | k?(S)
? ?(T )k2
Average linkage appears to be the most widely used of these; for instance, it is a standard
tool for analyzing gene expression data [1]. The three average linkage distance functions
are all trying to minimize something very much like our cost function. In particular, Ward?s
measure of the distance between two clusters is exactly the increase in k-means cost occasioned by merging those clusters. For our experimental comparisons, we have therefore
chosen Ward?s method.
3
Local moves
Each element of the search space is a tree structure in which the data points are leaves and
in which the interior nodes are ordered. A quick calculation shows that this space has size
n((n ? 1)!)2 /2n?1 (consider the sequence of n ? 1 merge operations which create the tree
from the data set). We consider two moves for navigating the space, along the lines of the
standard ?alternating optimization? paradigm of k-means and EM:
1. keep the structure fixed and reorder the internal nodes optimally;
2. keep the ordering of the internal nodes fixed and alter the structure by relocating
some subtree.
A key concern in the design of these local moves is efficiency. A k-means update takes
O(kn) time; in our situation the analogue would be O(n2 ) time since we are dealing with
all values of k. Ideally, however, we?d like a faster update. For our first move ? reordering
internal nodes ? we show that a previously-known scheduling algorithm [4] can be adapted
to solve this task (in the case of uniform weights) in just O(n log n) time. For the second
move, we show that any given subtree can be relocated optimally in O(n) time, using just
a single pass through the tree. These efficiency results are nontrivial; a crucial step in
obtaining them is to exploit special properties of squared Euclidean distance. In particular,
we write our cost function in three different, but completely equivalent, ways; and we
switch back and forth between these:
1. In the form given above (the definition).
P
2. We define the cost of a subtree Ti to be cost(Ti ) = x?Ti kx ? ?i k2 (where the
sum is over leaf nodes), that is, the cost of the single cluster rooted at point i. Then
the overall cost is a linear combination of subtree costs. Specifically, it is
n?1
X
W?(j),j ? cost(Tj ),
(1)
j=1
where ?(j) is the parent of node j and Wij = wi+1 + wi+2 + ? ? ? + wj .
3. We annotate each tree edge (i, j) (i is the parent of j > i) by k?i ? ?j k2 ; the
overall cost is also a linear combination of these edge weights, specifically,
X
Wk ? nl ? k?k ? ?l k2 ,
(2)
(k,l)?T
where Wk = w1 + w2 + ? ? ? + wk and nl is the number of leaves in subtree Tl .
All proofs are in a technical report [5] which can be obtained from the authors. To give a
hint for why these alternative formulations of the cost function are true, we briefly mention
a simple ?bias-variance? decomposition of squared Euclidean distance:
Suppose S is a set of points with mean ?S . Then for any ?,
X
X
kx ? ?k2 =
kx ? ?S k2 + |S| ? k? ? ?S k2 .
x?S
3.1
x?S
The graft
In a graft move, an entire subtree is moved to a different location, as shown below. The
letters a, b, i, . . . denote split numbers of interior nodes; here the subtree Tj is moved. The
only prerequisite (to ensure a consistent ordering) is a < i < b.
1
1
a
a
i
h
h
b
b
j
i
k
k
j
First of all, a basic sanity check: this move enables us to traverse the entire search space.
Claim. Any two hierarchical clusterings are connected by a sequence of graft operations.
It is important to find good grafts efficiently. Suppose we want to move a subtree T j ; what is
the best place for it? Evaluating the cost of a hierarchical clustering takes O(n) time using
equation (1) and doing a single, bottom-up pass. Since there are O(n) possible locations
for Tj , naively it seems like evaluating all of them would take O(n2 ) time. In fact, the best
relocation of Tj can be computed in just O(n) time, in a single pass over the tree.
To see why this is possible, notice that in the diagram above, the movement of T j affects
only the subtrees on the path between a and h. Some of these subtrees get bigger (T j is
added to them); others shrink (Tj is removed). The precise change in cost of any given
subtree Tl on this path is easy to compute:
Claim. If subtree Tj is merged into Tl , then the cost of Tl goes up by
nl nj
?+
? k?l ? ?j k2 .
l = cost(Tl ? Tj ) ? cost(Tl ) = cost(Tj ) +
nl + n j
Claim. If subtree Tj ? Tl is removed from Tl , then the cost of Tl changes by
ni nl
??
? k?l ? ?j k2 .
l = cost(Tl ? Tj ) ? cost(Tl ) = ?cost(Tj ) ?
nl ? n j
Using (1), the total change in cost from grafting Tj between a, b (as depicted above) can
be found by adding terms of the form W?(l),l ??
l , for nodes l on the path between j and
a. This suggests a two-pass algorithm for optimally relocating Tj : in the first pass over
the tree, for each Tl , the potential cost change from adding/removing Tj is computed. The
second pass finds the best location. In fact, these can be combined into a single pass [5].
3.2
Reordering internal nodes
Let Vint be the interior nodes of the tree; if there are n data points (leaves), then |Vint | =
n ? 1. For any x ? Vint , let Tx be the maximal subtree rooted at x, which contains all the
descendants of x. Let nx be the number of leaves in this subtree. If x has children y and z,
then the goodness of split at x is the reduction in cost obtained by splitting cluster T x ,
cost(Tx ) ? (cost(Ty ) + cost(Tz )),
which we henceforth denote g(x) (for leaves g(x) = 0). Again using properties of Euclidean distance, we can rewrite it thus:
g(x) = ny k?x ? ?y k2 + nz k?x ? ?z k2 .
function makequeue(x)
if x is a leaf return { }
let y, z be the children of x
Q ? union(makequeue(y), makequeue(z))
r ? ny k?x ? ?y k2 + nz k?x ? ?z k2
L ? [x]
Priority queue operations:
makequeue, max, deletemax,
union, insert.
Linked list operations:
? (concatenation)
while r < r(max(Q))
L0 ? deletemax(Q)
0
)?|L0 |
r ? r?|L|+r(L
|L|+|L0 |
L ? L ? L0
r(L) ? r
insert(Q, L)
return Q
procedure reorder(T )
u ? root of T
Q ? makequeue(u)
while Q is not empty
L ? deletemax(Q)
Output elements of list L, in order
Figure 1: The reordering move. Here Q is a priority queue of linked lists. Each list L has
a value r(L); and Q is ordered according to these.
We wish to find a numbering ? : Vint ? {1, 2, . . . , n ? 1} which
? respects the precedence constraints of the tree: if x is the parent of y then ?(x) < ?(y).
? minimizes the overall cost of the hierarchical clustering. Assuming uniform weights
wk = 1/n, this cost can be seen (by manipulating equation (2)) to be
1
n
X
?(x)g(x).
x?Vint
Notice that this is essentially a scheduling problem. There is a ?task? (a split) corresponding
to each x ? Vint . We would like to schedule the good tasks (with high g(x)) early on; in the
language of clustering, if there are particularly useful splits (which lead to well separated
clusters), we would like to perform them early in the hierarchy. And there are precedence
constraints which must be respected: certain splits must precede others.
The naive greedy solution ? always pick the node with highest g(x), subject to precedence
constraints ? doesn?t work. The reason: it is quite possible that a particular split has low
g(x)-value, but that it leads to other splits of very high value. A greedy algorithm would
schedule this split very late; an algorithm with some ?lookahead? capability would realize
the value of this split and schedule it early.
Horn[4] has a scheduling algorithm which obtains the optimal ordering, in the case where
all the weights wk are equal, and can be implemented in O(n log n) time. We believe it
can be extended to exponentially decaying, ?memoryless? weights, ie. wk = c ? ?k , where
? < 1 and c is some normalization constant.
We now present an overview of Horn?s algorithm. For each node x ? V , defineP
r(x) to be
the maximum, over all subtrees T (not necessarily maximal) rooted at x, of |T1 |
z?T g(z)
(in words, the average of g(?) over nodes of T ). This value r(x) is a more reliable indicator
of the utility of split x than the immediate return g(x). Once these r(x) are known, the
optimal numbering is easy to find: pick nodes in decreasing order of r(?) while respecting
the precedence constraints. So the main goal is to compute the r(x) for all x in the tree.
This can be done by a short divide-and-conquer procedure in O(n log n) time (Figure 1).
1
(a)
e
d
a
(b)
2
c
a
(c)
b
e
c
2
e
4
3
b
1
3
a
4
c
d
b
d
1
1
(d)
e
(e)
2
3
a
4
c
1
(f)
2
3
b
c
4
d
e
a
2
b
c
4
e
d
3
a
b
d
Figure 2: (a) Five data points. (b)?(f) Iteratively improving the hierarchical clustering.
(a)
1
(b)
a
c
b
1.0
0.8
d
1.0
3
b
1
d
2
a
(c)
2
a
3
b
c
d
c
Figure 3: (a) Four points on a line. (b) Average linkage. (c) Optimal tree.
4
Experiments
In the experiments, we used uniform weights wk = 1/n. In each iteration of our procedure,
we did a reordering of the nodes, and performed one graft ? by trying each possible subtree
(all O(n) of them), determining the optimal move for that subtree, and greedily picking
the best move. We would prefer a more efficient, randomized way to pick which subtree to
graft ? either completely randomly, or biased by a simple criterion like ?amount it deviates
from the center of its parent cluster?; this is future work.
Simple examples. To give some concrete intuition, Figure 2 shows the sequence of moves
taken on a toy example involving five data points in the plane. The initial tree (b) is random
and has a cost of 62.25. A single graft (c) reduces the cost to 27. A reordering (d), swapping
2 and 3, reduces the cost to 25.5, and a further graft (e) and reordering (f) result in the final
tree, which is optimal and has cost 21.
Figure 3 demonstrates a typical failing of average linkage. The initial greedy merger of
points b, c gives a small early benefit but later turns out to be a bad idea; yet the resulting
tree is only one graft away from being optimal. Really bad cases for average linkage can
be constructed by recursively compounding this simple instance.
A larger data set. Average linkage is often used in the analysis of gene expression data.
20
5600
18
5500
5400
14
12
5300
cost
% improvement over average linkage
16
10
5200
8
6
5100
4
5000
2
0
4900
0
50
100
150
200
250
k
300
350
400
450
500
0
10
20
30
40
50
iterations
60
70
80
Figure 4: (a) On the left, a comparison with average linkage. (b) On the right, the behavior
of the cost function over the 80 iterations required for convergence.
We tried our method on the yeast data of [1]. We randomly chose clean subsets (no missing
entries) of varying sizes from this data set, and tried the following on it: average linkage,
our method initialized randomly, and our method initialized with average linkage.
There were two clear trends. First of all: our method, whether initialized randomly or
with average linkage, systematically did better than average linkage, not only for the particular aggregate cost function we are using, but across the whole spectrum of values of
k. Figure 4(a), obtained on a 500-point data set, shows for each k, the percent by which
the (induced) k-clustering found in our method (initialized with average linkage) improved
upon that found by average linkage; the metric here is the k-means cost function. This is
a fair comparison because both methods are explicitly trying to minimize this cost. Notice
that an improvement in the aggregate (weighted average) is to be expected, since we are
hillclimbing based on this measure. What was reassuring to us was that this improvement
came across at almost all values of k (especially the smaller ones), rather than by negotiating some unexpected tradeoff between different values of k. This experiment also indicates
that, in general, the output of average linkage has real scope for improvement.
Second, our method often took an order of magnitude (ten or more times) longer to converge if initialized randomly than if initialized with average linkage, even though better
solutions were often found with random initialization. We therefore prefer starting with
average linkage. On the scant examples we tried, there was a period of rapid improvement involving grafts of large subtrees, followed by a long series of minor ?fixes?; see
Figure 4(b), which refers again to the 500-point data set mentioned earlier.
References
[1] T.L. Ferea et al. Systematic changes in gene expression patterns following adaptive
evolution in yeast. Proceedings of the National Academy of Sciences, 97, 1999.
[2] J.A. Hartigan. Clustering algorithms. Wiley, 1975.
[3] J.A. Hartigan. Statistical theory in clustering. Journal of Classification, 1985.
[4] W.A. Horn. Single-machine job sequencing with treelike precedence ordering and
linear delay penalties. SIAM Journal on Applied Mathematics, 23:189?202, 1972.
[5] D. Kauchak and S. Dasgupta. Manuscript, 2003.
90
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1,652 | 2,501 | Limiting form of the sample covariance
eigenspectrum in PCA and kernel PCA
David C. Hoyle & Magnus Rattray
Department of Computer Science,
University of Manchester,
Manchester M13 9PL, UK.
[email protected] [email protected]
Abstract
We derive the limiting form of the eigenvalue spectrum for sample covariance matrices produced from non-isotropic data. For the analysis of
standard PCA we study the case where the data has increased variance
along a small number of symmetry-breaking directions. The spectrum
depends on the strength of the symmetry-breaking signals and on a parameter ? which is the ratio of sample size to data dimension. Results are
derived in the limit of large data dimension while keeping ? fixed. As ?
increases there are transitions in which delta functions emerge from the
upper end of the bulk spectrum, corresponding to the symmetry-breaking
directions in the data, and we calculate the bias in the corresponding
eigenvalues. For kernel PCA the covariance matrix in feature space may
contain symmetry-breaking structure even when the data components are
independently distributed with equal variance. We show examples of
phase-transition behaviour analogous to the PCA results in this case.
1
Introduction
A number of data analysis methods are based on the spectral decomposition of large matrices. Examples include Principal Component Analysis (PCA), kernel PCA and spectral
clustering methods. PCA in particular is a ubiquitous method of data analysis [1]. The principal components are eigenvectors of the sample covariance matrix ordered according to
the size of the corresponding eigenvalues. In PCA the data is projected onto the subspace
corresponding to the n first principal components, where n is chosen according to some
model selection criterion. Most methods for model selection require only the eigenvalue
spectrum of the sample covariance matrix. It is therefore useful to understand how the
sample covariance spectrum behaves given a particular data distribution. Much is known
about the asymptotic properties of the spectrum in the case where the data distribution is
isotropic, e.g. for the Gaussian Orthogonal Ensemble (GOE), and this knowledge can be
used to construct model selection methods (see e.g. [2] and references therein). However, it
is also instructive to consider the limiting behaviour in the case where the data does contain
some low-dimensional structure. This is interesting as it allows us to understand the limits
of learnability and previous studies have already shown phase-transition behaviour in PCA
learning from data containing a single symmetry-breaking direction [3]. The analysis of
data models which include a signal component are also useful if we are to correct for bias
in the estimated eigenvalues corresponding to retained components.
PCA has limited applicability because it is a globally linear method. A promising nonlinear alternative is kernel PCA [4] in which data is projected into a high-dimensional
feature space and PCA is carried out in this feature space. The kernel trick allows all computations to be carried out efficiently so that the method is practical even when the feature
space has a very high, or even infinite, dimension. In this case we are interested in properties of the eigenvalue spectrum of the sample covariance matrix in feature space. The
covariance of the features will typically be non-isotropic even when the data itself has independently distributed components with equal variance. The sample covariance spectrum
will therefore show quite rich behaviour even when the data itself has no structure. It is important to understand the expected behaviour in order to develop model selection methods
for kernel PCA analogous to those used for standard PCA. Model selection methods based
on data models with isotropic noise (e.g. [2, 5]) are certainly not suitable for kernel PCA.
In this paper we apply methods from statistical mechanics and random matrix theory to determine the limiting form of eigenvalue spectrum for sample covariance matrices produced
from data containing symmetry-breaking structure. We first show how the replica method
can be used to derive the spectrum for Gaussian data with a finite number a symmetrybreaking directions. This result is confirmed and generalised by studying the Stieltjes
transform of the eigenvalue spectrum, suggesting that it may be insensitive to details of
the data distribution. We then show how the results can be used to derive the limiting form
of eigenvalue spectrum of the feature covariance matrix (or Gram matrix) in kernel PCA
for the case of a polynomial kernel.
2
Statistical mechanics theory for Gaussian data
We first consider a data set of N -dimensional data vectors {x? }p?=1 containing a signal
and noise component. Initially we restrict ourselves to the case where x? is drawn from a
Gaussian distribution whose covariance matrix C is isotropic except for a small number of
orthogonal symmetry-breaking directions, i.e.,
C = ?2 I + ?2
S
X
T
A m Bm Bm
,
BnT Bm = ?nm ,
Am > 0 .
(1)
m=1
? = p?1 P x? xT and study its eigenvalue spectrum
We define the sample covariance C
?
?
in the limit N ? ? when the ratio ? = p/N is held fixed and the number of symmetry? ?1
breaking directions S is finite. We work with the trace of the resolvent G(?) = (?I ? C)
from which the density of eigenvalues ?(?) can be calculated,
?(?) = lim+ (N ?)?1 Im trG(? ? i)
?0
where
trG(?) =
N
X
i=1
1
? ? ?i
? The trace of the resolvent can be represented as,
and ?i are eigenvalues of C.
?
? = ? log Z(?) .
trG(?) =
log det(?I ? C)
??
??
Using the standard representation of the determinant of a matrix,
Z
1
N
[det A]? 2 = (2?)? 2
exp ? 12 ?T A? d? ,
(2)
(3)
we have,
log Z(?) = N log 2? ? 2 log
Z
"
#
1 X
?
2
2
exp ? ||?|| +
(? ? x? ) d? .
2
2p ?
(4)
We assume that the eigenvalue spectrum is self-averaging, so that the calculation for a
specific realisation of the sample covariance can be replaced by an ensemble average for
large N that can be performed using the replica method (see e.g. [6]). Details are presented
elsewhere [7] and here we simply state the results. The calculation is similar to [3] where
Reimann et. al. study the performance of PCA on Gaussian data with a single symmetrybreaking direction, although there are also notable differences between the calculations.
We find the following asymptotic result for the spectral density,
?(?)
+
=
(1 ? ?)?(1 ? ?)?(?) +
S
1 X
?(? ? A?2
1?
m )
N m=1
where we have defined,
? 2
?)
!
S
1 X
?(? ? ?u (Am , ? 2 ))?(? ? A?2
m )
N m=1
? p
Max(0, (? ? ?min )(?max ? ?)) ,
2??? 2
(5)
1
).
(6)
?A
The first term in equation (5) sets a proportion 1 ? ? eigenvalues to zero when the rank of
? is less than N , i.e. when ? < 1. The last term represents the bulk of the spectrum and is
C
identical to the well-known Mar?cenko-Pastur law for isotropic data with variance ? 2 [8, 9].
In [7] we also give the O(1/N ) corrections to this term, but here we are mainly interested
in the leading order. The second term contains contributions due to the underlying structure
in the data. The mth symmetry-breaking term in the data covariance C only contributes
to the spectrum if ? > A?2
m . This transition must be exceeded before signals of a given
strength can be detected, i.e. the signal must be sufficiently strong or the data set sufficiently
large. This corresponds to the same learning transition point observed in studies of PCA
on Gaussian data with a single symmetry-breaking direction [3]. Above this transition
the sample covariance eigenvalue over-estimates the true variance corresponding to this
component by a factor 1 + 1/(?Am ) which indicates a significant bias when the data set is
small or the signal is relatively weak. Our result provides a method of bias correction for
the top eigenvalues in this case.
?max,min = ? 2 ??1 (1 ?
?u (A, ? 2 ) = ? 2 (1 + A)(1 +
In figure 1 we show results for Gaussian data with three symmetry-breaking directions,
each above the transition point. On the left we show how the top eigenvalues separate
from the bulk while the inset compares the density of the bulk with the theoretical result,
showing excellent agreement. On the right we show convergence to the theoretical result
for ?u (A, ? 2 ) in equation (6) as the data dimension N is increased for fixed ?.
3
Analysis of the Stieltjes transform
The statistical mechanics approach is useful because it allows the derivation of results from
first principles and it is possible to use this method to determine other self-averaging quantities of interest, e.g. the overlap between the leading eigenvectors of the sample and population covariances [3]. However, the method as presented here is restricted to Gaussian data.
A number of results from the statistics literature have been derived under much weaker and
often more explicit assumptions about the data distribution. It is therefore interesting to ask
whether equation (5) can also be derived from these results.
Mar?cenko and Pastur [8] studied the case of data with a general covariance matrix. The
limiting distribution was shown to satisfy,
Z
?(?)
?(?) = lim ? ?1 Im ?m? (? + i) where m? (z) = ??1 d?
.
(7)
??z
?0+
9
(a)
7
Log Fractional Error
Probability Density
Eigenvalue
8
0.2
0.15
0.1
0.05
0
6
5
0
2
4
6
log ??1
log ??2
log ??3
-2
0.3
0.25
0
8
1
10
Index
2
3
4
Eigenvalue
12
14
5
16
6
18
-3
(b)
-4
-5
20
-6
5
7
6
8
Log N
Figure 1: In (a) we show eigenvalues of the sample covariance matrix for Gaussian data
with ? 2 = 1, N = 2000 and ? = 0.5. The data contains three symmetry-breaking directions with strengths A21 = 20, A22 = 15 and A23 = 10 all above the transition point. The
inset shows the distribution of all non-zero eigenvalues except for the largest three with the
solid line showing the theoretical result. In (b) we show the fractional difference between
the three largest eigenvalues ?i and the theoretical value ?u (Ai , ? 2 ) for i = 1, 2, 3. We set
? = 0.2, averaged ?i over 1000 samples to get h?i i, set ??i = |1 ? h?i i/?u (Ai , ? 2 ))|
and set other values as in (a).
Here, m? (z) is the Stieltjes transform of ??1 ?(?) and is equal to ?p?1 trG(z). The above
equation is therefore exactly equivalent to equation (2) and we see that this approach starts
from the same point as the statistical mechanics theory. Mar?cenko and Pastur showed that
the Stieltjes transform satisfies the following relationship,
Z
1
dH(t)
z(m? ) = ?
+ ??1
.
(8)
?1
m?
t + m?
R
P
The measure H(t) is defined such that N ?1 i dki converges to tk dH(t) ?k where
di are the eigenvalues of C. An equivalent result is also derived by Wachter [10] and
more recently by Sengupta and Mitra using the replica method [11] (for Gaussian data).
Silverstein and Choi have shown that the support of ?(?) can be determined by the intervals
between extrema of z(m? ) [12] and this has been used to determine the signal component
of a spectrum when O(N ) equal strength symmetry-breaking directions are present [13].
Since C in equation (1) only contains a finite number of symmetry-breaking directions
then in the limit N ? ? these will have zero measure as defined by H. Thus, in this limit
the eigenvalue density would appear to be identical to the isotropic case. However, it is the
behaviour of the largest eigenvalues that we are most interested in, even though these may
have vanishing measure. For the case of a single symmetry-breaking direction (S = 1,
A1 = A) we take dH(t) = (1 ? )?(t ? ? 2 )dt + ?(t ? ? 2 (1 + A))dt, with ' 1/N . This
gives,
1
(1 ? )??1
??1
z(m? ) = ?
+ ?2
+ ?2
,
(9)
m?
? + m?
? (1 + A)?1 + m?
and stationary points satisfy,
0 =
(1 ? )??1
??1
1
?
?
.
2
?2
2
?2
m?
(? + m? )
(? (1 + A)?1 + m? )2
(10)
Since 1 we do not expect the behaviour of z(m? ) to be modified substantially in
the interval [?min , ?max ]. Therefore we look for additional stationary points close to the
singularity at m? = ?? ?2 (1+A)?1 . Setting m? = ?? ?2 (1+A)?1 +? and expanding (10)
p
1
yields ? = 2 /? 2 (1 + A) (? ? A?2 )+O(). Substituting this into (9) gives z(?? ?2 (1+
1
A)?1 + ?) = ? 2 (1 + A)(1 + (?A)?1 ) + O( 2 ). Thus, as N ? ?, if the stationary points
?2
?1
at ?? (1 + A) + ? exist they will define a small interval of z centred on ?u (A, ? 2 )
and so define an approximate contribution of N ?1 ?(? ? ?u (A, ? 2 )) to the spectrum, in
agreement with the previous calculations using replicas. We also see that for ? to be real
requires ? > A?2 , in agreement with our previous calculation for the learning transition
point. A similar perturbative analysis when C contains more than one symmetry-breaking
direction gives a set of contributions N ?1 ?(? ? ?u (Am , ? 2 )), m = 1, . . . , S, to ?(?).
Again this is in agreement with our previous replica analysis of the resolvent.
The relationship in equation (8) can be obtained with only relatively weak conditions on
?
the data distribution. One requirement is that the second moment of each element of C
exists. Bai has considered the case of data vectors with non-Gaussian i.i.d. components
(e.g. [14]) while Mar?cenko and Pastur show that the data vector components do not have
to be independently distributed for the relation to hold and they give sufficient conditions
on the 4th order cross-moments of the data vector components [8]. In [7] we study PCA
on some examples of non-Gaussian data with symmetry-breaking structure (non-Gaussian
signal and noise) and show that the separated eigenvalues behave similarly to figure 1.
4
Eigenvalue spectra for kernel PCA
Equation (8) holds under quite weak conditions on the data distribution. It is therefore
hoped that we can apply these results to the feature space of kernel PCA [4]. In kernel PCA the data x is transformed into a feature vector ?(x) and standard PCA is carried out in the feature space. The method requires that we can define a kernel function
k(x, y) = ?(x) ? ?(y) that allows efficient computation of the dot-product in a high, or
even infinite, dimensional space. The eigenvalues of the sample covariance in feature space
are identical to eigenvalues of the Gram matrix K?? with entries k(x? , x? ) and the eigenvalues can therefore be computed efficiently for arbitrary feature-space dimension as long
as the number of samples p is not too large (NB. The Gram matrix first has to be centred [4]
so that the data has zero mean in the feature space).
One common choice of kernel function is the polynomial kernel k(x, y) = (c + x ? y) d in
which case, for integer d, the features are all possible monomials up to order d involving
components of x. We limit our attention here to the quadratic kernel (d = 2). We consider data vectors with components that are independently and symmetrically
distributed
?
with equal variance ? 2 and choose a set of features ?(x) = ( 2cx, Vec[xxT ]) where
Vec[xxT ]j+N (i?1) = xi xj . The covariance in feature space is block diagonal,
2chxxT i
C=?
0
?
?
0
hVec[xxT ]Vec[xxT ]T i ?
?hVec[xxT ]ihVec[xxT ]T i
di
2c? 2
2? 4
2? 4 + ?i4
number
N
N (N ? 1)/2
N
where angled brackets denote expectations over the data distribution. The non-zero eigenvalues of C are shown on the right where ?i4 = hx4i i ? 3? 4 is the 4th cumulant of the
ith component of x. We see that although each component of the data is independently
distributed with equal variance, the covariance structure in feature space may be quite complex.
? Gaussian data, c = 0
For isotropic Gaussian data and c = 0 there is a single degenerate eigenvalue of C and the
asymptotic result for the spectrum is identical to the case of an isotropic distribution [8, 9]
with variance 2? 4 and ? defined as the ratio of the number of examples p to the effective
0.3
2
9
log( ?1 ? 5.8284)
1
0.2
8
?1
Probability Density
0.25
0.15
0
-1
-2
-3
7
-4
0.1
4
8
6
10
log p
0.05
6
0
0
1
2
3
4
Eigenvalue
5
6
7
0
10000
20000
p
30000
40000
Figure 2: On the left we show the Gram matrix eigenspectrum for a sample data set and
compare it to the theoretical result. The kernel is purely quadratic (c = 0) and we use
isotropic Gaussian data with 2? 4 = 1, N = 63 and p = 1000 so that ? ' 0.5. On the right
we show the averaged top eigenvalue against p for fixed ?. Each point is averaged over 100
samples except for the right-most which is averaged over 50. The dashed line shows the
theoretical result ?1 = 5.8284 and inset is a log-log plot of the same data.
dimension in the feature space N (N +1)/2 (i.e. the degeneracy of the non-zero eigenvalue)
so that ? = 2p/N (N + 1) and p = O(N 2 ) is the appropriate scaling.
On the left of figure 2 we compare the spectra for a single sample data set to the theory for
p = 1000 and N = 63 which corresponds to ? ' 0.50 and the theoretical curve is almost
identical to the one used in the inset to figure 1(a). The finite size effects are much larger
than would be observed for PCA with isotropic data and on the right of figure 2 we show
the average of the top eigenvalue for this value of ? as p is increased, showing a very slow
convergence to the asymptotic result.
? Gaussian data, c > 0
For isotropic Gaussian data and c > 0 there are two eigenvalues of C with degeneracy
N and N (N + 1)/2 respectively. For large N and c > ? 2 the top N eigenvalues play
an analogous role to the top S eigenvalues in the PCA data model defined in section 2.
A similar perturbative expansion to the one described in section 3 shows that when ? <
(c/? 2 ? 1)?2 (where ? ' 2p/N 2 is defined relative to the feature space) the distribution
is identical to the c = 0 case. For ? above this transition point the N top eigenvalues
separate from the bulk. In the limit N ? ? with p = O(N 2 ) the spread of the upper
N eigenvalues will tend to zero and they will become localised at ?u (c/? 2 ? 1, 2? 4 ) as
defined by equation (6). For finite N and when the two components of the spectra are well
separated, we can approximate the eigenvalue spectrum of the top N eigenvalues as though
the data only contains these components, i.e. we model this cluster as isotropic data with
? = p/N and variance 2c? 2 . We obtain an improved approximation by correcting the
mean of the separated cluster by the value predicted for the mean in the large N limit.
On the left of figure 3 we compare this approximation to the Gram matrix spectrum averaged over 300 data sets for large c, with the inset showing the separated cluster. The theory
is shown by the solid line and provides a good qualitative fit to the data although there are
significant discrepancies. For the bulk we believe these to be due to finite size effects but
the theory for the spread of the upper N eigenvalues is only approximate since the spread
of this cluster will vanish as N ? ? for fixed c and p = O(N 2 ). On the right of figure 3
we plot the average of the top N eigenvalues against c, showing good agreement with the
theory. The top eigenvalue of the population covariance is shown by the line and the theory
accurately predicts the bias in the sample estimate.
30
Average of top N Eigenvalues
0.002
Probability Density
0.25
0.0015
0.2
0.001
0.15
0.0005
0.1
0
10
20
15
30
25
40
35
0.05
0
0
2
4
20
15
10
5
0
8
6
Simulation
Theory
Unbiased Eigenvalue
25
0
10
5
20
15
c
Eigenvalue
Figure 3: On the left we show the Gram matrix eigenvalue spectrum averaged over 300
data sets and compare it to the theoretical result. The inset shows the density of the top N
eigenvalues
which are separated from the bulk. The kernel is quadratic with c = ? 2 (1 +
?
500) with other parameters as in figure 2. On the right we show the average of the top N
eigenvalues against the theoretical result as a function of c.
11
11
0.8
0.8
<?1>Theory
8
0.6
9
0.4
<?>
<?>
9
Probability Density
10
Probability Density
10
0.2
7
8
0.6
0.4
0.2
7
0
0
1
2
3
4
5
6
0
7
0
1
Eigenvalue
2
3
4
5
6
7
Eigenvalue
6
6
<?2>Theory
5
0
5
10
Rank
15
20
5
0
5
10
15
20
Rank
Figure 4: Results from a purely quadratic kernel (c = 0) on data containing a single dimension having positive kurtosis. We show the top 20 eigenvalues of the Gram matrix with the
bulk spectrum as an inset. On the left ?4 = 5 and we are above the transition where the top
eigenvalue is separated from the bulk. On the right ?4 = 1 is below the transition. Other
parameters were 2? 4 = 1, N = 70, p = 1500 and results were averaged over 25 data sets.
? Non-Gaussian data, c = 0
If the data has components with positive kurtosis then these will break the symmetry of the
covariance. This is analogous to the case for PCA studied in section 2 and the result for the
limiting spectrum carries over. We have ? ' 2p/N 2 defined with respect
? to the dimension
of the feature space. For each component of the data with ?i4 > 2? 4 / ? there will be a
delta function in the spectrum at ?u (?i4 /2? 4 , 2? 4 ) as defined by equation (6).
In figure 4 we show the Gram matrix eigenvalues for a data set containing a single dimension having positive kurtosis. On the left we have ?4 = 5 which is above the transition. We
have indicated with arrows the theoretical prediction for the top two eigenvalues and we
see that there is a significant difference, although the separation is quite well described by
the theory. We expect that these discrepancies are due to large finite size effects and further
simulations are required to verify this. On the right we have ?4 = 1 which is below the
transition and the spectrum is very similar to the case for isotropic Gaussian data.
5
Conclusion
We studied the asymptotic form of the sample covariance eigenvalue spectrum from data
with symmetry-breaking structure. For standard PCA the asymptotic results are very accurate even for moderate data dimension, but for kernel PCA with a quadratic kernel we
found that convergence to the asymptotic result was slow. The limiting form of sample covariance spectra has previously been studied in the neural networks literature where it can
be used in order to determine the optimal batch learning rate for large linear perceptrons.
Indeed, the results derived in section 2 for Gaussian data can also be derived by adapting an
elegant method developed by Sollich [15], without recourse to the replica method. Halkj?r
& Winther used this approach to compute the spectral density for the case of a single symmetry breaking direction and obtained a similar result to us, except that the position of the
separated eigenvalue was at ? 2 (1 + A) which differs from our result [16]. In fact they assumed a large signal in their derivation and their derivation can easily be adapted to obtain
an identical result to ours. However this method, as well as the replica approach used here,
is limited because it only applies to Gaussian data, while the Stieltjes transform relationship
in equation (8) has been derived under much weaker conditions on the data distribution.
Our current work is focussed on extending the analysis to more general kernels, such as
the radial basis function (RBF) kernel where the feature space dimension is infinite. In the
general case we find that the Stieltjes transform can be derived by a variational mean field
theory and therefore provides a principled approximation to the average spectral density.
Acknowledgments DCH was supported by a MRC(UK) Special Training Fellowship in
Bioinformatics. We would like to thank the anonymous reviewers for useful comments and
for pointing out references [15] and [16].
References
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1,653 | 2,502 | Nonlinear Filtering of Electron
Micrographs by Means of Support Vector
Regression
R. Vollgraf1 , M. Scholz1 , I. A. Meinertzhagen2 , K. Obermayer1
1
Department of Electrical Engineering and Computer Science
Berlin University of Technology, Germany
{vro,idefix,oby}@cs.tu-berlin.de
2
Dalhousie University, Halifax, Canada
[email protected]
Abstract
Nonlinear filtering can solve very complex problems, but typically
involve very time consuming calculations. Here we show that for
filters that are constructed as a RBF network with Gaussian basis
functions, a decomposition into linear filters exists, which can be
computed efficiently in the frequency domain, yielding dramatic
improvement in speed. We present an application of this idea to
image processing. In electron micrograph images of photoreceptor
terminals of the fruit fly, Drosophila, synaptic vesicles containing
neurotransmitter should be detected and labeled automatically. We
use hand labels, provided by human experts, to learn a RBF filter
using Support Vector Regression with Gaussian kernels. We will
show that the resulting nonlinear filter solves the task to a degree of
accuracy, which is close to what can be achieved by human experts.
This allows the very time consuming task of data evaluation to be
done efficiently.
1
Introduction
Using filters for image processing can be understood as a supervised learning method
for classification and segmentation of certain image elements. A given training image would contain a target that should be approximated by some filter at every
location. In principle, any kind of machine-learning techniques could be employed
to learn the mapping from the input receptive field of the filter to the target value.
The most simple filter is linear mapping. It has the advantage that it can be very
efficiently computed in the frequency domain. However linear filters may not be
complex enough for difficult problems. The complexity of nonlinear filters is in
principle unlimited (if we leave generalization issues aside), but the computation
of the filter output can be very time consuming, since usually there is no shortcut
in the frequency domain, as for linear filters. However, for nonlinear filters, that
are linear superpositions of Gaussian radial basis functions, there exists a decomposition into linear filters, allowing the filter output to be computed in reasonable
time. This sort of nonlinear filtering is for example obtained, when Support Vector
Machines (SVM) with a Gaussian kernel are used for learning. SVM have proved to
yield good performance on many applications [1]. This and the ability to compute
the filter output in an affordable time, make SVM interesting for nonlinear filtering
in image processing tasks. Here we apply this new method to the evaluation of electron micrograph images taken from the visual system of the fruit fly, Drosophila, as
a means to analyze morphological phenotypes of new genetic mutants. Genetically
manipulable organisms such as Drosophila provide means to address many current
questions in neuroscience. The action, even of lethal genes, can be uncovered in
photoreceptors by creating homozygous whole-eye mosaics in heterozygous flies [2].
Mutant synaptic phenotypes are then interpretable from detailed ultra-structural
knowledge of the photoreceptor terminals R1-R6 in the lamina [3]. Electron microscopy (EM) alone offers the resolution required to analyze sub-cellular structure,
even though this technique is tedious to undertake. In Drosophila genetics hundreds
of mutants of the visual system have been isolated, many even from a single genetic
screen. The task of analyzing each of these mutants manually is simply not feasible,
hence reliable automatic (computer assisted) methods are needed. The focus here
is just to count the number of synaptic vesicles, but in general the method proposed
in this report could be extended to the analysis of other structures as well.
As representative datasets showing the feasibility of the proposed method, we have
chosen two datasets from wild type Drosophila (ter01 for training and ter04 for
performance evaluation, cf. Fig. 1) and one from a visual system mutant (mutant,
also for performance evaluation, cf. Fig. 2, left).
2
Learning the RBF Filter
Given an image x, we want to find a RBF filter with Gaussian basis functions, the
output of which is closest to a target image y, in terms of some suitable distance
measure. The filter is constrained to some receptive field P , so that its output at
position r would be formulated in the most general form as
T
,
z(r) = fRBF (x(r)) = fRBF (x(r + ?r 1 ), . . . , x(r + ?r M ))
where P = {?r 1 , . . . , ?r M } is the neighborhood that forms the receptive field.
In the following we will continue using bold faced symbols to indicate a vector
containing the neighborhood (patch) at some location, while light faces indicate the
value of the image itself. Individual elements of patches are addressed by a subscript,
for example x?r (r) = x(r +?r). fRBF is a RBF network with M input dimensions.
It can be implemented as a feed forward net with a single hidden layer containing a
fixed number of RBF units and a linear output layer [4]. However we would rather
use the technique of Support Vector Regression (SVR) [5] as it has a number of
advantages over RBF feed forward networks. It offers adjustable model complexity
depending on the training data, thus providing good generalization performance.
The training of SVR is a quadratic, constrained optimization problem, which can
be solved efficiently without being trapped into local minima. In the linear case the
formulation of the ??SVR, as it was introduced in [6], would be
!
l
X
1
1
minimize
? (w, ? (?) , ?) = kwk2 + C ? ?? +
(?i + ?i? )
(1)
2
l i=1
s.t.
((w ? xi ) + b) ? yi ? ? + ?i ,
(?)
?i
? 0, ? ? 0
yi ? ((w ? xi ) + b) ? ? + ?i?
(2)
(3)
The constraints implement as a distance measure the ?-insensitive loss |y ? f (x)| ? =
max{0, |y ? f (x)| ? ?}, which is a basic feature of SVR, and has been shown to yield
robust estimation. The objective itself provides a solution of low complexity (small
kwk2 ) and, at the same time, low errors, balanced by C. In contrast to ??SVR, as
it was introduced at first in [5], parameterization with the hyper parameter ? also
allows optimization for the width ? of the insensitive region. Interacting with C, ?
controls the complexity of the model. It provides an upper bound on the fraction
of outliers (samples that do not fall into the epsilon tube) and a lower bound on
the fraction of support vectors (SV), see [6] and [1] for further details. As usual for
SVM, the system is transformed into a nonlinear regressor by replacing the scalar
product with a kernel, that fulfills Mercers condition [7]. With a Gaussian kernel
(RBF kernel) the regression function is
z(r) =
l
X
(?)
?i zi (r) + b ,
(4)
i=1
where
1 X
2
zi (r) = k(xi , x(r)) = exp ?
(xi,?r ? x(r + ?r))
?
?r?P
!
(5)
is the Gaussian- or RBF-kernel. The resulting SVs xi are a subset of the training
examples, for which one of the constraints (2) holds with equality. They correspond
(?)
to Lagrange multipliers having ?i = (?i ? ?i? ) 6= 0. In the analogy to a RBF
(?)
network, the SVs are the centers of the basis functions, while ?i are the weights
of the output layer.
3
RBF Filtering
To evaluate a RBF network filter at location r, all the basis functions have to
be evaluated for the neighborhood x(r). This calculation is computationally very
expensive when computed in the straightforward way given by (5). If the squared
sum is multiplied out, however, we can compute the kernel as
1
,
(6)
zi (r) = exp ? kxi k2 ? 2zi0 (r) + zi00 (r)
?
where
zi0 (r) =
X
xi,?r x(r + ?r)
?r?P
and
zi00 (r) =
X
x(r + ?r)2 .
(7)
?r?P
Now we are left with linear filtering operations only, the two cross correlations
z 0 and z 00 , which can be efficiently computed in the frequency domain, where the
cross correlation of a signal with some filter becomes a multiplication of the signal?s
spectrum with the conjugate complex spectrum of the filter. This operation is
so much faster that the additional computation cost of the Fourier transform is
neglectable. Note that in fact z 00 is the cross correlation of x2 with the filter o,
which is 1 for all ?r ? P . We need to compute the following Fourier transforms:
X(j?)
Xi (j?)
? F[x(r)] ,
? F[xi (r)] ,
X (2) (j?) ? F[x2 (r)] ,
O(j?) ? F[o(r)] .
(8)
xi (r) and o(r) are the filters xi and o, zero filled for r ?
/ P to the size of the
image. It is necessary to take care of the placement of the origin ?r = 0 and
the mapping of negative offsets in P , which depends on the implementation of the
Fourier transform. Now zi is easily computed as
h
i
1 T
zi (r) ? exp ?
xi xi ? F ?1 2XiC (j?)X(j?) ? O C (j?)X (2) (j?)
(9)
?
where (?)C indicates the conjugate complex. Using Fast Fourier Transform (FFT),
the speed improvement is much higher when the size of x is even in terms of powers
of 2 [8]. Thus one should consider enlarging the image size by adding the appropriate
number of zeros at the border. However this can lead to large overhead regions,
when the image size is not close to the next power of 2. For this reason we use a tiling
scheme, which processes the image in smaller parts of even size, which can cover
the entire image more closely. It is important to be aware of the distorted margins
of the image or its tiles, when filtering is done in the frequency domain. Because
the cross correlation in the frequency domain is cyclic, points at the margin, for
which the neighborhood P exceeds the image boundaries, have incorrect values in
the filter?s output. This is particularly important for the tiling scheme, which has to
provide sufficient overlap for the tiles, so that the image can be covered completely
with the uncorrupted inner parts of the tiles. Table 1 summarizes the speed-up
gain for the described filtering method. Most performance gain is obtained through
the filtering in the frequency domain. However, splitting the image into tiles of
appropriate size can improve speed even further.
Table 1: Computation time examples for different filtering methods.
filtering acc. to (5)
6d 10h
FFT filtering, whole image
55m
FFT filtering, tiles of 256 ? 256
24m
4
? image size 1686 ? 1681 pixel
? 200 SV of 50 ? 50 pixels size
? implementation in MATLAB
? SUN F6800 / 750MHz, 1 CPU
Experiments
To test the performance of the method we used two images of wild type and one
of mutant photoreceptor terminals. The profiles of the terminals contain typically
about 100 synaptic vesicles, the number of which could differ if the genes for membrane trafficking are mutated. Detecting such numerical differences is a simple but
tedious task best suited to a computational approach. The wild type images came
from electron micrographs of the same animal under the same physiological conditions. For all images visual identification and hand written labelings of the vesicles
were made. Image ter01 (Fig. 1, left) was used for training. The validation error
on ter04 (Fig. 1, right) was considered for model selection. Then the best model
was tested on the mutant image (Fig. 2).
4.1
Construction of the Target
ter01 contains 286 hand-labeled vesicles at discrete positions. To generate a smooth
target image y, circular gauss blobs with ? 2 = 40 and a peak value of 1 were
placed on every label. Now training examples x(r) where generated from ter01 by
Figure 1: EM images of photoreceptor terminals of the wild type fruit fly, Drosophila
melanogaster. The left image (ter01) was used for training, the right image (ter04)
for validation. Arrow: individual synaptic vesicle, 30nm in diameter.
taking square patches, centered around r. We have set the patch size P = 50 ? 50
pixels, to cover an entire vesicle plus a little surrounding. The corresponding values
y(r) of the target image where used as targets for regression. The most complete
training set would clearly contain patches from all locations, which however would be
computationally unfeasible. Instead we used patches from all hand-label positions
and additionally 2000 patches from random positions. No patches exceeded the
image boundaries. With these data the SVM was trained. We used the libsvm
implementation [9] which also contains, beside others, the ?-SVR. Mainly three
parameters have to be adjusted for training the ?-SVR: the width of the RBF kernel
? and the parameters ? and C. Since the training dataset is small compared to the
input dimensionality, the validation error on ter04 is subject to large variance.
Therefore we cannot give a complete parameter exploration here, but we would
expect a model with not too much complexity to give the best generalization. It
turned out that, for the given conditions, a kernel size of ? = 20.000 together with
a low value ? = 0.1 and C = 0.01 yield good validation results on ter04. The
optimization returned 245 SVs, 185 of which where outliers. The kernel width is
large compared with the average distance of the training examples in input space,
which was < 2.000. Because the computation time of the filter grows linearly with
the number of SVs, we are strongly interested in a solution with only few SVs. This
requires small values of ?, since it is a lower bound on the fraction of SVs. At the
same time, small ? values provide large ? and hence restrict the model complexity.
After filtering, the decision which point in z corresponds to a vesicle, has to be
made. Although the regions of high amplitude form sharp peaks, they still have
some spatial extension. Therefore we first discriminate for the peak locations and
then for the amplitude. In a first step, we determine those locations r, for which
z(r) is a local maximum in some neighborhood, which is determined roughly by the
size of a vesicle, i.e. we consider the set
Qd =
r : z(r) =
max
{?r:kr??rk?d}
z(r + ?r)
.
(10)
Then a threshold is applied to the candidates in Qd to yield the set of locations,
which are considered as detected vesicles,
Q? = {r ? Qd : z(r) > ?} .
(11)
We set the parameter d = 15 constant in our experiments, and will vary only the
threshold ?.
4.2
Performance Evaluation
To evaluate the performance of the method, the set of detected vesicles Q ? must be
compared with set QExp , which contains the locations detected by a human expert.
Clearly this is only meaningful when done on data which was not used to train the
SVM. We note that the location of the same vesicle may vary slightly in Q ? and
QExp , due to fluctuations in the manual labeling, for example. So we need to find
the set Qmatch , containing pairs (r1 , r2 ) with r1 ? Q? , r2 ? QExp , so that r1 and r2
are close to each other and describe the location of the same vesicle. We compute
this with a simple, greedy but fast algorithm:
? compute the matrix Dij = kri ? rj k for all ri ? Q? , rj ? QExp
? while Dij = min D ? dm
? put (ri , rj ) into Qmatch
? fill i-th row and j-th column of D with +?
The resulting pairs of matching locations are closer than dm , which should be set
approximately to the radius of a vesicle. This algorithm does not generally find
the global optimal assignment, which would be a NP-complete problem, but for
low point densities the error made by this algorithm is usually low. Now we can
evaluate the fraction of correctly detected and the fraction of false positives,
fc =
#Qmatch
,
#QExp
ff p = 1 ?
#Qmatch
,
#Q?
(12)
where # denotes the cardinality of the set. Depending on the threshold ?, #Q ?
may change and so does #Qmatch . So we get different values for fc and ff p . We
summarize these two rates in a diagram, which we call, following [10], Receiver
Operating Characteristic (ROC). In comparison to [10], fc represents the hit rate
and ff p represents the false alarm rate, cf. Fig. 3. However, our ROC differs in
some aspects. fc does not need to reach 1 for arbitrary low thresholds, as it is
restricted by the set Qd , which does not need to contain a match to all elements
of QExp . Furthermore, raising the threshold (decreasing #Q? ) may occasionally
increase #Qmatch due to the greedy matching algorithm. These artifacts yield
nonmonotonic parts in the ROC. If no a priori costs are assigned to fc and ff p ,
then a natural measure for quality is the area below the ROC, which would be close
to 1 at best, and 0 if no match would be contained in Qd .
4.3
Results
The ROC of the validation with ter04 and mutant is shown in Fig. 3. The rates
fc and ff p were computed for 50 different threshold values, covering the interval
[minr?Qd z(r), maxr?Qd z(r)]. For ter04 there exist four, and for mutant two, human expert labelings. Therefore we can plot either four or two curves, respectively,
and get an impression about the variance of our performance measure, the area
below the curve. Furthermore the multiple hand labelings allow us to plot them
Figure 2: left: Photoreceptor terminal a of mutant type (mutant). right: Close up
of the left panel, showing labels set by a human (+) and labels found by our method
(). Threshold ? was 0.3, which yields fc ? 1 ? ff p in this case.
against each other in the same figure (single crosses). They indicate what performance is achievable at best. A curve passing these points can be considered to
do the task as well on average as a human does. One can see that for the wild
type image the curve gets close to that region. For the mutant the performance is
slightly worse, in terms of the area. In mutants not only the number of vesicles, but
typically also their shape and appearance differ. This variability was not covered
by the training set and had to be generalized from the wild type data.
5
Discussion
We showed that SVR, used as a nonlinear filter, was able to detect synaptic vesicles
in electron micrographs with high accuracy. On the one hand, for good performance
the ability of the SVR to learn the input/output mapping properly is crucial. On
the other hand it is necessary that in the input image a small neighborhood contains
sufficient information about the target. Due to the ?curse of dimensionality? (cf.
[5]) the receptive field P must not be too large, unless there is a huge amount of
training data. A smaller input dimension P would make the learning easier, but if
P is too small the information that x(r) contains about y(r) may be too small and
the performance poor. For the presented application patch size P = 50 ? 50 was a
good tradeoff. Note that, since we do the filtering in the frequency domain, the size
of P has, in contrast to the number of SVs, no direct influence on the computation
time needed for filtering. Thus, we have a 2500 dimensional input space and only
286 points in this space, that describe a vesicle. Clearly, only a model with low
complexity would achieve acceptable generalization, and this is what we used. In
fact the best linear SVR, i.e. the best linear filter, which has an even much lower
complexity, still yields a performance of Ater04 = 0.82 and Amutant = 0.74 (cf.
Fig. 3, Ater04 = 0.85 . . . 0.89, Ater04 = 0.76 . . . 0.83). However, for future work we
plan to extend the training set significantly. To do so, we have access to hand
labelings for a broad variety of images of different mutants, also including slightly
different scalings. With such more training data the nonlinear SVR can get more
complex without loss of generalization performance. The capacity of the linear filter,
however, cannot grow any further. Thus we expect the performance gap between
nonlinear and linear filtering to grow significantly.
1
1
0.9
0.8
0.8
0.7
A1 = 0.863
A2 = 0.848
A3 = 0.838
A4 = 0.889
0.6
0.4
A1=0.826
A2=0.765
0.6
0.5
0.4
0.3
0.2
0.2
0
0
0.2
0.4
0.6
0.8
1
0.1
0
0.2
0.4
0.6
0.8
1
Figure 3: ROC of the validation with ter04 (left) and with mutant (right). For
various thresholds ?, fc is plotted on the x-axis versus 1 ? ff p on the y-axis. The
single crosses show the fraction of matching labels for every pair of hand labels of
ter04. For detailed explanation, see text.
Acknowledgments
Support Contributed By: BMBF grant 0311559 (R.V., M.S., K.O.) and NIH grant
EY-03592; Killam Trust (I.A.M.)
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[7] J. Mercer. Functions of positive and negative type and their connection with the theory of integral equations. Philosophical Transactions of the Royal Society of London
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[8] William H. Press, Saul A. Teukolsky, William T. Vetterling, and Brian P. Flannery.
Numerical Recipes in C. Cambridge University Press, 2nd. edition, 1992.
[9] Chih-Chung Chang and Chih-Jen Lin. LIBSVM ? A Library for Support Vector
Machines. http://www.csie.ntu.edu.tw/?cjlin/libsvm/, April 2003.
[10] L. O. Harvey, Jr. The critical operating characteristic and the evaluation of expert
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1992.
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1,654 | 2,503 | Extending Q-Learning to General Adaptive
Multi-Agent Systems
Gerald Tesauro
IBM Thomas J. Watson Research Center
19 Skyline Drive, Hawthorne, NY 10532 USA
[email protected]
Abstract
Recent multi-agent extensions of Q-Learning require knowledge of other
agents? payoffs and Q-functions, and assume game-theoretic play at
all times by all other agents. This paper proposes a fundamentally
different approach, dubbed ?Hyper-Q? Learning, in which values of
mixed strategies rather than base actions are learned, and in which other
agents? strategies are estimated from observed actions via Bayesian inference. Hyper-Q may be effective against many different types of adaptive agents, even if they are persistently dynamic. Against certain broad
categories of adaptation, it is argued that Hyper-Q may converge to exact optimal time-varying policies. In tests using Rock-Paper-Scissors,
Hyper-Q learns to significantly exploit an Infinitesimal Gradient Ascent
(IGA) player, as well as a Policy Hill Climber (PHC) player. Preliminary
analysis of Hyper-Q against itself is also presented.
1
Introduction
The question of how agents may adapt their strategic behavior while interacting with other
arbitrarily adapting agents is a major challenge in both machine learning and multi-agent
systems research. While game theory provides a pricipled calculation of Nash equilibrium
strategies, it is limited in practical use due to hidden or imperfect state information, and
computational intractability. Trial-and-error learning could develop good strategies by trying many actions in a number of environmental states, and observing which actions, in
combination with actions of other agents, lead to high cumulative reward. This is highly
effective for a single learner in a stationary environment, where algorithms such as QLearning [13] are able to learn optimal policies on-line without a model of the environment.
Straight off-the-shelf use of RL algorithms such as Q-learning is problematic, however, because: (a) they learn deterministic policies, whereas mixed strategies are generally needed;
(b) the environment is generally non-stationary due to adaptation of other agents.
Several multi-agent extensions of Q-Learning have recently been published. Littman [7]
developed a convergent algorithm for two-player zero-sum games. Hu and Wellman [5]
present an algorithm for two-player general-sum games, the convergence of which was
clarified by Bowling [1]. Littman [8] also developed a convergent many-agent ?friend-orfoe? Q-learning algorithm combining cooperative learning with adversarial learning. These
all extend the normal Q-function of state-action pairs Q(s, a) to a function of states and
joint actions of all agents, Q(s, a). These algorithms make a number of strong assumptions
which facilitate convergence proofs, but which may not be realistic in practice. These
include: (1) other agents? payoffs are fully observable; (2) all agents use the same learning
algorithm; (3) during learning, other agents? strategies are derivable via game-theoretic
analysis of the current Q-functions. In particular, if the other agents employ non-gametheoretic or nonstationary strategies, the learned Q-functions will not accurately represent
the expected payoffs obtained by playing against such agents, and the associated greedy
policies will not correspond to best-reponse play against the other agents.
The aim of this paper is to develop more general and practical extensions of Q-learning
avoiding the above assumptions. The multi-agent environment is modeled as a repeated
stochastic game in which other agents? actions are observable, but not their payoffs. Other
agents are assumed to learn, but the forms of their learning algorithms are unknown, and
their strategies may be asymptotically non-stationary. During learning, it is proposed to estimate other agents? current strategies from observation instead of game-theoretic analysis.
The above considerations lead to a new algorithm, presented in Section 2 of the paper,
called ?Hyper-Q Learning.? Its key idea is to learn the value of joint mixed strategies, rather
than joint base actions. Section 3 discusses the effects of function approximation, exploration, and other agents? strategy dynamics on Hyper-Q?s convergence. Section 4 presents
a Bayesian inference method for estimating other agents? strategies, by applying a recencyweighted version of Bayes? rule to the observed action sequence. Section 5 discusses implementation details of Hyper-Q in a simple Rock-Paper-Scissors test domain. Test results
are presented against two recent algorithms for learning mixed strategies: Infinitesimal
Gradient Ascent (IGA) [10], and Policy Hill Climbing (PHC) [2]. Preliminary results of
Hyper-Q vs. itself are also discussed. Concluding remarks are given in section 6.
2
General Hyper-Q formulation
An agent using normal Q-learning in a finite MDP repeatedly observes a state s, chooses
a legal action a, and then observes an immediate reward r and a transition to a new state
s . The Q-learning equation is given by: ?Q(s, a) = ?(t)[r + ? maxb Q(s , b) ? Q(s, a)],
where ? is a discount parameter, and ?(t) is an appropriate learning rate schedule. Given a
suitable method of exploring state-action pairs, Q-learning is guaranteed to converge to the
optimal value function Q? , and its associated greedy policy is thus an optimal policy ?? .
The multi-agent generalization of an MDP is called a stochastic game, in which each agent i
chooses an action ai in state s. Payoffs ri to agent i and state transitions are now functions
of joint actions of all agents. An important special class of stochastic games are matrix
games, in which |S| = 1 and payoffs are functions only of joint actions. Rather than
choosing the best action in a given state, an agent?s task in a stochastic game is to choose the
best mixed strategy xi = xi (s) given the expected mixed strategy x?i (s) of all other agents.
Here xi denotes a set a probabilities summing to 1 for selecting each of the Ni = Ni (s)
legal actions in state s. The space of possible mixed strategies is a continuous (Ni ? 1)
dimensional unit simplex, and choosing the best mixed strategy is clearly more complex
than choosing the best base action.
We now consider extensions of Q-learning to stochastic games. Given that the agent needs
to learn a mixed strategy, which may depend on the mixed strategies of other agents, an
obvious idea is to have the Q-function evaluate entire mixed strategies, rather than base
actions, and to include in the ?state? description an observation or estimate of the other
agents? current mixed strategy. This forms the basis of the proposed Hyper-Q learning
algorithm, which is formulated as follows. For notational simplicity, let x denote the HyperQ learner?s current mixed strategy, and let y denote an estimated joint mixed strategy of all
other agents (hereafter referred to as ?opponents?). At time t, the agent generates a base
action according to x, and then observes a payoff r, a new state s , and a new estimated
opponent strategy y . The Hyper-Q function Q(s, y, x) is then adjusted according to:
?Q(s, y, x) = ?(t)[r + ? max
Q(s , y , x ) ? Q(s, y, x)]
x
(1)
The greedy policy x
? associated with any Hyper-Q function is then defined by:
x
?(s, y) = arg max Q(s, y, x)
x
3
3.1
(2)
Convergence of Hyper-Q Learning
Function approximation
Since Hyper-Q is a function of continuous mixed strategies, one would expect it to require
some sort of function approximation scheme. Establishing convergence of Q-learning with
function approximation is substantially more difficult than for a normal Q-table for a finite
MDP, and there are a number of well-known counterexamples. In particular, finite discretization may cause a loss of an MDP?s Markov property [9].
Several recent function approximation schemes [11, 12] enable Q-learning to work well in
continuous spaces. There is a least one discretization scheme, Finite Difference Reinforcement Learning [9], that provably converges to the optimal value function of the underlying
continuous MDP. This paper employs a simple uniform grid discretization of the mixed
strategies of the Hyper-Q agent and its opponents. No attempt will be made to prove convergence under this scheme. However, for certain types of opponent dynamics described
below, a plausible conjecture is that a Finite-Difference-RL implementation of Hyper-Q
will be provably convergent.
3.2
Exploration
Convergence of normal Q-learning requires visiting every state-action pair infinitely often.
The clearest way to achieve this in simulation is via exploring starts, in which training consists of many episodes, each starting from a randomly selected state-action pair. For real
environments where this may not be feasible, one may utilize off-policy randomized exploration, e.g., -greedy policies. This will ensure that, for all visited states, every action will
be tried infinitely often, but does not guarantee that all states will be visited infinitely often
(unless the MDP has an ergodicity property). As a result one would not expect the trained
Q function to exactly match the ideal optimal Q? for the MDP, although the difference in
expected payoffs of the respective policies should be vanishingly small.
The above considerations should apply equally to Hyper-Q learning. The use of exploring starts for states, agent and opponent mixed strategies should guarantee sufficient exploration of the state-action space. Without exploring starts, the agent can use -greedy
exploration to at least obtain sufficient exploration of its own mixed strategy space. If
the opponents also do similar exploration, the situation should be equivalent to normal Qlearning, where some stochastic game states might not be visited infinitely often, but the
cost in expected payoff should be vanishingly small. If the opponents do not explore, the
effect could be a further reduction in effective state space explored by the Hyper-Q agent
(where ?effective state? = stochastic game state plus opponent strategy state). Again this
should have a negligible effect on the agent?s long-run expected payoff relative to the policy
that would have been learned with opponent exploration.
3.3
Opponent strategy dynamics
Since opponent strategies can be governed by arbitrarily complicated dynamical rules, it
seems unlikely that Hyper-Q learning will converge for arbitrary opponents. Nevertheless,
some broad categories can be identified under which convergence should be achievable.
One simple example is that of a stationary opponent strategy, i.e., y(s) is a constant. In this
case, the stochastic game obviously reduces to an equivalent MDP with stationary state
transitions and stationary payoffs, and with the appropriate conditions on exploration and
learning rates, Hyper-Q will clearly converge to the optimal value function.
Another important broad class of dynamics consists of opponent strategies that evolve according to a fixed, history-independent rule depending only on themselves and not on actions of the Hyper-Q player, i.e., yt+1 = f (s, yt ). This is a reasonable approximation for
many-player games in which any individual has negligible ?market impact,? or in which a
player?s influence on another player occurs only through a global summarization function
[6]. In such cases the relevant population strategy representation need only express global
summarizations of actitivy (e.g. averages), not details of which player does what. An example is the ?Replicator Dynamics? model from evolutionary game theory [14], in which
a strategy grows or decays in a population according to its fitness relative to the population average fitness. This leads to a history independent first order differential equation
y? = f (y) for the population average strategy. In such models, the Hyper-Q learner again
faces an effective MDP in which the effective state (s, y) undergoes stationary historyindependent transitions, so that Hyper-Q should be able to converge.
A final interesting class of dynamics occurs when the opponent can accurately estimate
the Hyper-Q strategy x, and then adapts its strategy using a fixed history-independent
rule: yt+1 = f (s, yt , xt ). This can occur if players are required to announce their mixed
strategies, or if the Hyper-Q player voluntarily announces its strategy. An example is the
Infinitesimal Gradient Ascent (IGA) model [10], in which the agent uses knowledge of the
current strategy pair (x, y) to make a small change in its strategy in the direction of the gradient of immediate payoff P (x, y). Once again, this type of model reduces to an MDP with
stationary history-independent transitions of effective state depending only on (s, y, x).
Note that the above claims of reduction to an MDP depend on the Hyper-Q learner being
able to accurately estimate the opponent mixed strategy y. Otherwise, the Hyper-Q learner
would face a POMDP situation, and standard convergence proofs would not apply.
4
Opponent strategy estimation
We now consider estimation of opponent strategies from the history of base actions. One
approach to this is model-based, i.e., to consider a class of explicit dynamical models of
opponent strategy, and choose the model that best fits the observed data. There are two
difficult aspects to this approach: (1) the class of possible dynamical models may need to
be extraordinarily large; (2) there is a well-known danger of ?infinite regress? of opponent
models if A?s model of B attempts to take into account B?s model of A.
An alternative approach studied here is model-free strategy estimation. This is in keeping
with the spirit of Q-learning, which learns state valuations without explicitly modeling the
dynamics of the underlying state transitions. One simple method used in the following
section is the well-known Exponential Moving Average (EMA) technique. This maintains
a moving average y? of opponent strategy by updating after each observed action using:
y?(t + 1) = (1 ? ?)?
y (t) + ?ua (t)
(3)
where ua (t) is a unit vector representation of the base action a. EMA assumes only that
recent observations are more informative than older observations, and should give accurate
estimates when significant strategy changes take place on time scales > O(1/?).
4.1
Bayesian strategy estimation
A more principled model-free alternative to EMA is now presented. We assume a discrete
set of possible values of y (e.g. a uniform grid). A probability for each y given the history
of observed actions H, P (y|H), can then be computed using Bayes? rule as follows:
P (y|H) =
P (H|y)P (y)
y P (H|y )P (y )
(4)
where P (y) is the prior probability of state y, and the sum over y extends over all strategy
grid points. The conditional probability of the history given the strategy, P (H|y), can
t
now be decomposed into a product of individual action probabilities k=0 P (a(k)|y(t))
assuming conditional independence of the individual actions. If all actions in the history
are equally informative regardless of age, we may write P (a(k)|y(t)) = ya(k) (t) for all
k. This corresponds to a Naive-Bayes equal weighting of all observed actions. However, it
is again reasonable to assume that more recent actions are more informative. The way to
implement this in a Bayesian context is with exponent weights wk that increase with k [4].
Within a normalization factor, we then write:
P (H|y) =
t
k=0
wk
ya(k)
(5)
A linear schedule wk = 1 ? ?(t ? k) for the weights is intuitively obvious; truncation of
the history at the most recent 1/? observations ensures that all weights are positive.
5
Implementation and Results
We now examine the performance of Hyper-Q learning in a simple two-player matrix game,
Rock-Paper-Scissors. A uniform grid discretization of size N = 25 is used to represent
mixed-strategy component probabilities, giving a simplex grid of size N (N + 1)/2 = 325
for either player?s mixed strategy, and thus the entire Hyper-Q table is of size (325)2 =
105625. All simulations use ? = 0.9, and for simplicity, a constant learning rate ? = 0.01.
5.1
Hyper-Q/Bayes formulation
Three different opponent estimation schemes were used with Hyper-Q learning: (1) ?Omniscient,? i.e. perfect knowledge of the opponent?s strategy; (2) EMA, using equation 3
with ? = 0.005; (3) Bayesian, using equations 4 and 5 with ? = 0.005 and a uniform
prior. Equations 1 and 2 were modified in the Bayesian case to allow for a distribution of
opponent states y, with probabilities P (y|H). The corresponding equations are:
?Q(y, x) = ?(t)P (y|H)[r + ? max
Q(y , x ) ? Q(y, x)]
x
? = arg max
x
x
P (y|H)Q(y, x)
(6)
(7)
y
A technical note regarding equation 6 is that, to improve tractability of the algorithm, an approximation P (y|H) ? P (y |H ) is used, so that the Hyper-Q table updates are performed
using the updated distribution P (y |H ).
5.2 Rock-Paper-Scissors results
We first examine Hyper-Q training online against an IGA player. Apart from possible state
observability and discretization issues, Hyper-Q should in principle be able to converge
against this type of opponent. In order to conform to the original implicit assumptions
underlying IGA, the IGA player is allowed to have omniscient knowledge of the Hyper-Q
player?s mixed strategy at each time step. Policies used by both players are always greedy,
apart from resets to uniform random values every 1000 time steps.
Figure 1 shows a smoothed plot of the online Bellman error, and the Hyper-Q player?s
average reward per time step, as a function of training time. The figure exhibits good
Hyper-Q vs. IGA: Online Bellman error
0.45
Hyper-Q vs. IGA: Avg. reward per time step
0.04
?Omniscient?
?EMA?
?Bayes?
0.4
0.03
0.02
0.35
0.01
0.3
0
0.25
-0.01
0.2
-0.02
0.15
-0.03
0.1
-0.04
0.05
-0.05
0
-0.06
0
400000
800000
1.2e+06
1.6e+06
?Omniscient?
?EMA?
?Bayes?
0
400000
Time Steps
800000
1.2e+06
1.6e+06
Time Steps
Figure 1: Results of Hyper-Q learning vs. an IGA player in Rock-Paper-Scissors, using
three different opponent state estimation methods: ?Omniscient,? ?EMA? and ?Bayes? as
indicated. Random strategy restarts occur every 1000 time steps. Left plot shows smoothed
online Bellman error. Right plot shows average Hyper-Q reward per time step.
Asymptotic IGA Trajectory
0.7
?IGA_Rock_Prob?
?IGA_Paper_Prob?
?HyperQ_Reward?
0.65
0.6
0.55
0.5
0.45
0.4
0.35
0.3
0.25
0.2
0
10000
20000
Time Steps
30000
40000
Figure 2: Trajectory of the IGA mixed strategy against the Hyper-Q strategy starting from
a single exploring start. Dots show Hyper-Q player?s cumulative (rescaled) reward.
progress toward convergence, as suggested by substantially reduced Bellman error and
substantial positive average reward per time step. Among the three estimation methods
used, Bayes reached the lowest Bellman error at long time scales. This is probably because
it updates many elements in the Hyper-Q table per time step, whereas the other techniques
only update a single element. Bayes also has by far the worst average reward at the start of
learning, but asymptotically it clearly outperforms EMA, and comes close to matching the
performance obtained with omniscient knowledge of opponent state.
Part of Hyper-Q?s advantage comes from exploiting transient behavior starting from a random initial condition. In addition, Hyper-Q also exploits the asymptotic behavior of IGA,
as shown in figure 2. This plot shows that the initial transient lasts at most a few thousand
time steps. Afterwards, the Hyper-Q policy causes IGA to cycle erraticly between two different probabilites for Rock and two different probabilities for Paper, thus preventing IGA
from reaching the Nash mixed strategy. The overall profit to Hyper-Q during this cycling
is positive on average, as shown by rising cumulative Hyper-Q reward. The observed cycling with positive profitability is reminiscent of an algorithm called PHC-Exploiter [3] in
play against a PHC player. An interesting difference is that PHC-Exploiter uses an explicit
model of its opponent?s behavior, whereas no such model is needed by a Hyper-Q learner.
Hyper-Q vs. PHC: Online Bellman error
0.4
Hyper-Q vs. PHC: Avg. reward per time step
0.2
?Omniscient?
?EMA?
?Bayes?
0.35
0.15
0.3
0.1
0.25
0.05
0.2
0
0.15
-0.05
?Omniscient?
?EMA?
?Bayes?
0.1
-0.1
0.05
400000
400000
800000
1.2e+06
800000
Time Steps
1.2e+06
Figure 3: Results of Hyper-Q vs. PHC in Rock-Paper-Scissors. Left plot shows smoothed
online Bellman error. Right plot shows average Hyper-Q reward per time step.
We now exmamine Hyper-Q vs. a PHC player. PHC is a simple adaptive strategy based
only on its own actions and rewards. It maintains a Q-table of values for each of its base
actions, and at every time step, it adjusts its mixed strategy by a small step towards the
greedy policy of its current Q-function. The PHC strategy is history-dependent, so that
reduction to an MDP is not possible for the Hyper-Q learner. Nevertheless Hyper-Q does
exhibit substantial reduction in Bellman error, and also significantly exploits PHC in terms
of average reward, as shown in figure 3. Given that PHC ignores opponent state, it should
be a weak competitive player, and in fact it does much worse in average reward than IGA.
It is also interesting to note that Bayesian estimation once again clearly outperforms EMA
estimation, and surprisingly, it also outperforms omniscient state knowledge. This is not
yet understood and is a focus of ongoing research.
Hyper-Q/Omniscient vs. itself: Online Bellman error
Hyper-Q/Bayes vs. itself: Online Bellman error
0.8
0.4
0.7
0.35
0.6
0.3
0.5
0.25
0.4
0.2
0.3
0.15
0.2
0.1
0.1
0.05
0
0
0
400000
800000
1.2e+06
1.6e+06
0
400000
800000
1.2e+06
1.6e+06
Figure 4: Smoothed online Bellman error for Hyper-Q vs. itself. Left plot uses Omniscient
state estimation; right plot uses Bayesian estimation.
Finally, we examine preliminary data for Hyper-Q vs. itself. The average reward plots
are uninteresting: as one would expect, each player?s average reward is close to zero. The
online Bellman error, shown in figure 4, is more interesting. Surprisingly, the plots are less
noisy and achieve asymptotic errors as low or lower than against either IGA or PHC. Since
Hyper-Q?s play is history-dependent, one can?t argue for MDP equivalence. However, it is
possible that the players? greedy policies x
?(y) and y?(x) simultaneously become stationary,
thereby enabling them to optimize against each other. In examining the actual play, it
does not converge to the Nash point ( 13 , 13 , 13 ), but it does appear to cycle amongst a small
number of grid points with roughly zero average reward over the cycle for both players.
Conceivably, Hyper-Q could have converged to a cyclic Nash equilibrium of the repeated
game, which would certainly be a nice outcome of self-play learning in a repeated game.
6
Conclusion
Hyper-Q Learning appears to be more versatile and general-purpose than any published
multi-agent extension of Q-Learning to date. With grid discretization it scales badly but
with other function approximators it may become practical. Some tantalizing early results
were found in Rock-Paper-Scissors tests against some recently published adaptive opponents, and also against itself. Research on this topic is very much a work in progress. Vastly
more research is needed, to develop a satisfactory theoretical analysis of the approach, an
understanding of what kinds of realistic environments it can be expcted to do well in, and
versions of the algorithm that can be successfully deployed in those environments.
Significant improvements in opponent state estimation should be easy to obtain. More
principled methods for setting recency weights should be achievable; for example, [4] proposes a method for training optimal weight values based on observed data. The use of
time-series prediction and data mining methods might also result in substantially better
estimators. Model-based estimators are also likely to be advantageous where one has a
reasonable basis for modeling the opponents? dynamical behavior.
Acknowledgements: The author thanks Michael Littman for many helpful discussions;
Irina Rish for insights into Bayesian state estimation; and Michael Bowling for assistance
in implementing the PHC algorithm.
References
[1] M. Bowling. Convergence problems of general-sum multiagent reinforcement learning. In
Proceedings of ICML-00, pages 89?94, 2000.
[2] M. Bowling and M. Veloso. Multiagent learning using a variable learning rate. Artificial Intelligence, 136:215?250, 2002.
[3] Y.-H. Chang and L. P. Kaelbling. Playing is believing: the role of beliefs in multi-agent learning.
In Proceedings of NIPS-2001. MIT Press, 2002.
[4] S. J. Hong, J. Hosking, and R. Natarajan. Multiplicative adjustment of class probability: educating naive Bayes. Technical Report RC-22393, IBM Research, 2002.
[5] J. Hu and M. P. Wellman. Multiagent reinforcement learning: theoretical framework and an
algorithm. In Proceedings of ICML-98, pages 242?250. Morgan Kaufmann, 1998.
[6] M. Kearns and Y. Mansour. Efficient Nash computation in large population games with bounded
influence. In Proceedings of UAI-02, pages 259?266, 2002.
[7] M. L. Littman. Markov games as a framework for multi-agent reinforcement learning. In
Proceedings of ICML-94, pages 157?163. Morgan Kaufmann, 1994.
[8] M. L. Littman. Friend-or-Foe Q-learning in general-sum games. In Proceedings of ICML-01.
Morgan Kaufmann, 2001.
[9] R. Munos. A convergent reinforcement learning algorithm in the continuous case based on a
finite difference method. In Proceedings of IJCAI-97, pages 826?831. Morgan Kaufman, 1997.
[10] S. Singh, M. Kearns, and Y. Mansour. Nash convergence of gradient dynamics in general-sum
games. In Proceedings of UAI-2000, pages 541?548. Morgan Kaufman, 2000.
[11] W. D. Smart and L. P. Kaelbling. Practical reinforcement learning in continuous spaces. In
Proceedings of ICML-00, pages 903?910, 2000.
[12] W. T. B. Uther and M. M. Veloso. Tree based discretization for continuous state space reinforcement learning. In Proceedings of AAAI-98, pages 769?774, 1998.
[13] C. Watkins. Learning from Delayed Rewards. PhD thesis, Cambridge University, 1989.
[14] J. W. Weibull. Evolutionary Game Theory. The MIT Press, 1995.
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1,655 | 2,504 | Measure Based Regularization
Olivier Bousquet, Olivier Chapelle, Matthias Hein
Max Planck Institute for Biological Cybernetics, 72076 T?
ubingen, Germany
{first.last}@tuebingen.mpg.de
Abstract
We address in this paper the question of how the knowledge of
the marginal distribution P (x) can be incorporated in a learning
algorithm. We suggest three theoretical methods for taking into
account this distribution for regularization and provide links to
existing graph-based semi-supervised learning algorithms. We also
propose practical implementations.
1
Introduction
Most existing learning algorithms perform a trade-off between fit of the data and
?complexity? of the solution. The way this complexity is defined varies from one
algorithm to the other and is usually referred to as a prior probability or a regularizer. The choice of this term amounts to having a preference for certain solutions
and there is no a priori best such choice since it depends on the learning problem
to be addressed. This means that the right choice should be dictated by prior
knowledge or assumptions about the problem or the class of problems to which
the algorithm is to be applied. Let us consider the binary classification setting. A
typical assumption that is (at least implicitly) used in many learning algorithms is
the following
Two points that are close in input space should have the same label.
One possible way to enforce this assumption is to look for a decision function which
is consistent with the training data and which does not change too much between
neighboring points. This can be done in a regularization setting, using the Lipschitz norm as a regularizer. For differentiable functions, the Lipschitz norm of a
function is the supremum of the norm of the gradient. It is thus natural to consider
algorithms of the form
min sup k?f (x)k under constraints yi f (xi ) ? 1.
f
(1)
x
Performing such a minimization on the set of linear functions leads to the maximum
margin solution (since the gradient x 7? hw, xi is w), whereas the 1-nearest neighbor
decision function is one of the solutions of the above optimization problem when
the set of functions is unconstrained [13].
Although very useful because widely applicable, the above assumption is sometimes
too weak. Indeed, most ?real-world? learning problems have more structure than
what this assumption captures. For example, most data is located in regions where
the label is constant (clusters) and regions where the label is not well-defined are
typically of low density. This can be formulated via the so-called cluster assumption:
Two points that are connected by a line that goes through high density regions should have the same label
Another related way of stating this assumption is to say that the decision boundary
should lie in regions of low density.
Our goal is to propose possible implementations of this assumption. It is important
to notice that in the context of supervised learning, the knowledge of the joint probability P (x, y) is enough to achieve perfect classification (taking arg maxy P (x, y)
as decision function), while in semi-supervised learning, even if one knows the distribution P (x) of the instances, there is no unique or optimal way of using it. We
will thus try to propose a principled approach to this problem. A similar attempt
was made in [10] but in a probabilistic context, where the decision function was
modeled by a conditional probability distribution, while here we consider arbitrary
real-valued functions and use the standard regularization approach.
We will use three methods for obtaining regularizers that depend on the distribution P (x) of the data. In section 2 we suggest to modify the regularizer in a
general way by weighting it with the data density. Then in section 3 we adopt a
geometric approach where we suggest to modify the distances in input space (in
a local manner) to take into account the density (i.e. we stretch or blow up the
space depending on the density). The third approach presented in section 4 builds
on spectral methods. The idea is to look for the analogue of graph-based spectral
methods when the amount of available data is infinite. We show that these three
approaches are related in various ways and in particular we clarify the asymptotic
behavior of graph-based regularization. Finally, in section 5 we give a practical
method for implementing one of the proposed regularizers and show its application
on a toy problem.
2
Density based regularization
The first approach we propose is to start with a gradient-based regularizer like
k?f k which penalizes large variations of the function. Now, to implement the
cluster assumption one has to penalize more the variations of the function in high
density regions and less in low density regions. A natural way of doing this is
to replace k?f k by kp?f k where p is the density of the marginal distribution P .
More generally, instead of the gradient, one can can consider a regularization map
X
L : RX 7? (R+ ) , where L(f )(x) is a measure of the smoothness of the function f
at the point x, and then consider the following regularization term
?(f ) = k L(f )?(p) k ,
(2)
where ? is a strictly increasing function.
An interesting case is when the norm in (2) is chosen as the L2 norm. Then, ?(f )
can be the norm of a Reproducing Kernel Hilbert Space (RKHS), which means that
there exist an Hilbert space H and a kernel function k : X 2 7? R such that
q
hf, f iH = ?(f ) and hf, k(x, ?)iH = f (x).
(3)
The reason for using an RKHS norm is the so-called representer theorem [5]: the
function minimizing the corresponding regularized loss can be expressed as a linear
combination of the kernel function evaluated at the labeled points.
However, it is not straightforward to find the kernel associated with an RKHS norm.
In general, one has to solve equation (3). For instance, in the case L(f ) = (f 2 +
2
k?f k )1/2 and without taking the density into account (? = 1), it has been shown in
[3] that the corresponding kernel is the Laplacian one, k(x, y) = exp(? kx ? ykL1 )
with associated inner product hf, giH = hf, giL2 + h?f, ?giL2 . Taking the density
into account, this inner product becomes
hf, giH = f, ?2 (p)g L2 + ?f, ?2 (p)?g L2 .
Plugging g = k(x, .) in above and expressing that (3) should be valid for all f ? H,
we find that k must satisfy
?2 (p)k(x, .) ? ?(?2 (p)?k(x, .)) = ?(x ? .),
where ? is the Dirac delta function. However, solving this differential equation is
not an easy task for arbitrary p.
Since finding the kernel function associated to a regularizer is, in general, a difficult
problem, we propose to perform the minimization of the regularized loss on a fixed
set of basis functions, i.e. f is expressed as a linear combination of functions ? i .
f (x) =
l
X
?i ?i (x) + b.
(4)
i=1
We will present in section 5 a practical implementation of this approach.
3
Density based change of geometry
We now try to adopt a geometric point of view. First we translate the cluster assumption into a geometric statement, then we explore how to enforce it by changing
the geometry of our underlying space. A similar approach was recently proposed by
Vincent and Bengio [12]. We will see that there exists such a change of geometry
which leads to the same type of regularizer that was proposed in section 2.
Recall that the cluster assumption states that points are likely to be in the same
class if they can be connected by a path through high density regions. Naturally
this means that we have to weight paths according to the density they are going
through. This leads to introducing a new distance measure on the input space
(typically Rd ) defined as the length of the shortest weighted path connecting two
points. With this new distance, we simply have to enforce that close points have
the same label (we thus recover the standard assumption).
Let us make this more precise. We consider the euclidean space Rd as a flat Riemannian manifold with metric tensor ?, denoted by (Rn , ?). A Riemannian manifold
(M, g) is also a metric space with the following path (or geodesic) distance:
d(x, y) = inf {L(?)|? : [a, b] ? M, ?(a) = x, ?(b) = y}
?
where ? is a piecewise smooth curve and L(?) is the length of the curve given by
Z bq
gij (?(t))?? i ?? j dt
(5)
L(?) =
a
We now want to change the metric ? of Rd such that the new distance is the weighted
path distance corresponding to the cluster assumption. The only information we
have is the local density p(x), which is a scalar at every point and as such can
only lead to an isotropic transformation in the tangent space Tx M. Therefore we
consider the following conformal transformation of the metric ?
1
?ij ? gij =
?ij
(6)
?(p(x))
where ? is a strictly increasing function. We denote by (Rd , g) the distorted euclidean
? space. Note that this kind of transformation also changes the volume element gdx1 . . . dxd , where g is the determinant of gij .
dx1 . . . dxd ?
?
gdx1 . . . dxd =
1
dx1 . . . dxd
?(p)d/2
(7)
In the following we will choose ?(x) = x, which is the simplest choice which gives
the desired properties.
The distance structure of the transformed space implements now the cluster assumption, since we see from (5) that all paths get weighted by the inverse density.
Therefore we can use any metric based classification method and it will automatically take into account the density of the data. For example the nearest neighbor classifier in the new distance is equivalent to the Lipschitz regularization (1)
weighted with the density proposed in the last section.
However, implementing such a method requires to compute the geodesic distance
in (Rd , g), which is non trivial for arbitrary densities p. We suggest the following
approximation which is similar in spirit to the approach in [11].
Since we have a global chart of Rd we can give for each neighborhood B (x) in the
euclidean space the following upper and lower bounds for the geodesic distance:
s
s
1
1
inf
kx ? yk ? d(x, y) ? sup
kx ? yk, ? y ? B (x)
(8)
p(z)
p(z)
z?B (x)
z?B (x)
Then we choose a real and set for each x the distance to all points in a p(x) ?1/2 ?1/2
neighborhood of x as d(x, y) = p( x+y
kx ? yk. The geodesic distance can then
2 )
be approximated by the shortest path along the obtained graph.
We now show the relationship to the the regularization based approach of the previous section. We denote by k?kL2 (Rd ,g,?) the L2 norm in (Rd , g) with respect to
the measure ? and by ? the standard Lebesgue measure on Rd . Let us consider
2
the regularizer k?f kL2 (Rd ,?,?) which is the standard L2 norm of the gradient. Now
modifying this regularizer according to section 2 (by changing the underlying mea2
sure) gives S(f ) = k?f kL2 (Rd ,?,P ) . On the distorted space (Rd , g) we keep the
Lebesgue measure ? which can be done by integrating on the manifold with respect to the density ? = ?1g = pd/2 , which cancels then with the volume element
?
?f ?f
? gdx1 . . . dxd = dx1 . . . dxd . Since we have on (Rd , g), k?f k2 = p(x)? ij ?x
i ?xj we
get equivalence of S(f ).
Z
?f ?f
S(f ) = k?f k2L2 (Rd ,?,P ) =
p(x)? ij i j dx1 . . . dxd = k?f k2L2 (Rd ,g,?)
(9)
?x
?x
d
R
This shows that modifying the measure and keeping the geometry, or modifying
the geometry and keeping the Lebesgue measure leads to the same regularizer S(f ).
However, there is a structural difference between the spaces (Rd , ?, P ) and (Rd , g, ?)
even if S(f ) is the same. Indeed, for regularization operators corresponding to
higher order derivatives the above correspondence is not valid any more.
4
Link with Spectral Techniques
Recently, there has been a lot of interest in spectral techniques for non linear dimension reduction, clustering or semi-supervised learning. The general idea of these
approaches is to construct an adjacency graph on the (unlabeled) points whose
weights are given by a matrix W . Then the first eigenvectors of a modified version
of W give a more suitable representation of the points (taking into account their
manifold and/or cluster structure). An instance of such an approach and related
references are given in [1] where the authors propose to use the following regularizer
m
1 X
(fi ? fj )2 Wij = f > (D ? W )f ,
2 i,j=1
(10)
where fi is the value of the function at point xi (the index
P ranges over labeled and
unlabeled points), D is a diagonal matrix with Dii = j Wij and Wij is chosen as
a function of the distance between xi and xj , for example Wij = K(kxi ? xj k /t).
Given a sample x1 , . . . , xm of m i.i.d. instances sampled according to P (x), it is
possible to rewrite (10) after normalization as the following random variable
X
1
Uf =
(f (xi ) ? f (xj ))2 K(kxi ? xj k /t) .
2m(m ? 1) i,j
Under the assumption that f and K are bounded, the result of [4] (see Inequality
(5.7) in this paper, which applies to U-statistics) gives
P [Uf ? E [Uf ] + t] ? e?mt
2
/C 2
,
where C is a constant which does not depend on n and t. This shows that for each
fixed function, the normalized regularizer Uf converges towards its expectation when
the sample size increases. Moreover, one can check that
Z Z
1
E [Uf ] =
(f (x) ? f (y))2 K(kx ? yk /t)dP (x)dP (y) .
(11)
2
This is the term that should be used as a regularizer if one knows the whole distribution since it is the limit of (10)1 .
The following proposition relates the regularizer (11) to the one defined in (2).
Proposition 4.1 If p is a density which is Lipschitz continuous and K is a continuous function on R+ such that x2+d K(x) ? L2 , then for any function f ? C 2 (Rd )
with bounded hessian
Z Z
d
lim
(f (x) ? f (y))2 K(kx ? yk /t)p(x)p(y)dxdy
(12)
t?0 C t2+d
Z
2
= k?f (x)k p2 (x)dx,
(13)
where C =
R
2
Rd
kxk K(kxk) dx.
Proof: Let?s fix x. Writing a Taylor-Lagrange expansion of f and p around x in
terms of h = (y ? x)/t gives
Z
kx ? yk
2
(f (x) ? f (y)) K
p(y)dy
t
Z
2
=
(t h?f (x), hi + O(t2 khk ))2 K(khk)(p(x) + O(t khk)td dh
Z
2
= td+2 p(x) h?f (x), hi K(khk)dh + O(td+3 ) ,
(14)
1
We have shown that the convergence of Uf towards E [Uf ] happens for each fixed f but
this convergence can be uniform over a set of functions, provided this set is small enough.
To
conclude
the
we
rewrite
this
last
integral
as
proof,
R
2
hh> K(khk)dh ?f (x) = k?f (x)k CdR . The last equality comes
?f (x)>
from the fact that, by symmetry considerations, hh> K(khk)dh is equal to a
constant (let?s call it C2 ) R times the identity
can be
matrixR and this constant
computed by C2 d = trace hh> K(khk)dh = trace h> hK(khk)dh = C.
Note that different K lead to different affinity matrices: if we choose K(x) =
exp(?x2 /2), we get a gaussian RBF affinity matrix as used in [7], whereas K(x) =
1x?1 leads to an unweighted neighboring graph (at size t) [1].
So we have proved that if one takes the limit of the regularizer (10) when the sample
size goes to infinity and the scale parameter t goes to 0 (with appropriate scaling),
one obtains the regularizer
Z
2
k?f (x)k p2 (x)dx = f, ?? Dp2 ?f ,
where ?? is the adjoint of ?, Dp is the diagonal operator that maps f to pf and
h., .i is the inner product in L2 .
In [2], the authors investigated the limiting behavior of the regularizer D ? W obtained from the graph and claimed that this is the empirical counterpart of the
Laplace operator defined on the manifold. However, this is true only if the distribution is uniform on the manifold. We have shown that, in the general case, the
continuous equivalent of the graph Laplacian is ?? Dp2 ?.
5
Practical Implementation and Experiments
As mentioned in section 2, it is difficult in general to find the kernel associated with
a given regularizer and instead, we decided to minimize the regularized loss on a
fixed basis of functions (?i )1?i?l , as expressed by equation (4).
The regularizer we considered is of the form (2) and is,
Z
? 2
?(f ) = k k?f k p kL2 = ?f (x) ? ?f (x)p(x)dx.
Thus, the coefficients ? and b in expansion (4) are found by minimizing the following
convex regularized functional
Z
l
n
X
1X
?i ?j ??i (x) ? ??j (x)p(x)dx .
`(f (xi ), yi ) +?
(15)
n i=1
i,j=1
|
{z
}
|
{z
}
? 2
Remp (f )
kL(f ) pkL2
R
Introducing the l ? l matrix Hij = ??i (x) ? ??j (x)p(x)dx and the n ? l matrix
K with Kij = ?j (xi ), the minimization of the functional (15) is equivalent to the
following one for the standard L1 -SVM loss:
n
X
min ?> H? + C
?i
?,b
Pl
i=1
under constraints ?i, yi ( j=1 Kij ?j + b) ? 1 ? ?i . The dual formulation of this
optimization problem turns out to be the standard SVM one with a modified kernel
function (see also [9]):
n
n
X
1 X
max
?i ?
?i ?j yi yj Lij ,
?
2 i,j=1
i=1
2.5
2
1.5
1
0.5
0
?0.5
?1
?1.5
?2
?2.5
?4
?3
?2
?1
0
1
2
3
4
Figure 1: Two moons toy problem: there are 2 labeled points (the cross and the
triangle) and 200 unlabeled points. The gray level corresponds to the output of the
function. The function was expanded on all unlabeled points (m=200 in (4)) and
the widths of the gaussians have been chosen as ? = 0.5 and ?p = 0.05.
P
under constraints 0 ? ?i ? C and
?i yi = 0, with L = KH ?1 K > .
Once the vector ? has been found, the coefficients ? of the expansion are given by
? = H ?1 K > diag(Y )?.
In order to calculate the Hij , one has to compute an integral. From now on, we
consider a special case where this integral can be computed analytically:
2
ik
? The basis functions are gaussian RBF, ?i (x) = exp ? kx?x
, where
2? 2
the points x1 , . . . , xl can be chosen arbitrarily. We decided to take the
unlabeled points (or a subset of them) for this expansion.
? The marginal density p is estimated
using
a Parzen window with a Gaussian
Pm
kx?xi k2
1
exp
?
kernel, p(x) = m
.
i=1
2? 2
p
Defining h = 1/? 2 and hp = 1/?p2 , this integral turns out to be, up to an irrelevant
constant factor,
!
m
2
2
2
X
h2 kxi ? xj k
hhp kxi ? xk k + kxj ? xk k
Hij =
exp ?
?
2h + hp
2
2h + hp
2
k=1
2
2
hp (xk ? xi ) ? (xk ? xj ) ? h(h + hp )(xi ? xj ) + d(2h + hp ) ,
where d is the dimension of the input space.
After careful dataset selection [6], we considered the two moons toy problem (see
figure 1). On this 2D example, the regularizer we suggested implements perfectly
the cluster assumption: the function is smooth on high density regions and the
decision boundary lies in a low density region.
We also tried some real world experiments but were not successful. The reason
might be that in dimension more than 2, the gradient does not yield a suitable
regularizer: there exists non continuous functions whose regularizer is 0. To avoid
this, from the Sobolev embedding lemma, we consider derivatives of order at least
d/2. More specifically, we are currently investigating the regularizer associated with
a Gaussian kernel of width ?r [8, page 100],
Z
?
X
?r2p
2
k?p f (x)k p(x)dx,
p
p!2
p=1
6
with ?2p ? ?p .
Conclusion
We have tried to make a first step towards a theoretical framework for semisupervised learning. Ideally, this framework should be based on general principles
which can then be used to derive new heuristics or justify existing ones.
One such general principle is the cluster assumption. Starting from the assumption
that the distribution P (x) of the data is known, we have proposed several ideas
to implement this principle and shown their relationships. In addition, we have
shown the relationship to the limiting behavior of an algorithm based on the graph
Laplacian.
We believe that this topic deserves further investigation. From a theoretical point
of view, other types of regularizers, involving, for example, higher order derivatives
should be studied. Also from a practical point of view, we should derive efficient
algorithms from the proposed ideas, especially by obtaining finite sample approximations of the limit case where P (x) is known.
References
[1] M. Belkin and P. Niyogi. Laplacian eigenmaps for dimensionality reduction and data
representation. Neural Computation, 15(6):1373?1396, 2003.
[2] M. Belkin and P. Niyogi. Semi-supervised learning on manifolds. Machine Learning
journal, 2003. to appear.
[3] F. Girosi, M. Jones, and T. Poggio. Priors, stabilizers and basis functions: From
regularization to radial, tensor and additive splines. Technical Report Artificial Intelligence Memo 1430, Massachusetts Institute of Technology, 1993.
[4] W. Hoeffding. Probability inequalities for sums of bounded random variables. Journal
of the American Statistical Association, 58:13?30, 1963.
[5] G. Kimeldorf and G. Wahba. Some results on tchebychean spline functions. Journal
of Mathematics Analysis and Applications, 33:82?95, 1971.
[6] Doudou LaLoudouana and Mambobo Bonouliqui Tarare. Data set selection. In
Advances in Neural Information Processing Systems, volume 15, 2002.
[7] A. Y. Ng, M. I. Jordan, and Y. Weiss. On spectral clustering: Analysis and an
algorithm. In Advances in Neural Information Processing Systems, volume 14, 2001.
[8] B. Sch?
olkopf and A. Smola. Learning with kernels. MIT Press, Cambridge, MA, 2002.
[9] A. Smola and B. Scholkopf. On a kernel-based method for pattern recognition, regression, approximation and operator inversion. Algorithmica, 22:211?231, 1998.
[10] M. Szummer and T. Jaakkola. Information regularization with partially labeled data.
In Advances in Neural Information Processing Systems, volume 15. MIT Press, 2002.
[11] J. B. Tenenbaum, V. de Silva, and J. C. Langford. A global geometric framework for
nonlinear dimensionality reduction. Science, 290(5500):2319?2323, 2000.
[12] P. Vincent and Y. Bengio. Density-sensitive metrics and kernels. Presented at the
Snowbird Learning Workshop, 2003.
[13] U. von Luxburg and O. Bousquet. Distance-based classification with lipschitz functions. In Proceedings of the 16th Annual Conference on Computational Learning
Theory, 2003.
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1,656 | 2,505 | Generalised Propagation for Fast Fourier
Transforms with Partial or Missing Data
Amos J Storkey
School of Informatics, University of Edinburgh
5 Forrest Hill, Edinburgh UK
[email protected]
Abstract
Discrete Fourier transforms and other related Fourier methods have
been practically implementable due to the fast Fourier transform
(FFT). However there are many situations where doing fast Fourier
transforms without complete data would be desirable. In this paper it is recognised that formulating the FFT algorithm as a belief
network allows suitable priors to be set for the Fourier coefficients.
Furthermore efficient generalised belief propagation methods between clusters of four nodes enable the Fourier coefficients to be
inferred and the missing data to be estimated in near to O(n log n)
time, where n is the total of the given and missing data points.
This method is compared with a number of common approaches
such as setting missing data to zero or to interpolation. It is tested
on generated data and for a Fourier analysis of a damaged audio
signal.
1
Introduction
The fast Fourier transform is a fundamental component in any numerical toolbox.
Commonly it is thought of as a deterministic transformation from data to Fourier
space. It relies on regularly spaced data, ideally of length 2hi for some hi in each
dimension i. However there are many circumstances where Fourier analysis would
be useful for data which does not take this form. The following are a few examples
of such situations:
?
?
?
?
?
?
There is temporary/regular instrument failure or interruption.
There are scratches on media, such as compact disks.
Missing packets occur in streamed data.
Data is not 2k in length or is from e.g. irregularly shaped image patches.
There is known significant measurement error in the data.
Data is quantised, either in Fourier domain (e.g. jpeg) or data domain (e.g.
integer storage).
Setting missing values to zeros or using interpolation will introduce various biases
which will also affect the results; these approaches can not help in using Fourier
information to help restore the missing data.
Prior information is needed to infer the missing data or the corresponding Fourier
components. However to be practically useful inference must be fast. Ideally we
want techniques which scale close to O(n log n).
The FFT algorithm can be described as a belief network with deterministic connections where each intermediate node has two parents and two children (a form
commonly called the butterfly net). The graphical structure of the FFT has been
detailed before in a number of places. See [1, 5] for examples. Prior distributions for
the Fourier coefficients can be specified. By choosing a suitable cluster set for the
network nodes and doing generalised propagation using these clusters, reasonable
inference can be achieved. In the case that all the data is available this approach is
computationally equivalent to doing the exact FFT.
There have been other uses of belief networks and Bayesian methods to improve
standard transforms. In [2], a hierarchical prior model of wavelet coefficients was
used with some success. Other authors have recognised the problem of missing data
in hierarchical systems. In [6] the authors specify a multiscale stochastic model,
which enables a scale recursive description of a random process. Inference in their
model is propagation within a tree structured belief network. FFT related Toeplitz
methods combined with partition inverse equations are applicable for inference in
grid based Gaussian process systems with missing data [9].
2
Fast Fourier Transform
2.1 The FFT Network
From this point forward the focus will be on the one dimensional fast Fourier
transform. The FFT utilises a simple recursive relationship in order to implement the discrete Fourier transform in O(n log n) time for n = 2h data points. For
W = exp(?2?i/n), the kth Fourier coefficient Fk is given by
def
Fk =
n?1
X
j=0
n/2?1
W kj xj =
X
j=0
n/2?1
W 2kj x2j +
X
W (2j+1)k x2j+1 = Fke + W k Fko
(1)
j=0
where Fke denotes the kth component of the length n/2 Fourier transform of the
even components of xj . Likewise Fko is the same for the odd components. The two
new shorter Fourier transforms can be split in the same way, recursively down to the
transforms of length 1 which are just the data points themselves. It is also worth
noting that Fke and Fko are in fact used twice, as Fk+n/2 = Fke ? W k Fko . The inverse
FFT uses exactly the same algorithm as the FFT, but with conjugate coefficients.
This recursive algorithm can be drawn as a network of dependencies, using the
inverse FFT as a generative model; it takes a set of Fourier components and creates
some data. The usual approach to the FFT is to shuffle the data into reverse bit
order (xi for binary i = 010111 is put in position i0 = 111010; see [8] for more
details). This places data which will be combined in adjacent positions. Doing this,
we get the belief network of Figure 1a as a representation of the dependencies. The
top row of this figure gives the Fourier components in order, and the bottom row
gives the bit reversed data. The intermediate nodes are the even and odd Fourier
coefficients at different levels of recursion.
2.2 A Prior on Fourier Coefficients
The network of Figure 1a, combined with (1), specifies the (deterministic) conditional distributions for all the nodes below the top layer. However no prior distribution is currently set for the top nodes, which denote the Fourier coefficients.
In general little prior phase information is known, but often there might be some
(a)
(b)
Figure 1: (a) The belief network corresponding to the fast Fourier transform. The
top layer are the Fourier components in frequency order. The bottom layer is the
data in bit reversed order. The intermediate layers denote the partial odd and even
transforms that the algorithm uses. (b) The moralised undirected network with
three clusters indicated by the boxes. All nodes surrounded by the same box type
form part of the same cluster.
expected power spectra. For example we might expect a 1/f power spectra, or in
some circumstances empirical priors may be appropriate. For simplicity we choose
independent complex Gaussian1 priors on each of the top nodes. Then the variance
of each prior will represent the magnitude of the expected power of that particular
coefficient.
3
Inference in the FFT Network
Suppose that some of the data which would be needed to perform the FFT is
missing. Then we would want to infer the Fourier coefficients based on the data that
was available. The belief network of Figure 1a is not singly connected and so exact
propagation methods are not appropriate. Forming the full Gaussian distribution of
the whole network and calculating using that is too expensive except in the smallest
situations. Using exact optimisation (eg conjugate gradient) in the conditional
Gaussian system is O(n2 ), although a smaller number of iterations of conjugate
gradient can provide a good approximation. Marrying parents and triangulating
the system will result in a number of large cliques and so junction tree methods will
not work in a reasonable time.
3.1
Loopy Propagation
Loopy propagation [7, 10, 3] in the FFT network suffers from some serious deficits.
Experiments with loopy propagation suggest that often there are convergence problems in the network, especially for systems of any significant size. Sometimes adding
additional jitter and using damping approaches (see e.g. [4]) can help the system to
converge, but convergence is then very slow. Intuitively the approximation given by
loopy propagation fails to capture the moralisation of the parents, which, given the
deterministic network, provides strong couplings. Note that when the system does
converge the mean inferred values are correct [11], but the variances are significantly
underestimated.
A complex Gaussian is of the form exp(?0.5xT C ?1 x)/Z where x is complex, and C
is positive (semi)definite. It is a more restrictive distribution than a general Gaussian in
the complex plane.
1
4
Generalised Belief Propagation for the FFT
In [14] the authors show that stationary points of loopy propagation between nodes
of a Markov network are minimisers of the Bethe free energy of the probabilistic
system. They also show that more general propagation procedures, such as propagation of information between clusters of a network correspond to the minimisation
of a more general Kikuchi free energy of which The Bethe free energy is a special
case.
To overcome the shortfalls of belief propagation methods, a generalised belief propagation scheme is used here. The basic problem is that there is strong dependence
between parents of a given node, and the fact that the values for those two nodes
are fully determined by the two children but undetermined by only one. Hence it
would seem sensible to combine these four nodes, the two parents and two children,
together into one cluster. This can be done for all nodes at all levels, and we find
that the cluster separator between any two clusters consists of at most one node.
At each stage of propagation between clusters only the messages (in each direction)
at single nodes need to be maintained.
The procedure can be summarised as follows: Start with the belief network of
Figure 1a and convert it to an undirected network by moralisation (Figure 1b).
Then we identify the clusters of the graph, which each consist of four nodes as
illustrated by the boxes in Figure 1b. Each cluster consists of two common parents
and their common children. Each node not in the extreme layers is also a separator
between two clusters. Building a network of clusters involves creating an edge for
each separator. From Figure 1 it can be seen that this network will have undirected
loops. Hence belief propagation in this system will not be exact. However it will be
possible to iteratively propagate messages in this system. Hopefully the iteration
will result in an equilibrium being reached which we can use as an approximate
inference for the marginals of the network nodes, although such convergence is not
guaranteed.
4.1
Propagation Equations
This section provides the propagation messages for the approach described above.
For simplicity, and to maintain symmetry we use an update scheme where messages
are first passed down from what were the root nodes (before moralisation) to the
leaf nodes, and then messages are passed up from the leaf to the root nodes. This
process is then iterated. The first pass down the network is data independent and
can be precomputed.
4.1.1
Messages Propagating Down
The Markov network derived from a belief network has the potentials of each cluster
defined by the conditional probability of all the child nodes in that cluster given
their parents. Two adjoining clusters of the network are illustrated in Figure 2a. All
the cluster interactions in the network have this form, and so the message passing
described below applies to all the nodes.
+
The message ?4 ? N (?+
4 , ?4 ) is defined to be that passed down from some cluster C 1
containing nodes y1 , y2 (originally the parents) and y3 , y4 (originally the children)
to the cluster below: C2 = (y4 , y5 , y6 , y7 ), with y6 and y7 the children. ?+
4 is the
message mean, and ?4+ is the covariance. The message is given by the marginal
of the cluster potential multiplied by the incoming messages from the other nodes.
The standard message passing scheme can be followed to get the usual form of
results for Gaussian networks [7, 11].
?
Suppose ?3 (y3 ) = N (y3 ; ??
3 , ?3 ) is the message passing up the network at node
+
+
3, whereas ?1 (y1 ) = N (y1 ; ?1 , ?1+ ) and ?2 (y2 ) = N (y2 ; ?+
2 , ?2 ) are the messages
passing down the network at nodes 1 and 2 respectively. Here we use the notation
of [7] and use ? to represent variances. Defining2
+
+
?1
0
?1
b31 b32
?
?A = B 1
B1 , ?A = B1
where B1 =
(2)
b41 b42
0 ?2+
?+
2
are the connection coefficients derived from (1), and
? ?
1/?3? 0
?3 /?3
?1
?1
??1
=
?
+
,
?
=
?
?
?
+
D
D
A
D
A
A
0
0
0
(3)
allows us to write the downward message as
+
?+
4 = (?D )2 and ?4 = (?D )22 .
4.1.2
(4)
Messages Propagating Up
In the same way we can calculate the messages which are propagated up the network.
?
The message ?4 = N (??
4 , ?4 ) passed up from cluster C2 to cluster C1 is given by
b64 b65
?
?
?4 = (?U )1 and ?4 = (?U )11 where B2 =
, and
(5)
b74 b75
?B = (B2?1 )diag(?6? , ?7? )(B2?1 )? , ?B = B2?1 (?6 , ?7 )T ,
??1
U
=
??1
B
+
diag(0, 1/?5+ ),
?U =
?U (??1
B ?B
+
T
diag(0, 1/?5+ )(0, ?+
5) )
(6)
(7)
All the other messages follow by symmetry.
4.1.3
Calculation of the Final Marginals
The approximate posterior marginal distributions are given by the product of the
? and ? messages. Hence the posterior marginal at each node k is also a Gaussian
distribution with variance and mean given by
?1
?
?k
?+
1
1
k
+ +
and ?k = ?k
+ +
respectively.
(8)
?k =
?k?
?k
?k?
?k
4.2 Initialisation
The network is initialised by setting the ? messages at the leaf nodes to be N (x, 0)
for a node known to take value x and N (0, ?) for the missing data. All the other
? messages are initialised to N (0, ?). The ? message at a given root node is set
to the prior at that root node. No other ? messages need to be initialised as they
are not needed before they are computed during the first pass. Computationally,
we usually have to add a small jitter term network noise, and represent the infinite
variances by large numbers to avoid numeric problems.
5
Demonstrations and Results
In all the tests in this section the generalised propagation converged in a small
number of iterations without the need to resort to damping. First we analyse the
simple case where the variances of the Fourier component priors have a 1/k form
where k is the component number (i.e., frequency). To test this scenario, a set of
2
The ?operator is used to denote the complex conjugate transpose (adjoint).
0.9
C1
y1
0.8
y2
y
y3
4
y5
Proportion of MSE
0.7
0.6
0.5
0.4
0.3
0.2
0.1
y6
C2
y7
0
?2
?1.5
?1
?0.5
0
0.5
Power
(a)
(b)
Figure 2: (a) Two clusters C1 and C2 . All the clusters in the network contain
four nodes. Each node is also common to one other cluster. Hence the interaction
between any two connected clusters is of the form illustrated in this figure. (b)
How the weighted mean square error varies for spectra with different power laws
(f Power ). The filled line is the belief network approach, the dashed line is linear
interpolation, the dotted line uses mean-valued data.
Mean fill Linear Spline
BN
MSE
0.072
0.045
9.9
0.037
WMSE
1.6
0.98
37.7
0.92
Table 1: Comparison of methods for estimating the FFT of a 1/f function. ?Zero
fill? replaces missing values by zero and then does an FFT. ?Linear? interpolates
linearly for the missing values. ?Spline? does the same with a cubic spline. ?BN? are
the results using the mean Fourier components produced by the method described
in this paper.
128 complex Fourier components are generated from the prior distribution. An inverse FFT is used to generate data from the Fourier components. A predetermined
set of elements is then ?lost?3 . The remaining data is then used with the algorithm
of this paper using 10 iterations of the down-up propagation. The resulting means
are compared with the components obtained by replacing the missing data with
zeros or with interpolated values and taking an FFT . Mean squared errors (MSE)
in the Fourier components are calculated for each approach over the 100 different
runs. Weighted mean squared errors (WMSE) are also calculated, where each frequency component is divided by its prior variance before averaging. The results are
presented in Table 1.
The generalised belief propagation produces better results than any of the other approaches. Similar results are achieved for a number of different spectral priors. The
benefits of interpolation are seen for situations where there are only low frequency
components, and the zeroing approach becomes more reasonable in white noise like
situations, but across a broad spread of spectral priors, the belief network approach
tends to perform better. Figure 2b illustrates of how the results vary for an average
of 100 runs as the power spectrum varies from f ?2 to f 0.5 . Note that the approach
is particularly good at the 1/f power spectra point, which corresponds to the form
of spectra in many real life problems.
3
Data in positions 3 4 5 6 8 11 13 15 18 21 22 24 25 27 28 29 30 32 33 34 35 36 42 47
51 55 58 61 65 67 71 73 75 77 78 79 81 84 86 94 97 101 102 103 104 114 115 116 117 118
119 120 121 122 123 124 125 126 127 are removed. This provides a mix of loss in whole
regions, but also at isolated points.
1/f 4
1/f 2
1/f
Linear
3.41 ? 10?8
3.33 ? 10?6
9.39 ? 10?5
Spline
1.72 ? 10?8
9.90 ? 10?6
5.15 ? 10?4
BN
8.51 ? 10?7
3.53 ? 10?6
5.52 ? 10?5
Table 2: Testing the MSE predictive ability of the belief network approach.
Zero fill Linear
Spline
BN
MSE
3.421
1.612
0.869
0.317
WMSE
1.96
0.883
0.465
0.125
MSEPRED
0.0033
0.0016 0.00085 0.00031
Table 3: Testing the ability of the belief network approach on real life audio data.
The BN approach performs better than all others for both prediction of the correct
spectrum and prediction of the missing data. MSE: mean squared error, WMSE:
weighted mean squared error, MSEPRED: Mean squared error of the data predictor.
Next we compare approaches for filling in missing data. This time 50 runs are
made on 1/f 4 , 1/f 2 and 1/f power spectra. Note that ignoring periodic boundary
constraints, a 1/f 2 power spectra produces a Brownian curve for which the linear
predictor is the optimal mean predictor. In this case the mean square error for
the belief network propagation approach (Table 2) is close to the linear error. On
smooth curves such as that produced by the 1/f 4 noise the predictive ability of the
approach (for small numbers of iterations) does not match interpolation methods.
The local smoothness information is not easily used in the belief network propagation, because neighbouring points in data space are only connected at the highest
level in the belief network. The approximations of loopy propagation methods do
not preserve enough information when propagated over these distances. However
for data such as that produced by the common 1/f power spectra, interpolation
methods are less effective, and the belief network propagation performs well. In
this situation the belief network approach outperforms interpolation. Calculations
using zero values or mean estimates also prove significantly worse.
Last, tests are made on some real world audio data. A 1024 point complex audio
signal is built up from a two channel sample from a short stretch of laughter.
Fourier power spectra of the mean of 15 other different sections of laughter are used
to estimate the prior power spectral characteristics. Randomly selected parts of the
data are removed corresponding to one tenth of the whole. A belief network FFT is
then calculated in the usual way, and compared with the true FFT calculated on the
whole data. The results are given in Table 3. The belief network approach performs
better than all other methods including linear and cubic spline interpolation.
6
Discussion
This paper provides a clear practical example of a situation where generalised propagation overcomes deficits in simpler propagation methods. It demonstrates how a
belief network representation of the fast Fourier transform allows Fourier approaches
to be used in situations where data is missing.
Kikuchi inference in the FFT belief network proves superior to many naive approaches for dealing with missing data in the calculation of Fourier transforms.
It also provides methods for inferring missing data. It does this while maintaining O(n log2 n) nature of the FFT algorithm, if we assume that the number of
iterations needed for convergence does not increase with data size. In practice, additional investigations have shown that this is not the case, but that the increase
in the number of iterations does not scale badly. Further investigation is needed
to show exactly what the scaling is, and further documentation of the benefits of
generalised propagation over loopy propagation and conjugate gradient methods is
needed beyond the space available here. It might be possible that variational approximation using clusters [12] could provide another approach to inference in this
system. This paper has also not considered the possibility of dependent or sparse
priors over Fourier coefficients, or priors over phase information, all of which would
be interesting. Formalising the extension to 2 dimensions would be straightforward
but valuable, as it is likely the convergence properties would be different.
In conclusion the tests done indicate that this is a valuable approach for dealing
with missing data in Fourier analysis. It is particularly suited to the types of spectra
seen in real world situations. In fact loopy propagation methods in FFT networks
are also valuable in many scenarios. Very recent work of Yedidia [13], shows that
discrete generalised belief propagation in FFT constructions may enable the benefits
of sparse decoders to be used for Reed-Solomon codes.
Acknowledgements
This work was funded by a research fellowship from Microsoft Research, Cambridge.
The author specifically thanks Erik Sudderth, Jonathan Yedidia, and the anonymous reviewers for their comments.
References
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[11] Y. Weiss and W. T. Freeman. Correctness of belief propagation in Gaussian models of arbitrary topology. Technical Report TR UCB//CSD-99-1046, University of
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1,657 | 2,506 | Learning with Local and Global Consistency
Dengyong Zhou, Olivier Bousquet, Thomas Navin Lal,
Jason Weston, and Bernhard Sch?olkopf
Max Planck Institute for Biological Cybernetics, 72076 Tuebingen, Germany
{firstname.secondname}@tuebingen.mpg.de
Abstract
We consider the general problem of learning from labeled and unlabeled
data, which is often called semi-supervised learning or transductive inference. A principled approach to semi-supervised learning is to design
a classifying function which is sufficiently smooth with respect to the
intrinsic structure collectively revealed by known labeled and unlabeled
points. We present a simple algorithm to obtain such a smooth solution.
Our method yields encouraging experimental results on a number of classification problems and demonstrates effective use of unlabeled data.
1
Introduction
We consider the general problem of learning from labeled and unlabeled data. Given a
point set X = {x1 , . . . , xl , xl+1 , . . . , xn } and a label set L = {1, . . . , c}, the first l points
have labels {y1 , . . . , yl } ? L and the remaining points are unlabeled. The goal is to predict
the labels of the unlabeled points. The performance of an algorithm is measured by the
error rate on these unlabeled points only.
Such a learning problem is often called semi-supervised or transductive. Since labeling
often requires expensive human labor, whereas unlabeled data is far easier to obtain, semisupervised learning is very useful in many real-world problems and has recently attracted
a considerable amount of research [10]. A typical application is web categorization, in
which manually classified web pages are always a very small part of the entire web, and
the number of unlabeled examples is large.
The key to semi-supervised learning problems is the prior assumption of consistency, which
means: (1) nearby points are likely to have the same label; and (2) points on the same structure (typically referred to as a cluster or a manifold) are likely to have the same label. This
argument is akin to that in [2, 3, 4, 10, 15] and often called the cluster assumption [4, 10].
Note that the first assumption is local, whereas the second one is global. Orthodox supervised learning algorithms, such as k-NN, in general depend only on the first assumption of
local consistency.
To illustrate the prior assumption of consistency underlying semi-supervised learning, let us
consider a toy dataset generated according to a pattern of two intertwining moons in Figure
1(a). Every point should be similar to points in its local neighborhood, and furthermore,
points in one moon should be more similar to each other than to points in the other moon.
The classification results given by the Support Vector Machine (SVM) with a RBF kernel
(a) Toy Data (Two Moons)
1.5
unlabeled point
labeled point ?1
labeled point +1
1
0.5
1
0.5
0
0
?0.5
?0.5
?1
?1
?1.5
?1.5
?1
?0.5
0
0.5
1
1.5
2
2.5
(c) k?NN
1.5
?1.5
1
1
0.5
0
0
?0.5
?0.5
?1
?1
?1.5
?1
?0.5
0
0.5
1
?1.5
?1
1.5
2
2.5
?1.5
?0.5
0
0.5
1
1.5
2
2.5
2
2.5
(c) Ideal Classification
1.5
0.5
?1.5
(b) SVM (RBF Kernel)
1.5
?1.5
?1
?0.5
0
0.5
1
1.5
Figure 1: Classification on the two moons pattern. (a) toy data set with two labeled points;
(b) classifying result given by the SVM with a RBF kernel; (c) k-NN with k = 1; (d) ideal
classification that we hope to obtain.
and k-NN are shown in Figure 1(b) & 1(c) respectively. According to the assumption of
consistency, however, the two moons should be classified as shown in Figure 1(d).
The main differences between the various semi-supervised learning algorithms, such as
spectral methods [2, 4, 6], random walks [13, 15], graph mincuts [3] and transductive SVM
[14], lie in their way of realizing the assumption of consistency. A principled approach to
formalize the assumption is to design a classifying function which is sufficiently smooth
with respect to the intrinsic structure revealed by known labeled and unlabeled points. Here
we propose a simple iteration algorithm to construct such a smooth function inspired by the
work on spreading activation networks [1, 11] and diffusion kernels [7, 8, 12], recent work
on semi-supervised learning and clustering [2, 4, 9], and more specifically by the work of
Zhu et al. [15]. The keynote of our method is to let every point iteratively spread its label
information to its neighbors until a global stable state is achieved.
We organize the paper as follows: Section 2 shows the algorithm in detail and also discusses
possible variants; Section 3 introduces a regularization framework for the method; Section
4 presents the experimental results for toy data, digit recognition and text classification,
and Section 5 concludes this paper and points out the next researches.
2
Algorithm
Given a point set X = {x1 , . . . , xl , xl+1 , . . . , xn } ? Rm and a label set L = {1, . . . , c},
the first l points xi (i ? l) are labeled as yi ? L and the remaining points xu (l+1 ? u ? n)
are unlabeled. The goal is to predict the label of the unlabeled points.
Let F denote the set of n ? c matrices with nonnegative entries. A matrix F =
[F1T , . . . , FnT ]T ? F corresponds to a classification on the dataset X by labeling each
point xi as a label yi = arg maxj?c Fij . We can understand F as a vectorial function
F : X ? Rc which assigns a vector Fi to each point xi . Define a n ? c matrix Y ? F with
Yij = 1 if xi is labeled as yi = j and Yij = 0 otherwise. Clearly, Y is consistent with the
initial labels according the decision rule. The algorithm is as follows:
1. Form the affinity matrix W defined by Wij = exp(?kxi ? xj k2 /2? 2 ) if i 6= j
and Wii = 0.
2. Construct the matrix S = D ?1/2 W D?1/2 in which D is a diagonal matrix with
its (i, i)-element equal to the sum of the i-th row of W.
3. Iterate F (t + 1) = ?SF (t) + (1 ? ?)Y until convergence, where ? is a parameter
in (0, 1).
4. Let F ? denote the limit of the sequence {F (t)}. Label each point xi as a label
yi = arg maxj?c Fij? .
This algorithm can be understood intuitively in terms of spreading activation networks
[1, 11] from experimental psychology. We first define a pairwise relationship W on the
dataset X with the diagonal elements being zero. We can think that a graph G = (V, E) is
defined on X , where the the vertex set V is just X and the edges E are weighted by W. In
the second step, the weight matrix W of G is normalized symmetrically, which is necessary
for the convergence of the following iteration. The first two steps are exactly the same as
in spectral clustering [9]. During each iteration of the third step each point receives the
information from its neighbors (first term), and also retains its initial information (second
term). The parameter ? specifies the relative amount of the information from its neighbors
and its initial label information. It is worth mentioning that self-reinforcement is avoided
since the diagonal elements of the affinity matrix are set to zero in the first step. Moreover,
the information is spread symmetrically since S is a symmetric matrix. Finally, the label of
each unlabeled point is set to be the class of which it has received most information during
the iteration process.
Let us show that the sequence {F (t)} converges and F ? = (1 ? ?)(I ? ?S)?1 Y. Without
loss of generality, suppose F (0) = Y. By the iteration equation F (t + 1) = ?SF (t) + (1 ?
?)Y used in the algorithm, we have
t?1
X
F (t) = (?S)t?1 Y + (1 ? ?)
(?S)i Y.
(1)
i=0
Since 0 < ? < 1 and the eigenvalues of S in [-1, 1] (note that S is similar to the stochastic
matrix P = D ?1 W = D?1/2 SD1/2 ),
t?1
X
t?1
(?S)i = (I ? ?S)?1 .
(2)
lim (?S)
= 0, and lim
t??
Hence
t??
i=0
F ? = lim F (t) = (1 ? ?)(I ? ?S)?1 Y,
t??
for classification, which is clearly equivalent to
F ? = (I ? ?S)?1 Y.
(3)
?
Now we can compute F directly without iterations. This also shows that the iteration
result does not depend on the initial value for the iteration. In addition, it is worth to notice
that (I ? ?S)?1 is in fact a graph or diffusion kernel [7, 12].
Now we discuss some possible variants of this method. The simplest modification is to
repeat the iteration after convergence, i.e. F ? = (I ? ?S)?1 ? ? ? (I ? ?S)?1 Y = (I ?
?S)?p Y, where p is an arbitrary positive integer. In addition, since that S is similar to P,
we can consider to substitute P for S in the third step, and then the corresponding closed
form is F ? = (I ? ?P )?1 Y. It is also interesting to replace S with P T , the transpose of P.
Then the classifying function is F ? = (I ??P T )?1 Y. It is not hard to see this is equivalent
to F ? = (D ? ?W )?1 Y. We will compare these variants with the original algorithm in the
experiments.
3
Regularization Framework
Here we develop a regularization framework for the above iteration algorithm. The cost
function associated with F is defined to be
2
n
n
X
1
2
1
1 X
?
p
Wij
F
?
Q(F ) =
F
+
?
F
?
Y
,
(4)
j
i
i
D i
2 i,j=1
Djj
ii
i=1
Where ? > 0 is the regularization parameter. Then the classifying function is
F ? = arg min Q(F ).
F ?F
(5)
The first term of the right-hand side in the cost function is the smoothness constraint, which
means that a good classifying function should not change too much between nearby points.
The second term is the fitting constraint, which means a good classifying function should
not change too much from the initial label assignment. The trade-off between these two
competing constraints is captured by a positive parameter ?. Note that the fitting constraint
contains labeled as well as unlabeled data.
We can understand the smoothness term as the sum of the local variations, i.e. the local
changes of the function between nearby points. As we have mentioned, the points involving
pairwise relationships can be be thought of as an undirected weighted graph, the weights
of which represent the pairwise relationships. The local variation is then in fact measured
on each edge. We do not simply define the local variation on an edge by the difference of
the function values on the two ends of the edge. The smoothness term essentially splits
the function value at each point among the edges attached to it before computing the local
changes, and the value assigned to each edge is proportional to its weight.
Differentiating Q(F ) with respect to F , we have
?Q
= F ? ? SF ? + ?(F ? ? Y ) = 0,
?F
?
F =F
which can be transformed into
?
1
SF ? ?
Y = 0.
1+?
1+?
Let us introduce two new variables,
?
1
, and ? =
.
?=
1+?
1+?
Note that ? + ? = 1. Then
(I ? ?S)F ? = ?Y,
Since I ? ?S is invertible, we have
F ? = ?(I ? ?S)?1 Y.
which recovers the closed form expression of the above iteration algorithm.
F? ?
(6)
Similarly we can develop the optimization frameworks for the variants F ? = (I ??P )?1 Y
and F ? = (D ? ?W )?1 Y . We omit the discussions due to lack of space.
4
Experiments
We used k-NN and one-vs-rest SVMs as baselines, and compared our method to its two
variants: (1) F ? = (I ? ?P )?1 Y ; and (2) F ? = (D ? ?W )?1 Y. We also compared to
Zhu et al.?s harmonic Gaussian field method coupled with the Class Mass Normalization
(CMN) [15], which is closely related to ours. To the best of our knowledge, there is no
reliable approach for model selection if only very few labeled points are available. Hence
we let all algorithms use their respective optimal parameters, except that the parameter ?
used in our methods and its variants was simply fixed at 0.99.
(a) t = 10
1.5
1
1
0.5
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0
0
?0.5
?0.5
?1
?1
?1.5
?1.5
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0
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(c) t = 100
1.5
1
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0
0
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2
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0
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1
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2
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2
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(d) t = 400
1.5
1
?1.5
(b) t = 50
1.5
?1.5
?1
?0.5
0
0.5
1
Figure 2: Classification on the pattern of two moons. The convergence process of our
iteration algorithm with t increasing from 1 to 400 is shown from (a) to (d). Note that the
initial label information are diffused along the moons.
Figure 3: The real-valued classifying function becomes flatter and flatter with respect to
the two moons pattern with increasing t. Note that two clear moons emerge in (d).
(a) SVM (RBF Kernel)
1.5
labeled point ?1
labeled point +1
1
0.5
1.5
1
0.5
0
0
?0.5
?0.5
?1
?1
?1.5
(b) Smooth with Global Consistency
?1.5
?1
?0.5
0
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1
1.5
2
2.5
?1.5
?1.5
?1
?0.5
0
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1
1.5
2
2.5
Figure 4: Smooth classification results given by supervised classifiers with the global consistency: (a) the classification result given by the SVM with a RBF kernel; (b) smooth the
result of the SVM using the consistency method.
4.1
Toy Problem
In this experiment we considered the toy problem mentioned in Section 1 (Figure 1).
The affinity matrix is defined by a RBF kernel but the diagonal elements are set to zero.
The convergence process of our iteration algorithm with t increasing from 1 to 400 is
shown in Figure 2(a)-2(d). Note that the initial label information are diffused along the
moons. The assumption of consistency essentially means that a good classifying function should change slowly on the coherent structure aggregated by a large amount of
data. This can be illustrated by this toy problem very clearly. Let us define a function
?
?
?
?
f (xi ) = (Fi1
? Fi2
)/(Fi1
+ Fi2
) and accordingly the decision function is sign(f (xi )),
which is equivalent to the decision rule described in Section 2. In Figure 3, we show that
f (xi ) becomes successively flatter with respect to the two moons pattern from Figure 3(a)3(d) with increasing t. Note that two clear moons emerge in the Figure 3(d).
The basic idea of our method is to construct a smooth function. It is natural to consider
using this method to improve a supervised classifier by smoothing its classifying result. In
other words, we use the classifying result given by a supervised classifier as the input of
our algorithm. This conjecture is demonstrated by a toy problem in Figure 4. Figure 4(a) is
the classification result given by the SVM with a RBF kernel. This result is then assigned
to Y in our method. The output of our method is shown in Figure 4(b). Note that the points
classified incorrectly by the SVM are successfully smoothed by the consistency method.
4.2
Digit Recognition
In this experiment, we addressed a classification task using the USPS handwritten 16x16
digits dataset. We used digits 1, 2, 3, and 4 in our experiments as the four classes. There
are 1269, 929, 824, and 852 examples for each class, for a total of 3874.
The k in k-NN was set to 1. The width of the RBF kernel for SVM was set to 5, and
for the harmonic Gaussian field method it was set to 1.25. In our method and its variants,
the affinity matrix was constructed by the RBF kernel with the same width used as in
the harmonic Gaussian method, but the diagonal elements were set to 0. The test errors
averaged over 100 trials are summarized in the left panel of Figure 5. Samples were chosen
so that they contain at least one labeled point for each class. Our consistency method and
one of its variant are clearly superior to the orthodox supervised learning algorithms k-NN
and SVM, and also better than the harmonic Gaussian method.
Note that our approach does not require the affinity matrix W to be positive definite. This
enables us to incorporate prior knowledge about digit image invariance in an elegant way,
e.g., by using a jittered kernel to compute the affinity matrix [5]. Other kernel methods are
0.35
0.3
k?NN (k = 1)
SVM (RBF kernel)
harmonic Gaussian
consistency method
variant consistency (1)
variant consistency (2)
0.65
0.6
0.55
test error
0.25
test error
0.7
k?NN (k = 1)
SVM (RBF kernel)
harmonic Gaussian
consistency method
variant consistency (1)
variant consistency (2)
0.2
0.15
0.5
0.45
0.4
0.35
0.1
0.3
0.05
0.25
0
4
10
15
20
25
# labeled points
30
40
50
0.2
4
10
15
20
25
# labeled points
30
40
50
Figure 5: Left panel: the error rates of digit recognition with USPS handwritten 16x16
digits dataset for a total of 3874 (a subset containing digits from 1 to 4). Right panel: the
error rates of text classification with 3970 document vectors in a 8014-dimensional space.
Samples are chosen so that they contain at least one labeled point for each class.
known to have problems with this method [5]. In our case, jittering by 1 pixel translation
leads to an error rate around 0.01 for 30 labeled points.
4.3
Text Classification
In this experiment, we investigated the task of text classification using the 20-newsgroups
dataset. We chose the topic rec which contains autos, motorcycles, baseball, and hockey
from the version 20-news-18828. The articles were processed by the Rainbow software
package with the following options: (1) passing all words through the Porter stemmer
before counting them; (2) tossing out any token which is on the stoplist of the SMART
system; (3) skipping any headers; (4) ignoring words that occur in 5 or fewer documents.
No further preprocessing was done. Removing the empty documents, we obtained 3970
document vectors in a 8014-dimensional space. Finally the documents were normalized
into TFIDF representation.
The distance between points xi and xj was defined to be d(xi , xj ) = 1?hxi , xj i/kxi kkxj k
[15]. The k in k-NN was set to 1. The width of the RBF kernel for SVM was set to 1.5, and
for the harmonic Gaussian method it was set to 0.15. In our methods, the affinity matrix
was constructed by the RBF kernel with the same width used as in the harmonic Gaussian
method, but the diagonal elements were set to 0. The test errors averaged over 100 trials
are summarized in the right panel of Figure 5. Samples were chosen so that they contain at
least one labeled point for each class.
It is interesting to note that the harmonic method is very good when the number of labeled
points is 4, i.e. one labeled point for each class. We think this is because there are almost
equal proportions of different classes in the dataset, and so with four labeled points, the proportions happen to be estimated exactly. The harmonic method becomes worse, however, if
slightly more labeled points are used, for instance, 10 labeled points, which leads to pretty
poor estimation. As the number of labeled points increases further, the harmonic method
works well again and somewhat better than our method, since the proportions of classes
are estimated successfully again. However, our decision rule is much simpler, which in
fact corresponds to the so-called naive threshold, the baseline of the harmonic method.
5
Conclusion
The key to semi-supervised learning problems is the consistency assumption, which essentially requires a classifying function to be sufficiently smooth with respect to the intrinsic
structure revealed by a huge amount of labeled and unlabeled points. We proposed a simple algorithm to obtain such a solution, which demonstrated effective use of unlabeled data
in experiments including toy data, digit recognition and text categorization. In our further
research, we will focus on model selection and theoretic analysis.
Acknowledgments
We would like to thank Vladimir Vapnik, Olivier Chapelle, Arthur Gretton, and Andre Elisseeff for their help with this work. We also thank Andrew Ng for helpful discussions about
spectral clustering, and the anonymous reviewers for their constructive comments. Special
thanks go to Xiaojin Zhu, Zoubin Ghahramani, and John Lafferty who communicated with
us on the important post-processing step class mass normalization used in their method and
also provided us with their detailed experimental data.
References
[1] J. R. Anderson. The architecture of cognition. Harvard Univ. press, Cambridge, MA,
1983.
[2] M. Belkin and P. Niyogi. Semi-supervised learning on manifolds. Machine Learning
Journal, to appear.
[3] A. Blum and S. Chawla. Learning from labeled and unlabeled data using graph mincuts. In ICML, 2001.
[4] O. Chapelle, J. Weston, and B. Sch?olkopf. Cluster kernels for semi-supervised learning. In NIPS, 2002.
[5] D. DeCoste and B. Sch?olkopf. Training invariant support vector machines. Machine
Learning, 46:161?190, 2002.
[6] T. Joachims. Transductive learning via spectral graph partitioning. In ICML, 2003.
[7] J. Kandola, J. Shawe-Taylor, and N. Cristianini. Learning semantic similarity. In
NIPS, 2002.
[8] R. I. Kondor and J. Lafferty. Diffusion kernels on graphs and other discrete input
spaces. In ICML, 2002.
[9] A. Y. Ng, M. I. Jordan, and Y. Weiss. On spectral clustering: analysis and an algorithm. In NIPS, 2001.
[10] M. Seeger. Learning with labeled and unlabeled data. Technical report, The University of Edinburgh, 2000.
[11] J. Shrager, T. Hogg, and B. A. Huberman. Observation of phase transitions in spreading activation networks. Science, 236:1092?1094, 1987.
[12] A. Smola and R. I. Kondor. Kernels and regularization on graphs. In Learning Theory
and Kernel Machines, Berlin - Heidelberg, Germany, 2003. Springer Verlag.
[13] M. Szummer and T. Jaakkola. Partially labeled classification with markov random
walks. In NIPS, 2001.
[14] V. N. Vapnik. Statistical learning theory. Wiley, NY, 1998.
[15] X. Zhu, Z. Ghahramani, and J. Lafferty. Semi-supervised learning using gaussian
fields and harmonic functions. In ICML, 2003.
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1,658 | 2,507 | Convex Methods for Transduction
Tijl De Bie
ESAT-SCD/SISTA, K.U.Leuven
Kasteelpark Arenberg 10
3001 Leuven, Belgium
[email protected]
Nello Cristianini
Department of Statistics, U.C.Davis
360 Kerr Hall One Shields Ave.
Davis, CA-95616
[email protected]
Abstract
The 2-class transduction problem, as formulated by Vapnik [1],
involves finding a separating hyperplane for a labelled data set
that is also maximally distant from a given set of unlabelled test
points. In this form, the problem has exponential computational
complexity in the size of the working set. So far it has been attacked
by means of integer programming techniques [2] that do not scale
to reasonable problem sizes, or by local search procedures [3].
In this paper we present a relaxation of this task based on semidefinite programming (SDP), resulting in a convex optimization
problem that has polynomial complexity in the size of the data set.
The results are very encouraging for mid sized data sets, however
the cost is still too high for large scale problems, due to the high dimensional search space. To this end, we restrict the feasible region
by introducing an approximation based on solving an eigenproblem.
With this approximation, the computational cost of the algorithm
is such that problems with more than 1000 points can be treated.
1
Introduction
The general transduction task is the following: given a training set of labelled data,
and a working set of unlabelled data (also called transduction samples), estimate the
value of a classification function at the given points in the working set. Statistical
learning results [1] suggest that this setting should deliver better results than the
traditional ?inductive? setting, where a function needs to be learned first and only
later tested on a test set of points chosen after the learning has been completed.
Different algorithms have been proposed so far to take advantage of this advance
knowledge of the test points (such as in [1], [2], [3], [4], [5], [6] and others).
Given this general task, much research has been concentrated on a specific approach
to transduction (first proposed by Vapnik [1]), based on the use of Support Vector
Machines (SVM?s). In this case, the algorithm is aimed at finding a separating
hyperplane for the training set that is also maximally distant from the (unlabelled)
working set. This hyperplane is used to predict the labels for the working set points.
In this form, the problem has exponential computational complexity, and several
approaches have been attempted to solve it. Generally they involve some form of
local search [3], or integer programming methods [2].
A recent development of convex optimization theory is Semi Definite Programming
(SDP), a branch of that field aimed at optimizing over the cone of semi positive definite (SPD) matrices. One of its main attractions is that it has proven successful in
constructing tight convex relaxations of hard combinatorial optimization problems
[7]. SDP has recently been applied successfully to machine learning problems [8].
In this paper we show how to relax the problem of transduction into an SDP problem, that can then be solved by (polynomial time) convex optimization methods.
Empirical results on mid-sized data sets are very promising, however, due to the
dimensionality of the feasible region of the relaxed parameters, still the algorithm
complexity appears too large to tackle large scale problems. Therefore, we subsequently shrink the feasible region by making an approximation that is based on a
spectral clustering method. Positive empirical results will be given.
Formal definition: transductive SVM. Based on the dual of the 1-norm soft
margin SVM with zero bias1 , the dual formulation of the transductive SVM optimization problem can be written as a minimization of the dual SVM cost function
(which is the inverse margin plus training errors) over label matrix ? ([1], p. 437):
min max
2?? e ? ?? (K ? ?)?
(1)
?
?
s.t.
C ? ?i ? 0
(2)
t t ?
y
y
?=
?
(3)
yw
yw
yiw ? {1, ?1}
(4)
The (symmetric) matrix ? is thus parameterized by the unknown working set label vector yw ? {?1, 1}nw (with nw the size of the working set). The vector
yt ? {?1, 1}nt (with nt the number of training points) is the given fixed vector containing the known labels for the training points. The (symmetric) matrix
K ? ?(nw +nt )?(nw +nt ) is the entire kernel matrix on the training set together with
the working set. The dual vector is denoted by ? ? ?nw +nt , and e is a vector of
appropriate size containing all ones. The symbol ? represents the elementwise matrix product. It is clear indeed this is a combinatorial problem. The computational
complexity scales exponentially in the size of the working set.
Further notation. Scalars are lower case; vectors boldface lower case; matrices
boldface upper case. The unit matrix is denoted by I. A pseudo-inverse is denoted
with a superscript ? , a transpose with a ? . For ease of notation, the training part
of the label matrix (and thus also of the kernel matrix) is always assumed to be
its upper nt ? nt block (as is assumed already in (3)). Furthermore, the nt+ positive training samples are assumed to correspond to the first entries in yt , the nt?
negative samples being at the end of this vector.
2
Relaxation to an SDP problem
In this section, we will gradually derive a relaxed version of the transductive SVM
formulation. To start with, we replace some of the constraints by an equivalent set:
Proposition 2.1 (3) and (4) are equivalent with the following set of constraints:
(5)
[?]i,j?{1:nt ,1:nt } = yit yjt
1
We do not include a bias term since this would make the problem too non-convex.
However this does not impair the result as is explained in [9].
diag(?) = e
rank(?) = 1
(6)
(7)
The values of ? will then indeed be equal to 1 or ?1. It is basically the rank constraint that makes the resulting constrained optimization problem combinatorial.
Note that these constraints imply that ? is semi positive definite (SPD): ? 0 (this
follows trivially from (3), or from (6) together with (7)). Now, in literature (see eg
[7]) it is observed that such an SPD rank one constraint can often be relaxed to only
the SPD constraint without sacrificing too much of the performance. Furthermore:
Proposition 2.2 If we relax the constraints by replacing (7) with
? 0,
(8)
the optimization problem becomes convex.
This follows from the fact that ? appears linearly in the cost function, and that the
constraints (2), (5), (6) and (8) consist of only linear equalities and linear (matrix)
inequalities in the variables. Further on it will be shown to be an SDP problem.
While this relaxation of the rank constraint makes the optimization problem convex,
the result will not be a rank one matrix anymore; it will only provide an approximation for the optimal rank one matrix. Thus the values of ? will not be equal to
1 or ?1 anymore. However, it is well known that:
Lemma 2.1 A principal submatrix of an SPD matrix is also SPD [10].
By applying this lemma on all 2 ? 2 principal submatrices of ?, it is shown that
Corollary 2.1 From constraints (6) and (8) follows: ?1 ? [?]i,j ? 1.
This is the problem will solve here: optimize (1) subject to (2), (5), (6) and (8). In
the remainder of this section we will reformulate the optimization problem into a
standard form of SDP, make further simplifications based on the problem structure,
and show how to extract an approximation for the labels from the result.
2.1
Formulation as a standard SDP problem
In the derivations in this subsection the equality constraints (5) and (6) will not
be stated for brevity. Their consequences will be treated further in the paper.
Furthermore, in the implementation, they will be enforced explicitly by the parameterization, thus they will not appear as constraints in the optimization problem.
Also the SPD constraint (8) is not written every time, it should be understood.
Let 2? ? 0 be the Lagrange dual variables corresponding to constraint ?i ? 0 and
2? ? 0 corresponding to constraint ?i ? C. Then, since the problem is convex and
thus the minimization and maximization are exchangeable (strong duality, see [8]
for a brief introduction to duality), the optimization problem is equivalent with:
min
max
?,? ?0,??0 ?
2?? (e + ? ? ?) ? ?? (K ? ?)? + 2C?? e
In case K ? ? is rank deficient, (e + ? ? ?) will be orthogonal to the null space
of K ? ? (otherwise, the object function could grow to infinity, and this while ?
and ? on the contrary are minimizing the objective). The maximum over ? is then
reached for ? = (K ? ?)? (e + ? ? ?). Substituting this in the object function gives:
min?,? ?0,??0 (e + ? ? ?)? (K ? ?)? (e + ? ? ?) + 2C?? e
or equivalently:
min
t
?,? ?0,??0,t
s.t.
t ? (e + ? ? ?)? (K ? ?)? (e + ? ? ?) + 2C?? e.
with as additional constraint that (e + ? ? ?) is orthogonal to the null space of
K ? ?. This latter constraint and the quadratic constraint can be reformulated
as one SPD constraint thanks to the following extension of the Schur complement
lemma [10] (the proof is omitted due to space restrictions):
Lemma 2.2 (Extended Schur complement lemma) For symmetric A 0
and C ? 0:
The column space of B ? the null space of A
A B
?
0.
B? C
C B ? A? B
Indeed, applying this lemma to our problem with A = K ? ?, B = e + ? ? ? and
C = t ? 2C?? e, leads to the problem formulation in the standard SDP form:
min
t
(9)
?,? ?0,??0,t
K??
(e + ? ? ?)
s.t.
0
(10)
(e + ? ? ?)? t ? 2C?? e
together with the constraints (5), (6) and (8). The relaxation for the hard margin
SVM is found by following a very similar derivation, or by just equating ? to 0.
The number of variables specifying ?, and the size of constraint (8) can be greatly
reduced due to structure in the problem. This is subject of what follows now.
2.2
Simplifications due to the problem structure
c
t t?
y
y
?
where we have a
The matrix ? can be parameterized as ? =
?c ? ?w
?
training block yt yt ? ?nt ?nt , cross blocks ?c ? ?nt ?nw and ?c ? , and a transduction
block ?w ? ?nw ?nw , which is a symmetric matrix with diagonal entries equal to 1.
We now use Lemma 2.1: by choosing a submatrix that contains all rows and columns
corresponding to the training block, and just one row and column corresponding to
the transduction part, the SPD constraint of ? is seen to imply that
?
yt yt ? ci
0
?=
? ci ?
1
where ? ci represents the ith column of ?c . Using the extended Schur complement
lemma 2.2, it follows that ? ci is proportional to yt (denoted by ? ci = gi yt ), and
?
?
t t?
?
yt yt
c
t? y y
t
2
1 ? ci ? yt yt
? ci = ? ci ? ky
t k4 ? i . This implies that 1 ? gi y
kyt k4 y gi = gi such
that ?1 ? gi ? 1. (Note that this is a corollary of the SPD constraint and does not
need to be imposed explicitly.) Thus, the parameterization of ? can be reduced to:
t t?
yy
yt g?
? =
with ?w
?
ii = 1
gyt
?w
where g is the vector with gi as ith entry. We can now show that:
Proposition 2.3 The constraint ? 0 is equivalent to (and can thus be replaced
e
by) the following SPD constraint on a smaller matrix ?:
?
1 g
e=
?
0.
g ?w
e is a principal submatrix of ? (assuming at least one training label is equal to
Since ?
e 0. On the other hand, note that
1), lemma 2.1 indeed shows that ? 0 implies ?
e
by adding a column and corresponding row to ?, the rank is not increased. Thus, an
eigenvalue equal to 0 is added. Due to the interlacing property for bordered matrices
e 0, we know this can only be the smallest eigenvalue of
[10] and the fact that ?
e 0 implies ? 0.
the resulting matrix. By induction this shows that also ?
This is the final formulation of the problem. For the soft margin case, the number of
n2 +5n
n2 +3n
parameters is now 1+2nt + w 2 w . For the hard margin case, this is 1+nt + w 2 w .
2.3
Extraction of an estimate for the labels from ?
In general, the optimal ? will of course not be rank one. We can approximate it by a
rank one matrix however, by taking g as an approximation for the labels optimizing
the unrelaxed problem. This is the approach we adopt: a thresholded value of the
entries of g will be taken as a guess for the labels of the working set.
Note that the minimum of the relaxed problem is always smaller than or equal to the
minimum of the unrelaxed problem. Furthermore, the minimum of the unrelaxed
problem is smaller than or equal to the value achieved by the thresholded relaxed
labels. Thus, we obtain a lower and an upper bound for the true optimal cost.
2.4
Remarks
The performance of this method is very good, as is seen on a toy problem (figure
1 shows an illustrative example). However, due to the (even though polynomial)
complexity of SDP in combination with the quadratic dependence of the number of
variables on the number of transduction points2 , problems with more than about
1000 training samples and 100 transduction samples can not practically be solved
with general purpose SDP algorithms. Especially the limitation on the working set
is a drawback, since the advantage of transduction becomes apparent especially for
a large working set as compared to the number of training samples. This makes the
applicability of this approach for large real life problems rather limited.
3
Subspace SDP formulation
However, if we would know a subspace (spanned by the d columns of a matrix
V ? ?(nt +nw )?d ) in which (or close to which) the label vector lies, we can restrict
the feasible region for ?, leading to a much more efficient algorithm. In the next
section a fast method to estimate such a space V will be provided. In this section
we assume V is known, and explain how to do the reduction of the feasible region.
If we know that the true label vector y lies in the column space of a matrix V,
we know the true label matrix can be written in the form ? = VMV? , with M a
symmetric matrix. The number of parameters is now only d(d + 1)/2. Furthermore,
constraint (8) that ? 0 is then equivalent to M 0, which is a cheaper constraint.
Note however that in practical cases, the true label vector will not lie within but
only close to the subspace spanned by the columns of V. Then the diagonal of the
label matrix ? can not always be made exactly equal to e as required by (6). We
thus relax this constraint to the requirement that the diagonal is not larger than
2
The worst case complexity for the problem at hand is O((nt +n2w )2 (nt +nw )2.5 ), which
is of order 6.5 in the number of transduction points nw .
1
1
1
0.5
0.5
0.5
0
0
0
?0.5
?0.5
?0.5
?1
?1
?1
0
1
0
20
40
60
?1
?1
0
1
Figure 1: The left picture shows 10 labelled samples represented by a ?o? or a ?+?,
depending on their class, together with 60 unlabelled samples represented by a ???.
The middle picture shows the labels for the working set as estimated using the
SDP method before thresholding: all are already invisibly close to 1 or ?1. The
right picture shows contour lines of the classification surface obtained by training
an SVM using all labels as found by the SDP method. The method clearly finds a
visually good label assignment that takes cluster structure in the data into account.
e. Similarly, the block in the label matrix corresponding to the training samples
may not contain 1?s and ?1?s exactly (constraint (5)). However, the better V is
chosen, the better this constraint will be met. Thus we optimize (9) subject to (10)
together with three constraints that replace the constraints (5), (6) and (8):
? = VMV?
diag(?) ? e
M 0
Thus we can approximate the relaxed transductive SVM using this reduced parameterization for ?. The number of effective variables is now only a linear function of
nw : 1 + nt + nw + d(d + 1)/2 for a hard margin and 1 + 2(nt + nw ) + d(d + 1)/2 for
a soft margin SVM. Furthermore, one of the SPD constraints is now a constraint
on a d ? d matrix instead of a potentially large (nw + 1) ? (nw + 1) matrix. For a
constant d, the worst case complexity is thus reduced to O((nt + nw )4.5 ).
The quality of the approximation can be determined by the user: the number
of components d can be chosen depending on the available computing resources,
however empirical results show a good performance already for relatively small d.
4
Spectral transduction to find the subspace
In this section we will discuss how to find a subspace V close to which the label
vector will lie. Our approach is based on the spectral clustering algorithm proposed
in [11]. They start with computing the eigenvectors corresponding to the largest
eigenvalues of D?1/2 KD?1/2 where d = Ke contains all row sums of K, and
D = diag(d). The dominant eigenvectors are shown to reflect the cluster structure
of the data. The optimization problem corresponding to this eigenvalue problem is:
e
max
v? D?1/2 KD?1/2 v = v? Kv
s.t. v? v = 1.
(11)
v
4.1
Constrained spectral clustering
We could apply this algorithm to the kernel matrix K, but we can do more since
we already know some of the labels: we will constrain the estimates of the labels
for the training samples that are known to be in the same class to be equal to each
other. Then we optimize the same object function with respect to these additional
constraints. This can be achieved by choosing the following parameterization for v:
?
? ?
?
?
ent+ / nt+
0
0
ht+
?
0
ent? / nt? 0 ? ? ? ht? ? = Lh
v=?
hw
0
0
I
where en+ and en? denote the vectors containing nt+ (the number of positive
training samples) and nt? (the number of negative training samples) ones. Then:
Proposition 4.1 Optimization problem (11) is equivalent with:
max
h
h? L? D?1/2 KD?1/2 Lh
s.t.
h? h = 1
which corresponds to the eigenvalue problem L? D?1/2 KD?1/2 Lh = ?h. Then v is
found as v = Lh.
This is an extension of spectral clustering towards transduction3 . We will use a
subscript i to denote the ith eigenvector and eigenvalue, where ?i ? ?j for i > j.
4.2
Spectral transduction provides a good V
By construction,
? all entries of vi corresponding to positive training samples will be
equal to ht+
i / nt+ ; entries corresponding to the negative ones will all be equal to
?
ht?
i / nt? . Furthermore, as in spectral clustering, the other entries of vectors vi
with large eigenvalue ?i will reflect the cluster structure of the entire data set, while
respecting the label assignment of the training points however4 . This means that
such a vi will provide a good approximation for the labels. More specifically, the
label vector will lie close to the column space of V, having d dominant ?centered? vi
as its columns; the larger d, the better the approximation. The way we ?center? vi
is by adding a constant so that entries for positive training samples become equal to
minus those for the negative ones. Since then the first nt columns of the resulting
? = VMV? will be equal up to a sign, we can adopt basically the same approach
as in section 2.3 to guess the labels: pick and threshold the first column of ?.
5
Empirical results
To show the potential of the method, we extracted data from the USPS data set
to form two classes. The positive class is formed by 100 randomly chosen samples representing a number 0, and 100 representing a 1; the negative class by 100
samples representing a 2 and 100 representing a 3. Thus, we have a balanced classification problem with two classes of each 200 samples. The training set is chosen
to contain only 10 samples from each of both classes, and is randomly drawn but
evenly distributed over the 4 numbers. We used a hard margin SVM with an rbf
kernel with ? = 7 (which is equal to the average distance of the samples to their
nearest neighbors, verified to be a good value for the induction as well as for the
3
We want to point out that the spectral transduction on its own is empirically observed
to significantly improve over standard spectral clustering algorithms, and compares favorably with a recently proposed [5] extension of spectral clustering towards transduction.
Furthermore, as also in [5] the method can be generalized towards a method for clustering
with side-information (where side-information consists of sets of points that are known to
be co-clustered). Space restrictions do not permit us to go into this in the current paper.
4
Note: to reduce the influence from outliers, large entries of the vi can be thresholded.
transduction case). The average ROC-score (area under the ROC-curve) over 10
randomizations is computed, giving 0.75 ? 0.03 as average for the inductive SVM,
and 0.959 ? 0.03 for the method developed in this paper (we chose d = 4). To
illustrate the scalability of the method, and to show that a larger working set is
effectively exploited, we used a similar setting (same training set size) but with 1000
samples and d = 3, giving an average ROC-score of 0.993 ? 0.004.
6
Conclusions
We developed a relaxation for the transductive SVM as first proposed by Vapnik.
It is shown how this combinatorial problem can be relaxed to an SDP problem.
Unfortunately, the number of variables in combination with the complexity of SDP is
too high for it to scale to significant problem sizes. Therefore we show how, based on
a new spectral method, the feasible region of the variables can be shrinked, leading
to an approximation for the original SDP method. The complexity of the resulting
algorithm is much more favorable. Positive empirical results are shown.
Acknowledgement
Tijl De Bie is a Research Assistant with the Fund for Scientific Research ? Flanders
(F.W.O.?Vlaanderen).
References
[1] V. N. Vapnik. Statistical Learning Theory. Springer, 1998.
[2] K. Bennett and A. Demiriz. Semi-supervised support vector machines. In M. S.
Kearns, S. A. Solla, and D. A. Cohn, editors, Advances in Neural Information
Processing Systems 11, Cambridge, MA, 1999. MIT Press.
[3] T. Joachims. Transductive inference for text classification using support vector machines. In Proc. of the International Conference on Machine Learning
(ICML), 1999.
[4] N. Cristianini, J. Kandola, A. Elisseeff, and J. Shawe-Taylor. On optimizing
kernel alignment. Submitted for publication, 2003.
[5] S. D. Kamvar, D. Klein, and C. D. Manning. Spectral learning. In Proc. of the
International Joint Conference on Artificial Intelligence (IJCAI), 2003.
[6] O. Chapelle, J. Weston, and B. Sch?olkopf. Cluster kernels for semi-supervised
learning. In S. Becker, S. Thrun, and K. Obermayer, editors, Advances in
Neural Information Processing Systems 15, Cambridge, MA, 2003. MIT Press.
[7] C. Helmberg. Semidefinite Programming for Combinatorial Optimization. Habilitationsschrift, TU Berlin, January 2000. ZIB-Report ZR-00-34, KonradZuse-Zentrum Berlin, 2000.
[8] G. Lanckriet, N. Cristianini, P. Bartlett, L. El Ghaoui, and M. I. Jordan.
Learning the kernel matrix with semidefinite programming. Journal of Machine
Learning Research (JMLR), 5:27?72, 2004.
[9] T. Poggio, S. Mukherjee, R. Rifkin, A. Rakhlin, and A. Verri. b. In Proceedings
of the Conference on Uncertainty in Geometric Computations, 2001.
[10] R. A. Horn and C. R. Johnson. Matrix Analysis. Cambridge University Press,
1985.
[11] J. Shi and J. Malik. Normalized cuts and image segmentation. IEEE Transactions on Pattern Analysis and Machine Intelligence, 22(8):888?905, 2000.
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1,659 | 2,508 | Parameterized Novelty Detection for
Environmental Sensor Monitoring
Cynthia Archer, Todd K. Leen, Antonio Baptista
OGI School of Science & Engineering
Oregon Health & Science University
20000 N. W. Walker Road
Beaverton, OR 97006
[email protected], [email protected], [email protected]
Abstract
As part of an environmental observation and forecasting system,
sensors deployed in the Columbia RIver Estuary (CORIE) gather
information on physical dynamics and changes in estuary habitat. Of these, salinity sensors are particularly susceptible to biofouling, which gradually degrades sensor response and corrupts critical data. Automatic fault detectors have the capability to identify
bio-fouling early and minimize data loss. Complicating the development of discriminatory classifiers is the scarcity of bio-fouling
onset examples and the variability of the bio-fouling signature. To
solve these problems, we take a novelty detection approach that
incorporates a parameterized bio-fouling model. These detectors
identify the occurrence of bio-fouling, and its onset time as reliably
as human experts. Real-time detectors installed during the summer of 2001 produced no false alarms, yet detected all episodes of
sensor degradation before the field staff scheduled these sensors for
cleaning. From this initial deployment through February 2003, our
bio-fouling detectors have essentially doubled the amount of useful
data coming from the CORIE sensors.
1
Introduction
Environmental observation and forecasting systems (EOFS) gather, process, and
deliver environmental information to facilitate sustainable development of natural resources. Our work is part of a pilot EOFS system being developed for the
Columbia River Estuary (CORIE) [1]. This system uses data from sensors deployed throughout the estuary (Figure 1) to calibrate and verify numerical models
of circulation and material transport. CORIE scientists use these models to predict
and evaluate the effects of development on the estuary environment (e.g. [2]).
CORIE salinity sensors deployed in the estuary lose several months of data every
year due to sensor degradation. Corrupted and missing field measurements compromise model calibration and verification, which can lead to invalid environmental
forecasts. The most common form of salinity sensor degradation is bio-fouling, a
reduction of the sensor response due to growth of biological material on the sensor.
Prior the deployment of the technology described here, on a yearly basis CORIE
salinity sensors suffered a 68% data loss due to bio-fouling. Although bio-fouling
degradation is a common problem for environmental sensors, there is apparently no
previous work that develops automatic detectors of such degradation.
Figure 1: Map of Columbia River estuary marked with locations of CORIE sensors.
Early bio-fouling detection is made difficult by the normal variability of salinity
measurements. Tides cause the measurements to vary from near river salinity to
near ocean salinity twice a day. The temporal pattern of salinity penetration varies
spatially in the estuary. In addition, upriver sites, such as AM169, show substantial
variability with the 14 and 28 day spring-neap tidal cycle. Changes in weather (e.g.
winds, precipitation) and ocean conditions cause additional variations in salinity.
To complicate bio-fouling detection further, the bio-fouling signature also varies
from episode to episode. The time from onset to complete bio-fouling can take
anywhere from 3 weeks to 5 months depending on the season and type of growth.
We observe two types of bio-fouling in the estuary, hard growth (e.g. barnacles)
characterized by quick linear degradation and soft growth (e.g. plant material)
characterized by slow linear degradation with occasional interruptions in the downtrend.
Figure 2 illustrates tidal variations in salinity and the effect that bio-fouling has on
these measurements. It contains salinity time series in practical salinity units (psu)
from two sensors mounted at the Red26 station, Figure 1. The upper trace, from
sensor CT1460, contains only clean measurements. The lower trace, from sensor
CT1448, contains both clean and bio-fouled measurements. The first half of the two
time series are similar, but beginning on September 28th , the salinity measurements
diverge. The CT1448 sensor exhibits typical hard-growth bio-fouling degradation.
The primary challenge to our work is to detect the degradation quickly, ideally
within several diurnal cycles. Early detection will limit the use of corrupted data in
on-line applications, and provide a basis to rapidly replace degrading sensors, and
thus drastically reduce data loss.
Although the CORIE data archives contain many months of bio-fouled data, there
are relatively few examples of the onset of degradation for most of the sensors
CT1460 salinity
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0
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Figure 2: Clean and bio-fouled salinity time series examples from Red26 station. The
upper time series is from clean instrument CT1460. The lower time series from instrument
CT1448 shows degradation beginning on September 28, 2001. On removal, CT1448 was
found to be bio-fouled.
deployed in the estuary, and it is this onset that we must detect. The dearth of
onset examples, and the observed variability of the bio-fouling signature spatially,
seasonally, and weekly (according to the spring/neap tidal cycle) prevents use of
classical discriminatory fault detectors. Instead we develop a parameterized novelty
detector to detect bio-fouling. This detector incorporates a parameterized model
of bio-fouling behavior. The parameters in the model of bio-fouled sensor behavior
are fit on-line by maximum-likelihood estimation. A model of the clean sensor
behavior is fit to archival data. These models are used in a sequential likelihood
test to provide detection of bio-fouling, and an estimation of the time at which the
degradation began.
Evaluations show that our detectors identify the onset of bio-fouling as reliably as
human experts, and frequently within fewer tidal cycles of the onset. Our deployment of sensors throughout the estuary has resulted in an actual reduction of the
error loss from 68% to 35%. However, this figure does not adequately reflect the
efficacy of the detectors. Were it economical to replace sensors immediately upon
detection of degradation, the data loss would have been reduced to 17%.
2
Salinity and Temperature
Our detectors monitor maximum diurnal (md) salinity, defined as the maximum
salinity near one of the two diurnal tidal floods. When the sensor is clean, the md
salinity stays close to some mean value, with occasional dips of several psu caused
by variations in the intrusion of salt water into the estuary. When the sensor biofouls, the md salinity gradually decreases to typically less than half its normal mean
value, as seen in the Figure 2 example.
Detectors that monitor salinity alone can not distinguish between normal decreases
in salinity and early bio-fouling. This results in a high false alarm rate 1 . Natural
salinity decreases can be recognized by monitoring a correlated source of information
that is not corrupted by bio-fouling.
Salinity and temperature at a station are products of the same mixing process of
ocean and river waters, so we expect these values will be correlated. Assuming
linear mixing of ocean and river waters, measured salinity Sm and temperature Tm
are linear functions of ocean {So , To } and river {Sr , Tr } values
Sm = ?(t)So + (1 ? ?(t))Sr
(1)
Tm = ?(t)To + (1 ? ?(t))Tr
(2)
where ?(t) is the mixing coefficient at time t. River salinity Sr is close to zero.
Consequently, the estimated mixing coefficient
?(t) =
Tr ? T m
Tr ? T o
(3)
should be well correlated with salinity, Sm ? ?So . The river temperature is measured at far upstream stations (Elliot or Woody). The ocean temperature is estimated from measurements at Sand Island, the outermost sensor station.
3
Bio-fouling Detection
Our early experiments with single-measurement detection suggested that we develop
detectors that accrue information over time - similar to the standard sequential
likelihood methods in classical pattern recognition. The is a natural framework for
detecting degradation that grows with time.
Assume a sequence of measurements (salinity and temperature) yn , n = 1, . . . , N
where N is the current time. We construct probability densities for such sequences
for both clean sensors p(y1 , . . . , yN | c), and for biofouled sensors p(y1 , . . . , yN | f ).
With these distributions, we construct a likelihood ratio test
f
h = ln
p(y1 , . . . , yN | f ) >
?
p(y1 , . . . , yN | c) <
c
(4)
where the threshold ? is chosen high enough to provide a specified false alarm rate
(Neyman-Pearson test).
We assume that the probability density for the measurement sequence for fouled
detectors is parameterized by a vector of unknown parameters ?. The model is
constructed such that at ? = 0 the density for the sequence assuming a fouled
detector is equal to the density of the sequence assuming a clean detector
p(y1 , . . . , yN | f, ? = 0) = p(y1 , . . . , yN | c)
(5)
Next, we suppose that a given sequence contains a bio-fouling event that is initiated
at the unknown time ? . Under our density models (below), consecutive measurements in the sequence are independent conditioned on the state of the detector.
1
Equivalently, if the alarm threshold is increased to maintain a low false alarm rate,
the rate of proper detections is decreased.
Consequently, the likelihood ratio for the sequence (4) reduces to
h =
=
ln
p(y1 , . . . , y? ?1 | c) p(y? , . . . , yN | ?, ?, f )
p(y1 , . . . , yN | f, ?, ?)
= ln
p(y1 , . . . , yN | c)
p(y1 , . . . , yN | c)
N
X
n=?
f
ln
p(yn | ?, ?, f ) >
?
<
p(yn | c)
c
(6)
Finally, we fit the fouling model parameters ? and the onset time ? , by maximizing
the log-likelihood ln p(y1 , . . . , yN | f, ?, ?) with respect to ? and ? . Since the clean
detector model is independent of ? and ?, this is equivalent to maximizing the
log-likelihood ratio in (6). Hence, we replace the latter with
h = max
?,?
N
X
n=?
f
ln
p(yn |?, ?, f ) >
?
<
p(yn |c)
c
(7)
If the sequence is coming from a clean sensor, the fit should give ? ? 0 and hence
h ? 0 (cf 5), and we will detect no event (assuming ? > 0). This construction is a
variant of the type of signal change detection discussed by Basseville [3].
3.1
Bio-fouling Fault Model
By parameterizing the bio-fouling model, we are able to develop detectors using
only clean example data. In this parameterized novelty detector, the bio-fouled
parameters ? are fit on-line to the data under test. To develop our classifier, we
first define models of the clean and bio-fouled data. We model the true salinity, s,
and temperature-based mixing coefficient, ?, as jointly Gaussian,
2
?s
?s ?s?
p(s, ?|c) = N (?, ?) where ? =
and ? =
.
(8)
??
?s? ??2
This provides a regression of the salinity on ?. The probability of md salinity measurement conditioned on temperature when the sensor is clean is Gaussian N (?, ? 2 ),
with conditional mean
E[s|?, c] ? ? = ?s + (?s? /??2 ) (? ? ?? )
(9)
and conditional variance
2
var[s|?, c] ? ?2 = ?s2 ? ?s?
/??2
(10)
When bio-fouling occurs, the salinity measurement is suppressed relative to the
true value. We model this suppression as a linear downtrend with (unknown) rate
(slope) m that begins at (unknown) time ? . The model of the measured md salinity
value for a fouled detector is
xn = g(n)sn
(11)
where the suppression factor, g(n), is
1
g(n) =
(1 ? m(n ? ? ))
n<?
n??
(12)
and m is the bio-fouling rate (1/sec). Using this suppression factor g(n) (12), the
probability of the salinity measurement, x, conditioned on temperature is
p(xn |?n , m, ?, f ) = N (g(n)?n , g 2 (n)?2 )
(13)
Note that since the temperature sensor is not susceptible to bio-fouling, we need
not consider the case of both sensors degrading at the same time.
The discriminant function in (7) depends on the parameters of the clean model (9)
and (10) which are estimated from historical data. It also depends on the slope
parameter ? = m of the fouling model, and the onset time ? which are fit online as
per (7).
Applying our Gaussian models in (8) and 13) to (7) gives us
N
X
1
(xn ? ?n )2
(xn ? (1 ? m(n ? ? ))?n )2
h = max
ln
+
?
2
?,m
1 ? m(n ? ? )
2?
2(1 ? m(n ? ? ))2 ?2
n=?
(14)
When h is above our chosen threshold, the detector signals a biofouled sensor. The
threshold ? is set to provide a maximum false alarm rate on historical data.
3.2
Model Fitting
We find maximum likelihood estimates for ? and ? from clean archival timeP
series
data. For yn = [sn , ?n ]T and N training values, the mean is given by ? = N1 n yn
P
and the covariance matrix by ? = N1 n (yn ? ?)(yn ? ?)T . All other classifier
parameter values, such as ?s or E[s|?], can be extracted or calculated from ? and
?.
At each time step N , we determine the maximum likelihood estimate of onset time
? and bio-fouling rate m from the data under test. We find the maximum likelihood
estimate of bio-fouling rate m, for some onset time ? , by setting the first derivative
of (14) with respect to m equal to zero. This operation yields the relation
N
N
X
X
(k ? ? )2 2
k ? ? (xk ? ?k )?k
(xk ? ?k ?k )2
2
m
(15)
?k =
?? +
?k2
?k
?k
?k2
k=? +1
k=? +1
where ?k = 1 ? m(k ? ? ) and N is the current time. Note that m appears both at
the beginning of (15) and in the definition of ?, so we do not have a closed form
solution for m. However, the ? values act as weights that increase the importance
of most recent measurements. This weighting accounts for the expected decrease in
measurement variance as bio-fouling progresses. To estimate m we take an iterative
approach. First, initialize m to its minimum mean-squared error value given by
PN
+1 (k ? ? )(xk ? ?k )?k
(0)
m = ? k=?
(16)
PN
2 2
k=? +1 (k ? ? ) ?k
Second, repeatedly solve (15) for m(i) with ? calculated using the previous value
m(i?1) . The estimated rate value stops changing when h reaches a maximum.
If we set the window length N ? k to maximize the log likelihood ratio, h, the best
estimate of onset time is ? . To determine the onset time estimate, ? , we search
over over all past time for the value of k that maximizes h (14). For each possible
window length, that is k = 3 . . . N , we determine the maximum likelihood estimate
for m and then calculate the corresponding discriminant h. The estimated onset
time ? is the window length N ? k that gives the largest value of h. If this h is
above our threshold, the current measurement is classified as bio-fouled.
4
On-line Bio-fouling Detectors
To see how well our classifiers worked in practice, we implemented versions that operated on real-time salinity and temperature measurements. For all four instances
35
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30
Max Sal
Max Sal
of sensor degradation (three bio-fouling incidents and one instrument failure that
mimicked bio-fouling) that occurred in the summer 2001 test period, our classifiers
correctly indicated a sensor problem before the field staff was aware of it. In addition, the real-time classifiers produced no false alarms during the summer test
period. More in-depth discussion of the detector suite is given by Archer et al in
[4].
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10/05
10/10
30
SLR
15
SLR
20
15
9/05
20
10
20
10
5
0
25
9/05
9/10
9/15
9/20
9/25
Date
(a) Red26
9/30
10/05
10/10
0
(b) Tansy Point
Figure 3: Bio-fouling Indicators Red26 and Tansy Point. Top plots show maximum
diurnal salinity. Dotted lines indicate historical no false alarm (lower) and 10% false
alarm rate (upper). Field staff schedule sensors for cleaning when the maximum salinity
drops ?too low?, roughly the no false alarm level. Bottom plots show the sequential
likelihood discriminant for forty days of salinity and temperature measurements. Dotted
lines indicate historical no false alarm (upper) and 10% false alarm rate (lower). The ?
indicates the estimated bio-fouling onset time.
The on-line monitor displays a bio-fouling indicator for the previous forty days of
data. Figure 3 shows the on-line bio-fouling monitor during incidents at the Red26
CT1448 sensor and the Tansy Point CT1462 sensor. Since we had another sensor
mounted at the Red26 site that did not bio-foul, Figure 2, we were able to estimate
the bio-fouling time as September 28th . Our detector discriminant passed the no
false alarm threshold five days after onset and roughly three days before the field
staff decided the instrument needed cleaning. This reduction in time to detection
corresponds to reduced data loss of over 30%. In addition, the onset time estimate
of September 29th was within a day of the true onset time.
The Tansy Point CT1462 sensor began to bio-foul a few days after the Red26
CT1448 sensor. Our detector indicated that the Tansy Point sensor was bio-fouling
on October 9th . Since neighboring sensor Red26 was being replaced on October
11th , the field staff decided to retrieve the Tansy Point sensor as well. On removal,
this sensor was found to be in the early stages of bio-fouling. In this case, indications
from our classifier permitted the sensor to be replaced before the field staff would
normally have scheduled it for retrieval. Experience with our on-line bio-fouling
indicators demonstrates that these methods substantially reduce time from biofouling onset to detection.
In addition to the events described above, we have fairly extensive experience with
the online detectors since their initial deployment in the Spring of 2001. At this
writing we have bio-fouling detectors at all observing stations in the estuary and
experience with events throughout the year. Near the end of October, 2001 we
experienced a false alarm in a sensor near the surface in the lower estuary. In
this case, a steady downward trend in surface salinity, caused by several days of
rain triggered a detector response. Following cessation of the precipitation, the
discriminant function h returned back to sub-threshold levels.
In a recent (February 2003) study of five sensor stations in the estuary we compared
data loss prior to the deployment of bio-fouling detectors, with data loss postdeployment. The pre-deployment period included approximately four years of data
from 1997 through the summer of 2001. The post-deployment period ran from
spring/summer of 2001 through February 2003.
Neglecting seasonal variation, prior to the deployment of our detectors, 68% of all
the sensor data was corrupted by bio-fouling. Following deployment, the rate of
data loss due to bio-fouling dropped to 35%. This is the actual data loss, and
includes delay in responding to the event detection. Were it economical to replace
the sensors immediately upon detection of bio-fouling, the data loss rate would have
been dropped farther to 17%. Even with the delay in responding to event detection,
the detectors have more than doubled the amount of reliable data collected from the
estuary.
5
Discussion
CORIE salinity sensors lose several months of data every year due to sensor biofouling. Developing discriminatory fault detectors for these sensors is hampered by
the variability of the bio-fouling time-signature, and the dearth of bio-fouling onset
example data for training. To solve this problem, we built parameterized novelty
detectors. Clean sensor models were developed based on archive data, while biofouled sensor models are given a simple parametric form that is fit online. On-line
bio-fouling detectors deployed during the summer of 2001 detected all episodes
of sensor degradation several days before the field staff without generating any
false alarms. Expanded installation of a suite of detectors throughout the estuary
continue to successfully detect bio-fouling with minimal false alarm intrusion. The
detector deployment has effectively doubled the amount of clean data available from
the estuary salinity sensors.
Acknowledgements
We thank members of the CORIE team, Arun Chawla and Charles Seaton, for
their help in acquiring appropriate sensor data, Michael Wilkin for his assistance
in labeling the sensor data, and Haiming Zheng for carrying forward the sensor
development and deployment and providing the comparison of data loss rates before
and after the detector deployment.. This work was supported by the National
Science Foundation under grants ECS-9976452 and CCR-0082736.
References
[1] A. Baptista, M. Wilkin, P. Pearson, P. Turner, C. McCandlish, and P. Barrett. Costal
and estuarine forecast systems: A multipurpose infrastructure for the Columbia river.
Earth System Monitor, 9(3), 1999.
[2] U.S. Army Corps of Engineers. Biological asssessment - Columbia river channel improvements project. Technical report, USACE Portland District, December 2001.
[3] M. Basseville. Detecting changes in signals and systems - a survey. Automatica,
24(3):309?326, 1988.
[4] C. Archer, A. Baptista, and T.K. Leen. Fault detection for salinity sensors in the
Columbia River Estuary. Water Resources Research, 39, 2003.
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1,660 | 2,509 | Learning Non-Rigid 3D Shape from 2D Motion
Lorenzo Torresani
Stanford University
[email protected]
Aaron Hertzmann
University of Toronto
[email protected]
Christoph Bregler
New York University
[email protected]
Abstract
This paper presents an algorithm for learning the time-varying shape of a
non-rigid 3D object from uncalibrated 2D tracking data. We model shape
motion as a rigid component (rotation and translation) combined with a
non-rigid deformation. Reconstruction is ill-posed if arbitrary deformations are allowed. We constrain the problem by assuming that the object
shape at each time instant is drawn from a Gaussian distribution. Based
on this assumption, the algorithm simultaneously estimates 3D shape and
motion for each time frame, learns the parameters of the Gaussian, and
robustly fills-in missing data points. We then extend the algorithm to
model temporal smoothness in object shape, thus allowing it to handle
severe cases of missing data.
1
Introduction
We can generally think of a non-rigid object?s motion as consisting of a rigid component
plus a non-rigid deformation. For example, a person?s head can move rigidly (e.g. turning
left or right) while deforming (due to changing facial expressions). If we view this non-rigid
motion from a single camera view, the shape and motion are ambiguous: for any hypothetical rigid motion, a corresponding 3D shape can be devised that fits the image observations.
Even if camera calibration and rigid motion are known, a depth ambiguity remains. Despite
this apparent ambiguity, humans interpret the shape and motion of non-rigid objects with
relative ease; clearly, more assumptions about the nature of the deformations are used by
humans.
This paper addresses the question: how can we resolve the ambiguity, with as weak assumptions as possible? We argue that, by assuming that the 3D shape is drawn from some
non-uniform PDF, we can reconstruct 3D non-rigid shape from 2D motion unambiguously.
Moreover, we show that this can be done without assuming that the parameters of the PDF
are known in advance. The use of a proper PDF makes the technique robust to noise and
overfitting. We demonstrate this approach by modeling the PDF as a Gaussian distribution
(more specifically, as a factor analyzer), and describe a novel EM algorithm for simultaneously learning the 3D shapes, the rigid motion, and the parameters of the Gaussian. We
also generalize this approach by modeling the shape as a Linear Dynamical System (LDS).
Our algorithm can be thought of as a structure-from-motion (SFM) algorithm with a learning component: we assume that a set of labeled point tracks have been extracted from a raw
video sequence, and the goal is to estimate 3D shape, camera motion, and a deformation
PDF. Our algorithm is well-suited to reconstruction in the case of missing data, such as
due to occlusions and other tracking outliers. However, we show significant improvements
over previous algorithms even when all tracks are visible.
Our work may also be seen as unifying Active Shape Models [1, 2, 5] with SFM, where both
are estimated jointly from an image sequence. Our methods are closely related to factor
analysis, probabilistic PCA, and linear dynamical systems. Our missing-data technique can
be viewed as generalizing previous algorithms for SFM with missing data (e.g. [8, 9]) to the
nonrigid case. In work concurrent to our own, Gruber and Weiss [7] also apply EM to SFM;
their work focuses on the rigid case with known noise, and applies temporal smoothing to
rigid motion parameters rather than shape.
2
Deformation, Shape, and Ambiguities
We now formalize the problem of interpreting non-rigid shape and motion. We assume
that a scene consists of J scene points sj,t , where j is an index over scene points, and t
is an index over image frames. The 2D projections pj,t of these points are imaged under
orthographic projection:
pj,t
= Rt (sj,t + dt ) + n
(1)
where pj,t is the 2D projection of scene point j at time t, dt is a 2 ? 1 translation vector,
Rt is a 2 ? 3 matrix that combines rotation with orthographic projection [12], and n is
zero-mean Gaussian noise with variance ? 2 . Collecting the projected points into a 2 ? J
matrix Pt = [p1,t , ..., pJ,t ] and the 3D shape into a 3 ? J matrix St = [s1,t , ...sJ,t ] gives
the equivalent form
Pt
= Rt (St + Dt ) + N
(2)
where Dt = dt 1T contains J copies of the translation matrix dt . Note that rigid motion of
the object and rigid motion of the camera are interchangeable. Our goal is to estimate the
time-varying shape St and motion (Rt , Dt ) from the observed projections Pt . Without any
constraints on the 3D shape sj,t , this problem is extremely ambiguous [11]. For example,
given a shape St and motion (Rt , Dt ) and an arbitrary orthonormal matrix At , we can
produce a new shape At St and motion (Rt A?1
t , At Dt ) that together give identical 2D
projections as the original model, even if a different matrix At is applied in every frame.
A common way to model non-rigid deformations is to assume that the shape is produced
?
by adding deformations to a shape average S:
St
=
?+
S
K
X
Vk zk,t
(3)
k=1
where zk,t are scalar per-frame weights that indicate the contributions of the deformations
?
to each shape; these weights are combined in a vector zt = [z1,t , ..., zK,t ]T . Together, S
and Vk are referred to as the shape basis. Equivalently, the space of possible shapes may be
described by linear combinations of basis shapes, by selecting K + 1 linearly independent
points in the space. This model was first applied to non-rigid SFM by Bregler et al. [4].
However, this model contains ambiguities, since, for some 3D shape and motion, there
will still be ways to combine different weights and a different rigid motion to produce the
same 3D shape. Since we are performing a 2D projection, an additional
P depth ambiguity
occurs. For example, whenever there exist weights wk such that Rt Vk wk = 0 and
P
Vk wk 6= 0, these weights define a linear space of distinct 3D shapes (with weights
zt,k + ?wk ) that give identical 2D projections. (When the number of basis shapes is small,
these ambiguities are rarer and may not make a dramatic impact.) Furthermore, a leastsquares fit may overfit noise, especially with many basis shapes. As the number of basis
shapes grows, the problem is more likely to become unconstrained, eventually approaching
the totally unconstrained case described above.
The ambiguity and overfitting may be resolved by introducing regularization terms that penalize large deformations, and then solving for 3D shape
sense. Soatto
P in a least-squares
? 2 . However, this reguand Yezzi [11] use a regularization term equivalent to t ||St ? S||
larization may be too restrictive in many cases and too loose in others. For example, when
tracking a face, deformations of the jaw are much more likely than deformations of the
nose. Moreover, the weight for this regularization term must be specified by hand 1 . Alternatively, Brand [3] proposes placing a user-specified Gaussian prior on the deformation
basis and a prior on the deformations based on an initial estimate.
In order to motivate our approach, we can restate the above techniques as follows. Suppose
we assume that shapes St are drawn from a probabilitity distribution p(St |?) with known
parameters ?. The non-rigid shape and motion are estimated by maximizing
p(S, R, D|P, ?, ? 2 )
? p(P|S, R, D, ?, ? 2 )p(S, R, D|?, ? 2 )
Y
?
p(Pt |St , Rt , Dt , ? 2 )p(St |?)
(4)
(5)
t
assuming uniform priors on Rt , and Dt . The projection likelihood p(Pt |St , Rt , Dt , ? 2 )
is a spherical Gaussian (Equation 2). The negative log-posterior ? ln p(S, R, D, ?|P)
corresponds to a standard least-squares formulation for SFM, plus a regularization term
? ln p(St |?). If we set p(St |?) to be a uniform distribution, then we get the highly underconstrained case described above. If we set p(St |?) to be a spherical Gaussian with
? ? 2 I)) then we obtain the simple regularization
a specified variance (e.g. p(St |?) = N (S;
used previously ? the problem is constrained, but by a weak regularization term with a
user-specified weight (variance).
Our approach. Our approach is to simultaneously estimate the rigid motion and learn
the shape PDF. In other words, we estimate R, D, ?, and ? 2 to maximize
Z
p(R, D, ?, ? 2 |P) =
p(R, D, ?, S, ? 2 |P)dS
(6)
Z
?
p(P|R, D, S, ? 2 )p(S|?)dS
(7)
The key idea is that we can estimate shape and motion while learning the parameters of the
PDF p(S|?) over shapes. (Our method marginalizes over the unknown shapes S t , rather
than solving for estimates of shape.) In effect, the regularization terms (i.e. the PDF) are
learned simultaneously with the rest of SFM. This means that the regularization terms need
not be set manually, and can thus be much more sophisticated and have many more parameters than previous methods. In practice, we find that this leads to significantly improved
reconstructions over user-specified shape PDFs. We demonstrate the approach by modeling the shape PDF as a general Gaussian. We reduce the dimensionality of the Gaussian
by representing it as a factor analyzer. In this case, the factors Vk may be interpreted as
basis deformations. We later generalize this approach to model shape as an LDS, leading
to temporal correlations in the shape PDF.
It might seem that, since the parameters of the PDF are not known a priori, the algorithm
could estimate wildly varying shapes, and then learn a correspondingly spread-out PDF.
1
In their work, Soatto and Yezzi address a slightly simpler problem where the 3D data is observed
without noise or projection, and thus there are no weights to specify in this case
However, such a spread-out PDF would assign very low likelihood to the solution and thus
be suboptimal; this is a typical case of Bayesian learning naturally balancing the desire to
fit the data with the desire for a ?simple? model. One way to see this is to consider the
terms of ? ln p(R, D, ?|P) in the case of the Gaussian prior PDF: in addition to the datafitting term and the regularization term, there is a ?normalization constant? term of T ln |?|,
where T is the number of frames and ? is the covariance of the shape PDF. This term
directly penalizes spread-out Gaussians. Hence, the optimal solution trades-off between
(a) fitting the projection data, (b) fitting the shapes St to the shape PDF (regularizing),
and (c) minimizing the variance of the shape PDF as much as possible. The algorithm
simultaneously regularizes and learns the regularization.
3
Learning a Gaussian shape distribution
We now describe our algorithm in detail. We model p(St |?) as a factor analyzer [6]. In
this setting, the factors of the Gaussian can be interpreted as basis deformations ? shape
is modeled by Equation 3 ? but the weights zt are now hidden variables, with zero-mean
Gaussian priors with unit variance for each:
zt
?
N (0; I)
(8)
The shape and projection model is then completely specified by Equations 2, 3,
and 8. The problem of non-rigid SFM is now to solve for the maximum like? V, and ? 2 , i.e. maximize p(Rt , Dt , S,
? V, ? 2 |Pt ) ?
lihood
estimates of Rt , Dt , S,
Q
Q R
2
2
?
?
p(P
|R
,
D
,
S,
V,
?
)
=
p(P
,
z
|R
,
D
,
S,
V,
?
)p(z
)dz
t
t
t
t t
t
t
t
t
t
t
3.1
Vectorized form.
For later computations, it is useful to rewrite the model in a vectorized form. First, define
ft to be the vector of point tracks ft = vec(Pt ) = [x1,t , y1,t , ..., xJ,t , yJ,t ]T . Note that ft
is the same variable as Pt , but written as a vector rather than a matrix2 . Expanding ft we
have
ft
= vec(Pt ) = vec(Rt St + Rt Dt + Nt )
=
K
X
(9)
? + vec(Rt Dt ) + vec(Nt )
vec(Rt Vk )zk,t + vec(Rt S)
(10)
k=1
= M t zt + ?
ft + Tt + vec(Nt )
(11)
?
?
where Mt = [vec(Rt V1 ), ..., vec(Rt VK )], zt = [z1,t , ..., zK,t ] , ft = vec(Rt S) and
Tt = vec(Rt Dt ) = [(Rt dt )T , ..., (Rt dt )T ]T = [tTt , ..., tTt ]T . Note that the marginal
distribution over shape ? as well as its projection ? is Gaussian:
Z
p(ft |?) =
p(ft |zt , ?)p(zt |?)dzt
(12)
T
= N (ft |Tt + ?
ft ; Mt MTt + ? 2 I)
? Vk , Rt , Dt and ? 2 .
where ? encapsulates the model parameters S,
(13)
? = [vec(S),
? vec(V1 ), ..., vec(VK )] and ?
Let H
zt = [1, zTt ]T . We can also rewrite the
? zt , by using the identity
shape equation as vec(Rt St ) = (I ? Rt )vec(St ) = (I ? Rt )H?
T
vec(ABC) = (C ? A)vec(B). The symbol ? denotes Kronecker product.
a0 a2
=
a1 a3
[a0 , a1 , a2 , a3 ]T . The operator is linear: vec(A + B) = vec(A) + vec(B), vec(?A) = ?vec(A)
for any matrices A and B and scalar ?.
2
The vec operator stacks the columns of a matrix into a vector, e.g. vec
3.2
Generalized EM algorithm.
Given a set of point tracks P (equivalently, f ), we can estimate the motion and deformation
model using EM; the algorithm is similar to EM for factor analysis [6].
The E-step. We estimate the distribution over zt given the current motion and shape
estimates, for each frame t. Defining q(zt ) to be the distribution to be estimated in frame
t, it can be computed as
q(zt )
?
= p(zt |ft , ?)
= N (zt |?(ft ? ?
ft ? Tt ); I ? ?Mt )
(14)
(15)
= MTt (Mt MTt + ? 2 I)?1
(16)
The matrix inversion lemma may be used to accelerate the computation of ?. We define
the expectations ?t ? Eq [zt ] and ?t ? Eq [zt zTt ] and compute them as:
?t
?t
= ?(ft ? ?
ft ? T t )
= I ? ?Mt +
(17)
?t ?Tt
(18)
? = E[?
We also define ?
?t = E[?
zt ] = [1, ?Tt ]T and ?
zt ?
zTt ] =
1
?t
?Tt
?t
.
The M-step. We estimate the motion parameters by minimizing
Q(P, ?)
= Eq(z1 ),...,q(zT ) [? log p(P|?)]
(19)
X
?
2
2
=
Eq(zt ) [||ft ? vec(Rt St ) ? Tt )|| /(2? )] + 2JT log 2?? 2 (20)
t
? Vk ), in the rigid motion parameters
This function is quadratic in the shape parameters (S,
(Rt , Tt ) and in the gaussian noise variance parameter ? 2 . To update each of these parameters we compute the corresponding partial derivative of the expected log likelihood, set it
to zero and solve it. The parameter update rules are:
? Shape basis:
? ?
vec(H)
X
t
?t ? (I ? RTt Rt ))
(?
!?1
X
vec
t
(I ? Rt )T (ft ? Tt )?
?Tt
!
(21)
? Noise variance:
1 X
?2 ?
(||ft ? ?
ft ? Tt ||2 ? 2(ft ? ?
ft ? Tt )T Mt ?t + tr(MTt Mt ?t )) (22)
2JT t
? Translation:
Tt ? (1 ? I)
? Rotation:
Rt ? arg min ||Rt
Rt
X
1X
?j +
(ftj ? Rt (S
Vkj ?tk ))
J j
(23)
k
X
j
?t H
? j?
? Tj ) ?
(H
X
? = [H
? T1 , ..., H
? T ]T and ft = [ft1 , ..., ftJ ].
where H
J
j
? Tj )||
((ftj ? tt )?
?Tt H
(24)
Since the system of equations in Equation 21 is large and sparse, we solve it using conjugate
gradient. In Equation 24, we enforce orthonormality of rotations by parameterizing R t
with exponential coordinates. We linearize the equation with respect to the exponential
coordinates, and solve the resulting quadratic.
If any of the point tracks are missing, they are also filled in during the M-step. Let f t?
denote the elements of a frame of tracking data that are not observed; they are estimated as
ft? ? ?
ft? + M?t ?t + T?t
(25)
where (? ) indicates rows that correspond to the missing data.
In our M-step, we apply each of these updates once, although they could also be alternated.
?
Once
P EM has converged, the maximum likelihood shapes may be computed as S t = S +
V
?
.
k t,k
k
4
Learning dynamics
Many real deformations contain some temporal smoothness. We model temporal behavior
of deformations using a Linear Dynamical System (LDS). In this model, Equation 8 is
replaced with
z0
zt
? N (0; I)
= ?zt?1 + n,
n ? N (0; Q)
(26)
(27)
where ? is an arbitrary unknown K ? K matrix, and Q is a K ? K covariance matrix. For
certain estimates of ?, this model corresponds to an assumption of continuously or slowly
changing shape. Since our model is a special form of Shumway and Stoffer?s algorithm
for LDS learning with EM [10], it is straightforward to adapt it to our needs. In the Estep, we apply Shumway and Stoffer?s E-step to estimate ?t , ?t , and E[zt zTt?1 ], based on
? Mt , ?, Q, and ? 2 . In the M-step, we apply the same shape and motion updates as
Pt , S,
in the previous section; additionally, we update ? and Q in the same way as in Shumway
and Stoffer?s algorithm. In other words, this reconstruction algorithm learns 3D shape with
temporal smoothing, while learning the temporal smoothness term.
5
Experiments
We compared our algorithm with the iterative SFM algorithm presented by Torresani et
al. [13], which we will refer to as ILSQ (iterative least-squares) in the following discussion3 . ILSQ optimizes Equations 2 and 3 by alternating optimization of each of the unknowns (rotation, basis shapes, and coefficients). We also improved the algorithm by updating the translations as well. When some data is missing, ILSQ optimizes with respect
to the available data. For both algorithms, the rigid motion is initialized by Tomasi-Kanade
[12], and random initialization of the shape basis and coefficients. For the algorithm presented in section 3, we adopted an annealing scheme that forces ? 2 to remain large in
the initial steps of the optimization. We refer to our new algorithms as EM-Gaussian and
EM-LDS.
We tested the algorithms on a synthetic animation of a deforming shark in Figure 1. The
motion consists of rigid rotation plus deformations generated by K = 2 basis shapes. The
average reconstruction errors in Z for ILSQ and EM-Gaussian are respectively 7.10% and
2.50% on this sequence after 100 parameter updates.4 By enforcing temporal smoothness
3
In our experience, ILSQ always performs better than the algorithm of Bregler et al. [4].
All errors are computed in percentage points: the average distance of the reconstructed point to
the correct point divided by the size of the shape.
4
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Figure 1: Reconstructions of the shark sequence using the three algorithms. Each algorithm was
given 2D tracks as inputs; reconstructions are shown here from a different viewpoint than the inputs
to the algorithm. Ground-truth features are shown as blue dots; reconstructions are red circles. Note
that, although ILSQ gets approximately the correct shape in most cases, it misses details, whereas
EM gives very accurate results most of the time. Some of the deformation errors of EM-Gaussian
(e.g. for t=148) are corrected by EM-LDS through temporal smoothing.
EM-LDS was able to correct some of the deformation errors of EM-Gaussian. The average
Z error for EM-LDS on the shark sequence after 100 EM iterations is 1.24%. Videos of
the shark reconstructions and the Matlab software used for these experiments are available
from http://movement.stanford.edu/learning-nr-shape/ .
In highly-constrained cases ? low-rank motion, no image noise, and no missing data ?
ILSQ achieved reasonably good results. However, EM-Gaussian gave better results in
nearly every case, and dramatically better results in underconstrained cases. Figure 2(a)
and (b) show experimental results on another set of artificial data consisting of random
basis shapes. Figure 2(a) shows the results of reconstruction with missing data; the ILSQ
results degrade much faster as the percentage of missing data increases. Figure 2(b) shows
the effect of changing the complexity of the model, while leaving the complexity of the
data fixed. ILSQ yields poor results when the model complexity does not closely match the
data complexity, but EM-Gaussian yields reasonable results regardless.
6
Discussion and future work
We have described an approach to non-rigid structure-from-motion with a probabilistic
deformation model, and demonstrated its usefulness in the case of a Gaussian deformation
model. We expect that more sophisticated distributions can be used to model more complex
non-rigid shapes in video. More general graphical models with other correlations (such
as from audio data) could be built from this method. Our method is also applicable to
3
2.5
10
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ILSQ
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% z error
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(b)
Figure 2: Error comparison between ILSQ and EM-Gaussian on random basis shapes. (a) Increasing
missing data. As the percentage of missing feature tracks per frame increases, ILSQ degenerates
much more rapidly than EM-Gaussian. (b) ILSQ gives poor results when the model complexity does
not match the actual data complexity, whereas EM-Gaussian is relatively robust to this.
separating rigid from non-rigid motion in fully-observed data, as in Soatto and Yezzi?s
work [11]. Our models could easily be generalized to perspective projection, although the
optimization may be more difficult.
Acknowledgements. Thanks to Hrishikesh Deshpande for assisting with an early version of this
project, and to Stefano Soatto for discussing deformation ambiguities. Portions of this work were
performed while LT was visiting New York University, AH was at University of Washington, and
while CB was at Stanford University. LT and CB were supported by ONR grant N00014-01-1-0890
under the MURI program. AH was supported in part by UW Animation Research Labs, NSF grant
IIS-0113007, the Connaught Fund, and an NSERC Discovery Grant.
References
[1] A. Blake and M. Isard. Active Contours. Springer-Verlag, 1998.
[2] V. Blanz and T. Vetter. A Morphable Model for the Synthesis of 3D Faces. In Proceedings of
SIGGRAPH 99, Computer Graphics Proceedings, pages 187?194, Aug. 1999.
[3] M. Brand. Morphable 3D models from video. In Proc. CVPR 2001, 2001.
[4] C. Bregler, A. Hertzmann, and H. Biermann. Recovering Non-Rigid 3D Shape from Image
Streams. In Proc. CVPR 2000, 2000.
[5] T. F. Cootes and C. J. Taylor. Statistical models of appearance for medical image analysis and
computer vision. In Proc. SPIE Medical Imaging, 2001.
[6] Z. Ghahramani and G. E. Hinton. The EM Algorithm for Mixtures of Factor Analyzers. Technical Report CRG-TR-96-1, University of Toronto, 1996.
[7] A. Gruber and Y. Weiss. Factorization with Uncertainty and Missing Data: Exploiting Temporal
Coherence. In Proc. NIPS 2003, 2003. In these proceedings.
[8] D. W. Jacobs. Linear Fitting with Missing Data for Structure-From-Motion. Computer Vision
and Image Understanding, 82:57?82, 2001.
[9] H. Shum, K. Ikeuchi, and R. Reddy. Principal Component Analysis with Missing Data and Its
Applications to Polyhedral Object Modeling. IEEE Trans. PAMI, 17(9):854?867, 1995.
[10] R. H. Shumway and D. S. Stoffer. An approach to time series smoothing and forecasting using
the em algorithm. J. Time Series Analysis, 3(4):253?264, 1982.
[11] S. Soatto and A. J. Yezzi. Deformotion: Deforming Motion, Shape Averages, and the Joint
Registration and Segmentation of Images. In Proc. ECCV 2002, May 2002.
[12] C. Tomasi and T. Kanade. Shape and motion from image streams under orthography: A factorization method. Int. J. of Computer Vision, 9(2):137?154, 1992.
[13] L. Torresani, D. Yang, G. Alexander, and C. Bregler. Tracking and Modeling Non-Rigid Objects
with Rank Constraints. In Proc. CVPR, 2001.
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1,661 | 251 | 332
Hormel
A Sell-organizing Associative
Memory System lor Control
Applications
Michael Bormel
Department of Control Theory and Robotics
Technical University of Darmstadt
Schlossgraben 1
6100 Darmstadt/W.-Ger.any
ABSTRACT
The CHAC storage scheme has been used as a basis
for a software implementation of an associative
.emory system AHS, which itself is a major part
of the learning control loop LERNAS. A major
disadvantage of this CHAC-concept is that the
degree of local generalization (area of interpolation) is fixed. This paper deals with an algorithm for self-organizing variable generalization for the AKS, based on ideas of T. Kohonen.
1 INTRODUCTION
For several years research at the Department of Control Theory and Robotics at the Technical University of Darmstadt
has been concerned with the design of a learning real-time
control loop with neuron-like associative memories (LERNAS)
A Self-organizing Associative Memory System for Control Applications
for the control of unknown, nonlinear processes (Ersue,
Tolle, 1988). This control concept uses an associative memory system AHS, based on the cerebellar cortex model CHAC by
Albus (Albus, 1972), for the storage of a predictive nonlinear process model and an appropriate nonlinear control strategy (Fig. 1).
e&;ected process response
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Figure 1: The learning control loop LERNAS
One problem for adjusting the control loop to a process is,
however, to find a suitable set of parameters for the associative memory. The parameters in question determine the
degree of generalization within the memory and therefore
have a direct influence on the number of training steps required to learn the process behaviour. For a good performance of the control loop it ? is desirable to have a very
small generalization around a given setpoint but to have a
large generalization elsewhere. Actually, the amount of collected data is small during the transition phase between two
333
334
Hormel
setpoints but is large during setpoint control. Therefore a
self-organizing variable generalization, adapting itself to
the amount of available data would be very advantageous.
Up to now, when working with fixed generalization, finding
the right parameters has meant to find the best compromise
between performance and learning time required to generate a
process model. This paper will show a possibility to introduce a self-organizing variable generalization capability
into the existing AMS/CMAC algorithm.
2 THE AMS-CONCEPT
The associative memory syste. AMS is based on the "Cerebellar Model Articulation Controller CMAC" as presented by J.S.
Albus. The information processing structure of AMS can be
divided into three stages.
1.) Each component of a n-dimensional input vector (stimulus) activates a fixed number p of sensory cells, the
receptive fields of which are overlapping. So n?p sensory cells become active.
2.) The active sensory cells are grouped to form p n-dimensional vectors. These vectors are mapped to p association cells. The merged receptive fields of the sensory
cells described by one vector can be seen as a hypercube
in the n-dimensional input space and therefore as the
receptive field of the association cell. In normal applications the total number of available association
cells is about 100?p.
3.) The association cells are connected to the output cells
by modifiable synaptic weights. The output cell computes
the mean value of all weights that are connected to active association cells (active weights).
Figure 2 shows the basic principle of the associative memory
system AMS.
A Self-organizing Associative Memory System for Control Applications
output value
input space
adjustable weights
Figure 2: The basic aechanism of AMS
During training the generated output is compared with a desired output, the error is computed and equally distributed
over all active weights. For the mapping of sensory cells to
association cells a hash-coding mechanism is used.
3 THE SELF-ORGANIZING FEATURE MAP
An approach for explaining the self-organizing capabilities
of the nervous system has been presented by T. Kohonen (Kohonen, 1988).
In his "self-organizing feature mapft a network of laterally
interconnected neurons can adapt itself according to the
density of trained points in the input space. Presenting a
n-diaensional input vector to the network causes every neuron to produce an output signal which is correlated with the
similarity between the input vector and a "template vector"
which may be stored in the synaptic weights of the neuron.
Due to the "mexican-hat" coupling function between the neurons, the one with the maximum output activity will excite
its nearest neighbours but will inhibit neurons farther away, therefore generating a localized response in the network. The active cells can now adapt their input weights in
order to increase their similarity to the input vector. If
we define the receptive field of a neuron by the number of
input vectors for which the neurons activity is greater than
335
336
Hormel
that of any other neuron in the net, this yields the effect
that in areas with a high density of trained points the receptive fields become small whereas in areas with a low density of trained points the size of the receptive fields is
large. Is mentioned above this is a desired effect when
workin; with a learning control loop.
4 SELF-ORGANIZING VARIABLE GENERALIZATION
Both of the approaches above have several advantages and
disadvantages when using them for real-time control applications.
In the AKS algorithm one does not have to care for predefining a network and the coupling functions or coupling matrices among the elements of the network. Association and
weight cells are generated when they are needed during
training and can be adressed very quietly to produce a memory response. One of the disadvantages is the fixed generalization once the parameters of a .eaory unit have been
chosen.
Unlike AHS, the feature map allows the adaption of the network according to the input data. This advantage has to be
payed for by extensive search for the best matching neuron
in the network and therefore the response time of the network aay be too large for real-tiae control when working
with big networks.
These problems can be overcome when allowing that the mapping of sensory cells to association cells in AKS is no
longer fixed but can be changed during training.
To accomplish this a template vector t is introduced for
every association cell. This vector i serves as an indicator
for the stimuli by which the association cell has been accessed previously. During an associative recall for a stimulus !o a preliminary set of p association cells is activated
by the hash coding mechanism. Due to the self-organizing
process during training the template vectors do not need to
correspond to the input vector !o. For the search for the
A Self-organizing Associative Memory System for Control Applications
best aatching cell the template vector 10 of the accessed
association cell is compared to the stiaulus and a difference vector is calculated.
6.
,
n
=."
t.
i = O, ??? ,n
- L.v
(1)
s
number of searching steps
s
This vector can now be used to compute a virtual stimulus
which compensates the mapping errors of the hash-coding
mechanism.
~+1 = ~ -
-4
i=O, ??? ,n
(2)
s
The best matching cell is found for
j =
i
ain
II '1.
6. "
.
= O, ??? ,ns
(3)
1
and can be adressed by the virtual stimulus ~j when using
the hash coding mechanism. This search mechanism ensures
that the best matching cell is found even if self organization is in effect.
During training the template
cells are updated by
t(t+l)
d
= a(k,d) ?(!(k)
-let?~
vectors of
the
association
+ t(k)
(4)
lateral distance of neurons in the network
where t(k) denotes the value of the teaplate vector at time
k and ~(k) denotes the stimulus. a(t,d) is a monotonic decreasing function of time and the lateral distance between
neurons in the network.
6 SIMULATION RESULTS
Figure 3 and 4 show some simulation results of the presented
algorithm for the dase of a two dimensional stimulus vector.
337
338
Hormel
Figure J shows the expected positions in input space of the
untrained template vectors ( x denotes untrained association
cells).
? ? ? ?
? ? ? ?
? ? ? ?
? ? ? ?
? ? ? ?
? ? ?
? ? ?
? ? ?
? ? ?
? ? ?
? ? ? ? ?
? ? ? ? ?
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? ? ? ? ?
? ? ? ? ?
? ?
? ?
? ?
? ?
?
?
?
?
? ? ?
?? .? .? .? .? .? .? .? .? .? . . .? .? .?
?? .? .? .? .? .? .? .? .? .? . . .. .. ..
? ? ? ? ? ? ? ? ? ? ? ? ? ? ?
? ? ? ? ? ? ? ? ? ? ? ? ? ? ?
? ? ? ? ? ? ? ? ? ? ? ? ? ? ?
?? .? .? .? .? .? .? .? .? .? .? .? .? .? .?
? ? ? ? ? ? ? ? ? ? ? ? ? ? ?
Figure J: Untrained .etwork
Figure 4 shows the network after 2000 training steps with
stimuli of gaussian distribution in input space. The position of the template vectors of trained cells has shifted
into the direction of the better trained areas, so that more
association cells are used to represent this area than before. Therefore the stored information will be more exact in
this area.
? ? ? ?
? ? ? ?
? ? ? ?
? ? ? ?
? ? ?
? ? ? ?
? ?
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? ?
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?
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Figure 4: Network after 2000 training steps
A Self-organizing Associative Memory System for Control Applications
6 CONCLUSION
The ney algorithm presented above introduces the capability
to adapt the storage mechanisms of a CMAC-type associative
memory according to the arriving stimuli. This will result
in various degrees of generalization depending on the number
of trained points in a given area. It therefore will make it
unnecessary to choose a generalization factor as a compromise between several constraints when representing nonlinear
functions by storing them in this type of associative memory. Some results on tests will be presented together with a
comparison on respective results for the original AMS.
Acknowledgements
This work was sponsored by the German !inistry for
and Technology (BMFT) under grant no. ITR 8800 B/5
Research
References
E.
Ersue, H. Tolle. (1988) Learning Control Structures with
Neuron-Like Associative memories. In: v. Seelen, Shaw, Leinhos (Eds.) Organization of Neural Networks, VCH Verlagsgesellschaft, Weinheim, FRG, 1988
J.S. llbu~ (1972) Theoretical and experimental aspects of a
cerebellar model, PhD thesis, University of Maryland, USA
E. Ersue, X. Mao (1983) Control of pH by Use of a Self-organizing Concept with Associative Memories. ACI'83, Kopenhagen, Denmark
E. Ersue, J. Militzel (1984) Real-tiae Implementation of an
Associative Memory-based Learning Control Scheme for Nonlin-ear Jfultivariable Systems. SymposiuDl on "Applications of
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T. Kohonen. (1988) Self-Organization and Associative Memory,
2nd Ed., Springer Verlag
339
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1,662 | 2,510 | Fast Embedding of
Sparse Music Similarity Graphs
John C. Platt
Microsoft Research
1 Microsoft Way
Redmond, WA 98052 USA
[email protected]
Abstract
This paper applies fast sparse multidimensional scaling (MDS) to a large
graph of music similarity, with 267K vertices that represent artists, albums, and tracks; and 3.22M edges that represent similarity between
those entities. Once vertices are assigned locations in a Euclidean space,
the locations can be used to browse music and to generate playlists.
MDS on very large sparse graphs can be effectively performed by a
family of algorithms called Rectangular Dijsktra (RD) MDS algorithms.
These RD algorithms operate on a dense rectangular slice of the distance
matrix, created by calling Dijsktra a constant number of times. Two RD
algorithms are compared: Landmark MDS, which uses the Nystr?m approximation to perform MDS; and a new algorithm called Fast Sparse
Embedding, which uses FastMap. These algorithms compare favorably
to Laplacian Eigenmaps, both in terms of speed and embedding quality.
1
Introduction
This paper examines a general problem: given a sparse graph of similarities between a set of
objects, quickly assign each object a location in a low-dimensional Euclidean space. This
general problem can arise in several different applications: the paper addresses a specific
application to music similarity.
In the case of music similarity, a set of musical entities (i.e., artists, albums, tracks) must
be placed in a low-dimensional space. Human editors have already supplied a graph of
similarities, e.g., artist A is similar to artist B. There are three good reasons to embed a
musical similarity graph:
1. Visualization ? If a user?s musical collection is placed in two dimensions, it can
be easily visualized on a display. This visualization can aid musical browsing.
2. Interpolation ? Given a graph of similarities, it is simple to find music that
?sounds like? other music. However, once music is embedded in a lowdimensional space, new user interfaces are enabled. For example, a user can specify a playlist by starting at song A and ending at song B, with the songs in the
playlist smoothly interpolating between A and B.
3. Compression ? In order to estimate ?sounds like? directly from a graph of music
similarities, the user must have access to the graph of all known music. However,
once all of the musical entities are embedded, the coordinates for the music in a
user?s collection can be shipped down to the user?s computer. These coordinates
are much smaller than the entire graph.
It is important to have algorithms that exploit the sparseness of similarity graphs because
large-scale databases of similarities are very often sparse. Human editors cannot create a
dense N ? N matrix of music similarity for large values of N. The best editors can do is
identify similar artists, albums, and tracks. Furthermore, humans are poor at accurately
estimating large distances between entities (e.g., which is farther away from The Beatles:
Enya or Duke Ellington?)
Hence, there is a definite need for an scalable embedding algorithm that can handle a sparse
graph of similarities, generalizing to similarities not seen in the training set.
1.1
Structure of Paper
The paper describes three existing approaches to the sparse embedding problem in section
2 and section 3 describes a new algorithm for solving the problem. Section 4.1 verifies
that the new algorithm does not get stuck in local minima and section 4.2 goes into further
detail on the application of embedding musical similarity into a low-dimensional Euclidean
space.
2
Methods for Sparse Embedding
Multidimensional scaling (MDS) [4] is an established branch of statistics that deals with
embedding objects in a low-dimensional Euclidean space based on a matrix of similarities.
More specifically, MDS algorithms take a matrix of dissimilarities ?rs and find vectors ~xr
whose inter-vector distances drs are well matched to ?rs . A common flexible algorithm is
called ALSCAL [13], which encourages the inter-vector distances to be near some ideal
values:
2
2 2
min ?(drs
? d?rs
) ,
(1)
~xr
rs
where d? are derived from the dissimilarities ?rs , typically through a linear relationship.
There are three existing approaches for applying MDS to large sparse dissimilarity matrices:
1. Apply an MDS algorithm to the sparse graph directly.
Not all MDS algorithms require a dense matrix ?rs . For example, ALSCAL can operate on
a sparse matrix by ignoring missing terms in its cost function (1). However, as shown in
section 4.1, ALSCAL cannot reconstruct the position of known data points given a sparse
matrix of dissimilarities.
2. Use a graph algorithm to generate a full matrix of dissimilarities.
The Isomap algorithm [14] finds an embedding of a sparse set of dissimilarities into a lowdimensional Euclidean space. Isomap first applies Floyd?s shortest path algorithm [9] to
find the shortest distance between any two points in the graph, and then uses these N ? N
distances as input to a full MDS algorithm. Once in the low-dimensional space, data can
easily be interpolated or extrapolated. Note that the systems in [14] have N = 1000.
For generalizing musical artist similarity, [7] also computes an N ? N matrix of distances
between all artists in a set, based on the shortest distance through a graph. The sparse
graph in [7] was generated by human editors at the All Music Guide. [7] shows that human
perception of artist similarity is well modeled by generalizing using the shortest graph
distance. Similar to [14], [7] projects the N ? N set of artist distances into a Euclidean
space by a full MDS algorithm. Note that the MDS system in [7] has N = 412.
The computational complexity for these methods inhibit their use on large data sets. Let us
analyze the complexity for each portion of this method.
For finding all of the minimum distances, Floyd?s algorithm operates on a dense matrix of
distances and has computational complexity O(N 3 ). A better choice is to run Dijkstra?s
algorithm [6], which finds the minimum distances from a single vertex to all other vertices
in the graph. Thus, Dijkstra?s algorithm must be run N times. The complexity of one invocation of Dijkstra?s algorithm (when implemented with a binary heap [11]) is O(M log N)
where M is the number of edges in the graph.
Running a standard MDS algorithm on a full N ? N matrix of distances requires O(N 2 Kd)
computation, where K is the number of iterations of the MDS algorithm and d is the dimensionality of the embedding. Therefore, the overall computational complexity of the
approach is O(MN log N + N 2 Kd), which can be prohibitive for large N and M.
3. Use a graph algorithm to generate a thin dense rectangle of distances.
One natural way to reduce the complexity of the graph traversal part of Isomap is to not run
Dijkstra?s algorithm N times. In other words, instead of generating the entire N ? N matrix
of dissimilarities, generate an interesting subset of n rows, n << N.
There are a family of MDS algorithms, here called Rectangular Dijkstra (RD) MDS algorithms. RD algorithms operate on a dense rectangle of distances, filled in by Dijkstra?s
algorithm. The first published member of this family was Landmark MDS (LMDS) [5].
Bengio, et al.[2] show that LMDS is the Nystr?m approximation [1] combined with classical MDS [4] operating on the rectangular distance matrix. (See also [10] for Nystr?m
applied to spectral clustering).
LMDS operates on a number of rows proportional to the embedding dimensionality, d.
Thus, Dijkstra gets called O(d) times. LMDS then centers the n ? n distance submatrix,
converting it into a kernel matrix K. The top d column eigenvectors (~vi ) and eigenvalues ?i
of K are then computed. The embedding coordinate for the mth point is thus
1
~xm = ? Mi j (A j ? D jm ),
(2)
2 j
?
where Mi j = ~vTi / ?i , A j is the average distance in the jth row of the rectangular distance
matrix and D jm is the distance between the mth point and the jth point ( j ? [1..n]). Thus,
the computational complexity of LMDS is O(Md log N + Nd 2 + d 3 ).
3
New Algorithm: Fast Sparse Embedding
LMDS requires the solution of an n ? n eigenproblem. To avoid this eigenproblem, this
paper presents a new RD MDS algorithm, called FSE (Fast Sparse Embedding). Instead of
a Nystr?m approximation, FSE uses FastMap [8]: an MDS algorithm that takes a constant
number of rows of the dissimilarity matrix. FastMap iterates over the dimensions of the
projection, fixing the position of all vertices in each dimension in turn. FastMap thus
approximates the solution of the eigenproblem through deflation.
Consider the first dimension. Two vertices (~xa ,~xb ) are chosen and the dissimilarity from
these two vertices to all other vertices i are computed: (?ai , ?bi ). In FSE, these dissimilarities are computed by Dijkstra?s algorithm. During the first iteration (dimension), the
distances (dai , dbi ) are set equal to the dissimilarities.
The 2N distances can determine the location of the vertices along the dimension up to a
shift, through use of the law of cosines:
xi =
2 ? d2
dai
bi
.
2dab
(3)
For each subsequent dimension, two new vertices are chosen and new dissimilarities
(?ai , ?bi ) are computed by Dijkstra?s algorithm. The subsequent dimensions are assumed to
be orthogonal to previous ones, so the distances for dimension N are computed from the
dissimilarities via:
2
+
?ai2 = dai
N?1
N?1
n=1
n=1
? (xan ? xin )2 ? dai2 = ?ai2 ? ? (xan ? xin )2 .
(4)
Thus, each dimension accounts for a fraction of the dissimilarity matrix, analogous to PCA.
Note that, except for dab , all other distances are needed as distance squared, so only one
square root for each dimension is required. The distances produced by Dijkstra?s algorithm
are the minimum graph distances modified by equation (4) in order to reflect the projection
used so far.
For each dimension, the vertices a and b are heuristically chosen to be as far apart as
possible. In order to avoid an O(N 2 ) step in choosing a and b, [8] recommends starting
with an arbitrary point, finding the point furthest away from the current point, then setting
the current point to the farthest point and repeating.
The work of each Dijkstra call (including equation (4)) is O(M log N + Nd), so the complexity of the entire algorithm is O(Md log N + Nd 2 ).
4
Experimental Results
4.1
Artificial Data
Output of FSE
Output of ALSCAL
6.4
10
6.2
9
6
8
5.8
7
5.6
6
5.4
5
5.2
4
5
3
4.8
2
4.6
4.4
4.5
1
5
5.5
6
6.5
2
4
6
8
10
Figure 1: Reconstructing a grid of points directly from a sparse distance matrix. On the
left, ALSCAL cannot reconstruct the grid, while on the right, FSE accurately reconstructs
the grid.
An MDS algorithm needs to be tested on distance matrices that are computed from distances between real points, in order to verify that the algorithm quickly produces sensible
results.
FSE and ALSCAL were both tested on a set of 100 points in a 10 ? 10 2D grid with unit
spacing. The distance from each point to a random 10 of the nearest 20 other points were
presented to each algorithm. The results are shown in Figure 1. Procrustes analysis [4] is
applied to output of each algorithm; the output is shown after the best orthogonal affine
projection between the algorithm output and the original data.
Figure 1 shows that ALSCAL does a very poor job of reconstructing the locations of the
data points, while FSE accurately reconstructs the grid locations. ALSCAL?s poor performance is caused by performing optimization on a non-convex cost function. When the
dissimilarity matrix is very sparse, there are not enough constraints on the final solution,
so ALSCAL gets stuck in a local minimum. Similar results were seen from Sammon?s
method [4].
These results show that FSE (and other RD MDS algorithms) are preferable to using sparse
MDS algorithms. FSE does not solve an optimization problem, hence does not get stuck in
a local minimum.
4.2
Application: Generalizing Music Similarity
This section presents the results of using RD MDS algorithms to project a large music
dissimilarity graph into low-dimensional Euclidean space. This projection enables visualization and interpolation over music collections.
The dissimilarity graph was derived from a music metadata database. The database consists
of 10289 artists, 67799 albums, and 188749 tracks. Each track has subjective metadata
assigned to it by human editors: style (specific style), subgenre (more general style), vocal
code (gender of singer), and mood. See [12] for more details on the metadata. The database
contains which tracks occur on which albums and which artists created those albums.
Relationship Between Entities
Two tracks have same style, vocal code, mood
Two tracks have same style
Two tracks have same subgenre
Track is on album
Album is by artist
Edge Distance in Graph
1
2
4
1
2
Table 1: Mapping of relationship to edge distance.
A sparse similarity graph was extracted from the metadata database according to Table 1.
Every track, album, and artist are represented by a vertex in the graph. Every track was
connected to all albums it appeared on, while each album was connected to its artist. The
track similarity edges were sampled randomly, to provide an average of 7 links of edges of
distance 1, 2, and 4. The final graph contained 267K vertices and 3.22M edges. RD MDS
enabled this experiment: the full distance matrix would have taken days to compute with
267K calls to Dijkstra. Also, the graph distances were derived after some tuning (not on
the test set): the speed of RD MDS enabled this tuning.
One advantage of the music application is that the quality of the embedding can be tested
externally. A test set of 50 playlists, with 444 pairs of sequential songs was gathered from
real users who listened to these playlists. An embedding is considered good if sequential
songs in the playlists are frequently closer to each other than random songs in the database.
Table 2 shows the quality of the embedding as a fraction of random songs that are closer
than sequential songs. The lower the fraction, the better the embedding, because the embedding more accurately reflects users? ideas of music similarity. This fraction is computed
by treating the pairwise distances as scores from a classifier, computing an ROC curve, then
computing 1.0-the area under the ROC curve [3].
Algorithm
n
FSE
LMDS
LMDS
LMDS
LMDS
Laplacian Eigenmaps
60
60
100
200
400
N/A
Average % of
Random Songs Closer
than Sequential Songs
5.0%
4.5%
4.1%
3.3%
3.2%
13.0%
CPU time
(sec)
52.8
52.7
87.4
175.0
355.1
8003.4
Table 2: Speed and accuracy of music embedding for various algorithms.
All embeddings are 20-dimensional (d = 20). The CPU time was measured on a 2.4 GHz
Pentium 4. FSE uses a fixed rectangle size n = 3d, so has one entry in the table. For the
same n, FSE and LMDS are competitive. However, LMDS can trade off speed for accuracy
by increasing n.
A Laplacian Eigenmap applied to the entire sparse similarity matrix was much slower than
either of the RD MDS algorithms, and did not perform as well for this problem. A Gaussian
kernel with ? = 2 was used to convert distances to similarities for the Laplacian Eigenmap.
The slowness of the Laplacian eigenmap prevented extensive tuning of the parameters.
Bob Dylan
Cat Stevens
2.5
The Eagles The Beatles
The Who
Led Zeppelin
The Doors
2
1.5
Aerosmith
Jimi Hendrix
Talking Heads
The Police
Bryan Ferry
Fleetwood Mac
Dire Straits
The Rolling Stones
1
Kate Bush
Genesis
Sheryl Crow
Suzanne Vega
Alanis Morissette
0.5
Peter Gabriel
Sarah McLachlan
0
?0.5
0
0.5
1
Tori Amos
1.5
2
2.5
Figure 2: LMDS Projection of the entire music dissimilarity graph into 2D. The coordinates
of 23 artists are shown.
Given that LMDS outperforms FSE for large n, this paper now presents qualitative results
from the LMDS n = 400 projection. First, the top two dimensions are plotted to form a
visualization of music space. This visualization is shown in Figure 4.2, which shows the
coordinates of 23 artists that occur near the center of the space. Even restricted to the top
two dimensions, the projection is sensible. For example, Tori Amos and Sarah McLachlan
are mapped to be very close.
Artist 1
Jimi Hendrix
Jimi Hendrix
Jimi Hendrix
Jimi Hendrix
Jimi Hendrix
Jimi Hendrix
Doors
Doors
Doors
Doors
Cat Stevens
Cat Stevens
Cat Stevens
Cat Stevens
The Beatles
The Beatles
The Beatles
The Beatles
The Beatles
The Beatles
Track 1
Purple Haze
Fire
Red House
I Don?t Live Today
Foxey Lady
3rd Stone from the Sun
Waiting for the Sun
LA Woman
Riders on the Storm
Love her Madly
Ready
Music
Jesus
King of Trees
Octopus?s Garden
I?m So Tired
Revolution 9
Sgt. Pepper?s Lonely
Please Please Me
Eleanor Rigby
Artist 2
Alanis
Alanis
Alanis
Alanis
Alanis
Alanis
Sarah McLachlan
Sarah McLachlan
Sarah McLachlan
Sarah McLachlan
Sarah McLachlan
Blondie
Sarah McLachlan
Sarah McLachlan
Fiona Apple
Fiona Apple
Fiona Apple
Blondie
Blondie
Blondie
Track 2
Hand In My Pocket
All I Really Want
You Oughta Know
Right Through You
You Learn
Ironic
Full of Grace
Hold On
Good Enough
The Path of Thorns
Possession
Tide is High
Ice Cream
Fumbling Towards Ecstasy
Limp
Paper Bag
Fast As You Can
Call Me
Hanging on the Telephone
Rapture
Table 3: Two playlists produced by the system. Each playlist reads top to bottom. The
playlists interpolate between the first and last songs.
The main application for the music graph projection is the generation of playlists. There
are several different possible objectives for music playlists: background listening, dance
mixes, music discovery. One of the criteria for playlists is that they play similar music
together (i.e., avoid distracting jumps, like New Age to Heavy Metal). The goal for this
paper is to generate playlists for background listening. Therefore, the only criterion we
use for generation is smoothness and playlists are generated by linear interpolation in the
embedding space.
However, smoothness is not the only possible playlist generation mode: other criteria can
be added (such as matching beats or artist self-avoidance or minimum distance between
songs). These criteria can be added on top of the smoothness criteria. Such criteria are a
matter of subjective musical taste and are beyond the scope of this paper.
Table 3 shows two background-listening playlists formed by interpolating in the projected
space. The playlists were drawn from a collection of 3920 songs. Unlike the image interpolation in [14], not every point in the 20-dimensional space has a valid song attached
to it. The interpolation was performed by first computing the line segment connecting the
first and last song, and then placing K equally-spaced points along the line segment, where
K is the number of slots in the playlist. For every slot, the location of the previous song
is projected onto a hyperplane normal to the line segment that goes through the ith point.
The projected location is then moved halfway to the ith point, and the nearest song to
the moved location is placed into the playlist. This method provides smooth interpolation
without large jumps, as can be seen in Table 3.
5
Discussion and Conclusions
Music playlist generation and browsing can utilize a large sparse similarity graph designed
by editors. In order to allow tractable computations on this graph, its vertices can be projected into a low-dimensional space. This projection enables smooth interpolation and
two-dimensional display of music.
Music similarity graphs are amongst the largest graphs ever to be embedded. Rectangular
Dijkstra MDS algorithms can be used to efficiently embed these large sparse graphs. This
paper showed that FSE and the Nystr?m (LMDS) technique are both efficient and have
comparable performance for the same size of rectangle. Both algorithms are much more
efficient than Laplacian Eigenmaps. However, LMDS permits an accuracy/speed trade-off
that makes it preferable. Using LMDS, a music graph with 267K vertices and 3.22M edges
can be embedded in approximately 6 minutes.
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1,663 | 2,511 | Learning curves for stochastic gradient descent
in linear feedforward networks
Justin Werfel
Dept. of EECS
MIT
Cambridge, MA 02139
[email protected]
Xiaohui Xie
Dept. of Molecular Biology
Princeton University
Princeton, NJ 08544
[email protected]
H. Sebastian Seung
HHMI
Dept. of Brain & Cog. Sci.
MIT
Cambridge, MA 02139
[email protected]
Abstract
Gradient-following learning methods can encounter problems of implementation in many applications, and stochastic variants are frequently
used to overcome these difficulties. We derive quantitative learning
curves for three online training methods used with a linear perceptron:
direct gradient descent, node perturbation, and weight perturbation. The
maximum learning rate for the stochastic methods scales inversely with
the first power of the dimensionality of the noise injected into the system; with sufficiently small learning rate, all three methods give identical learning curves. These results suggest guidelines for when these
stochastic methods will be limited in their utility, and considerations for
architectures in which they will be effective.
1
Introduction
Learning in artificial systems can be formulated as optimization of an objective function
which quantifies the system?s performance. A typical approach to this optimization is to
follow the gradient of the objective function with respect to the tunable parameters of the
system. Frequently this is accomplished directly, by calculating the gradient explicitly and
updating the parameters by a small step in the direction of locally greatest improvement.
In many circumstances, however, attempts at direct gradient-following can encounter problems. In VLSI and other hardware implementations, computation of the gradient may be
excessively unwieldy, if not impossible due to unavoidable imperfections in manufacturing
[1]-[5]. In some cases, as with many where the reinforcement learning framework is used,
there may be no explicit form for the objective function and hence no way of calculating its
gradient [6]. And in biological systems, any argument that direct gradient calculation might
be what the system is actually doing typically encounters severe obstacles. For instance,
backpropagation, the standard method for training artificial neural networks, requires twoway, multipurpose synapses, units with global knowledge about the system that are able to
recognize different kinds of signals and treat them in very different ways, and (in the case
of trajectory learning) the ability to run backwards in time, all of which strain the bounds
of biological plausibility [1, 7]. For reasons such as these, there has been broad interest in
stochastic methods which approximate the gradient on average.
Compared to a method that follows the true gradient directly, we would intuitively expect
a stochastic gradient-following approach to learn more slowly. The stochastic algorithms
in this study use a reinforcement-learning framework with a single reward signal, which is
assigned based on the contributions of all the tunable parameters of the system; that single reward is all that is available to evaluate how every one of the parameters should be
updated, in contrast to a true-gradient method where the optimal updates are all specified.
Moreover, if the network is made larger and the number of parameters thereby increased,
this credit assignment problem becomes still more difficult; thus we expect the performance
of stochastic gradient methods to scale up with network size more poorly than deterministic methods. However, under some circumstances stochastic methods can be equally as
effective as direct ones in training even large networks, generating near-identical learning
curves (see, e.g., Fig. 2 below). Under what circumstances, then, will stochastic gradient
descent have performance comparable to that of the deterministic variety? And how good
can that performance be?
In this paper, we investigate these issues quantitatively by calculating the learning curves
for a linear perceptron using a direct gradient method and two stochastic methods, node
perturbation and weight perturbation. We find that the maximum learning speed for each
algorithm scales inversely with the first power of the dimensionality of the noise injected
into the system; this result is in contradiction to previous work, which reported maximum
learning speed scaling inversely with the square root of the dimensionality of the injected
noise [4]. Additionally, when learning rates are chosen to be very low, and such that the
weight updates prescribed by each method are equal on average, we find that all three
methods give identical learning curves.
2
Perceptron comparison
Direct and stochastic gradient approaches are general classes of training methods. We
study the operation of exemplars of both on a feedforward linear perceptron, which has the
advantage over the nonlinear case that the learning curves can be calculated exactly [8]. We
have N input units and M output units, connected by a weight matrix w of M N elements;
outputs in response to an input x are given by y = wx. For the ensemble of possible inputs,
we want to train the network to produce desired corresponding outputs y = d; in order to
ensure that this task is realizable by the network, we assume the existence of a teacher
network w ? such that d = w ? x. We use the squared error function
1
1
1
E = |y ? d|2 = |(w ? w? )x|2 = |W x|2
(1)
2
2
2
where we have defined the matrix W ? w ? w ? . We train the network with an online
approach, choosing at each time step an input vector x with components drawn from a
Gaussian distribution with mean 0 and unit variance, and using it to construct a weight
update according to one of the three prescriptions below.
The online gradient-following approach explicitly uses the gradient of the error function
for a given input to determine the weight update:
?WOL = ???E
where ? > 0 is the learning rate. This is the approach taken, e.g., by backpropagation.
In the stochastic algorithms, the gradient is not calculated directly; instead, some noise is
introduced into the system, affecting its error for a given input, and the difference between
the error with and without noise is used to estimate the gradient. The simplest case is when
noise is added directly to the weight matrix:
1
0
EWP
= |(W + ?)x|2
2
Such an approach is sometimes termed ?weight perturbation? [2, 4]. We choose each element of the noise matrix ? from a Gaussian distribution with mean 0 and variance ? 2 .
Intuitively, if the addition of the noise lowers the error, that perturbation to the weight matrix is retained, which will mean lower error for that input in future. Conversely, if the noise
leads to an increase in error, the opposite change is made to the weights; the effect of small
noise on error can be approximated as linear, and the opposite change in weights will lead
to the opposite change in error, again decreasing error for that input in future. These two
cases can be combined into the single weight update
?
0
?WWP = ? 2 (EWP
? E)?
?
A more subtle way to introduce stochasticity involves adding the noise to the output of each
output unit rather than to every weight:
1
0
ENP
= |W x + ?|2
2
Such an approach is sometimes called ?node perturbation? [1, 3]. Here if the noise leads
to a decrease in error, the weights are adjusted in such a way as to move the outputs in the
direction of that noise. The degree of freedom for each output unit corresponds to the adjustment of its threshold, making the unit more or less responsive to a given pattern of input
activity. The elements of ? are again chosen independently from a Gaussian distribution
with variance ? 2 ; here ? has M elements, whereas ? in the previous case had M N . The
REINFORCE framework [9] gives for the weight update
?
0
?WNP = ? 2 (ENP
? E)?xT
?
These stochastic frameworks produce weight updates identical to that of direct gradient
descent on the error function when averaged over all values of the noise [4, 9], which is the
sense in which they constitute stochastic gradient descent. This result is easy to verify in
the particular forms taken by ?WNP and ?WWP here, shown below.
2.1
Online gradient method
Taking the gradient of the error function of Eq. 1 gives
?WOL = ??W xxT
(2)
as the individual weight update for particular values of W and x. This rule lets us calculate
a recursion relation specifying how kW k2 changes from one time step to the next:
X
X (t?1) 2
(t)
h(Wij )2 it = (1 ? 2? + (N + 2)? 2 )
Wij
(3)
ij
ij
where the parenthesized superscript is a time index, and the subscripted angle brackets
denote an average over the ensemble of all inputs at that time. Applying this recursion relation gives an expression for the average error as a function of time, where the unsubscripted
brackets indicate a mean taken over all inputs at every time step:
(t)
hEOL i = (1 ? 2? + (N + 2)? 2 )t E (0)
In a single online learning run, E (t) would depend on the particular values of x that were
randomly chosen; averaging over the ensemble of possible inputs x removes this variation.
We therefore use this averaged error hE (t) i as the learning curve measuring the performance of the system.
We have the condition for convergence of the average error
2
?<
N +2
The limit on ? has this dependence on N because of the randomness inherent in an online
training regimen; the exact gradient for error due to a given single input x will not in general
match that for error averaged over the entire ensemble of inputs. We can write an expression
for the ij-component of the weight update, explicitly in terms of ?gradient signal? (term
multiplying Wij ) plus ?gradient noise? [1] (contamination from other components of W
due to projection onto x):
?
?
X
?Wij = ?? ?Wij x2j +
Wik xk xj ?
k6=j
We can similarly rewrite Eq. 3 as
X (1)2
X (0)2
X (0)
hWij i =
Wij (1 ? 2? + 3? 2 ) + ? 2 (N ? 1)
Wij
ij
ij
ij
where the first term is due entirely to the gradient signal and the second to the gradient
noise; choosing ? . 1/N allows the signal to be revealed via averaging over & N samples
(see also the Discussion). This gradient noise is common to all three algorithms considered
here.
2.2
Node perturbation
Here averages are taken at each step not only over the inputs x but also over the noise ?.
The weight update, recursion relation, learning curve, and convergence condition are
1
?
?WNP = ? 2 (? T W x + ? T ?)?xT
?
2
X (t)2
X (t?1)2
hWij it =
Wij
(1 ? 2? + ? 2 (M + 2)(N + 2))
ij
ij
1
+ ? 2 ? 2 M N (M + 2)(M + 4))
4
?? 2 (M + 2)(M + 4)M N/8
(t)
(1 ? 2? + (M + 2)(N + 2)? 2 )t
hENP i = E (0) ?
2 ? (N + 2)(M + 2)?
?? 2 (M + 2)(M + 4)M N/8
+
2 ? (M + 2)(N + 2)?
2
?<
(M + 2)(N + 2)
In this case the recursion relation has not only a multiplicative term as before but also an
additive one. The latter is a result of the noise ?; when W is far from the minimum of the
objective function, ? will typically be small in comparison to W x and the additive term will
be negligible, but close to the minimum the noise will prevent the system from attaining
arbitrarily low error. This effect appears also in the learning curve. The limit on ? is stricter
by a factor of M , the dimensionality of the noise, as discussed below.
2.3
Weight perturbation
The same approach as before gives in this case
?
1
?WWP = ? 2 (xT ? T W x + xT ? T ?x)?
?
2
X (t)2
X (t?1)2
(1 ? 2? + ? 2 (M N + 2)(N + 2))
hWij it =
Wij
ij
ij
1
+ ? 2 ? 2 (M 3 N 3 + 2M 2 N 3 + 2M 3 N 2 + 16M 2 N 2 + 24M N )
4
?? 2 M N (M N (M + 2)(N + 2) + 12(M N + 2))
(t)
)
hEWP i = (E (0) ?
8(2 ? (N + 2)(M N + 2)?)
?(1 ? 2? + (N + 2)(M N + 2)? 2 )t
?? 2 M N (M N (M + 2)(N + 2) + 12(M N + 2))
+
8(2 ? (N + 2)(M N + 2)?)
2
?<
(M N + 2)(N + 2)
As with node perturbation, the recursion relation involves both multiplicative and additive
terms, and the learning curve shows nonzero residual error even at infinite time. The limit
on ? is a further factor of N smaller, corresponding to the greater dimensionality of ?
compared to ?.
3
Comparison of learning curves
t
?
All three of the above learning curves hE (t) i take the form E(a(?))
+ b(?, ?), where b
is the residual error which the network will approach as t ? ? if learning converges,
? ? E (0) ? b is the transient error, and a is a multiplicative factor by which E
? changes
E
at each time step. The magnitude of a, which depends on the parameter ? but not on ?,
determines whether the average error will converge and the rate at which it will do so.
For the online gradient method, b = 0; a network trained this way, if it converges, will
approach zero error as t ? ?. The stochastic algorithms have positive residual noise b,
which depends on both ? and ?; in the limit ? ? 0, this residual error vanishes. Of course,
? cannot be set directly to 0 or the stochastic algorithms will cease to function.
3.1
Maximal learning rates
The analysis of the previous section suggests at least two reasonable ways to compare these
different algorithms with respect to performance. One is to choose the optimal learning rate
for each, that value of ? for which the average error converges most quickly. The learning
curves, to highest order in ?, M , and N , then become
t
1
(t)
?
hEOL i = E 1 ?
N
t
1
(t)
? 1? 1
hENP i = E
+ ?2 M 2
MN
8
t
1
(t)
? 1? 1
hEWP i = E
+ ?2 M 2 N
MN2
8
Direct gradient descent, then, can train a network faster than can node perturbation, which
in turn is faster than weight perturbation.
The noise takes different forms in the two stochastic variants. For node perturbation, ? i is
added directly P
to the ith output unit; for weight perturbation, the quantity added to the same
output unit is ij ?ij xj . By the central limit theorem, the latter approaches a Gaussian
with mean 0 and variance N ? 2 for large N . For most direct
?comparison of the two stochastic variants, therefore, ? for ? should be chosen a factor N larger than for ?. With this
choice, the residual error for the two stochastic variants becomes identical, and the learning
curves differ only in their rates of convergence.
3.2
Equal average updates
A second way to compare the algorithms is to choose learning rates such that all three have
the same average weight update. As noted above, choosing the same value of ? in all three
cases will ensure this condition. That common value of ? must be small enough that all
three algorithms converge; if we take ? M1N 2 , the learning curves become
(t)
hEOL i
=
? ? 2?)t
E(1
? (1 ? 2?)t + 1 ?? 2 M 3 N
E
16
(t)
t
? (1 ? 2?) + 1 ?? 2 M 3 N 3
hEWP i = E
16
We began by saying that, because of the credit assignment problem of choosing updates
to many parameters based on a single reward signal, intuition is that a stochastic gradientfollowing approach should learn more slowly than a direct one. However, for equal small
?, the average error for all three algorithms converges at the same rate. Weight perturbation
approaches a larger value of residual error than does node perturbation; however, in the
? ? 0 limit, the residual error vanishes for both.
(t)
hENP i
4
=
Discussion
In a linear feedforward network of N input and M output units, in terms of the maximum
possible rate of convergence of average error, online gradient descent on a squared error
function is faster by a factor of M than node perturbation, which in turn is faster by a factor
of N than weight perturbation. The difference in the rate of convergence is the dimensionality of the noise. Weight perturbation operates by explicit exploration of the entire
M N -dimensional weight space; only one component of a particular update will be in the
direction of the true gradient for a given input, while the other components can be viewed as
noise masking that signal. That is, an update can be written as ?W = h?W i (the ?learning
signal?, the actual gradient) + (?W ? h?W i) (the ?learning noise?), where the
? average is
taken over all values of ?. This learning noise will typically have magnitude M N larger
than the learning signal, and so M N samples are required in order to average it away. Direct gradient descent gives weight updates that are purely signal in this sense; while still
occurring in an M N -dimensional space, they are by definition exactly in the direction of
the gradient for a given input. Thus no exploration of the weight space nor averaging over
multiple samples is necessary, and the maximum learning speed is correspondingly greater.
Node perturbation is a stochastic algorithm like weight perturbation, but it explores the M dimensional output space rather than the larger weight space; the learning noise is of lower
dimension, and correspondingly fewer samples need to be averaged to reveal a learning
signal of a given size.
It has previously been argued that the maximum learning rate should scale, not with the
dimensionality of the update as shown here, but with the square root of that dimensionality
[4]. That claim is based on the fact that the squared magnitude of the update goes as the
number of dimensions, and for a given error landscape and position in parameter space,
there will be a maximum update size, greater than which instability will result. However,
a more quantitative approach is to examine the conditions under which error will decrease,
as we have done above. Rather than stopping with the statement that the size of the weight
update scales as the square root of the number of dimensions, we have shown that this fact
implies that the restriction on convergence scales with the first power of the dimensionality.
Numerical simulations of error curves, averaged over many individual trials with online
updating, support these conclusions with respect to both the quantitative shapes of the
learning curves and the scaling behavior of the conditions on convergence (Fig. 1).
Online gradient method
Weight perturbation
Error (arbitrary units)
Node perturbation
0
10
2
10
# of examples
0
10
2
10
# of examples
4
10
0
10
5
# of examples
10
Figure 1: Sample learning curves for the three algorithms applied to a linear feedforward
network as described in the text, showing the agreement between theory (black) and experiment (gray). In each case, a network of linear units with N = 20, M = 25, ? = 10?3 , and
optimal ? was trained on successive input examples for the number of iterations shown.
100 such runs were averaged together in each case; the three gray lines show the mean
(solid) and standard deviation (dashed) of squared error among those runs.
This scaling result means that, for these stochastic methods, there is no net advantage in
speed of training when all degrees of freedom are varied at the same time, compared to
when they are varied sequentially, in terms of scaling with M and N . For instance, in the
case of weight perturbation, varying only one weight at a time would allow the learning
rate to be increased by a factor on the order of M N ; but each of the M N weights would
need to be trained in this way, so that the total training time required would scale in the
same way as if all were varied at once. (The speed of learning for parallel vs. sequential
variation, however, can differ by a constant ratio, though we do not pursue this issue here.)
The analysis here describes the behavior in a worst case of sorts, where the objective function and distribution of inputs are isotropic. In the anisotropic case, where the problem is
effectively lower-dimensional, the scaling behavior of all three methods can be correspondingly more favorable than that derived here, and the relative performance of the stochastic
methods can be better.
The results described in this paper extend at least qualitatively to more complicated networks and architectures. For instance, Fig. 2 shows learning curves that result from applying the three algorithms to a two-layer feedforward network of nonlinear units. All three
algorithms give identical learning curves if the learning rate is set small enough; as ? is
increased, the weight perturbation curve fails to converge to low error, while the other two
curves continue to match; increasing ? further leads to the node perturbation curve also
failing to converge.
In the above, we have shown that stochastic gradient descent techniques can be expected
to scale with increasing network size more poorly than direct ones, in terms of maximum
learning rate. This may serve as a caution regarding the size of networks they may usefully be applied to. However, with learning rates in the regime where error converges,
equal learning curves in each of the three will follow from equal learning rates, although
individual weight updates will typically be considerably different. This is because for correspondingly small adjustments to the weights, only the component parallel to the gradient
will have a significant effect on error; orthogonal components will not affect the error to
first order. Moreover, node perturbation can have performance comparable to that of direct gradient descent even in training very large networks, so long as the number of output
units is small [6]. Thus these stochastic methods may be of considerable utility for training networks in some situations, particularly in reinforcement learning frameworks and
those where the gradient of the objective function is difficult or impossible to calculate, for
mathematical or practical reasons.
?4
? = 4 ? 10
?2
? = 4 ? 10
?3
Error (arbitrary units)
? = 4 ? 10
0
10
5
# of examples
10
0
10
5
# of examples
10
10
0
2
10
# of examples
Figure 2: Sample learning curves for the three algorithms applied to a two-layer nonlinear
feedforward network (gradient descent, black dotted; node perturbation, dark gray dashed;
weight perturbation, light gray solid). The input, hidden, and output layers each had 10
units, whose output was equal to the hyperbolic tangent of their weighted input. Inputs and
noises were drawn from the same distributions as in the linear case; ? = 10?3 , ? had the
value shown for all three algorithms in each panel. In each case, the network was trained
on successive input examples for the number of iterations shown; curves show single representative runs. Error was evaluated based on the total squared difference between the
output of the network and that of a teacher network with randomly chosen weights; the test
error shown was the mean of that for 100 random inputs not used in training.
Acknowledgments
We thank Ila Fiete and Gert Cauwenberghs for useful discussions and comments. This
work was supported in part by a Packard Foundation Fellowship (to H.S. Seung) and NIH
grants (GM07484 to MIT and MH60651 to H.S. Seung).
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[6] Fiete, I. Private communication.
[7] Bartlett, P. & Baxter, J. Hebbian synaptic modifications in spiking neurons that learn. Technical
report, November 27 1999.
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[9] Williams, R.J. Simple statistical gradient-following algorithms for connectionist reinforcement
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1,664 | 2,512 | Computing Gaussian Mixture Models with EM
using Equivalence Constraints
Noam Shental
Computer Science & Eng.
Center for Neural Computation
Hebrew University of Jerusalem
Jerusalem, Israel 91904
[email protected]
Aharon Bar-Hillel
Computer Science & Eng.
Center for Neural Computation
Hebrew University of Jerusalem
Jerusalem, Israel 91904
[email protected]
Tomer Hertz
Computer Science & Eng.
Center for Neural Computation
Hebrew University of Jerusalem
Jerusalem, Israel 91904
[email protected]
Daphna Weinshall
Computer Science & Eng.
Center for Neural Computation
Hebrew University of Jerusalem
Jerusalem, Israel 91904
[email protected]
Abstract
Density estimation with Gaussian Mixture Models is a popular generative technique used also for clustering. We develop a framework to
incorporate side information in the form of equivalence constraints into
the model estimation procedure. Equivalence constraints are defined on
pairs of data points, indicating whether the points arise from the same
source (positive constraints) or from different sources (negative constraints). Such constraints can be gathered automatically in some learning problems, and are a natural form of supervision in others. For the
estimation of model parameters we present a closed form EM procedure
which handles positive constraints, and a Generalized EM procedure using a Markov net which handles negative constraints. Using publicly
available data sets we demonstrate that such side information can lead to
considerable improvement in clustering tasks, and that our algorithm is
preferable to two other suggested methods using the same type of side
information.
1 Introduction
We are used to thinking about learning from labels as supervised learning, and learning
without labels as unsupervised learning, where ?supervised? implies the need for human
intervention. However, in unsupervised learning we are not limited to using data statistics
only. Similarly supervised learning is not limited to using labels. In this work we focus
on semi-supervised learning using side-information, which is not given as labels. More
specifically, we use unlabeled data augmented by equivalence constraints between pairs
of data points, where the constraints determine whether each pair was generated by the
same source or by different sources. Such constraints may be acquired without human
intervention in a broad class of problems, and are a natural form of supervision in other
scenarios. We show how to incorporate equivalence constraints into the EM algorithm [1],
in order to fit a generative Gaussian mixture model to the data.
Density estimation with Gaussian mixture models is a popular generative technique, mostly
because it is computationally tractable and often produces good results. However, even
when the approach is successful, the underlying assumptions (i.e., that the data is generated by a mixture of Gaussian sources) may not be easily justified. It is therefore important
to have additional information which can steer the GMM estimation in the ?right? direction. In this paper we propose to incorporate equivalence constraints into an EM parameter
estimation algorithm. One added value may be a faster convergence to a high likelihood
solution. Much more importantly, the constraints change the GMM likelihood function and
therefore may lead the estimation procedure to choose a better solution which would have
otherwise been rejected (due to low relative likelihood in the unconstrained GMM density
model). Ideally the solution obtained with side information will be more faithful to the
desired results. A simple example demonstrating this point is shown in Fig. 1.
Unconstrained
(a)
constrained
(b)
unconstrained
constrained
(c)
(d)
Figure 1: Illustrative examples for the importance of equivalence constraints. Left: the data set consists of 2 vertically aligned classes - (a) given no additional information, the EM algorithm identifies
two horizontal classes, and this can be shown to be the maximum likelihood solution (with log likelihood of ?3500 vs. log likelihood of ?2800 for the solution in (b)); (b) additional side information in
the form of equivalence constraints changes the probability function and we get a vertical partition as
the most likely solution. Right: the dataset consists of two classes with partial overlap - (c) without
constraints the most likely solution includes two non-overlapping sources; (d) with constraints the
correct model with overlapping classes was retrieved as the most likely solution. In all plots only the
class assignment of novel un-constrained points is shown.
Equivalence constraints are binary functions of pairs of points, indicating whether the two
points come from the same source or from two different sources. We denote the first case
as ?is-equivalent? constraints, and the second as ?not-equivalent? constraints. As it turns
out, ?is-equivalent? constraints can be easily incorporated into EM, while ?not-equivalent?
constraints require heavy duty inference machinery such as Markov networks. We describe
the derivations in Section 2.
Our choice to use equivalence constraints is motivated by the relative abundance of equivalence constraints in real life applications. In a broad family of applications, equivalence
constraints can be obtained without supervision. Maybe the most important source of unsupervised equivalence constraints is temporal continuity in data; for example, in video
indexing a sequence of faces obtained from successive frames in roughly the same location
are likely to contain the same unknown individual. Furthermore, there are several learning
applications in which equivalence constraints are the natural form of supervision.
One such scenario occurs when we wish to enhance a retrieval engine using supervision
provided by its users. The users may be asked to help annotate the retrieved set of data
points, in what may be viewed as ?generalized relevance feedback?. The categories given
by the users have subjective names that may be inconsistent. Therefore, we can only extract
equivalence constraints from the feedback provided by the users. Similar things happen in
a ?distributed learning? scenario, where supervision is provided by several uncoordinated
teachers. In such scenarios, when equivalence constraints are obtained in a supervised
manner, our method can be viewed as a semi-supervised learning technique. Most of the
work in the field of semi-supervised learning focused on the case of partial labels augmenting a large unlabeled data set [4, 8, 5].
A few recent papers use side information in the form of equivalence constraints [6, 7, 10].
In [9] equivalence constraints were introduced into the K-means clustering algorithm. The
algorithm is closely related to our work since it allows for the incorporation of both ?isequivalent? and ?not-equivalent? constraints. In [3] equivalence constraints were introduced into the complete linkage clustering algorithm. In comparison with both approaches,
we gain significantly better clustering results by introducing the constraints into the EM algorithm. One reason may be that the EM of a Gaussian mixture model is preferable as
a clustering algorithm. More importantly, the probabilistic semantics of the EM procedure allows for the introduction of constraints in a principled way, thus overcoming many
drawbacks of the heuristic approaches. Comparative results are given in Section 3, demonstrating the very significant advantage of our method over the two alternative constrained
clustering algorithms using a number of data sets from the UCI repository and a large
database of facial images [2].
2 Constrained EM: the update rules
A Gaussian mixture model (GMM) is a parametric statistical model which assumes that the
data originates from a weighted sum of several Gaussian sources. More formally, a GMM
is given by p(x|?) = ?M
l=1 ?l p(x|?l ), where ?l denotes the weight of each Gaussian, ?l its
respective parameters, and M denotes the number of Gaussian sources in the GMM. EM
is a widely used method for estimating the parameter set of the model (?) using unlabeled
data [1]. Equivalence constraints modify the ?E? (expectation computation) step, such that
the sum is taken only over assignments which comply with the given constraints (instead
of summing over all possible assignments of data points to sources).
It is important to note that there is a basic difference between ?is-equivalent? (positive)
and ?not-equivalent? (negative) constraints: While positive constraints are transitive (i.e.
a group of pairwise ?is-equivalent? constraints can be merged using a transitive closure),
negative constraints are not transitive. The outcome of this difference is expressed in the
complexity of incorporating each type of constraint into the EM formulation. Therefore, we
begin by presenting a formulation for positive constraints (Section 2.1), and then present a
different formulation for negative constraints (Section 2.2). A unified formulation for both
types of constraints immediately follows, and the details are therefore omitted.
2.1 Incorporating positive constraints
Let a chunklet denote a small subset of data points that are known to belong to a single
unknown class. Chunklets may be obtained by applying the transitive closure to the set of
?is-equivalent? constraints.
Assume as given a set of unlabeled data points and a set of chunklets. In order to write
down the likelihood of a given assignment of points to sources, a probabilistic model of
how chunklets are obtained must be specified. We consider two such models:
1. Chunklets are sampled i.i.d, with respect to the weight of their corresponding
source (points within each chunklet are also sampled i.i.d).
2. Data points are sampled i.i.d, without any knowledge about their class membership, and only afterwards chunklets are selected from these points.
The first assumption may be appropriate when chunklets are automatically obtained using
temporal continuity. The second sampling assumption is appropriate when equivalence
constraints are obtained using distributed learning. When incorporating these sampling
assumptions into the EM formulation for GMM fitting, different algorithms are obtained:
Using the first assumption we obtain closed-form update rules for all of the GMM parameters. When the second sampling assumption is used there is no closed-form solution for the
sources? weights. In this section we therefore restrict the discussion to the first sampling
assumption only; the discussion of the second sampling assumption, where generalized EM
must be used, is omitted.
PM
More specifically, let p(x) = l=1 ?l pl (x|?l ) denote our GMM. Each pl (x|?l ) term is a
Gaussian parameterized by ?l = (?l , ?l ) with a mixing coefficient ?l . Let X denote the
L
set of all data points, X = {xi }N
i=1 . Let {Xj }j=1 , L ? N denote the distinct chunklets,
SL
N
where each Xj is a set of points xi such that j=1 Xj = {xi }i=1 (unconstrained data
points appear as chunklets of size one). Let Y = {yi }N
i=1 denote the source assignment
|Xj |
1
of the respective data-points, and Yj = {yj . . . yj } denote the source assignment of the
chunklet Xj . Finally, let E? denote the event {Y complies with the constraints}.
The expectation of the log likelihood is the following:
X
new
old
E[log(p(X, Y|?
where
PM
, E? ))|X ?
log(p(X, Y|?new , E? )) ?p(Y|X, ?old , E? )
, E? ] =
Y
P
stands for a summation over all assignments of points to
Y
PM
. In the following discussion we shall also reorder the
y1 =1 . . . PyN =1 P
P
P
P
P
chunklets:
Y
?
Y1
...
YL ,
where
Yj
stands for
y1j
???
j
y|X
j|
(1)
P
sources:
Y ?
sum according to
.
First, using Bayes rule and the independence of chunklets, we can write
p(E? |Y, X, ?old ) p(Y|X, ?old )
p(Y|X, ?old , E? ) = P
old ) p(Y|X, ?old )
Y p(E? |Y, X, ?
QL
old
)
j=1 ?Yj p(Yj |Xj , ?
(2)
= P
P QL
old )
Y1 . . .
YL
j=1 ?Yj p(Yj |Xj , ?
where ?Yj ? ?yj ,...,yj equals 1 if all the points in chunklet i have the same label, and 0
1
|Xj |
otherwise.
Next, using chunklet independence and the independence of points within a chunklet we
see that
p(X, Y|?new , E? ) = p(Y|?new , E? ) p(X|Y, ?new , E? )
=
L
Y
?yj
j=1
Hence the log-likelihood is:
log p(X, Y|?new , E? ) =
L
X
X
N
Y
p(xi |yi , ?new )
i=1
log p(xi |yi , ?new ) +
j=1 xi ?Xj
L
X
log(?yj )
(3)
j=1
Finally, we substitute (3) and (2) into (1); after some manipulations, we obtain the following
expression:
M X
L
X
X
E(LogLikelihood) =
log p(xi |l, ?new ) ? p(Yj = l|Xj , ?old )
l=1 j=1 xi ?Xj
+
M X
L
X
l=1 j=1
log ?l ? p(Yj = l|Xj , ?old )
where the chunklet posterior probability is:
Q
old
?l
p(Yj = l|Xj , ?old ) = PM
m=1
xi ?Xj
old
?m
Q
p(xi |yij = l, ?old )
xi ?Xj
p(xi |yij = m, ?old )
To find the update rule for each parameter, we differentiate (4) with respect to ?l , ?l and
?l . We get the following rules:
?lnew
=
L
1 X
p(Yj = l|Xj , ?old )
L j=1
PL
?new
l
=
j=1
PL
? j p(Yj = l|Xj , ?old )|Xj |
X
j=1
p(Yj = l|Xj , ?old )|Xj |
PL
?new
l
=
new
j=1 ?jl p(Yj
PL
j=1 p(Yj =
= l|Xj , ?old )|Xj |
l|Xj , ?old )|Xj |
? j denotes the sample mean of the points in chunklet j, |Xj | denotes the number of
where X
points in chunklet j, and ?new
denotes the sample covariance matrix of the jth chunklet of
jl
the lth class.
As can be readily seen, the update rules above effectively treat each chunklet as a single
data point weighed according to the number of elements in it.
2.2 Incorporating negative constraints
The probabilistic description of a data set using a GMM attaches to each data point two
random variables: an observable and a hidden. The hidden variable of a point describes its
source label, while the data point itself is an observed example from the source. Each pair
of observable and hidden variables is assumed to be independent of the other pairs. However, negative equivalence constraints violate this assumption, as dependencies between
the hidden variables are introduced.
Specifically, assume we have a group ? = {(a1i , a2i )}P
i=1 of index pairs corresponding to P pairs of points that are negatively constrained, and define the event E? =
{Y complies with the constraints}. Now
p(X, Y|?, E? ) = p(X|Y, ?, E? ) p(Y|?, E? ) =
p(X|Y, ?) p(E? |Y) p(Y|?)
p(E? |?)
Let Z denote the constant p(E? |?). Assuming sample independence, it follows that
QN
p(X|Y, ?) ? p(Y|?) = i=1 p(yi |?)p(xi |yi , ?). By definition p(E? |Y) = 1Y?E? ,
hence
N
Y
1
p(yi |?)p(xi |yi , ?)
(4)
p(X, Y|?, E? ) = 1Y?E?
Z
i=1
Expanding 1Y?E? gives the following expression
p(X, Y|?, E? ) =
N
Y
1 Y
(1 ? ?ya1 ,ya2 )
p(yi |?)p(xi |yi , ?)
Z 1 2
i
i
i=1
(5)
(ai ,ai )
As a product of local components, the distribution in (5) can be readily described using a
Markov network. The network nodes are the hidden source variables and the observable
data point variables. The potential p(xi |yi , ?) connects each observable data point, in a
Gaussian manner, to a hidden variable corresponding to the label of its source. Each hidden
source node holds an initial potential of p(yi |?) reflecting the prior of the cluster weights.
Negative constraints are expressed by edges between hidden variables which prevent them
from having the same value. A potential over an edge (a1i ? a2i ) is expressed by 1 ? ?ya1 ,ya2
i
(see Fig. 2).
i
Figure 2: An illustration of the Markov network required for incorporating ?not-equivalent? constraints. Data points 1 and 2 have a negative constraint, and so do points 2 and 3.
We derived an EM procedure which maximizes log(p(X|?, E? )) entailed by this distribution. The update rules for ?l and ?l are still
PN c
PN
old
old
, E? )
, E? )
new
i=1 xi p(yi = l|X, ?
i=1 ?i lp(yi = l|X, ?
,
?
=
?new
=
PN
P
l
l
N
old , E )
old , E )
?
?
i=1 p(yi = l|X, ?
i=1 p(yi = l|X, ?
ci l = (xi ? ?new )(xi ? ?new )T denotes the sample covariance matrix. Note,
where ?
l
l
however, that now the vector of probabilities p(yi = l|X, ?old , E? ) is inferred using the
net.
The update rule of ?l = p(yi = l|?new , E? ) is more intricate, since this parameter appears
in the normalization factor Z in the likelihood expression (4):
Z = p(E? |?) =
X
p(Y|?)p(E? |Y) =
Y
X
y1
...
N
XY
yN i=1
?yi
Y
(1 ? ?ya1 ,ya2 )
(a1i ,a2i )
i
(6)
i
This factor can be calculated using a net which is similar to the one discussed above but
lacks the observable nodes. We use such a net to calculate Z and differentiate it w.r.t ?l ,
after which we perform gradient ascent. Alternatively, we can approximate Z by assuming
that the pairs of negatively constrained points are disjoint. Using such an assumption, Z is
PM
reduced to the relatively simple expression: Z = (1 ? i=1 ?i2 )P . This expression for Z
can be easily differentiated, and can be used in the Generalized EM scheme. Although the
assumption is not valid in most cases, it is a reasonable approximation in sparse networks,
and our empirical tests show that it gives good results.
3 Experimental results
In order to evaluate the performance of our EM derivations and compare it to the constrained K-means [9] and constrained complete linkage [3] algorithms, we tested all 3 algorithms using several data sets from the UCI repository and a real multi-class facial image
database [2]. We simulated a ?distributed learning? scenario in order to obtain side information. In this scenario equivalence constraints are obtained by employing N uncoordinated
teachers. Each teacher is given a random selection of K data points from the data set, and is
then asked to partition this set of points into equivalence classes. The constraints provided
BALANCE N=625 d=4 C=3
BOSTON N=506 d=13 C=3
"much"
"little"
IONOSPHERE N=351 d=34 C=2
"much"
"little"
1
1
0.9
0.9
0.9
0.8
f1/2
1
f1/2
f1/2
"little"
0.8
0.8
0.7
0.7
0.7
0.6
0.6
0.6
0.5
a b c d e f g h i
0.5
a b c d e f g h i
a b c d e f g h i
PROTEIN N=116 d=20 C=6
0.5
a b c d e f g h i
"much"
a b c d e f g h i
a b c d e f g h i
"little"YaleB N=640 d=60 C=10"much"
WINE N=168 d=12 C=3
0.9
"little"
"much"
"little"
1
1
0.9
0.9
"much"
0.8
0.8
0.6
f1/2
f1/2
f1/2
0.7
0.8
0.5
0.4
0.7
0.7
0.6
0.6
0.3
0.2
0.1
0.5
a b c d e f g h i
a b c d e f g h i
0.5
a b c d e f g h i
a b c d e f g h i
a b c d e f g h i
a b c d e f g h i
Figure 3: Combined precision and recall scores (f 1 ) of several clustering algorithms over 5 data
2
sets from the UCI repository, and 1 facial image database (YaleB). The YaleB dataset contained a
total of 640 images including 64 frontal pose images of 10 different subjects. In this dataset the variability between images of the same person was due mainly to different lighting conditions. Results
are presented for the following algorithms: (a) K-means, (b) constrained K-means using only positive constraints, (c) constrained K-means using both positive and negative constraints, (d) complete
linkage, (e) complete linkage using positive constraints, (f) complete linkage using both positive and
negative constraints, (g) regular EM, (h) EM using positive constraints, and (i) EM using both positive and negative constraints. In each panel results are shown for two cases, using 15% of the data
points in constraints (left bars) and 30% of the points constrained (right bars). The results were averaged over 100 realizations of constraints for the UCI datasets, and 1000 realizations for the YaleB
dataset. Also shown are the names of the data sets used and some of their parameters: N - the size of
the data set; C - the number of classes; d - the dimensionality of the data.
by the teachers are gathered and used as equivalence constraints. Each of the 3 algorithms
(constrained EM, constrained K-means, and constrained complete linkage) was tested in
three modes: (i) basic algorithm without using any side information, (ii) constrained version using only positive equivalence constraints, and (iii) constrained version using both
positive and negative equivalence constraints. The results of the 9 algorithmic variants are
compared in Fig. 3.
In the simulations, the number of constrained points was determined by the number of
teachers N and the size of the subset K given to each. By controlling the product N K
we controlled the amount of side information provided to the learning algorithms. We
experimented with two conditions: using ?little? side information (approximately 15% of
the data points are constrained) and using ?much? side information (approximately 30%
of the points are constrained). All algorithms were given initial conditions that did not
take into account the available equivalence constraints. The results were evaluated using a
2P R
.
combined measure of precision P and recall R scores: f 21 = R+P
Several effects can clearly be seen in the results reported in Fig. 3:
? The constrained EM outperformed the two alternative algorithms in almost all
cases, while showing substantial improvement over the baseline EM. The one
case where constrained complete linkage outperformed all other algorithms involved the ?wine? dataset. In this dataset the data lies in a high-dimensional space
(R12 ) and therefore the number of model parameters to be estimated by the EM
algorithm is relatively large. The EM procedure was not able to fit the data well
even with constraints, probably due to the fact that only 168 data points were
available for training.
? Introducing side information in the form of equivalence constraints clearly improves the results of both K-means and the EM algorithms. This is not always
true for the constrained complete linkage algorithm. As the amount of sideinformation increases, performance typically improves.
? Most of the improvement can be attributed to the positive constraints, and can be
achieved using our closed form EM version. In most cases adding the negative
constraints contributes a small but significant improvement over results obtained
when using only positive constraints.
References
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and T. Petsche, editors, NIPS 11, page 803ff. MIT Press, 1998.
[7] N. Shental, T. Hertz, D. Weinshall, and M. Pavel. Adjustment learning and relevant component
analysis. In A. Heyden, G. Sparr, M. Nielsen, and P. Johansen, editors, Computer Vision ECCV 2002, volume 4, page 776ff, 2002.
[8] M. Szummer and T. Jaakkola. Partially labeled classification with markov random walks. In
NIPS, volume 14. The MIT Press, 2001.
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background knowledge. In Proc. 18th International Conf. on Machine Learning, pages 577?
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1,665 | 2,513 | Kernel Dimensionality Reduction for Supervised
Learning
Kenji Fukumizu
Institute of Statistical
Mathematics
Tokyo 106-8569 Japan
[email protected]
Francis R. Bach
CS Division
University of California
Berkeley, CA 94720, USA
[email protected]
Michael I. Jordan
CS Division and Statistics
University of California
Berkeley, CA 94720, USA
[email protected]
Abstract
We propose a novel method of dimensionality reduction for supervised
learning. Given a regression or classification problem in which we wish
to predict a variable Y from an explanatory vector X, we treat the problem of dimensionality reduction as that of finding a low-dimensional ?effective subspace? of X which retains the statistical relationship between
X and Y . We show that this problem can be formulated in terms of
conditional independence. To turn this formulation into an optimization
problem, we characterize the notion of conditional independence using
covariance operators on reproducing kernel Hilbert spaces; this allows us
to derive a contrast function for estimation of the effective subspace. Unlike many conventional methods, the proposed method requires neither
assumptions on the marginal distribution of X, nor a parametric model
of the conditional distribution of Y .
1
Introduction
Many statistical learning problems involve some form of dimensionality reduction. The
goal may be one of feature selection, in which we aim to find linear or nonlinear combinations of the original set of variables, or one of variable selection, in which we wish to select
a subset of variables from the original set. Motivations for such dimensionality reduction
include providing a simplified explanation and visualization for a human, suppressing noise
so as to make a better prediction or decision, or reducing the computational burden.
We study dimensionality reduction for supervised learning, in which the data consists of
(X, Y ) pairs, where X is an m-dimensional explanatory variable and Y is an -dimensional
response. The variable Y may be either continuous or discrete. We refer to these problems
generically as ?regression,? which indicates our focus on the conditional probability density
pY |X (y|x). Thus, our framework includes classification problems, where Y is discrete.
We wish to solve a problem of feature selection in which the features are linear combinations of the components of X. In particular, we assume that there is an r-dimensional
subspace S ? Rm such that the following equality holds for all x and y:
(1)
pY |X (y|x) = pY |?S X (y|?S x),
where ?S is the orthogonal projection of Rm onto S. The subspace S is called the effective subspace for regression. Based on observations of (X, Y ) pairs, we wish to re-
cover a matrix whose columns span S. We approach the problem within a semiparametric statistical framework?we make no assumptions regarding the conditional distribution
pY |?S X (y|?S x) or the distribution pX (x) of X. Having found an effective subspace, we
may then proceed to build a parametric or nonparametric regression model on that subspace. Thus our approach is an explicit dimensionality reduction method for supervised
learning that does not require any particular form of regression model; it can be used as a
preprocessor for any supervised learner.
Most conventional approaches to dimensionality reduction make specific assumptions regarding the conditional distribution pY |?S X (y|?S x), the marginal distribution pX (x), or
both. For example, classical two-layer neural networks can be seen as attempting to estimate an effective subspace in their first layer, using a specific model for the regressor.
Similar comments apply to projection pursuit regression [1] and ACE [2], which assume
T
an additive model E[Y |X] = g1 (?1T X) + ? ? ? + gK (?K
X). While canonical correlation
analysis (CCA) and partial least squares (PLS, [3]) can be used for dimensionality reduction in regression, they make a linearity assumption and place strong restrictions on the
allowed dimensionality. The line of research that is closest to our work is sliced inverse regression (SIR, [4]) and related methods including principal Hessian directions (pHd, [5]).
SIR is a semiparametric method that can find effective subspaces, but only under strong
assumptions of ellipticity for the marginal distribution pX (x). pHd also places strong restrictions on pX (x). If these assumptions do not hold, there is no guarantee of finding the
effective subspace.
In this paper we present a novel semiparametric method for dimensionality reduction that
we refer to as Kernel Dimensionality Reduction (KDR). KDR is based on a particular class
of operators on reproducing kernel Hilbert spaces (RKHS, [6]). In distinction to algorithms
such as the support vector machine and kernel PCA [7, 8], KDR cannot be viewed as a ?kernelization? of an underlying linear algorithm. Rather, we relate dimensionality reduction
to conditional independence of variables, and use RKHSs to provide characterizations of
conditional independence and thereby design objective functions for optimization. This
builds on the earlier work of [9], who used RKHSs to characterize marginal independence
of variables. Our characterization of conditional independence is a significant extension,
requiring rather different mathematical tools?the covariance operators on RKHSs that we
present in Section 2.2.
2
2.1
Kernel method of dimensionality reduction for regression
Dimensionality reduction and conditional independence
The problem discussed in this paper is to find the effective subspace S defined by Eq. (1),
given an i.i.d. sample {(Xi , Yi )}ni=1 , sampled from the conditional probability Eq. (1) and
a marginal probability pX for X. The crux of the problem is that we have no a priori
knowledge of the regressor, and place no assumptions on the conditional probability pY |X
or the marginal probability pX .
We do not address the problem of choosing the dimensionality r in this paper?in practical
applications of KDR any of a variety of model selection methods such as cross-validation
can be reasonably considered. Rather our focus is on the problem of finding the effective
subspace for a given choice of dimensionality.
The notion of effective subspace can be formulated in terms of conditional independence.
Let Q = (B, C) be an m-dimensional orthogonal matrix such that the column vectors of
B span the subspace S (thus B is m ? r and C is m ? (m ? r)), and define U = B T X
and V = C T X. Because Q is an orthogonal matrix, we have pX (x) = pU,V (u, v) and
pX,Y (x, y) = pU,V,Y (u, v, y). Thus, Eq. (1) is equivalent to
pY |U,V (y|u, v) = pY |U (y|u).
(2)
Y
Y
Y
X
V
U
X
V |U
X = (U,V)
Figure 1: Graphical representation of dimensionality reduction for regression.
This shows that the effective subspace S is the one which makes Y and V conditionally
independent given U (see Figure 1).
Mutual information provides another viewpoint on the equivalence between conditional
independence and the effective subspace. It is well known that
(3)
I(Y, X) = I(Y, U ) + EU I(Y |U, V |U ) ,
where I(Z, W ) is the mutual information between Z and W . Because Eq. (1) implies
I(Y, X) = I(Y, U ), the effective subspace S is characterized as the subspace which retains
the entire mutual information between X and Y , or equivalently, such that I(Y |U, V |U ) =
0. This is again the conditional independence of Y and V given U .
2.2
Covariance operators on kernel Hilbert spaces and conditional independence
We use cross-covariance operators [10] on RKHSs to characterize the conditional independence of random variables. Let (H, k) be a (real) reproducing kernel Hilbert space of
functions on a set ? with a positive definite kernel k : ? ? ? ? R and an inner product
?, ?H . The most important aspect of a RKHS is the reproducing property:
f, k(?, x)H = f (x)
for all x ? ? and f ? H.
In this paper we focus on the Gaussian kernel k(x1 , x2 ) = exp ?x1 ? x2 2 /2? 2 .
(4)
Let (H1 , k1 ) and (H2 , k2 ) be RKHSs over measurable spaces (?1 , B1 ) and (?2 , B2 ), respectively, with k1 and k2 measurable. For a random vector (X, Y ) on ?1 ? ?2 , the
cross-covariance operator ?Y X from H1 to H2 is defined by the relation
g, ?Y X f H2 = EXY [f (X)g(Y )] ? EX [f (X)]EY [g(Y )]
(= Cov[f (X), g(Y )]) (5)
for all f ? H1 and g ? H2 . Eq. (5) implies that the covariance of f (X) and g(Y ) is given
by the action of the linear operator ?Y X and the inner product. Under the assumption that
EX [k1 (X, X)] and EY [k2 (Y, Y )] are finite, by using Riesz?s representation theorem, it is
not difficult to see that a bounded operator ?Y X is uniquely defined by Eq. (5). We have
??Y X = ?XY , where A? denotes the adjoint of A. From Eq. (5), we see that ?Y X captures
all of the nonlinear correlations defined by the functions in HX and HY .
Cross-covariance operators provide a useful framework for discussing conditional probability and conditional independence, as shown by the following theorem and its corollary1 :
Theorem 1. Let (H1 , k1 ) and (H2 , k2 ) be RKHSs on measurable spaces ?1 and ?2 , respectively, with k1 and k2 measurable, and (X, Y ) be a random vector on ?1 ??2 . Assume
that EX [k1 (X, X)] and EY [k2 (Y, Y )] are finite, and for all g ? H2 the conditional expectation EY |X [g(Y ) | X = ?] is an element of H1 . Then, for all g ? H2 we have
?XX EY |X [g(Y ) | X = ?] = ?XY g.
1
Full proofs of all theorems can be found in [11].
(6)
? ?1 be the right inverse of ?XX on (Ker?XX )? . Under the same
Corollary 2. Let ?
XX
assumptions as Theorem 1, we have, for all f ? (Ker?XX )? and g ? H2 ,
? ?1 ?XY gH = f, EY |X [g(Y ) | X = ?]H .
f, ?
1
1
XX
1/2
(7)
1/2
Sketch of the proof. ?XY can be decomposed as ?XY = ?XX V ?Y Y for a bounded oper? ?1 ?XY is well-defined, because Range?XY ?
ator V (Theorem 1, [10]). Thus, we see ?
XX
?
Range?XX = (Ker?XX ) . Then, Eq. (7) is a direct consequence of Theorem 1.
Given that ?XX is invertible, Eq. (7) implies
EY |X [g(Y ) | X = ?] = ??1
XX ?XY g
for all g ? H2 .
(8)
This can be understood by analogy to the conditional expectation of Gaussian random
variables. If X and Y are Gaussian random variables, it is well-known that the conditional
expectation is given by EY |X [aT Y | X = x] = xT ??1
XX ?XY a for an arbitrary vector a,
where ?XX and ?XY are the variance-covariance matrices in the ordinary sense.
Using cross-covariance operators, we derive an objective function for characterizing conditional independence. Let (H1 , k1 ) and (H2 , k2 ) be RKHSs on measurable spaces ?1
and ?2 , respectively, with k1 and k2 measurable, and suppose we have random variables
U ? H1 and Y ? H2 . We define the conditional covariance operator ?Y Y |U on H1 by
? ?1 ?U Y .
?Y Y |U := ?Y Y ? ?Y U ?
UU
(9)
Corollary 2 easily yields the following result on the conditional covariance of variables:
Theorem 3. Assume that EX [k1 (X, X)] and EY [k2 (Y, Y )] are finite, and that
EY |X [f (Y )|X] is an element of H1 for all f ? H2 . Then, for all f, g ? H2 , we have
g, ?Y Y |U f H2 = EY [f (Y )g(Y )] ? EU EY |U [f (Y )|U ]EY |U [g(Y )|U ]
(10)
= EU CovY |U f (Y ), g(Y ) | U .
As in the case of Eq. (8), Eqs. (9) and (10) can be viewed as the analogs of the well-known
equality for Gaussian variables: Cov[aT Y, bT Y |U ] = aT (?Y Y ? ?Y U ??1
U U ?U Y )b.
From Theorem 3, it is natural to use minimization of ?Y Y |U as a basis of a method for
finding the most informative U , which gives the least VarY |U [f (Y )|U ]. The following
definition is needed to justify this intuition. Let (?, B) be a measurable space, let (H, k) be
a RKHS over ? with k measurable and bounded, and let M be the set of all the probability
measures on (?, B). The RKHS H is called probability-determining, if the map
MP
?
(f ? EX?P [f (X)]) ? H?
(11)
?
is one-to-one, where H is the dual space of H. The following theorem can be proved
using a argument similar to that used in the proof of Theorem 2 in [9].
Theorem 4. For an arbitrary ? > 0, the RKHS with Gaussian kernel k(x, y) = exp(?x?
y2 /2? 2 ) on Rm is probability-determining.
Recall that for two RKHSs H1 and H2 on ?1 and ?2 , respectively, the direct product
H1 ?H2 is the RKHS on ?1 ??2 with the kernel k1 k2 [6]. The relation between conditional
independence and the conditional covariance operator is given by the following theorem:
Theorem 5. Let (H11 , k11 ), (H12 , k12 ), and (H2 , k2 ) be RKHSs on measurable spaces
?11 , ?12 , and ?2 , respectively, with continuous and bounded kernels. Let (X, Y ) =
(U, V, Y ) be a random vector on ?11 ? ?12 ? ?2 , where X = (U, V ), and let H1 =
H11 ? H12 be the direct product. It is assumed that EY |U [g(Y )|U = ?] ? H11 and
EY |X [g(Y )|X = ?] ? H1 for all g ? H2 . Then, we have
?Y Y |U ? ?Y Y |X ,
(12)
where the inequality refers to the order of self-adjoint operators. If further H2 is
probability-determining, in particular, for Gaussian kernels, we have the equivalence:
?Y Y |X = ?Y Y |U
??
Y?
?V | U.
(13)
Sketch of the proof. Taking the
of the well-known equality VarY |U [g(Y )|U ] =
expectation
EV |U Var
EY |U,V [g(Y
[g(Y
)|U,
V
]
+
Var
Y
|U,V
V
|U
)|U, V ] with respect to U , we
obtain EU VarY |U [g(Y )|U ] ?EX VarY |X [g(Y )|X] = EU VarV |U [EY |X [g(Y )|X]] ? 0,
which implies Eq. (12). The equality holds iff EY |X [g(Y )|X] = EY |U [g(Y )|U ] for a.e. X.
Since H2 is probability-determining, this means PY |X = PY |U , that is, Y ?
?V | U .
From Theorem 5, for probability-determining kernel spaces, the effective subspace S can
be characterized in terms of the solution to the following minimization problem:
min ?Y Y |U ,
S
2.3
subject to
U = ?S X.
(14)
Kernel generalized variance for dimensionality reduction
To derive a sampled-based objective function from Eq. (14) for a finite sample, we have to
estimate the conditional covariance operator with given data, and choose a specific way to
evaluate the size of self-adjoint operators. Hereafter, we consider only Gaussian kernels,
which are appropriate for both continuous and discrete variables.
For the estimation of the operator, we follow the procedure in [9] (see also [11] for further
details), and use the centralized Gram matrix [9, 8], which is defined as:
? Y = In ? 1 1n 1T GY In ? 1 1n 1T , K
? U = In ? 1 1n 1T GU In ? 1 1n 1T (15)
K
n
n
n
n
n
n
n
n
where 1n = (1, . . . , 1)T , (GY )ij = k1 (Yi , Yj ) is the Gram matrix of the samples of Y ,
and (GU )ij = k2 (Ui , Uj ) is given by the projection Ui = B T Xi . With a regularization
? Y Y |U is then defined by
constant ? > 0, the empirical conditional covariance matrix ?
2
?2 ? ?
?Y Y ??
?Y U?
?
? ? ?
? Y Y |U := ?
? ?1 ?
?
?
KU KY . (16)
U U U Y = (KY + ?In ) ? KY KU (KU + ?In )
? Y Y |U in the ordered set of positive definite matrices can be evaluated by its
The size of ?
? Y Y |U , such as
determinant. Although there are other choices for measuring the size of ?
the trace and the largest eigenvalue, we focus onthe determinant
in this paper. Using the
A B
?1 T
Schur decomposition, det(A ? BC B ) = det B T C /detC, we have
? [Y U ][Y U ] / det ?
? UU ,
? Y Y |U = det ?
(17)
det ?
?
2
? [Y U ][Y U ] = ?? Y Y ?? Y U = (KY +?In ) K? Y K? U 2 .
? [Y U ][Y U ] is defined by ?
where ?
? U +?In )
? UY ?
? UU
?U K
?Y
(K
?
K
? Y Y , which yields
We symmetrize the objective function by dividing by the constant det ?
min
m?r
B?R
? [Y U ][Y U ]
det ?
,
? Y Y det ?
? UU
det ?
where U = B T X.
(18)
We refer to this minimization problem with respect to the choice of subspace S or matrix
B as Kernel Dimensionality Reduction (KDR).
Eq. (18) has been termed the ?kernel generalized variance? (KGV) by Bach and Jordan [9].
They used it as a contrast function for independent component analysis, in which the goal
is to minimize a mutual information. They showed that KGV is in fact an approximation
of the mutual information among the recovered sources around the factorized distributions.
In the current setting, on the other hand, our goal is to maximize the mutual information
R(b1 )
R(b2 )
SIR(10)
0.987
0.421
SIR(15)
0.993
0.705
SIR(20)
0.988
0.480
SIR(25)
0.990
0.526
pHd
0.110
0.859
KDR
0.999
0.984
Table 1: Correlation coefficients. SIR(m) indicates the SIR method with m slices.
I(Y, U ), and with an entirely different argument, we have shown that KGV is an appropriate objective function for the dimensionality reduction problem, and that minimizing
Eq. (18) can be viewed as maximizing the mutual information I(Y, U ).
Given that the numerical task that must be solved in KDR is the same as the one to be
solved in kernel ICA, we can import all of the computational techniques developed in [9]
for minimizing KGV. In particular, the optimization routine that we use is gradient descent
with a line search, where we exploit incomplete Cholesky decomposition to reduce the
n ? n matrices to low-rank approximations. To cope with local optima, we make use of an
annealing technique, in which the scale parameter ? for the Gaussian kernel is decreased
gradually during the iterations of optimization. For a larger ?, the contrast function has
fewer local optima, and the search becomes more accurate as ? is decreased.
3
Experimental results
We illustrate the effectiveness of the proposed KDR method through experiments, comparing it with several conventional methods: SIR, pHd, CCA, and PLS.
The first data set is a synthesized one with 300 samples of 17 dimensional X and one
dimensional Y , which are generated by Y ? 0.9X1 + 0.2/(1 + X17 ) + Z, where Z ?
N (0, 0.012 ) and X follows a uniform distribution on [0, 1]17 . The effective subspace is
given by b1 = (1, 0, . . . , 0) and b2 = (0, . . . , 0, 1). We compare the KDR method with
SIR and pHd only?CCA and PLS cannot find a 2-dimensional subspace, because Y is onedimensional. To evaluate estimation accuracy, we use the multiple correlation coefficient
R(b) = max??S ? T ?XX b/(? T ?XX ? ? b T ?XX b)1/2 , which is used in [4]. As shown
in Table 1, KDR outperforms the others in finding the weak contribution of b2 .
Next, we apply the KDR method to classification problems, for which many conventional
methods of dimensionality reduction are not suitable. In particular, SIR requires the dimensionality of the effective subspace to be less than the number of classes, because SIR uses
the average of X in slices along the variable Y . CCA and PLS have a similar limitation
on the dimensionality of the effective subspace. Thus we compare the result of KDR only
with pHd, which is applicable to general binary classification problems.
We show the visualization capability of the dimensionality reduction methods for the Wine
dataset from the UCI repository to see how the projection onto a low-dimensional space realizes an effective description of data. The Wine data consists of 178 samples with 13 variables and a label with three classes. Figure 2 shows the projection onto the 2-dimensional
subspace estimated by each method. KDR separates the data into three classes most completely. We can see that the data are nonlinearly separable in the two-dimensional space.
In the third experiment, we investigate how much information on the classification is preserved in the estimated subspace. After reducing the dimensionality, we use the support
vector machine (SVM) method to build a classifier in the reduced space, and compare its
accuracy with an SVM trained using the full-dimensional vector X. We use three data sets
from the UCI repository. Figure 3 shows the classification rates for the test set for subspaces of various dimensionality. We can see that KDR yields good classification even in
low-dimensional subspaces, while pHd is much worse in small dimensionality. It is noteworthy that in the Ionosphere data set the classifier in dimensions 5, 10, and 20 outperforms
20
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Figure 2: Projections of Wine data: ?+?, ???, and gray ?? represent the three classes.
the classifier in the full-dimensional space. This is caused by suppressing noise irrelevant
to explain Y . These results show that KDR successfully finds an effective subspace which
preserves the class information even when the dimensionality is reduced significantly.
4
Extension to variable selection
The KDR method can be extended to variable selection, in which a subset of given explanatory variables {X1 , . . . , Xm } is selected. Extension of the KGV objective function
to variable selection is straightforward. We have only to compare the KGV values for all
the subspaces spanned by combinations of a fixed number of selected variables. We of
course do not avoid the combinatorial problem of variable selection; the total number of
combinations may be intractably large for a large number of explanatory variables m, and
greedy or random search procedures are needed.
We first apply this kernel method to the Boston Housing data (506 samples with 13 dimensional X), which has been used as a typical example of variable selection. We select
four variables that attain the smallest KGV value among all the combinations. The selected
variables are exactly the same as the ones selected by ACE [2]. Next, we apply the method
to the leukemia microarray data of 7129 dimensions ([12]). We select 50 effective genes
to classify two types of leukemia using 38 training samples. For optimization of the KGV
value, we use a greedy algorithm, in which new variables are selected one by one, and
subsequently a variant of genetic algorithm is used. Half of the 50 genes accord with 50
genes selected by [12]. With the genes selected by our method, the same classifier as that
used in [12] classifies correctly 32 of the 34 test samples, for which, with their 50 genes,
Golub et al. ([12]) report a result of classifying 29 of the 34 samples correctly.
5
Conclusion
We have presented KDR, a novel method of dimensionality reduction for supervised learning. One of the striking properties of this method is its generality. We do not place any
strong assumptions on either the conditional or the marginal distribution, in distinction to
(a) Heart-disease
(b) Ionosphere
(c) Wisconsin Breast Cancer
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60
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PHD
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50
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Classification rate (%)
Classification rate (%)
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Dimensionality
34
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Kernel
PHD
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Figure 3: Classification accuracy of the SVM for test data after dimensionality reduction.
essentially all existing methods for dimensionality reduction in regression, including SIR,
pHd, CCA, and PPR. We have demonstrating promising empirical performance of KDR,
showing its practical utility in data visualization and feature selection for prediction. We
have also discussed an extension of KDR method to variable selection.
The theoretical basis of KDR lies in the nonparametric characterization of conditional independence that we have presented in this paper. Extending earlier work on the kernel-based
characterization of marginal independence [9], we have shown that conditional independence can be characterized in terms of covariance operators on a kernel Hilbert space.
While our focus has been on the problem of dimensionality reduction, it is also worth noting that there are many possible other applications of this result. In particular, conditional
independence plays an important role in the structural definition of graphical models, and
our result may have implications for model selection and inference in graphical models.
References
[1] Friedman, J.H. and Stuetzle, W. Projection pursuit regression. J. Amer. Stat. Assoc., 76:817?
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[2] Breiman, L. and Friedman, J.H. Estimating optimal transformations for multiple regression and
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[3] Wold, H. Partial least squares. in S. Kotz and N.L. Johnson (Eds.), Encyclopedia of Statistical
Sciences, Vol. 6, Wiley, New York. pp.581?591. 1985.
[4] Li, K.-C. Sliced inverse regression for dimension reduction (with discussion). J. Amer. Stat.
Assoc., 86:316?342, 1991.
[5] Li, K.-C. On principal Hessian directions for data visualization and dimension reduction: Another application of Stein?s lemma. J. Amer. Stat. Assoc., 87:1025?1039, 1992.
[6] Aronszajn, N. Theory of reproducing kernels. Trans. Amer. Math. Soc., 69(3):337?404, 1950.
[7] Sch?olkopf, B., Burges, C.J.C., and Smola, A. (eds.) Advances in Kernel Methods: Support
Vector Learning. MIT Press. 1999.
[8] Sch?olkopf, B., Smola, A and M?uller, K.-R. Nonlinear component analysis as a kernel eigenvalue
problem. Neural Computation, 10:1299?1319, 1998.
[9] Bach, F.R. and Jordan, M.I. Kernel independent component analysis. JMLR, 3:1?48, 2002.
[10] Baker, C.R. Joint measures and cross-covariance operators. Trans. Amer. Math. Soc., 186:273?
289, 1973.
[11] Fukumizu, K., Bach, F.R. and Jordan, M.I. Dimensionality reduction for supervised learning
with reproducing kernel Hilbert spaces. JMLR, 5:73?99, 2004.
[12] Golub T.R. et al. Molecular classification of cancer: Class discovery and class prediction by
gene expression monitoring. Science, 286:531?537, 1999.
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1,666 | 2,514 | Eye micro-movements improve stimulus
detection beyond the Nyquist limit in the
peripheral retina
Matthias H. Hennig and Florentin W?org?otter
Computational Neuroscience
Psychology
University of Stirling
FK9 4LR Stirling, UK
{hennig,worgott}@cn.stir.ac.uk
Abstract
Even under perfect fixation the human eye is under steady motion
(tremor, microsaccades, slow drift). The ?dynamic? theory of vision [1, 2] states that eye-movements can improve hyperacuity. According to this theory, eye movements are thought to create variable spatial
excitation patterns on the photoreceptor grid, which will allow for better
spatiotemporal summation at later stages. We reexamine this theory using a realistic model of the vertebrate retina by comparing responses of a
resting and a moving eye. The performance of simulated ganglion cells
in a hyperacuity task is evaluated by ideal observer analysis. We find that
in the central retina eye-micromovements have no effect on the performance. Here optical blurring limits vernier acuity. In the retinal periphery however, eye-micromovements clearly improve performance. Based
on ROC analysis, our predictions are quantitatively testable in electrophysiological and psychophysical experiments.
1
Introduction
Normal visual acuity is limited by the photoreceptor distance on the retina to about 10 of
visual angle, which is imposed by the neural nyquist sampling limit. The human visual
system, however, is capable of resolving certain stimuli (e.g. vernier stimuli) at a much
higher resolution of < 500 . This effect, called hyperactuity, has given rise to a large number
of psychophysical studies and several qualitative theories about perception as well as the
underlying neuronal properties. Most notably are the so-called ?dynamic? and ?static?
theories of vision [3], which claim that hyperacuity would require eye-micromovements
(microtremor, microsaccades) or not. Along the dynamic theory it has been suggested by
Averill and Weymouth [1] and later by Marshall and Talbot [2] that small eye-movements
would shift the photoreceptor grid across the stimulus leading to a better discriminability
when appropriate spatiotemporal integration is used.
In a previous study we had designed a realistic and detailed model of the vertebrate
retina [4]. This allows us for the first time to quantitatively test the Marshall-Talbot
Figure 1: Overview of the model. A, Structure of the retina model. Photoreceptors (P)
connect to horizontal (H) and bipolar cells (B). Horizontal cells antagonize bipolar cells.
Bipolar cells provide the center input to ganglion cells (G) and the surround is mediated by a
Type 1 (1) amacrine cell [4]. B, Scaling of optical point spread functions (top curves), photoreceptor (upper lines, values shown, data from [5]) and ganglion cell separation (lower
lines, values shown, data from [6, 7]) at different retinal eccentricities. PSF?s are shown for
the constant (straight lines) and scaled case (dashed lines). C, Spatial layout of the stimulus
(S) and the photoreceptor (P) and ganglion cell (G) grids. D, Nyquist frequencies for photoreceptors, P ganglion cells and the scaled PSF as a function of the eccentricity. Aliasing
occurs in the shaded region.
theory under different experimental conditions. We will show that the presence of eyemicromovements indeed improves hyperacuity. Contrary to earlier assumptions we find
that eye micromovements have no effect in the central part of the retina, where optical
blurring defines the limit for hyperacuity tasks. At above 5? retinal eccentricity, eyemicromovements are clearly improving hyperacuity. Our approach relies on a model free
(receiver-operator characteristic, ROC) analysis, and the reported results should be directly
measurable in retinal ganglion cells and psychophysically.
2
MATERIALS AND METHODS
The model used in this study is based on a previously described model of the light adapted
retina. In this section, we only mention aspects which are important in the context of this
study. For a detailed discussion of the model, see [4].
Briefly, the model consists of cone photoreceptors, horizontal and bipolar, amacrine and
ganglion cells (Fig. 1A). Neurons are arranged on homogeneous two-dimensional hexago-
nal grids (Fig. 1C). Ganglion cells are shifted randomly by 12% of their separation to account for the non-ideal distribution on the hexagonal grid. Cones, bipolar and ganglion cells
form the feed-forward path and horizontal and amacrine cells two lateral layers. Densities
and receptive field sizes of photoreceptors and ganglion cells were adjusted to the anatomical data available for the human retina at the different eccentricities studied (Fig.1B). The
separation of horizontal, bipolar and amacrine cells was scaled proportional to the cone
density.
Eccentricity
[deg]
0
5
10
15
20
PSF scaling
1.00
2.51
2.98
3.31
3.52
Vernier offset
[arcsec]
7
46
83
92
98
Table 1: Spatial scaling of the PSF that simulates the optical blurring and of the vernier
offset as a function of the eccentricity.
The photoreceptor model is a slightly modified version of the mathematical description
given by Hennig et al. [4]. It is originally based on a description by Schnapf et al. [8].
The voltage responses were tested against experimental data from the macaque monkey
by Schneeweis and Schnapf [9]. To account for the sustained responses for strong, but
brief stimuli, the single initial activation stage [4] was replaced by three cascaded lowpass filters. This study focuses on human P On-center cells (or ?midget? cells). Receptive
field sizes and densities were chosen according to anatomical data (Fig. 1). The center and
surround input of both cell types is weighted by overlapping Gaussian profiles [10], where
the surround extends > 3.8 times the center input [11].
Ocular optical blurring has been accounted for by convolving the stimulus with the pointspread function (PSF) given by Westheimer et al. [12] for the fovea:
1.36
P SF (?) = 0.933 ? e?2.59??
1.74
+ 0.047 ? e?2.34??
(1)
? is the radius in arcmin. For higher eccentricities two sets of simulations were performed,
one with a constant and one with a scaled PSF (Fig.1B). The first case is an approximation
of the case when off-axis refractory errors of the ocular optics are corrected. Then aliasing occurs already at the level of the cone mosaic. The more realistic case corresponds
to a scaled PSF because off-axis astigmatism and increasing cone aperture increase the
amount of blurring at higher eccentricities. Scaling factors were chosen to fit experimental
data (Tab. 1, [13]). Under these conditions, aliasing on the ganglion cell layer begins at
5? (Fig.1D).
Eye micromovements where modeled by shifting the retina randomly relative to the stimulus by using a data fit by Eizenman et al. (Fig. 2A,B, [14]). They include the ocular microtremor and fast and slow microsaccades (Fig. 2B). Two types of micromovements were used in the simulations in this work: slow and fast microsaccades and the
microtremor (MT) and only fast microsaccades and the tremor (FMT).
A typical vernier stimulus has been used in the simulations. To remove the effect of the
stimulus size, we used a bipartite field of 100% contrast with a small horizontal displacement in the vertical half (Fig.1C). Simulations were carried out at five different retinal
eccentricities: in the fovea and at 5, 10, 15 and 20 deg. The vernier offset was scaled
with increasing eccentricity proportional to the ratio of the cone to ganglion cell separation (Tab.1).
Figure 2: Characteristics of the simulated eye-micromovements. A, Traces of the horizontal retinal displacement for the two tremor spectra used (top: MT, bottom: FMT, see
Methods). B, Power spectra of the two cases from part A (dashed line: MT, dotted line:
FMT) and the full spectrum given by Eigenman et al. (straight line, [14]). C, Responses
of P-ganglion cells to a contrast step (100% contrast) without tremor (solid line) and with
eye micromovements (MT, dotted line). Horizontal alignment corresponds to the location
of the cell relative to the stimulus (location of contrast step indicated by dotted line).
3
Results
Fig. 2 summarizes the characteristics of simulated eye-micromovements. In part A an
example for the horizontal displacement of the retina is shown for the two types of micromovements included in the model (MT and FMT, see Methods). Part B shows the corresponding power spectra. Fig. 2C shows the membrane potential of a simulated ganglion
cell at different locations relative to a contrast step with and without eye micromovements.
When the cell is located in the dark half of the contrast step, moving the light half of
the stimulus into its receptive field causes frequent strong depolarizations. For the reverse
case, when the dark half of the stimulus moves into the receptive field of a cell which was
previously excited, the membrane potential hyperpolarizes. These hyperpolarizations are
weaker than the depolarizations in the former case because the photoreceptor response is
asymmetric with respect to the to on- and offset of light. Light onset leads a to brief, strong
transient hyperpolarization whereas offset causes a slower response decay and a weaker
phasic depolarization [4, 9].
Fig. 3A,E show the spatial response distribution on the ganglion cell layer 30ms after
stimulus onset for two retinal eccentricities for the constant PSF. At 5? eccentricity the
vernier offset is well visible by eye by comparing the upper and lower half of the responses.
At 10? however, upper and lower half look very similar, implying that vernier detection is
not possible.
To quantify the detectability of a vernier stimulus we performed a ROC analysis of the
spatial response profiles. This procedure is shown in Fig.3: First a horizontal cross-section
of the spatial response profile on the ganglion cell layer is taken for the upper and lower
part of the stimulus (B, F). The detectability of a vernier stimulus should be reflected in the
population average of the ganglion cell responses for upper and lower part of the stimulus.
This assumption reflects the known convergence properties of the primary visual pathway,
where each cortical cell receives input (via the LGN) from many ganglion cells. We used
Figure 3: Spatial analysis of the vernier stimuli. A, Spatial response profiles of the ganglion cells to a vernier stimulus 30ms after stimulus onset (5? retinal eccentricity, vernier
offset 4500 ). The membrane potential is coded by gray levels. B, Spatial response profile
for the upper (black) and lower half (grey) of the responses in A (average over four rows).
C, Spatial derivative of the curves in B, rectified at zero. D, ROC curve calculated from the
curves in C. Value of the integral of the ROC curve (shaded gray) is shown for each curve
(detectability index). E-H The same analysis at 10? retinal eccentricity and a vernier offset
of 9200 .
an average of four rows of the ganglion cells for analysis. The resulting profiles closely
fit cumulative Difference of Gaussians functions, which is a consequence of the ganglion
cell receptive field structure. In the next step, the spatial derivative of the response profile
is calculated and rectified at the resting potential (C, G). This operation is similar to a
cortical edge detection mechanism [15] and leads to Gaussian-like distributions. From
these curves it is possible to directly compute a ROC-curve (D, H). The integral of the
ROC curve, ranging from 0.5 to 1, is then taken as a direct measure of the detectability of
the vernier offset. This method combines the standard, model-free ROC-type analysis with
basic assumptions about the convergence properties in the primary visual pathway.
Eye-movements lead to temporal changes of the detectability. Thus, the integral of the
ROC curve, which we will call the ?detectability index? (DI), then varies over time. Fig.
4A shows this effect for the five different retinal eccentricities studied and different types
of micromovements using the scaled PSF. For each eccentricity, the stimulus has been
placed at five different locations relative to the ganglion cell receptive fields. We found
that, without eye-micromovements and increasing eccentricities, the detectability strongly
depends on the location of the stimulus in the receptive field. This is not surprising when
one considers that spatial undersampling of the stimulus occurs at the ganglion cell layer.
At the fovea visual resolution is limited by the optics of the eye. At > 5? eccentricity,
there are substantial ?gaps? in the ganglion cell representation of the stimulus (see Fig.1B)
which cause aliasing effects. Aliasing effects in the periphery due to undersampling has
been reported in psychophysics [16].
Ocular micromovements leads to clearly visible effects (Fig. 4A). The noisy curves are
Figure 4: Temporal analysis of the ROC curves. A, Detectability index as function
of time at different retinal eccentricities and different stimulus displacements relative to
the ganglion cell positions (thick curves: resting eye, thin curves: slow+fast microsaccades+tremor, grey curves: fast microsaccades+tremor). Stimulus offsets are shown above
the traces. B, Maximum of the curves in A at each eccentricity and location for the scaled
PSF on a noisy ganglion cells grid. Only values are considered as a maximum where the
DI stays above the mean for > 10ms. C, Maximal DI for the constant PSF.
now randomly oscillating across the smooth curves without micromovements. We note for
most curves obtained with tremor there is an interval of at least 10ms where the DI is
substantially above its mean and equal or above the noise-free equivalent. Psychophysical
evidence shows that detection tasks may require only short periods of as little as 5-10ms
where the detectability must exceed threshold [17]. Thus in the retinal periphery the eye
micromovements have a beneficial effect on the detectability by reducing aliasing.
In Fig. 4B, the maximum of DI at different stimulus locations is plotted as function of
the stimulus position. The maximum is defined as the largest value of the detectability
index within a > 10ms transient. The curves show the same effects as described above:
Performance remains the similar in the central and improves in the peripheral retina. If the
mean value of DI instead of the maximum is considered, the effect is similar in the fovea,
but no performance increase can be observed in the periphery (not shown). Fig. 4C shows
the same analysis of responses for a constant PSF on a regular ganglion cell grid (see Fig.
1B), where aliasing occurs already at the photoreceptor level. The effect is very similar to
that of the scaled PSF with stronger aliasing at higher eccentricities. However, at 10 and 15
deg, DI is lower for all cases because the disarray of the ganglion cells allows for improved
spatial averaging.
To summarize the previous results, the mean value of each curve in Fig. 4B and C is calculated. This can be interpreted as the psychophysical performance of a subject after many
stimulus repetitions. They are shown in Fig. 5A for the scaled and Fig. 5B for the constant
Figure 5: Mean detectability index (DI) for the experiments in Fig.3A (left, constant PSF)
and B (right, PSF scaled proportional to cone-ganglion cell convergence ratio) as function
of the retinal eccentricity.
PSF. The differences in DI at different eccentricities is a result of the stimulus scaling. For
both cases, eye micromovements increases the detectability at all eccentricities except in
the fovea. For the two types of eye micromovements, the maximal relative improvement
of DI happens at different eccentricities. The first type, comprising microsaccades and
tremor, frequently shifts the stimulus across adjoining ganglion cells at eccentricities 20? .
The second type has a smaller amplitude, thus the excitation of nearby ganglion cells is
most efficient at 10? . Thus, the effect depends on the spatial extend of the eye movements.
At 20? , DI is much lower for the scaled PSF on a noisy ganglion cell grid than for the
constant PSF on the regular grid. Because DI is consistently lower in the latter case for
the other eccentricities, this indicates that here the effect of the spatial disarray can not be
countered by spatial averaging of just four rows of ganglion cells.
Taken together, the results from the simulations shown here imply that a complex interplay
of different factors affect the detectability of hyperacuity stimuli. Indeed the quantitative
results from the model are very sensitive to changes of certain parameters (e.g. cell density).
Equally, a great variability in human psychophysical performance exists. However, the
effect of eye micromovements is consistent across the two cases shown here.
4
Discussion
Our results suggest that eye-micromovements contribute to visual hyperacuity in the peripheral visual field. By simulating ganglion cell responses for vernier stimuli using a
realistic model and applying model-free ideal observer analysis, we show that in the retinal
periphery eye-micromovements reduce the effect of aliasing due to neural undersampling.
This leads to a higher detectability of hyperacuity stimuli. There has been a successful
attempt to use small, continuous ?scanning? movements to increase the resolution of a
low resolution sensor array as a technical application [18]. We show that this principle
can indeed be employed by vertebrates to improve visual acuity in certain (hyperacuity)
tasks. However, eye movements have the reverse effect on detection tasks that require
aliasing. Packer and Williams [19] have shown that in a high frequency (aliasing) grating
detection task contrast thresholds are low for very brief and long presentation durations.
For intermediate presentation times the threshold increases substantially. Because detection relies on aliasing, it requires a resting eye. This is more likely for very brief and
long presentation times. For intermediate intervals, motion prevents aliasing. In hyperacuity, eye-micromovements increase detectability and we expect an asymptotic decrease of
thresholds as function of the presentation time.
The question arises how eye-micromovements affect human psychophysical performance.
We predict an influence of the effect of stimulus presentation time for vernier targets between the central and peripheral retina. We would also expect an increase of detection
thresholds under stabilized eye conditions in the periphey. This and further experiments
also suggest that eye micromovements generally influence detection tasks that are performed close to the psychophysical threshold. It is further possible to directly apply the
experimental procedure that was used in this work in an electrophysiological study. Specifically, it is possible to record from one ganglion cell with many different stimulus locations.
These responses can then be used to reconstruct a spatial response profile equivalent to our
simulated activity distribution (Fig.3B, F) and ROC analysis can be applied.
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1,667 | 2,515 | A Computational Geometric Approach to Shape
Analysis in Images
Washington Mio
Department of Mathematics
Florida State University
Tallahassee, FL 32306
[email protected]
Anuj Srivastava
Department of Statistics
Florida State University
Tallahassee, FL 32306
[email protected]
Xiuwen Liu
Department of Computer Science
Florida State University
Tallahassee, FL 32306
[email protected]
Eric Klassen
Department of Mathematics
Florida State University
Tallahassee, FL 32306
[email protected]
Abstract
We present a geometric approach to statistical shape analysis of closed
curves in images. The basic idea is to specify a space of closed curves
satisfying given constraints, and exploit the differential geometry of this
space to solve optimization and inference problems. We demonstrate this
approach by: (i) defining and computing statistics of observed shapes, (ii)
defining and learning a parametric probability model on shape space, and
(iii) designing a binary hypothesis test on this space.
1
Introduction
An important goal in image understanding is to detect, track and label objects of interest
present in observed images. Imaged objects can be characterized in many ways: according
to their colors, textures, shapes, movements, and locations. The past decade has seen significant advances in the modeling and analysis of pixel values or textures to characterize
objects in images, albeit with limited success. On the other hand, planar curves that represent contours of objects have been studied independently for a long time. An emerging
opinion in the vision community is that global features such as shapes of contours should
also be taken into account for the successful detection and recognition of objects. A common approach to analyzing curves in images is to treat them as level sets of functions,
and algorithms involving such active contours are governed usually by partial differential
equations (PDEs) driven by appropriate data terms and smoothness penalties (see for example [10]). Regularized curve evolutions and region-based active contours offer alternatives
in similar frameworks. This remarkable body of work contains various studies of curve
evolution, each with relative strengths and drawbacks.
In this paper, we present a framework for the algorithmic study of curves, their variations and statistics. In this approach, a fundamental element is a space of closed curves,
with additional constraints to impose equivalence of shapes under rotation, translation, and
scale. We exploit the geometry of these spaces using elements such as tangents, normals,
geodesics and gradient flows, to solve optimization and statistical inference problems for
a variety of cost functions and probability densities. This framework differs from those
employed in previous works on ?geometry-driven flows? [8] in the sense that here both the
geometry of the curves and the geometry of spaces of curves are utilized. Here the dynamics of active contours is described by vector fields on spaces of curves. It is important to
emphasize that a shape space is usually a non-linear, infinite-dimensional manifold, and
its elements are the individual curves of interest. Several interesting applications can be
addressed in this formulation, including: 1) Efficient deformations between any two curves
are generated by geodesic paths connecting the elements they represent in the shape space.
Geodesic lengths also provide a natural metric for shape comparisons. 2) Given a set of
curves (or shapes), one can define the concepts of mean and covariance using geodesic
paths, and thus develop statistical frameworks for studying shapes. Furthermore, one can
define probabilities on a shape space to perform curve (or shape) classification via hypothesis testing. While these problems have been studied in the past with elegant solutions
presented in the literature (examples include [9, 11, 7, 2, 5]), we demonstrate the strength
of the proposed framework by addressing these problems using significantly different ideas.
Given past achievements in PDE-based approaches to curve evolution, what is the need
for newer frameworks? The study of the structure of the shape space provides new insights
and solutions to problems involving dynamic contours and problems in quantitative shape
analysis. Once the constraints are imposed in definitions of shape spaces, the resulting
solutions automatically satisfy these constraints. It also complements existing methods of
image processing and analysis well by realizing new computational efficiencies. The main
strength of this approach is its exploitation of the differential geometry of the shape space.
For instance, a geodesic or gradient flow Xt of an energy function E can be generated as a
solution of an ordinary differential equation of the type
dXt
= ?(?E(Xt )) ,
dt
(1)
where ? denotes an appropriate projection onto a tangent space. This contrasts with the
nonlinear PDE-based curve evolutions of past works. The geometry of shape space also
enables us to derive statistical elements: probability measures, means and covariances;
these quantities have rarely been treated in previous studies. In shape extraction, the main
focus in past works has been on solving PDEs driven by image features under smoothness
constraints, and not on the statistical analysis of shapes of curves. The use of geodesic
paths, or piecewise geodesic paths, has also seen limited use in the past.
We should also point out the main limitations of the proposed framework. One drawback
is that curve evolutions can not handle certain changes in topology, which is one of the
key features of level-set methods; a shape space is purposely setup to not allow curves to
branch into several components. Secondly, this idea does not extend easily to the analysis
of surfaces in R3 . Despite these limitations, the proposed methodology provides powerful
algorithms for the analysis of planar curves as demonstrated by the examples presented
later. Moreover, even in applications where branching appears to be essential, the proposed
methods may be applicable with additional developments.
This paper is laid out as follows: Section 2 studies geometric representations of constrained
curves as elements of a shape space. Geometric analysis tools on the shape space are
presented in Section 3. Section 4 provides examples of statistical analysis on the shape
space, while Section 5 concludes the paper with a brief summary.
2
Representations of Shapes
In this paper we restrict the discussion to curves in R2 although curves in R3 can be handled
similarly. Let ? : R 7? R2 denote the coordinate function of a curve parametrized by arclength, i.e., satisfying k?(s)k
?
= 1, for?every s. A direction function ?(s) is a function
satisfying ?(s)
?
= ej ?(s) , where j = ?1. ? captures the angle made by the velocity
vector with the x-axis, and is defined up to the addition of integer multiples of 2?. The
?
curvature function ?(s) = ?(s)
can also be used to represent a curve.
Consider the problem of studying shapes of contours or silhouettes of imaged objects as
closed, planar curves in R2 , parametrized by arc length. Since shapes are invariant to rigid
motions (rotations and translations) and uniform scaling, a shape representation should be
insensitive to these transformations. Scaling can be resolved by fixing the length of ?
to be 2?, and translations by representing curves via their direction functions. Thus, we
consider the space L2 of all square integrable functions ?: [0, 2?] ? R, with the usual inner
R 2?
product hf, gi = 0 f (s)g(s) ds. To account for rotations and ambiguities on the choice
of ?, we restrict direction functions to those having a fixed average, say, ?. For ? to be
R 2?
closed, it must satisfy the closure condition 0 ej?(s) ds = 0. Thus, we represent curves
by direction functions satisfying the average-? and closure conditions; we call this space of
direction functions D. Summarizing, D is the subspace of L2 consisting of all (direction)
functions satisfying the constraints
Z 2?
Z 2?
Z 2?
1
?(s) ds = ? ;
cos(?(s)) ds = 0 ;
sin(?(s)) ds = 0 .
(2)
2? 0
0
0
It is still possible to have multiple continuous functions in D representing the same shape.
This variability is due to the choice of the reference point (s = 0) along the curve. For
x ? S1 and ? ? D, define (x ? ?) as a curve whose initial point (s = 0) is changed by a
distance of x along the curve. We term this a re-parametrization of the curve. To remove
the variability due to this re-parametrization group, define the quotient space C ? D/S 1 as
the space of continuous, planar shapes. For details, please refer to the paper [4].
3
Geometric Tools for Shape Analysis
The main idea in the proposed framework is to use the geometric structure of a shape space
to solve optimization and statistical inference problems on these spaces. This approach often leads to simple formulations of these problems and to more efficient vision algorithms.
Thus, we must study issues related to the differential geometry and topology of a shape
space. In this paper we restrict to the tangent and normal bundles, exponential maps, and
their inverses on these spaces.
3.1
Tangents and Normals to Shape Space
The main reason for studying the tangential and normal structures is the following: We
wish to employ iterative numerical methods in the simulation of geodesic and gradient
flows on the shape space. At each step in the iteration, we first flow in the linear space L 2
using standard methods, and then project the new point back onto the shape space using
our knowledge of the normal structure.
For technical reasons, it is convenient to reduce optimization and inference problems on C
to problems on the manifold D, so we study the latter. It is difficult to specify the tangent
spaces to D directly, because they are infinite-dimensional. When working with finitely
many constraints, as is the case here, it is easier to describe the space of normals to D in
L2 instead. It can be shown that a vector f ? L2 is tangent to D at ? if and only if f
is orthogonal to the subspace spanned by {1, sin ?, cos ?}. Hence, these three functions
span the normal space to D at ?. Implicitly, the tangent space is given as: T? (D) =
{f ? L2 |f ? span{1, cos ?, sin ?}} . Thus, the projection ? in Eqn. 1 can be specified
by subtracting from a function (in L2 ) its projection onto the space spanned by these three
elements.
3.2
Exponential Maps
We first describe the computation of geodesics (or, one-parameter flows) in D with prescribed initial conditions. Geodesics on D are realized as exponential maps from tangent
spaces to D. The intricate geometry of D disallows explicit analytic expressions. Therefore, we adopt an iterative strategy, where in each step, we first flow infinitesimally in the
prescribed tangent direction in the space L2 , and then project the end point of the path to D.
Next, we parallel transport the velocity vector to the new point by projecting the previous
velocity orthogonally onto the tangent space of D at the new point. Again, this is done
by subtracting normal components. The simplest implementation is to use Euler?s method
in L2 , i.e., to move in each step along short straight line segments in L2 in the prescribed
direction, and then project the path back onto D. Details of this numerical construction of
geodesics are provided in [4].
A geodesic can be specified by an initial condition ? ? D and a direction f ? T ? (D),
the space of all tangent directions at ?. We will denote the corresponding geodesic by
?(?, t, f ), where t is the time parameter. The technique just described allows us to compute
? numerically. For t = 1, ?(?, 1, f ) is the exponential map from f ? T? D to D.
3.3
Shape Logarithms
Next, we focus on the problem of finding a geodesic path between any two given shapes
?1 , ?2 ? D. This is akin to inverting the exponential map. The main issue is to find that
appropriate direction f ? T?1 (D) such that a geodesic from ?1 in that direction passes
through ?2 at time t = 1. In other words, the problem is to solve for an f ? T?1 (D) such
that ?(?1 , 0, f ) = ?1 and ?(?1 , 1, f ) = ?2 . One can treat the search for this direction as
an optimization problem over the tangent space T?1 (D). The cost to be minimized is given
by the functional H[f ] = k?(?1 , 1, f ) ? ?2 k2 , and we are looking for that f ? T?1 (C)
for which: (i) H[f ] is zero, and (ii) kf k is minimum among all such tangents. Since the
space T?1 (D) is infinite dimensional, this optimization is not straightforward. However,
since f ? L2 , it has a Fourier decomposition, and we can solve the optimization problem
over a finite number of Fourier coefficients. For any two shapes ?1 , ?2 ? D, we have used
a shooting method to find the optimal f [4]. The basic idea is to choose an initial direction
f specified by its Fourier coefficients and then use a gradient search to minimize H as a
function of the Fourier coefficients.
Finally, to find the shortest path between two shapes in C, we compute the shortest geodesic
connecting representatives of the given shapes in D. This is a simple numerical problem,
because C is the quotient of D by the 1-dimensional re-parametrization group S 1 . Shown
in Figure 1 are three examples of geodesic paths in C connecting given shapes. Drawn in
between are shapes corresponding to equally spaced points along the geodesic paths.
4
Statistical Analysis on Shape Spaces
Our goal is to develop tools for statistical analysis of shapes. Towards that goal, we develop
the following ideas.
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Figure 1: Top panels show examples of shapes manually extracted from the images. Bottom
panels show examples of evolving one shape into another via a geodesic path. In each case,
the leftmost shape is ?1 , rightmost curves are ?2 , and intermediate shapes are equi-spaced
points along the geodesic.
4.1
Sample Means on Shape Spaces
Algorithms for finding geodesic paths on the shape space allow us to compute means and
covariances in these spaces. We adopt a notion of mean known as the intrinsic mean or
the Karcher mean ([3]) that is quite natural in our geometric framework. Let d( , ) be
the shortest-path metric on C. To calculateP
the Karcher mean of shapes {?1 , . . . , ?n } in C,
n
define a function V : C ? R by V (?) = i=1 d(?, ?i )2 . Then, define the Karcher mean
of the given shapes to be any point ? ? C for which V (?) is a local minimum.
Pn In the case
of Euclidean spaces this definition agrees with the usual definition ? = n1 i=1 pi . Since
C is complete, the intrinsic mean as defined above always exists. However, there may be
collections of shapes for which ? is not unique. An iterative algorithm for finding a Karcher
mean of given shapes is given in [4] and see [4] for details.
4.2
Shape Learning
Another important problem in statistical analysis of shapes is to ?learn? probability models
from the observed shapes. Once the shapes are clustered, we assume that elements in
the same cluster are (random) samples from the same probability model, and try to learn
this model. These models can then be used for future Bayesian discoveries of shapes or
for classification of new shapes. To learn a probability model amounts to estimating a
probability density function on the shape space, a task that is rather difficult to perform
precisely. The two main difficulties are: nonlinearity and infinite-dimensionality of C, and
they are handled here as follows.
1. Tangent Space: Since C is a nonlinear manifold, we impose a probability density
on a tangent space rather than on C directly. For a mean shape ? ? C, the space of
all tangents to the shape space at ?, T? (C) ? L2 , is an infinite-dimensional vector
space. Similar to the ideas presented in [1], we impose a probability density
function f on T? (C) in order to avoid dealing with the nonlinearity of C. The
basic assumption here is that the support of f in T? (C) is sufficiently small so that
the exponential map between the support and C has a well-defined inverse.
2. Finite-Dimensional Representation: Assume that the covariance operator of the
probability distribution on T? (C) has finite spectrum, and thus admits a finite representation. We approximate a tangent function by a truncated Fourier series to
obtain a finite-dimensional representation. We thus characterize a probability distribution on T? (C) as that on a finite-dimensional vector space.
Let
Pma tangent element g ? T? (C) be represented by its Fourier expansion: g(s) m=
i=1 xi ei (s), for a large positive integer m. Using the identification g ? x = {x i } ? R ,
one can define a probability distribution on elements of T? (C) via a probability distribution
on the coefficients x.
We still have to decide what form does the resulting probability distribution takes. One
common approach is to assume a parametric form so that learning is reduced to an estimation of the relevant parameters. As an example, a popular idea is to assume a Gaussian
distribution on the underlying space. The variations of x as mostly restricted to an m 1 dimensional subspace of Rm , called the principal subspace, for some m1 ? m. On this
subspace we adopt a multivariate normal with mean ? ? Rm1 and variance K ? Rm1 ?m1 .
Estimation of ? and K from the observed shapes follows the usual procedures. Computation of the mean shape ? is described in [4]. Using ? and any observed shapes ? j , we find
the tangent vectors gj ? T? (C) such that the geodesic from ? in the direction gj passes
through ?j in unit time. This tangent vector is actually computed via its finite-dimensional
representation and results in the corresponding vector of coefficients xj . From the observed
values of xj ? Rm , one can estimate the principal subspace and the covariance matrix. Extracting the dominant eigenvectors of the estimated covariance matrix, one can capture the
dominant modes of variations. The density function associated with this family of shapes
is given by:
1
T
exp(?(x ? ?) K ?1 (x ? ?)/2) ,
(2?)m/2 det(K)1/2
Pm 1
where ?(?, g, 1) = ? and g = i=1
(xi ? ?i )ei (s).
h(?; ?, K) ?
(3)
An example of this shape learning is shown in Figure 2. The top panels show infrared
pictures of tanks, followed by their extracted contours in the second row of images. These
contours are then used in analyzing shapes of tanks. As an example, the 12 panels in
bottom left show the observed contours of a tank when viewed from a variety of angles,
and we are interested in capturing this shape variation. Repeating earlier process, the mean
shape is shown in the top middle panel and the eigen values are plotted in the bottom middle
panel. Twelve panels on the right show shape generated randomly from a parametric model
h(?; ?, ?).
In Figure 3 we present an interesting example of samples from three different shape models.
Let the original model be h(?; ?, K) where ? and K are as shown in Figure 2. Six samples
from this model are shown in the left of Figure 3. The middle shows samples from a
probability density h(?; ?, 0.2K) to demonstrate a smaller covariance; the samples here
seem much closer to the mean shape. The right shows samples from a density where the
covariance is equivariant in principal subspace, i.e. the covariance is given by 0.4kKk 2
times a matrix whose top left is a 12 ? 12 identity matrix and remaining entries are zero.
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10
12
Figure 2: Top two rows: Infrared images and extracted contours of two tanks M60 and T72
at different viewing angles. Bottom row: For the 12 observed M60 shapes shown in left,
the middle panels show the mean shape and the principal eigenvalues of covariance, and
the right panels show 12 random samples from Gaussian model h(?; ?, K).
Figure 3: Comparison of samples from three families: (i) h(?; ?, K), (ii) h(?; ?, 0.2K),
and (iii) h(?; ?, 0.4kKk2 I12 ).
4.3
Hypothesis Testing
This framework of shape representations and statistical models on shape spaces has important applications in decision theory. One is to recognize an imaged object according
to the shape of its boundary. Statistical analysis on shape spaces can be used to make a
variety of decisions such as: Does this shape belong to a given family of shapes? Does
these two families of shapes have similar means and variances? Given a test shape and two
competing probability models, which one explains the test shape better?
We restrict to the case of binary hypothesis testing since for multiple hypotheses, one can
find the best hypothesis using a sequence of binary hypothesis tests. Consider two shape
families specified by their probability models: h1 and h2 . For an observed shape ? ? C, we
are interested in selecting one of two following hypotheses: H0 : ? ? h1 or H1 : ? ? h2 .
>
We will select a hypothesis according to the likelihood ratio test: l(?) ? log( hh12 (?)
(?) ) < 0 .
Substituting for the normal distributions (Eqn. 3) for h1 ? h(?; ?1 , ?1 ) and h2 ?
h(?; ?2 , ?2 ), we can obtain sufficient statistics for this test. Let x1 be the vector of Fourier
coefficients that encode the tangent direction from
Pm ?1 to ?, and x2 be the
Pmsame for direction
from ?2 to ?. In other words, if we let g1 = i=1 x1,i ei and g2 = i=1 x2,i ei , then we
have ? = ?(?1 , g1 , 1) = ?(?2 , g2 , 1). It follows that
1
?
l(?) = (xT1 ??
(4)
1 x1 ? x2 ?2 x2 ) ? (log(det(?2 )) ? log(det(?1 )))
2
In case the two covariances are equal to ?, the hypothesis test reduces to l(?) =
(xT1 ?? x1 ? x2 ?? x2 ) >
< 0 , and when ? is identity, the log-likelihood ratio is given
by l(?) = kx1 k2 ? kx2 k2 . The curved nature of the shape space C makes the analysis
of this test difficult. For instance, one may be interested in probability of type one error
but that calculation requires a probability model on x2 when H0 is true. As a first order
approximation, one can write x2 ? N (?
x, ?1 ), where x
? is the coefficient vector of tangent
direction in T?2 (C) that corresponds to the geodesic from ?2 to ?1 . However, the validity
of this approximation remains to be tested under experimental conditions.
5
Conclusion
We have presented an overview of an ambitious framework for solving optimization and
inference problems on a shape space. The main idea is to exploit the differential geometry
of the manifold to obtain simpler solutions as compared to those obtained with PDE-based
methods. We have presented some applications of this framework in image understanding.
In particular, these ideas lead to a novel statistical theory of shapes of planar objects with
powerful tools for shape analysis.
Acknowledgments
This research was supported in part by grants NSF (FRG) DMS-0101429, NMA 201-012010, and NSF (ACT) DMS-0345242.
References
[1] I. L. Dryden and K. V. Mardia. Statistical Shape Analysis. John Wiley & Son, 1998.
[2] N. Duta, M. Sonka, and A. K. Jain. Learning shape models from examples using
automatic shape clustering and Procrustes analysis. In Proceedings of Information
in Medical Image Processing, volume 1613 of Lecture Notes in Computer Science,
pages 370?375. Springer, 1999.
[3] H. Karcher. Riemann center of mass and mollifier smoothing. Communications on
Pure and Applied Mathematics, 30:509?541, 1977.
[4] E. Klassen, A. Srivastava, W. Mio, and S. Joshi. Analysis of planar shapes using
geodesic paths on shape spaces. IEEE Pattern Analysis and Machiner Intelligence,
26(3):to appear, March, 2004.
[5] H. Le. Locating frechet means with application to shape spaces. Advances in Applied
Probability, 33(2):324?338, 2001.
[6] W. Mio, A. Srivastava, and E. Klassen. Interpolation by elastica in Euclidean spaces.
Quarterly of Applied Mathematics, to appear, 2003.
[7] D. Mumford. Elastica and computer vision, pages 491?506. Springer, New York,
1994.
[8] Editor: B. Romeny. Geometry Driven Diffusions in Computer Vision. Kluwer, 1994.
[9] T. B. Sebastian, P. N. Klein, and B. B. Kimia. On aligning curves. IEEE Transactions
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[10] J. Sethian. Level Set Methods: Evolving Interfaces in Geometry, Fluid Mechanics,
Computer Vision, and Material Science. Cambridge University Press, 1996.
[11] E. Sharon, A. Brandt, and R. Basri. Completion energies and scale. IEEE Transactions on Pattern Analysis and Machine Intelligence, 22(10):1117?1131, 2000.
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1,668 | 2,516 | Dynamical Modeling with Kernels for Nonlinear
Time Series Prediction
Liva Ralaivola
Laboratoire d?Informatique de Paris 6
Universit?e Pierre et Marie Curie
8, rue du capitaine Scott
F-75015 Paris, FRANCE
[email protected]
Florence d?Alch?e?Buc
Laboratoire d?Informatique de Paris 6
Universit?e Pierre et Marie Curie
8, rue du capitaine Scott
F-75015 Paris, FRANCE
[email protected]
Abstract
We consider the question of predicting nonlinear time series. Kernel Dynamical Modeling (KDM), a new method based on kernels, is proposed
as an extension to linear dynamical models. The kernel trick is used
twice: first, to learn the parameters of the model, and second, to compute
preimages of the time series predicted in the feature space by means of
Support Vector Regression. Our model shows strong connection with the
classic Kalman Filter model, with the kernel feature space as hidden state
space. Kernel Dynamical Modeling is tested against two benchmark time
series and achieves high quality predictions.
1
Introduction
Prediction, smoothing and filtering are traditional tasks applied to time series. The machine
learning community has recently paid a lot of attention to these problems and especially
to nonlinear time series prediction in various areas such as biological signals, speech or
financial markets. To cope with non linearities, extensions of the Kalman filter [5, 4] have
been proposed for filtering and smoothing while recurrent artificial neural networks [2] and
support vector regressors [7, 8] have been developed for prediction purposes. In this paper,
we focus on prediction tasks and introduce a powerful method based on the kernel trick
[1], which has been successfully used in tasks ranging from classification and regression
to data analysis (see [13, 15] for details). Time series modeling is addressed by extending
the framework of observable linear dynamical systems [12] to the feature space defined
by a kernel. The predictions are realized in the feature space and are then transformed to
obtain the corresponding preimages in the input space. While the proposed model could
be used for smoothing as well as filtering, we here focus on the prediction task. A link to
the Kalman filter can be drawn by noticing that given the efficiency of our model for the
prediction task it can be used as a hidden transition process in the Kalman filter setting.
The paper is organized as follows. In the next section, we describe how the modeling of
a time series can take place in the feature space and explain how to solve the preimage
problem by a learning strategy. In the third section, we present prediction results achieved
by our model In the fourth section, the estimation algorithm is discussed and its link to the
Kalman filter is highlighted. We finally conclude by giving some perspectives to our work.
2
2.1
Principles of Dynamical Modeling with Kernels
Basic Formulation
The problem we address is that of modeling d-dimensional nonlinear real-valued time series defined as
xt+1 = h(xt ) + u
(1)
from an observed sequence x1:T = {x1 , . . . , xT } produced by this model, where h is a
(possibly unknown) nonlinear function and u a noise vector.
Modeling such a series can be done with the help of recurrent neural networks [2] or
support vector machines [7]. In this work, we instead propose to deal with this problem by extending linear dynamical modeling thanks to the kernel trick. Instead of considering the observation sequence x1:T = {x1 , . . . , xT }, we consider the sequence
x?1:T = {?(x1 ), . . . , ?(xT )}, where ? is a mapping from Rd to H and k its associated
kernel function [15] such that k(v1 , v2 ) = h?(v1 ), ?(v2 )i ?v1 , v2 ? Rd , h?, ?i being the
inner product of H. The Kernel Dynamical Model (KDM) obtained can be written as:
x?t+1 = A? x?t + ?? + ? ?
(2)
where A? is the process transition matrix, ?? an offset vector, ? ? ? H a gaussian isotropic
noise of magnitude ? 2 and x?t stands for ?(xt ).
We are going to show that it is possible to apply the maximum likelihood principle to identify ? 2 , A? and ?? and come back to the input space thanks to preimages determination.
2.2
Estimation of the Model Parameters
Learning the parameters of the model (2) by maximum likelihood given an observation
sequence x?1:T merely consists in optimizing the associated log-likelihood L? (x?1:T , ? ? )1 :
!
T
Y
?
? ?
?
? ?
L (x1:T ; ? ) = ln P (x1 )
P (xt |xt?1 )
t=2
= g(??1 , ??1 ) ?
T
1 X ?
1
kxt ? A? x?t?1 ? ?? k2 ? p(T ? 1) ln ? 2
2
2? t=2
2
where p is the dimension of H, g(??1 , ??1 ) is a function straightforward to compute which
we let aside as it does not add any complexity in setting the gradient of L? to 0. Indeed,
performing this task leads to the equations:
A
?
=
T
X
0
x?t x?t?1
t=2
T
X
t=2
??
=
?2
=
T
T
1 X ?X ? 0
?
xt?1
xt
T ? 1 t=2
t=2
0
x?t?1 x?t?1
!
T
T
1 X ? X ? 0
?
xt?1
xt?1
T ? 1 t=2
t=2
(3)
!?1
T
?
1 X? ?
xt ? A? x?t?1
T ? 1 t=2
T
X
1
kx? ? A? x?t?1 ? ?? k2
p(T ? 1) t=2 t
? := {A? , ?? , ? 2 , ??1 , ??1 }, and ??1 and ??1 are the parameters of the gaussian vector x?1 .
1 ?
(4)
(5)
which require to address two problems: inverting a matrix which could be of infinite
dimension (e.g., if a gaussian kernel is used) and/or singular (equation (3)) and making a
division by the dimension of the feature space (p in equation (5)).
A general solution to circumvent these problems is to introduce an orthonormal basis U =
{u?1 , . . . , u?m } for the subspace Hx of H spanned by x?1:T . For instance, U can be obtained
by computing the set of principal components with non-zero eigenvalues of x ?1:T following
the procedure proposed in [6]. Once such a set of vectors is available, trying to find good
parameters for the model (2) is equivalent to finding an m-dimensional linear dynamical
model for the sequence z1:T = {z1 , . . . , zT } where zt is the vector of coordinates of x?t
with respect to U , i.e.:
i0
h
(6)
zt = hx?t , u?1 i hx?t , u?2 i ? ? ? hx?t , u?m i ?t = 1, . . . , T.
Given z1:T , the following linear dynamical model has to be considered:
zt+1 = Az zt + ?z + ? z
(7)
2
where ? z is again a gaussian noise vector of variance ? . Determining a basis of Hx allows
to learn the linear dynamical model (7). As it is based on the coordinates of the observed
vectors x?1 , . . . , x?T with respect to the basis, it is equivalent to learning (2). The parameters
are estimated thanks to equations (3), (4) and (5) where x?t is replaced with zt and p with
m.
For the sake of generalization ability, it might be useful to choose Az as simple as possible
[15]. To do this, we put a penalization on matrices Az having large values, by imposing
a prior distribution pA on Az defined as: pA (Az ) ? exp(? ?2 trace (A0z Az )), ? > 0. The
computation of the maximum a posteriori values for A, ? and ? 2 is very similar to (3), (4)
and (5) except that a few iterations of gradient ascent have to be done.
2.3
Back to the Input Space: the Preimage Problem
The problem Predicting the future observations with model (7) gives vectors in the feature space H while vectors from the input space Rd are needed. Given a vector z? in H,
finding a good vector x in Rd such that ?(x) is as close as possible to z? is known as the
preimage problem.
Mika et al. [6] propose to tackle this problem considering the optimization problem:
min k?(x) ? z? k2 .
x
This problem can be solved efficiently by gradient descent techniques for gaussian kernels.
Nevertheless, it may require several optimization phases with different starting points to be
ran when other kernels are used (e.g. polynomial kernels of some particular degree).
Here, we propose to use Support Vector Regression (SVR) to solve the preimage problem.
This avoids any local minimum problem and allows to benefit from the fact that we have to
work with vectors from the inner product space H. In addition, using this strategy, there is
no need to solve an optimization problem each time a preimage has to be computed.
SVR and Preimages Learning Given a sample dataset S = {(z1 , y1 ), . . . , (z` , y` )}
with pairs in Z ? R, the SVR algorithm assumes a structure on Z given by a kernel k z
and its associated mapping ? and feature space H (see [15]). It proceeds as follows (see
[14] and [15] for further details). Given a real positive value ?, the algorithm determines
a function f such that (a) it maps each zi to a value not having deviation larger than ?
from y , and (b) it is as flat as possible. This function computes its output as f (z) =
P` i ?
?
i=1 (?i ? ?i )kz (zi , z) + b where the vectors ? and ? are the solutions of the problem
max
?? ,?
s.t.
1
1
??0 (?? + ?) + y0 (?? ? ?) ? ((?? ? ?)KZ (?? ? ?) + (?? 0 ?? + ?0 ?))
2
C
0 ?
1 (? ? ?) = 0
?? ? 0, ? ? 0
The vectors involved in this program are of dimension `, with 1 = [1 ? ? ? 1]0 , 0 = [0 ? ? ? 0]0 ,
? = [? ? ? ? ?]0 , y = [y1 ? ? ? y` ]0 and KZ is the Gram matrix KZij = kz (zi , zj ). Here, ? is
the parameter of the Vapnik?s ?-insensitive quadratic loss function and C is a user-defined
constant penalizing data points which fail to meet the ?-deviation constraint.
In our case, we are interested in learning the mapping from Hx to Rd . In order to learn this
mapping, we construct d (the dimension of input space) SVR machines f1 , . . . , fd . Each fi
is trained to estimate the ith coordinate of the vector xt given the coordinates vector zt of
xt with respect to U . Denoting by zu the function which maps a vector x to its coordinate
vector z in U , the d machines provide the mapping ?:
? : H x ? Rd
0
x ?
7
[f1 (zu (x)) ? ? ? fd (zu (x))]
(8)
which can be used to estimate the preimages. Using ?, and noting that the program involved
by the SVR algorithm is convex, the estimation of the preimages does not have to deal with
any problem of local minima.
3
Numerical Results
In this section we present experiments on highly nonlinear time series prediction with
Kernel Dynamical Modeling. As the two series we consider are one dimensional we
use the following setup. Each series of length T is referred to as x1:T . In order to
model it, we introduce an embedding dimension d and a step size ? such that vectors
xt = (xt , xt?? , . . . , xt?(d?1)? )0 are considered. We compare the perfomances of KDM
to the performances achieved by an SVR for nonlinear time series analysis [7, 8], where
the mapping associating xt to xt+? is learned. The hyperparameters (kernel parameter
and SVR penalization constant C) are computed with respect to the one-step prediction
error measured on a test set, while the value of ? is set to 1e-4. Prediction quality is assessed on an independent validation sequence on which root mean squared error (RMSE)
is computed.
Two kinds of prediction capacity are evaluated. The first one is a one-step prediction when
after a prediction has been made, the true value is used to estimate the next time series
output. The second one is a multi-step or trajectory prediction, where the prediction made
by a model serves as a basis for the future predictions.
In order to make a prediction for a time t > T , we suppose that we are provided with the
vector xt?1 , which may have been observed or computed. We determine the coordinates
zt?1 of x?t?1 with respect to U and infer the value of zt by zt = Az zt?1 + ?z (see
equation (7)); ? is then used to recover an estimation of xt+1 (cf. equation (8)). In all our
experiments we have made the crude ?yet efficient? choice of the linear kernel for k z .
1.3
250
1.2
200
1.1
1
150
0.9
0.8
100
0.7
0.6
50
0.5
0.4
0
0
20
40
60
80
100
0
50
100
150
200
250
300
350
Figure 1: (left) 100 points of the Mackey-Glass time series M G17 , (right) the first 350
points of the Laser time series.
Table 1: Error (RMSE) of one-step and trajectory predictions for gaussian and polynomial
kernels for the time series M G17 . The regularizing values used for KDM are in subscript.
The best results are italicized.
Gaussian
3.1
Polynomial
Algo.
1S
100S
1S
100S
SVR
KDM0
KDM0.1
KDM1
KDM10
KDM100
0.0812
0.0864
0.0863
0.0859
0.0844
0.0899
0.2361
0.2906
0.2893
0.2871
0.2140
0.1733
0.1156
0.1112
0.1112
0.1117
0.1203
0.0970
0.2975
0.2775
0.2956
0.1964
0.1744
Mackey-Glass Time Series Prediction
The Mackey-Glass time series comes from the modeling of blood cells production evolution. It is a one-dimensional signal determined by
dx(t)
0.2x(t ? ? )
= ?0.1x(t) +
dt
1 + x(t ? ? )10
which, for values of ? greater than 16.8, shows some highly nonlinear chaotic behavior
(see Figure 1 left).
We focus on M G17 , for which ? = 17, and construct embedding vectors of size d = 6
and step size ? = 6. As xt is used to predict xt+? , the whole dataset can be divided
into six ?independent? datasets, the first one S1 containing x1+(d?1)? , the second one S2 ,
x2+(d?1)? , . . . , and the sixth one S6 , xd? . Learning is done as follows. The first 100
points of S1 are used to learning, while the first 100 points of S2 serve to choose the
hyperparameters. The prediction error is measured with respect to the points in the range
201 to 300 of S1 .
Table 1 reports the RMSE error obtained with gaussian and polynomial kernels, where 1S
and 100S respectively stand for one-step prediction and multi-step prediction over the 100
future observations.
SVR one-step prediction with gaussian kernel gives the best RMSE. None of the tested
regularizers allows KDM to perform better, even if the prediction error obtained with them
is never more than 10% away from SVR error.
Table 2: Error (RMSE) of one-step and trajectory predictions for gaussian and polynomial
kernels for the time series Laser. The regularizing values used for KDM are in subscript.
Gaussian
Polynomial
Algo.
1S
100S
1S
100S
SVR
KDM0
KDM0.1
KDM1
KDM10
KDM100
15.81
67.95
16.59
13.96
15.18
18.65
67.57
416.2
69.65
70.16
66.82
56.53
18.14
43.92
22.37
18.13
17.39
17.61
66.73
68.90
69.60
70.65
69.43
53.84
KDM trajectory prediction with gaussian kernel and regularizer ? = 100 leads to the best
error. It is around 17% lower than that of SVR multi-step prediction while KDM with no
regularizer gives the poorest prediction, emphasizing the importance of the regularizer.
Regarding one-step prediction with polynomial kernel, there is no significant difference
between the performance achieved by SVR and that of KDM, when regularizer is 0, 0.1, 1
or 10. For a regularizer ? = 100, KDM however leads to the best one-step prediction error,
around 16% lower than that obtained by SVR prediction.
The dash ?-? appearing in the first line of the table means that the trajectory prediction made
by the SVR with a polynomial kernel has failed to give finite predictions. On the contrary,
KDM never shows this kind of behavior. For a regularizer value of ? = 100, it even gives
the best trajectory prediction error.
3.2
Laser Time Series Prediction
The Laser time series is the dataset A from the Santa Fe competition. It is a univariate time
series from an experiment conducted in a physics laboratory (Figure 1 (right) represents
the first 350 points of the series). An embedding dimension d = 3 and a step size ? = 1
are used. The dataset is divided as follows. The first 100 points are used for training,
whereas the points in the range 201 to 300 provide a test set to select hyperparameters. The
validation error (RMSE) is evaluated on the points in the range 101 to 200.
Table 2 reports the validation errors obtained for the two kinds of prediction. The most
striking information provided by this table is the large error archieved by KDM with no
regularizer when a gaussian kernel is used. Looking at the other RMSE values corresponding to different regularizers, the importance of penalizing transition matrices with large
entries is underlined.
Besides, when the regularizer ? is appropriately chosen, we see that KDM with a gaussian
kernel can achieve very good predictions, for the one-step prediction and the multi-step
prediction as well. KDM one-step best prediction error is however not as far from SVR
one-step prediction (about 10% lower) than KDM multi-step is from its SVR counterpart
(around 16% lower).
When a polynomial kernel is used, we observe that KDM with no regularizer provides poor
results with regards to the one-step prediction error. Contrary to what occurs with the use
of a gaussian kernel, KDM with no regularization does not show bad multi-step prediction
ability. Looking at the other entries of this table once again shows that KDM can give
very good predictions when a well-suited regularizer is chosen. Hence, we notice that the
best multi-step prediction error of KDM is above 19% better than that obtained by SVR
multi-step prediction.
4
4.1
Discussion
Another Way of Choosing the Parameters
The introduction of a basis U allows to find the parameters of KDM without computing
any inversion of infinite dimensional matrices or division by the dimension of H. There
is, however a more elegant way to find these parameters when ? 2 is assumed to be known.
In this case, equation (5) needs not to be considered any longer. Considering the prior
0
pA (A? ) ? exp(? 2?? 2 trace (A? A? )), for a user defined ?, the maximum a posteriori for
A? is obtained as:
!
A?
=
T
X
0
x?t x?t?1 ?
t=2
?I +
T
X
T
T
1 X ?X ? 0
xt
xt?1
T ? 1 t=2
t=2
0
x?t?1 x?t?1
t=2
T
T
1 X ? X ? 0
?
xt?1
xt?1
T ? 1 t=2
t=2
!?1
.
Introducing the matrix X ? = [x?1 ? ? ? x?T ], the T -dimensional vectors f := [0 1 ? ? ? 1]0 ,
g := [1 ? ? ? 1 0], the T ? T matrix P = (Pij ) = (?i,j+1 ) defining J = P ? f g/(T ? 1)0
and M = diag (g) ? gg0 /(T ? 1), A? can be rewritten as
0
0 ?1
A? =
X ? JX ?
?I + X ? M X ?
1 ?
1
1
0
=
X J I ? KM (I + M KM )?1 M X ?
?
?
?
thanks to the Sherman-Woodbury formula, K being the Gram matrix associated to x ?1:T . It
is thus possible to directly determine the matrix A? when ? 2 is known, the same holding
for ?? since equation (5) remains unchanged.
4.2
Link to Kalman Filtering
The usual way to recover a noisy nonlinear signal is to use the Extended Kalman Filter
(EKF) or the Unscented Kalman Filter (UKF) [4]. The use of these algorithms involves
two steps. First, the clean dynamics, as given by h in equation (1) is learned by a regressor,
e.g., a multilayer perceptron. Given a noisy time series from the same driving process h,
EKF and UKF then process that series by respectively a first-order linearization of h and
an efficient ?sampling? method to determine the clean signal. Apart from the latter essential
approximations done by these algorithms, the core of EKF and UKF resembles that of
classical Kalman filtering (and smoothing).
Regarding the performances of KDM to learn a complex dynamics, it could be directly used
to model the process h. In addition, its matricial formulation is suitable to the traditional
matrices computations involved by the filtering task (see [5, 11] for details). Hence, a link
between KDM and Kalman filtering has been the purpose of [9, 10] where a nonlinear
Kalman filter based on the use of kernels is proposed: the ability of the proposed model to
address the modeling of nonlinear dynamics is demonstrated, while the classical procedures
(even the EM algorithm) associated to linear dynamical systems remain valid.
5
Conclusion and Future Work
Three main results are presented: first, we introduce KDM, a kernel extension of linear
dynamical models and show how the kernel trick allows to learn a linear model in a feature
space associated to a kernel. Second, an original and efficient solution based on learning
has been applied for the preimage problem. Third, Kernel Dynamical Model can be linked
to the Kalman filter model with a hidden state process living in the feature space.
In the framework of time series prediction, KDM proves to work very well and to compete
with the best time series predictors particularly on long time range prediction.
To conclude, this work can lead to several future directions. All classic tasks involving a
dynamic setting such as filtering/predicting (e.g., tracking) and smoothing (e.g., time series
denoising) can be tackled by our approach and have to be tested. As pointed out by [9, 10],
the kernel approach can also be applied to linear dynamical models with hidden states to
provide a kernelized version of the Kalman filter, particularly allowing the implementation
of an exact nonlinear EM procedure (involving closed form equations as the method proposed by [3]). Besides, the use of kernel opens the door to dealing with structured data,
making KDM a very attractive tool in many areas such as bioinformatics, texts and video
application. Lastly, from the theoretical point of view, a very interesting issue is that of the
actual noise corresponding to a gaussian noise in a feature space.
References
[1] B. Boser, I. Guyon, and V. Vapnik. A Training Algorithm for Optimal Margin Classifiers. In
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[2] G. Dorffner. Neural networks for time series processing. Neural Network World, 6(4):447?468,
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[3] Z. Ghahramani and S. Roweis. Learning nonlinear dynamical systems using an em algorithm. In
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[4] S. Julier and J. Uhlmann. A New Extension of the Kalman Filter to Nonlinear Systems. In Int.
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[5] R. E. Kalman. A New Approach to Linear Filtering and Prediction Problems. Transactions of
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[6] S. Mika, B. Sch?olkopf, A. J. Smola, K.-R. M?uller, M. Scholz, and G. R?atsch. Kernel PCA and
De-Noising in Feature Spaces. In NIPS. MIT Press, 1999.
[7] S. Mukherjee, E. Osuna, and F. Girosi. Nonlinear prediction of chaotic time series using support
vector machines. In Proc. of IEEE NNSP?97, 1997.
[8] K. M?uller, A. Smola, G. R?atsch, B. Sch?olkopf, J. Kohlmorgen, and V. Vapnik. Predicting
Time Series with Support Vector Machines. In W. Gerstner, A. Germond, M. Hasler, and J.-D.
Nicoud, editors, Artificial Neural Networks - ICANN?97, pages 999?1004. Springer, 1997.
[9] L. Ralaivola. Mod?elisation et apprentissage de concepts et de syst`emes dynamiques. PhD thesis,
Universit?
e Paris 6, France, 2003.
[10] L. Ralaivola and F. d?Alch?
e-Buc. Filtrage de Kalman non lin?
eaire a` l?aide de noyaux. In Actes
du 19eme Symposium GRETSI sur le traitement du signal et des images, 2003.
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[12] S. Roweis and Z. Ghahramani. A unifying review of linear Gaussian models. Neural Computation, 11(2):305?345, 1997.
[13] B. Sch?olkopf and A. J. Smola. Learning with Kernels, Support Vector Machines, Regularization, Optimization and Beyond. MIT University Press, 2002.
[14] A. Smola and B. Sch?olkopf. A Tutorial on Support Vector Regression. Technical Report NC2TR-1998-030, NeuroCOLT2, 1998.
[15] V. Vapnik. Statistical Learning Theory. John Wiley and Sons, inc., 1998.
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1,669 | 2,517 | Extreme Components Analysis
Max Welling
Department of Computer Science
University of Toronto
10 King?s College Road
Toronto, M5S 3G5 Canada
[email protected]
Felix Agakov, Christopher K. I. Williams
Institute for Adaptive and Neural Computation
School of Informatics
University of Edinburgh
5 Forrest Hill, Edinburgh EH1 2QL, UK
{ckiw,felixa}@inf.ed.ac.uk
Abstract
Principal components analysis (PCA) is one of the most widely used
techniques in machine learning and data mining. Minor components
analysis (MCA) is less well known, but can also play an important role
in the presence of constraints on the data distribution. In this paper we
present a probabilistic model for ?extreme components analysis? (XCA)
which at the maximum likelihood solution extracts an optimal combination of principal and minor components. For a given number of components, the log-likelihood of the XCA model is guaranteed to be larger or
equal than that of the probabilistic models for PCA and MCA. We describe an efficient algorithm to solve for the globally optimal solution.
For log-convex spectra we prove that the solution consists of principal
components only, while for log-concave spectra the solution consists of
minor components. In general, the solution admits a combination of both.
In experiments we explore the properties of XCA on some synthetic and
real-world datasets.
1 Introduction
The simplest and most widely employed technique to reduce the dimensionality of a data
distribution is to linearly project it onto the subspace of highest variation (principal components analysis or PCA). This guarantees that the reconstruction error of the data, measured
with L2 -norm, is minimized. For some data distributions however, it is not the directions of
large variation that are most distinctive, but the directions of very small variation, i.e. constrained directions. In this paper we argue that in reducing the dimensionality of the data,
we may want to preserve these constrained directions alongside some of the directions of
large variability.
The proposed method, termed ?extreme components analysis? or XCA, holds the middle
ground between PCA and MCA (minor components analysis?the method that projects on
directions of low variability). The objective that determines the optimal combination of
principal and minor components derives from the probabilistic formulation of XCA, which
neatly generalizes the probabilistic models for PCA and MCA. For a fixed number of components, the XCA model will always assign higher probability to the (training) data than
PCA or MCA, and as such be more efficient in encoding the data. We propose a very
simple and efficient algorithm to extract the optimal combination of principal and minor
components and prove some results relating the shape of the log-spectrum to this solution.
The XCA model is inspired by Hinton?s ?product of experts? (PoE) model [1]. In a
PoE, linear combinations of an input vector are penalized according to their negative logprobability and act as constraints. Thus, configurations of high probability have most of
their constraints approximately satisfied. As we will see, the same is true for the XCA
model which can therefore be considered as an under-complete product of Gaussians
(PoG).
2 Variation vs. Constraint: PCA vs. MCA
Consider a plane embedded in 3 dimensions that cuts through the origin. There are 2
distinct ways to mathematically describe points in that plane:
x = Ay
? y ? R2 ,
or
? x ? R3
s.t. wT x = 0
(1)
where A is a 3 ? 2 matrix, the columns of which form a basis in the plane, and w is a vector
orthogonal to the plane. In the first description we parameterize the modes of variation,
while in the second we parameterize the direction of no variation or the direction in which
the points are constrained. Note that we only need 3 real parameters to describe a plane in
terms of its constraint versus 6 parameters to describe it in terms of its modes of variation.
More generally, if we want to describe a d-dimensional subspace in D dimensions we may
use D ? d constraint directions or d subspace directions.
Next consider the stochastic version of the above problem: find an accurate description
of an approximately d-dimensional data-cloud in D dimensions. The solution that probabilistic PCA (PPCA) [3, 4] provides is to model those d directions using unit vectors ai
(organized as columns of a matrix A) while adding isotropic Gaussian noise in all directions,
x = Ay + n
y ? N [0, Id ]
n ? N [0, ?02 ID ]
(2)
The probability density of x is Gaussian with covariance
CPCA = hxxT i = ?02 ID + AAT .
(3)
In [4] it was shown that at the maximum likelihood solution the columns
of A are given
p
2
by the first d principal components of the data with length ||ai || = ?i ? ?02 where ?i2 is
the i0 th largest eigenvalue of the sample covariance matrix and ?02 is equal to the average
variance in the directions orthogonal to the hyperplane.
Alternatively, one may describe the data as D ? d approximately satisfied constraints, embedded in a high variance background model. The noisy version of the constraint wT x = 0
is given by z = wT x where z ? N [0, 1]. The variance of the constrained direction,
1/||w||2 , should be smaller than that of the background model. By multiplying D ? d of
these ?Gaussian pancake? models [6] a probabilistic model for MCA results with inverse
covariance given by,
ID
?1
(4)
CMCA
= 2 + WTW
?0
where wT form the rows of W . It was shown that at the maximum likelihood solution the
rows of W are given by the first D ? d minor components of the data with length ||wi || =
p
1/?i2 ? 1/?02 where ?i2 is the i0 th smallest eigenvalue of the sample covariance matrix
and ?02 is equal to the average variance in the directions orthogonal to the hyperplane. Thus,
while PPCA explicitly models the directions of large variability, PMCA explicitly models
the directions of small variability.
3 Extreme Components Analysis (XCA)
Probabilistic PCA can be interpreted as a low variance data cloud which has been stretched
out in certain directions. Probabilistic MCA on the other hand can be thought of as a
large variance data cloud which has been pushed inward in certain directions. Given the
Gaussian assumption, the approximation that we make is due to the fact that we replace the
variances in the remaining directions by their average. Intuitively, better approximations
may be obtained by identifying the set of eigenvalues which, when averaged, induces the
smallest error. The appropriate model, to be discussed below, will both have elongated and
contracted directions in its equiprobable contours, resulting in a mix of principal and minor
components.
3.1 A Probabilistic Model for XCA
The problem can be approached by either starting at the PPCA or PMCA model. The
restricting aspect of the PPCA model is that the noise n is added in all directions in input
space. Since adding random variables always results in increased variance, the directions
modelled by the vectors ai must necessarily have larger variance than the noise directions,
resulting in principal components. In order to remove that constraint we need to add the
noise only in the directions orthogonal to the ai ?s. This leads to the following ?causal
generative model? model1 for XCA,
?
x = Ay + PA
n
n ? N [0, ?02 ID ]
y ? N [0, Id ]
(5)
?
where PA
= ID ?A(AT A)?1 AT is the projection operator on the orthogonal complement
of the space spanned by the columns of A. The covariance of this model is found to be
?
CXCA = ?02 PA
+ AAT .
(6)
Approaching the problem starting at the PMCA model we start with d components {wi }
(organized as rows in W ) and add isotropic noise to the remaining directions,
z1 = W x
z1 ? N [0, Id ]
z2 ? N [0, ?02 I(D?d) ]
z2 = V x
(7)
where the rows of V form an orthonormal basis in the orthogonal complement of the space
spanned by {wi }. Importantly, we will not impose any constraints on the norms of {wi }
or ?0 , i.e. the components are allowed to model directions of large or small variance. To
derive the PDF we note that ({z1i }, {z2i }) are independent random variables implying that
P (z1 , z2 ) is a product of marginal distributions. This is then p
converted to P (x) by taking
into account the Jacobian of the transformation J(z1 ,z2 )?x = det(W W T ). The result is
that x has a Gaussian distribution with with inverse covariance,
?1
CXCA
=
1 ?
P + WTW
?02 W
(8)
?
where PW
= ID ? W T (W W T )?1 W is the projection operator on the orthogonal com2(d?D)
?1
plement of W . Also, det(CXCA
) = det(W W T )?0
.
def
It is now not hard to verify that by identifying A = W # = W T (W W T )?1 (the pseudoinverse of W ) the two models defined through eqns. 6 and 8 are indeed identical. Thus,
by slightly changing the noise model, both PPCA and PMCA result in XCA (i.e. compare
eqns.3,4,6,8).
1
Note however that the semantics of a two-layer directed graphical model is problematic since
p(x|y) is improper.
3.2 Maximum Likelihood Solution
For a centered (zero mean) dataset {x} of size N the log-likelihood is given by,
? ?
?
1
N ? ?1
ND
N
N (D ? d)
T
log(2?)+ log det(W W )+
log
? tr CXCA
L=?
S (9)
2
2
2
2
?0
2
PN
1
T
D?D
is the covariance of the data. To solve for the stationwhere S = N i=1 xi xi ? R
ary points of L we take derivatives w.r.t W T and 1/?02 and equate them to zero. Firstly, for
W we find the following equation,
1 ?
SW # = 0.
W # ? SW T + 2 PW
(10)
?0
Let W T = U ?RT be the singular value decomposition (SVD) of W T , so that U ? RD?d
forms an incomplete orthonormal basis, ? ? Rd?d is a full-rank diagonal matrix, and
R ? Rd?d is a rigid rotation factor. Inserting this into eqn. 10 we find,
1
U ??1 RT ? SU ?RT + 2 (ID ? U U T )SU ??1 RT = 0.
(11)
?0
Next we note that the projections of this equation on the space spanned by W and its
orthogonal complement should hold independently. Thus, multiplying equation 11 on the
?
left by either PW or PW
, and multiplying it on the right by R??1 , we obtain the following
two equations,
U ??2 = U U T SU,
?
?
?
?
??2
??2
T
= U U SU Id ? 2 .
SU Id ? 2
?0
?0
(12)
(13)
Inserting eqn. 12 into eqn. 13 and right multiplying with (Id ? ??2 /?02 )?1 we find the
eigenvalue equation2 ,
SU = U ??2 .
(14)
Inserting this solution back into eqn. 12 we note that it is satisfied as well. We thus conclude
that U is given by the eigenvectors of the sample covariance matrix S, while the elements
of the (diagonal) matrix ? are given by ?i = 1/?i with ?i2 the eigenvalues of S (i.e. the
spectrum).
Finally, taking derivatives w.r.t. 1/?02 we find,
? ? ?
?
1 ?
1 X 2
1
tr PW
S =
tr(S) ? tr(U ??2 U T ) =
?i
?02 =
D?d
D?d
D?d
(15)
i?G
where G is the set of all eigenvalues of S which are not represented in ??2 . The above
equation expresses the fact that these eigenvalues are being approximated through their
average ?02 .
Inserting the solutions 14 and 15 back into the log-likelihood (eqn. 9) we find,
?
!
X
1
ND
NX
N
(D
?
d)
log(2?e) ?
log(?i2 ) ?
log
?i2
L=?
2
2
2
D?d
i?C
(16)
i?G
where C is the set of retained eigenvalues. The log-likelihood has now been reduced to a
function of the discrete set of eigenvalues {?i2 } of S.
2
As we will see later, the left-out eigenvalues have to be contiguous in the spectrum, implying
that the matrix (Id ? ??2 /?02 )?1 can only be singular if there is a retained eigenvalue that is equal
to all left-out eigenvalues. This is clearly an uninteresting case, since the likelihood will not decrease
if we leave this component out as well.
3.3 An Algorithm for XCA
To optimize 16 efficiently we first note that the sum of the eigenvalues {?i2 } is constant:
P
2
i?C?G ?i = tr(S). We may use this to rewrite L in terms of the retained eigenvalues
only. We define the following auxiliary cost to be minimized which is proportional to ?L
up to irrelevant constants,
X
X
K=
log ?i2 + (D ? d) log(tr(S) ?
?i2 ).
(17)
i?C
i?C
Next we recall an important result that was proved in [4]: the minimizing solution has
eigenvalues ?i2 , i ? G which are contiguous in the (ordered) spectrum, i.e. the eigenvalues which are averaged form a ?gap? in the spectrum. With this result, the search for the
optimal solution has been reduced from exponential to linear in the number of retained dimensions d. Thus we obtain the following algorithm for determining the optimal d extreme
components: (1) Compute the first d principal components and the first d minor components, (2) for all d possible positions of the ?gap? compute the cost K in eqn. 17, and (3)
select the solution that minimizes K.
It is interesting to note that the same equations for the log-likelihood (L, eqn.16) and cost
(K, eqn.17) appear in the analysis of PPCA [4] and PMCA [6]. The only difference being
that certain constraints forcing the solution to contain only principal or minor components
are absent in eqn. 16. For XCA, this opens the possibility for mixed solutions with both
principal and minor components. From the above observation we may conclude that the
optimal ML solution for XCA will always have larger log-likelihood on the training data
then the optimal ML solutions for PPCA and PMCA. Moreover, when XCA contains only
principal (or minor) components, it must have equal likelihood on the training data as PPCA
(or PMCA). In this sense XCA is the natural extension of PPCA and PMCA.
4 Properties of the Optimal ML Solution
We will now try to provide some insight into the nature of the the optimal ML solutions.
First we note that the objective K is shifted by a constant if we multiply all variances by
a factor ?i2 ? ??i2 , which leaves its minima invariant. In other words, the objective is
only sensitive to changing ratios between eigenvalues. This property suggests to use the
logarithm of the eigenvalues of S as the natural quantities since multiplying all eigenvalues
with a constant results in a vertical shift of the log-spectrum. Consequently, the properties
of the optimal solution only depend on the shape of the log-spectrum. In appendix A we
prove the following characterization of the optimal solution,
Theorem 1
? A log-linear spectrum has no preference for principal or minor components.
? The extreme components of log-convex spectra are principal components.
? The extreme components of log-concave spectra are minor components.
Although a log-linear spectrum with arbitrary slope has no preference for principal or minor
components, the slope does have an impact on the accuracy of the approximation because
the variances in the gap are approximated by their average value. A spectrum that can be
exactly modelled by PPCA with sufficient retained directions is one which has a pedestal,
i.e. where the eigenvalues become constant beyond some value. Similarly PMCA can
model exactly a spectrum which is constant and then drops off while XCA can model
exactly a spectrum with a constant section at some arbitrary position. Some interesting
examples of spectra can be obtained from the Fourier (spectral) representation of stationary
Gaussian processes. Processes with power-law spectra S(?) ? ? ?? are log convex. An
example of a spectrum which is log linear is obtained from the RBF covariance function
Table 1: Percent classification error of noisy sinusoids as a function of g = D ? d.
g
?XCA
?M CA
?P CA
2
1.88
2.37
1.88
3
1.91
3.10
2.50
4
2.35
4.64
12.21
5
1.88
4.06
14.57
6
2.37
2.37
19.37
7
3.27
3.27
32.99
8
28.24
28.24
30.14
with a Gaussian weight function, [7]. The RBF covariance function on the circle will give
2
rise to eigenvalues ?i ? e??i , i.e. a log-concave spectrum.
Both PCA and MCA share the convenient property that a solution with d components is
contained in the solution with d + 1 components. This is not the case for XCA: the solution
with d + 1 components may look totally different than the solution with d components (see
inset in Figure 1c), in fact they may not even share a single component!
5 Experiments
Small Sample Effects
When the number of data cases is small relative to the dimensionality of the problem, the
log-spectrum tends to bend down on the MC side producing ?spurious? minor components
in the XCA solution. Minor components that result from finite sample effects, i.e. that do
not exist in the infinite data limit, have an adverse effect on generalization performance.
This is shown in Figure 1a for the ?Frey-Faces? dataset, where we plot the log-likelihood
for (centered) training and test data for both PCA and XCA. This dataset contains 1965
images of size 20 ? 28, of which we used 1000 for training and 965 for testing. Since
the number of cases is small compared to the number of dimensions, both PCA and XCA
show a tendency to overfit. Note that at the point that minor components appear in the XCA
solution (d = 92) the log-likelihood of the training data improves relative to PCA, while
the log-likelihood of the test data suffers.
Sinusoids in noise
Pp
Consider a sum of p sinusoids Y (t) =
i=1 Ai cos(?i t + ?i ) sampled at D equallyspaced time points. If each ?i is random in (0, 2?) then the covariance hY (t)Y (t0 )i =
P
p
0
2
i=1 Pi cos ?i (t ? t ) where Pi = Ai /2. This signal defines a 2p-dimensional linear
manifold in the D-dimensional space (see [2] ?12.5). By adding white noise to this signal
we obtain a non-singular covariance matrix. Now imagine we have two such signals, each
described by p different powers and frequencies. Instead of using the exact covariance
matrix for each we approximate the covariance matrix using either XCA, PMCA or PPCA.
We then compare the accuracy of a classification task using either the exact covariance
matrix, or the approximations. (Note that although the covariance can be calculated exactly
the generating process is not in fact a Gaussian process.) By adjusting p, the powers and
the frequencies of the two signals, a variety of results can be obtained. We set D = 9 and
p = 4. The first signal had P = (1.5, 2.5, 3, 2.5) and ? = (1.9, 3.5, 4.5, 5), and the second
P = (3, 2, 1.8, 1) and ? = (1.7, 2.9, 3.3, 5.3). The variance of the background noise was
0.5. Table 1 demonstrates error rates on 10000 test cases obtained for XCA, PMCA and
PPCA using g = D ? d approximated components. For all values of g the error rate for
XCA is ? than that for PPCA and PMCA. For comparison, the optimal Gaussian classifier
has an error rate of 1.87%. For g = 2 the XCA solution for both classes is PPCA, and for
g = 6, 7, 8 it is PMCA; in between both classes have true XCA solutions. MCA behaviour
is observed if ?02 is low.
2-D Positions of Face Features
671 cases were extracted from a dataset containing 2-D coordinates of 6 features on frontal
Frey?Faces
2
10
?80
?82
?82
0
?84
?84
10
10
Spectrum
?88
?6
1000
?86
?86
?4
0
200
400
600
eigendirection
900
6
10
?90
4
10
variance
?92
800
?94
?88
Extreme Components
12
?90
10
?92
nr. PCs / MCs
1100
?2
10
log?probability
variance
10
1200
log?probability
1300
log?probability
Test Data
Training Data
?80
Spectrum
1400
?94
2
10
700
?96
?96
600
?98
10
500
0
100
200
300
400
nr. retained dimensions
500
(a)
?100
0
?98
0
0
5
5
10
nr. retained dimensions
(b)
10
eigendirection
?100
0
6
4
2
0
0
15
15
8
5
5
10
nr. retained dimensions
10
nr. retained dimensions
15
(c)
Figure 1: (a) Log-likelihood of the ?Frey-faces? training data (top curves) and test data (bottom
curves) for PCA (dashed lines) and XCA (solid lines) as a function of the number of components.
Inset: log-spectrum of training data.(b) Log-likelihood of training data for PCA (dash), MCA (dashdot) and XCA (solid) as a function of the number of components. Inset: log-spectrum of training
data. (c) Log-likelihood of test data. Inset: number of PCs (dash) versus number of MCs (dash-dot)
as a function of the number of components.
faces3 . To obtain a translation and orientation invariant representation, we computed the 15
squared (Euclidean) distances between the features and removed their mean. In Figures 1b
and 1c we show the log-likelihood for PCA, MCA and XCA of 335 training cases and 336
test cases respectively. Clearly, XCA is superior even on the test data. In the inset of Figure
1c we depict the number of PCs and MCs in the XCA solution as we vary the number of
retained dimensions. Note the irregular behavior when the number of components is large.
6 Discussion
In this paper we have proposed XCA as the natural generalization of PCA and MCA for the
purpose of dimensionality reduction. It is however also possible to consider a model with
non-Gaussian components. In [5] the components were distributed according to a Student-t
distribution resulting in a probabilistic model for undercomplete independent components
analysis (UICA).
There are quite a few interesting questions that remain unanswered in this paper. For instance, although we have shown how to efficiently find the global maximum of the loglikelihood, we haven?t identified the properties of the other stationary points. Unlike PPCA
we expect many local maxima to be present. Also, can we formulate a Bayesian version of
XCA where we predict the number and nature of the components supported by the data?
Can we correct the systematic under-estimation of MCs in the presence of relatively few
data cases? There are a number of extensions of the XCA model worth exploring: XCA
with multiple noise models (i.e. multiple gaps in the spectrum), mixtures of XCA and so
on.
A Proof of Theorem 1
Using the fact that the sum and the product of the eigenvalues are constant we can rewrite
the cost eqn.17 (up to irrelevant constants) in terms of the left-out eigenvalues of the spectrum only. We will also use the fact that the left-out eigenvalues are contiguous in the
3
The dataset was obtained by M. Weber at the computational vision lab at Caltech and contains
the 2-D coordinates of 6 features (eyes, nose, 3 mouth features) of unregistered frontal face images.
def
spectrum, and form a ?gap? of size g = D ? d,
??
?
i X
+g?1
i? X
+g?1
fi ?
?
C = g log
e
?
fi
i=i?
(18)
i=i?
where fi are the log-eigenvalues and i? is the location of the left hand side of the gap. We
are interested in the change of this cost ?C if we shift it one place to the right (or the left).
This can be expressed as
!
?
efi? +g ? efi?
? (f (i? + g) ? f (i? )) .
(19)
?C = g log 1 + Pi? +g?1
fi
e
?
i=i
Pg?1
Inserting a log-linear spectrum: fi = b + a ? i with a < 0 and using the result i=0 ea?i =
(eag ? 1)/(ea ? 1) we find that the change in C vanishes for all log-linear spectra. This
establishes the first claim. For the more general case we define corrections ci to the loglinear spectrum that runs through the points fi? and fi? +g , i.e. fi = b + a ? i + ci . First
consider the case of a convex spectrum between i? and i? +g, which implies that all ci < 0.
Inserting this into 19 we find after some algebra
!
?
eag ? 1
?C = g log 1 + Pg?1 a?i0 +c 0 ? ? ag.
(20)
[i +i ]
i0 =0 e
Because all ci < 0, the first term must be smaller (more negative) than the corresponding
term in the linear case implying that ?C < 0 (the second term is unchanged w.r.t the
linear case). Thus, if the entire spectrum is log-convex the gap will be located on the right,
resulting in PCs. A similar argument shows that for log-concave spectra the solutions
consist of MCs only. In general log-spectra may have convex and concave pieces. The cost
18 is minimized when some of the ci are positive and some negative in such a way that,
Pg?1 a?i0
Pg?1 a?i0 +c 0 ?
[i +i ] ?
Note that due to the exponent in this sum, positive ci
i0 =0 e
i0 =0 e
have a stronger effect than negative ci .
Acknowledgements
We?d like to thank the following people for their invaluable input into this paper: Geoff Hinton, Sam
Roweis, Yee Whye Teh, David MacKay and Carl Rasmussen. We are also very grateful to Pietro
Perona and Anelia Angelova for providing the ?feature position? dataset used in this paper.
References
[1] G.E. Hinton. Products of experts. In Proceedings of the International Conference on Artificial
Neural Networks, volume 1, pages 1?6, 1999.
[2] J.G. Proakis and D.G. Manolakis. Digital Signal Processing: Principles, Algorithms and Applications. Macmillan, 1992.
[3] S.T. Roweis. Em algorithms for pca and spca. In Advances in Neural Information Processing
Systems, volume 10, pages 626?632, 1997.
[4] M.E. Tipping and C.M. Bishop. Probabilistic principal component analysis. Journal of the Royal
Statistical Society, Series B, 21(3):611?622, 1999.
[5] M. Welling, R.S. Zemel, and G.E. Hinton. A tractable probabilistic model for projection pursuit.
In Proceedings of the Conference on Uncertainty in Artificial Intelligence, 2003. accepted for
publication.
[6] C.K.I. Williams and F.V. Agakov. Products of gaussians and probabilistic minor components
analysis. Neural Computation, 14(5):1169?1182, 2002.
[7] H. Zhu, C. K. I. Williams, R. J. Rohwer, and M. Morciniec. Gaussian regression and optimal finite dimensional linear models. In C. M. Bishop, editor, Neural Networks and Machine Learning.
Springer-Verlag, Berlin, 1998.
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1,670 | 2,518 | AUC Optimization vs. Error Rate Minimization
Corinna Cortes? and Mehryar Mohri
AT&T Labs ? Research
180 Park Avenue, Florham Park, NJ 07932, USA
{corinna, mohri}@research.att.com
Abstract
The area under an ROC curve (AUC) is a criterion used in many applications to measure the quality of a classification algorithm. However,
the objective function optimized in most of these algorithms is the error
rate and not the AUC value. We give a detailed statistical analysis of the
relationship between the AUC and the error rate, including the first exact
expression of the expected value and the variance of the AUC for a fixed
error rate. Our results show that the average AUC is monotonically increasing as a function of the classification accuracy, but that the standard
deviation for uneven distributions and higher error rates is noticeable.
Thus, algorithms designed to minimize the error rate may not lead to
the best possible AUC values. We show that, under certain conditions,
the global function optimized by the RankBoost algorithm is exactly the
AUC. We report the results of our experiments with RankBoost in several
datasets demonstrating the benefits of an algorithm specifically designed
to globally optimize the AUC over other existing algorithms optimizing
an approximation of the AUC or only locally optimizing the AUC.
1 Motivation
In many applications, the overall classification error rate is not the most pertinent performance measure, criteria such as ordering or ranking seem more appropriate. Consider for
example the list of relevant documents returned by a search engine for a specific query.
That list may contain several thousand documents, but, in practice, only the top fifty or so
are examined by the user. Thus, a search engine?s ranking of the documents is more critical
than the accuracy of its classification of all documents as relevant or not. More generally, for a binary classifier assigning a real-valued score to each object, a better correlation
between output scores and the probability of correct classification is highly desirable.
A natural criterion or summary statistic often used to measure the ranking quality of a classifier is the area under an ROC curve (AUC) [8].1 However, the objective function optimized by most classification algorithms is the error rate and not the AUC. Recently, several
algorithms have been proposed for maximizing the AUC value locally [4] or maximizing
some approximations of the global AUC value [9, 15], but, in general, these algorithms do
not obtain AUC values significantly better than those obtained by an algorithm designed to
minimize the error rates. Thus, it is important to determine the relationship between the
AUC values and the error rate.
?
This author?s new address is: Google Labs, 1440 Broadway, New York, NY 10018,
[email protected].
1
The AUC value is equivalent to the Wilcoxon-Mann-Whitney statistic [8] and closely related to
the Gini index [1]. It has been re-invented under the name of L-measure by [11], as already pointed
out by [2], and slightly modified under the name of Linear Ranking by [13, 14].
True positive rate
ROC Curve. AUC=0.718
(1,1)
True positive rate =
(0,0)
False positive rate =
False positive rate
correctly classified positive
total positive
incorrectly classified negative
total negative
Figure 1: An example of ROC curve. The line connecting (0, 0) and (1, 1), corresponding to random
classification, is drawn for reference. The true positive (negative) rate is sometimes referred to as the
sensitivity (resp. specificity) in this context.
In the following sections, we give a detailed statistical analysis of the relationship between
the AUC and the error rate, including the first exact expression of the expected value and
the variance of the AUC for a fixed error rate.2 We show that, under certain conditions, the
global function optimized by the RankBoost algorithm is exactly the AUC. We report the
results of our experiments with RankBoost in several datasets and demonstrate the benefits
of an algorithm specifically designed to globally optimize the AUC over other existing
algorithms optimizing an approximation of the AUC or only locally optimizing the AUC.
2 Definition and properties of the AUC
The Receiver Operating Characteristics (ROC) curves were originally developed in signal
detection theory [3] in connection with radio signals, and have been used since then in many
other applications, in particular for medical decision-making. Over the last few years, they
have found increased interest in the machine learning and data mining communities for
model evaluation and selection [12, 10, 4, 9, 15, 2].
The ROC curve for a binary classification problem plots the true positive rate as a function
of the false positive rate. The points of the curve are obtained by sweeping the classification threshold from the most positive classification value to the most negative. For a fully
random classification, the ROC curve is a straight line connecting the origin to (1, 1). Any
improvement over random classification results in an ROC curve at least partially above
this straight line. Fig. (1) shows an example of ROC curve. The AUC is defined as the area
under the ROC curve and is closely related to the ranking quality of the classification as
shown more formally by Lemma 1 below.
Consider a binary classification task with m positive examples and n negative examples.
We will assume that a classifier outputs a strictly ordered list for these examples and will
denote by 1X the indicator function of a set X.
Lemma 1 ([8]) Let c be a fixed classifier. Let x1 , . . . , xm be the output of c on the positive
examples and y1 , . . . , yn its output on the negative examples. Then, the AUC, A, associated
to c is given by:
Pm Pn
i=1
j=1 1xi >yj
(1)
A=
mn
that is the value of the Wilcoxon-Mann-Whitney statistic [8].
Proof. The proof is based on the observation that the AUC value is exactly the probability
P (X > Y ) where X is the random variable corresponding to the distribution of the outputs for the positive examples and Y the one corresponding to the negative examples [7].
The Wilcoxon-Mann-Whitney statistic is clearly the expression of that probability in the
discrete case, which proves the lemma [8].
Thus, the AUC can be viewed as a measure based on pairwise comparisons between classifications of the two classes. With a perfect ranking, all positive examples are ranked higher
than the negative ones and A = 1. Any deviation from this ranking decreases the AUC.
2
An attempt in that direction was made by [15], but, unfortunately, the authors? analysis and the
result are both wrong.
Threshold ?
k ? x Positive examples
x Negative examples
n ? x Negative examples m ? (k ? x) Positive examples
Figure 2: For a fixed number of errors k, there may be x, 0 ? x ? k, false negative examples.
3 The Expected Value of the AUC
In this section, we compute exactly the expected value of the AUC over all classifications
with a fixed number of errors and compare that to the error rate.
Different classifiers may have the same error rate but different AUC values. Indeed, for a
given classification threshold ?, an arbitrary reordering of the examples with outputs more
than ? clearly does not affect the error rate but leads to different AUC values. Similarly,
one may reorder the examples with output less than ? without changing the error rate.
Assume that the number of errors k is fixed. We wish to compute the average value of the
AUC over all classifications with k errors. Our model is based on the simple assumption
that all classifications or rankings with k errors are equiprobable. One could perhaps argue
that errors are not necessarily evenly distributed, e.g., examples with very high or very low
ranks are less likely to be errors, but we cannot justify such biases in general.
For a given classification, there may be x, 0 ? x ? k, false positive examples. Since the
number of errors is fixed, there are k ? x false negative examples. Figure 3 shows the corresponding configuration. The two regions of examples with classification outputs above
and below the threshold are separated by a vertical line. For a given x, the computation of
the AUC, A, as given by Eq. (1) can be divided into the following three parts:
A1 + A2 + A3
A=
,
with
(2)
mn
A1 = the sum over all pairs (xi , yj ) with xi and yj in distinct regions;
A2 = the sum over all pairs (xi , yj ) with xi and yj in the region above the threshold;
A3 = the sum over all pairs (xi , yj ) with xi and yj in the region below the threshold.
The first term, A1 , is easy to compute. Since there are (m ? (k ? x)) positive examples
above the threshold and n ? x negative examples below the threshold, A1 is given by:
A1 = (m ? (k ? x))(n ? x)
(3)
To compute A2 , we can assign to each negative example above the threshold a position
based on its classification rank. Let position one be the first position above the threshold
and let ?1 < . . . < ?x denote the positions in increasing order of the x negative examples
in the region above the threshold. The total number of examples classified as positive is
N = m ? (k ? x) + x. Thus, by definition of A2 ,
x
X
A2 =
(N ? ?i ) ? (x ? i)
(4)
i=1
where the first term N ? ?i represents the number of examples ranked higher than the ith
example and the second term x ? i discounts the number of negative examples incorrectly
ranked higher than the ith example. Similarly, let ?01 < . . . < ?0k?x denote the positions of
the k ? x positive examples below the threshold, counting positions in reverse by starting
from the threshold. Then, A3 is given by:
0
A3 =
x
X
(N 0 ? ?0j ) ? (x0 ? j)
(5)
j=1
with N 0 = n ? x + (k ? x) and x0 = k ? x. Combining the expressions of A1 , A2 , and
A3 leads to:
Px
Px0
0
A1 + A2 + A3
(k ? 2x)2 + k ( i=1 ?i + j=1 ?j )
=1+
?
(6)
A=
mn
2mn
mn
Lemma 2 For a fixed x, the average value of the AUC A is given by:
x
n
< A >x = 1 ?
+ k?x
m
2
(7)
Px
Proof. The proof is based on the computation of the average values of i=1 ?i and
Px0
0
j=1 ?j for a given x. We start by computing the average value < ?i >x for a given
i, 1 ? i ? x. Consider all the possible positions for ?1 . . . ?i?1 and ?i+1 . . . ?x , when the
value of ?i is fixed at say ?i = l. We have i ? l ? N ? (x ? i) since there need to be at
least i ? 1 positions before ?i and N ? (x ? i) above. There are l ? 1 possible positions for
?1 . . . ?i?1 and N ? l possible positions for ?i+1 . . . ?x . Since the total number of ways
of choosing the x positions for ?1 . . . ?x out of N is N
x , the average value < ?i >x is:
PN ?(x?i) l?1 N ?l
l i?1 x?i
l=i
< ?i >x =
(8)
N
x
Thus,
<
x
X
?i >x =
i=1
Px
i=1
PN ?(x?i)
l=i
N
x
l
l?1
i?1
u
p1 +p2 =p p1
v
p2
P
=
PN
u+v
p
l=1
l
Px
l?1
i=1 i?1
N
x
N ?l
x?i
, we can write:
P
N
N ?1
N ?1
x
X
N (N + 1) x?1
x(N + 1)
l=1 l x?1
<
?i >x =
=
=
N
N
2
2
x
x
i=1
Using the classical identity:
N ?l
x?i
=
(9)
(10)
Similarly, we have:
0
<
x
X
j=1
?0j >x =
x0 (N 0 + 1)
2
(11)
Px
Px0
0
Replacing <
i=1 ?i >x and <
j=1 ?j >x in Eq. (6) by the expressions given by
Eq. (10) and Eq. (11) leads to:
(k ? 2x)2 + k ? x(N + 1) ? x0 (N 0 + 1)
=1?
2mn
which ends the proof of the lemma.
< A >x = 1 +
x
n
+ k?x
m
2
(12)
Note that Eq. (7) shows that the average AUC value for a given x is simply one minus the
average of the accuracy rates for the positive and negative classes.
Proposition 1 Assume that a binary classification task with m positive examples and n
negative examples is given. Then, the expected value of the AUC A over all classifications
with k errors is given by:
Pk?1 m+n !
k
(n ? m)2 (m + n + 1)
k
x
< A >= 1 ?
?
? Pk x=0 m+n+1
(13)
m+n
4mn
m+n
x=0
x
Proof. Lemma 2 gives the average value of the AUC for a fixed value of x. To compute
the average over all possible values of x, we need to weight the expression
of Eq. (7) with
the total number of possible classifications for a given x. There are N
ways of
x possible
N0
choosing the positions of the x misclassified negative examples, and similarly x0 possible
ways of choosing the positions of the x0 = k ? x misclassified positive examples. Thus, in
view of Lemma 2, the average AUC is given by:
k?x
x
Pk
N N0
n+ m
)
x=0 x
x0 (1 ?
2
< A >=
(14)
Pk
N N0
x=0
x
x0
r=0.05
r=0.01 r=0.1 r=0.25
0.0
0.1
0.2
r=0.5
0.3
Error rate
0.4
0.5
.00 .05 .10 .15 .20 .25
0.5 0.6 0.7 0.8 0.9 1.0
Mean value of the AUC
Relative standard deviation
r=0.01
r=0.05
r=0.1
0.0
0.1
r=0.25
0.2
0.3
Error rate
r=0.5
0.4
0.5
Figure 3: Mean (left) and relative standard deviation (right) of the AUC as a function of the error rate.
Each curve corresponds to a fixed ratio of r = n/(n + m). The average AUC value monotonically
increases with the accuracy. For n = m, as for the top curve in the left plot, the average AUC
coincides with the accuracy. The standard deviation decreases with the accuracy, and the lowest
curve corresponds to n = m.
This expression can be simplified into Eq. (13)3 using the following novel identities:
k
X
N
x
x=0
k
X
N
x
x
x=0
!
N0
x0
!
!
N0
x0
!
=
=
!
k
X
n+m+1
x
x=0
!
k
X
(k ? x)(m ? n) + k n + m + 1
2
x
x=0
(15)
(16)
that we obtained by using Zeilberger?s algorithm4 and numerous combinatorial ?tricks?.
From the expression of Eq. (13), it is clear that the average AUC value is identical to the
accuracy of the classifier only for even distributions (n = m). For n 6= m, the expected
value of the AUC is a monotonic function of the accuracy, see Fig. (3)(left). For a fixed
ratio of n/(n + m), the curves are obtained by increasing the accuracy from n/(n + m)
to 1. The average AUC varies monotonically in the range of accuracy between 0.5 and
1.0. In other words, on average, there seems nothing to be gained in designing specific
learning algorithms for maximizing the AUC: a classification algorithm minimizing the
error rate also optimizes the AUC. However, this only holds for the average AUC. Indeed,
we will show in the next section that the variance of the AUC value is not null for any ratio
n/(n + m) when k 6= 0.
4 The Variance of the AUC
2
Px
Px0
+k
, a = i=1 ?i , a0 = j=1 ?0j , and ? = a + a0 . Then, by
Let D = mn + (k?2x)
2
Eq. (6), mnA = D ? ?. Thus, the variance of the AUC, ? 2 (A), is given by:
(mn)2 ? 2 (A)
=
< (D ? ?)2 ? (< D > ? < ? >)2 >
=
< D2 > ? < D >2 + < ?2 > ? < ? >2 ?2(< ?D > ? < ? >< D >)
(17)
As before, to compute the average of a term X over all classifications, we can first determine its average < X >x for a fixed x, and then use the function F defined by:
Pk
N N0
Y
0
x=0 x
F (Y ) = Pk
(18)
x
N N0
x=0
x
x0
and < X >= F (< X >x ). A crucial step in computing the exact value of
variance of
Pthe
x
the AUC is to determine the value of the terms of the type < a2 >x =< ( i=1 ?i )2 >x .
3
An essential difference between Eq. (14) and the expression given by [15] is the weighting by
the number of configurations. The authors? analysis leads them to the conclusion that the average
AUC is identical to the accuracy for all ratios n/(n + m), which is false.
4
We thank Neil Sloane for having pointed us to Zeilberger?s algorithm and Maple package.
Px
Lemma 3 For a fixed x, the average of ( i=1 ?i )2 is given by:
x(N + 1)
< a2 > x =
(3N x + 2x + N )
12
Proof. By definition of a, < a2 >x = b + 2c with:
x
x
X
X
b =<
?2i >x
c =<
(19)
?i ?j >x
(20)
1?i<j?x
i=1
Reasoning as in the proof of Lemma 2, we can obtain:
Px PN ?(x?i) 2 l?1 N ?l
N ?1
N
X
l i?1 x?i
(N + 1)(2N + 1)x
i=1
l=i
2 x?1
b=
=
l
=
N
N
6
x
x
(21)
l=1
To compute c, we start by computing the average value of < ?i ?j >x , for a given pair (i, j)
with i < j. As in the proof of Lemma 2, consider all the possible positions of ?1 . . . ?i?1 ,
?i+1 . . . ?j?1 , and ?j+1 . . . ?x when ?i is fixed at ?i = l, and ?j is fixed at ?j = l0 .
There are l ? 1 possible positions for the ?1 . . . ?i?1 , l0 ? l ? 1 possible positions for
?i+1 . . . ?j?1 , and N ? l0 possible positions for ?j+1 . . . ?x . Thus, we have:
0
P
0
0 l?1 l ?l?1 N ?l
i?l<l0 ?N ?(x?j) ll i?1 j?i?1
x?j
(22)
< ?i ?j >x =
N
x
and
l?1 l0 ?l?1 N ?l0
m1 +m2 +m3 =x?2 m1
m2
m3
c=
N
x
l0 ?l?1
N ?l0
P
?2
identity m1 +m2 +m3 =x?2 l?1
= N
m1
m2
m3
x?2 , we
P
Using the
l<l0
ll0
P
(N + 1)(3N + 2)x(x ? 1)
24
Combining Eq. (21) and Eq. (24) leads to Eq. (19).
c=
(23)
obtain:
(24)
Proposition 2 Assume that a binary classification task with m positive examples and n
negative examples is given. Then, the variance of the AUC A over all classifications with
k errors is given by:
x
x
+ k?x
+ k?x
m 2
m
) ) ? F ((1 ? n
))2 +
(25)
? 2 (A) = F ((1 ? n
2
2
2
2
mx + n(k ? x) + (m(m + 1)x + n(n + 1)(k ? x)) ? 2x(k ? x)(m + n + 1)
F(
)
12m2 n2
Proof. Eq. (18) can be developed and expressed in terms of F , D, a, and a0 :
(mn)2 ? 2 (A) = F ([D? < a + a0 >x ]2 ) ? F (D? < a + a0 >x )2 +
F (< a2 >x ? < a >2x ) + F (< a02 >x ? < a0 >2x ) (26)
The expressions for < a >x and < a0 >x were given in the proof of Lemma 2, and
that of < a2 >x by Lemma 3. The following formula can be obtained in a similar
0
0
way: < a02 >x = x (N12+1) (3N 0 x0 + 2x0 + N 0 ). Replacing these expressions in Eq. (26)
and further simplifications give exactly Eq. (25) and prove the proposition.
The expression of the variance is illustrated by Fig. (3)(right) which shows the value of
one standard deviation of the AUC divided by the corresponding mean value of the AUC.
This figure is parallel to the one showing the mean of the AUC (Fig. (3)(left)). Each line
is obtained by fixing the ratio n/(n + m) and varying the number of errors from 1 to the
size of the smallest class. The more uneven class distributions have the highest variance,
the variance increases with the number of errors. These observations contradict the inexact
claim of [15] that the variance is zero for all error rates with even distributions n = m. In
Fig. (3)(right), the even distribution n = m corresponds to the lowest dashed line.
Dataset
Breast-Wpbc
Credit
Ionosphere
Pima
SPECTF
Page-blocks
Yeast (CYT)
Size
194
653
351
768
269
5473
1484
# of
Attr.
33
15
34
8
43
10
8
n
n+m
(%)
23.7
45.3
35.9
34.9
20.4
10.2
31.2
AUCsplit[4]
Accuracy (%)
AUC (%)
69.5 ? 10.6
59.3 ? 16.2
89.6 ? 5.0
72.5 ? 5.1
89.7 ? 6.7
76.7 ? 6.0
96.8 ? 0.2
71.1 ? 3.6
95.1 ? 6.9
73.3 ? 4.0
RankBoost
Accuracy (%)
AUC (%)
65.5 ? 13.8
80.4 ? 8.0
81.0 ? 7.4
94.5 ? 2.9
83.6 ? 10.9
98.0 ? 3.3
69.7 ? 7.6
84.8 ? 6.5
67.3
93.4
92.0 ? 2.5
98.5 ? 1.5
45.3 ? 3.8
78.5 ? 3.0
Table 1: Accuracy and AUC values for several datasets from the UC Irvine repository. The values
for RankBoost are obtained by 10-fold cross-validation. The values for AUCsplit are from [4].
5 Experimental Results
Proposition 2 above demonstrates that, for uneven distributions, classifiers with the same
fixed (low) accuracy exhibit noticeably different AUC values. This motivates the use of
algorithms directly optimizing the AUC rather than doing so indirectly via minimizing the
error rate. Under certain conditions, RankBoost [5] can be viewed exactly as an algorithm
optimizing the AUC. In this section, we make the connection between RankBoost and
AUC optimization, and compare the performance of RankBoost to two recent algorithms
proposed for optimizing an approximation [15] or locally optimizing the AUC [4].
The objective of RankBoost is to produce a ranking that minimizes the number of incorrectly ordered pairs of examples, possibly with different costs assigned to the mis-rankings.
When the examples to be ranked are simply two disjoint sets, the objective function minimized by RankBoost is
m X
n
X
1 1
rloss =
1x ?y
(27)
m
n i j
i=1 j=1
which is exactly one minus the Wilcoxon-Mann-Whitney statistic. Thus, by Lemma 1, the
objective function maximized by RankBoost coincides with the AUC.
RankBoost?s optimization is based on combining a number of weak rankings. For our
experiments, we chose as weak rankings threshold rankers with the range {0, 1}, similar
to the boosted stumps often used by AdaBoost [6]. We used the so-called Third Method of
RankBoost for selecting the best weak ranker. According to this method, at each step, the
weak threshold ranker is selected so as to maximize the AUC of the weighted distribution.
Thus, with this method, the global objective of obtaining the best AUC is obtained by
selecting the weak ranking with the best AUC at each step.
Furthermore, the RankBoost algorithm maintains a perfect 50-50% distribution of the
weights on the positive and negative examples. By Proposition 1, for even distributions,
the mean of the AUC is identical to the classification accuracy. For threshold rankers like
step functions, or stumps, there is no variance of the AUC, so the mean of the AUC is equal
to the observed AUC. That is, instead of viewing RankBoost as selecting the weak ranker
with the best weighted AUC value, one can view it as selecting the weak ranker with the
lowest weighted error rate. This is similar to the choice of the best weak learner for boosted
stumps in AdaBoost. So, for stumps, AdaBoost and RankBoost differ only in the updating scheme of the weights: RankBoost updates the positive examples differently from the
negative ones, while AdaBoost uses one common scheme for the two groups.
Our experimental results corroborate the observation that RankBoost is an algorithm optimizing the AUC. RankBoost based on boosted stumps obtains AUC values that are substantially better than those reported in the literature for algorithms designed to locally or
approximately optimize the AUC. Table 1 compares the results of RankBoost on a number
of datasets from the UC Irvine repository to the results reported by [4]. The results for
RankBoost are obtained by 10-fold cross-validation. For RankBoost, the accuracy and the
best AUC values reported on each line of the table correspond to the same boosting step.
RankBoost consistently outperforms AUCsplit in a comparison based on AUC values, even
for the datasets such as Breast-Wpbc and Pima where the two algorithms obtain similar accuracies. The table also lists results for the UC Irvine Credit Approval and SPECTF heart
dataset, for which the authors of [15] report results corresponding to their AUC optimization algorithms. The AUC values reported by [15] are no better than 92.5% for the Credit
Approval dataset and only 87.5% for the SPECTF dataset, which is substantially lower.
From the table, it is also clear that RankBoost is not an error rate minimization algorithm.
The accuracy for the Yeast (CYT) dataset is as low as 45%.
6 Conclusion
A statistical analysis of the relationship between the AUC value and the error rate was
given, including the first exact expression of the expected value and standard deviation of
the AUC for a fixed error rate. The results offer a better understanding of the effect on the
AUC value of algorithms designed for error rate minimization. For uneven distributions
and relatively high error rates, the standard deviation of the AUC suggests that algorithms
designed to optimize the AUC value may lead to substantially better AUC values. Our
experimental results using RankBoost corroborate this claim.
In separate experiments we have observed that AdaBoost achieves significantly better error rates than RankBoost (as expected) but that it also leads to AUC values close to those
achieved by RankBoost. It is a topic for further study to explain and understand this property of AdaBoost. A partial explanation could be that, just like RankBoost, AdaBoost
maintains at each boosting round an equal distribution of the weights for positive and negative examples.
References
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1,671 | 2,519 | Simplicial Mixtures of Markov Chains:
Distributed Modelling of Dynamic User Profiles
Mark Girolami
Department of Computing Science
University of Glasgow
Glasgow, UK
[email protected]
Ata Kab?an
School of Computer Science
University of Birmingham
Birmingham, UK
[email protected]
Abstract
To provide a compact generative representation of the sequential activity of a number of individuals within a group there is a tradeoff between
the definition of individual specific and global models. This paper proposes a linear-time distributed model for finite state symbolic sequences
representing traces of individual user activity by making the assumption that heterogeneous user behavior may be ?explained? by a relatively
small number of common structurally simple behavioral patterns which
may interleave randomly in a user-specific proportion. The results of an
empirical study on three different sources of user traces indicates that
this modelling approach provides an efficient representation scheme, reflected by improved prediction performance as well as providing lowcomplexity and intuitively interpretable representations.
1 Introduction
The now commonplace ability to accurately and inexpensively log the activity of individuals in a digital environment makes available a variety of traces of user activity and with
it the necessity to develop efficient representations, or profiles, of individuals. Most often, such recordings take the form of streams of discrete symbols ordered in time. The
modelling of time dependent sequences of discrete symbols employing n?th order Markov
chains has been extensively studied in a number of domains. The representation provided
by such models is global in the sense that it is assumed that one global generating process underlies all observed sequences. To capture the possible heterogeneous nature of the
observed sequences a model with a number of differing generating processes needs to be
considered. Indeed the notion of a heterogeneous population, characterized for example by
occupational mobility and consumer brand preferences, has been captured in the MoverStayer model [3]. This model is a discrete time stochastic process that is a two component
mixture of first order Markov chains, one of which is degenerate and possesses an identity
transition matrix characterizing the stayers in the population. The original notion of a twocomponent mixture of Markov chains has recently been extended to the general form of a
mixture model of Markov chains in [2]. Whilst the main motivation was the visualization
of the class structure inherent in the browsing patterns of visitors to a commercial website,
each class of users being characterized by their global behavior, such mixture models will
not be appropriate for identifying the shared behavioral patterns which are the basis of multiple relationships between users and groups of users and which may yield a more realistic
model of the population.
The purpose of this paper is to develop a dynamic user model for individuals within a
group that explicitly captures the assumption of the existence of a common set of behavioral patterns which can be estimated from all observed users along with their user-specific
proportion of participation and these form the basis of individual profiles within a group.
This is also a computationally attractive model, as simple structural characteristics may be
assumed at the generative level, while allowing them to interleave randomly can account
for more complex individual behavior. The resulting model is thus a distributed dynamic
model which benefits from the recent technical developments in distributed parts based
modelling of static vectorial data [7, 9, 5, 1, 8], with various applications including image decomposition, document modelling, information retrieval and collaborative filtering.
Consistent generative semantics similar to the recently introduced latent Dirichlet allocation (LDA) [1] will be adopted and by analogy with [8] the resulting model will be referred
to as a simplicial mixture.
2 Simplicial Mixtures of Markov Chains
Assume that a sequence of L symbols sL sL?1 , ? ? ? , s0 , denoted by s, can be drawn
from a dictionary S by a process k, which has initial state probability P1 (k) and has
|S|m+1 state transition probabilities denoted by T (sm , ? ? ? , s1 ? s0 |k). The number
of times that the symbol s0 follows from the state defined by the m-tuple of symbols
sm , ? ? ? , s1 within the n-th sequence is denoted as rnsm ,??? ,s1 ?s0 and so the probability of the sequence of symbols under the k?th m-th order Markov process is P (s|k) =
Q|S|
Q|S|
sm ,??? ,s1 ?s0
. To introduce a more comP1 (k) sm =1 ? ? ? s0 =1 T (sm , ? ? ? , s1 ? s0 |k)r
pact notation we represent the elements of the state transition matrix for the k?th Markov
process by Tm???0,k and the counts rsm ,??? ,s1 ?s0 within the n?th observed sequence as
rnm???0 . In addition, we employ Start and Stop states in each symbol sequence sn and
incorporate the initial state distribution of the Start state as the transition probabilities
from this state within the state transition matrix Tk . We denote the set of all state transition matrices {T1 , ? ? ? , Tk , ? ? ? , TK } as T. Suppose that we are given a set of symbolic
trajectories {sn }n=1:N over a common finite state space, each having length Ln . As opposed and somewhat complementary to cluster models for trajectories which try to model
inter-sequence heterogeneities, our intuition is that sequences over a common finite state
space, provided they are sufficiently long and possibly non-stationary, could have several
randomly interleaved generator processes, some of which might be common to several
sequences. To account for this idea, we will adopt a similar modelling strategy to LDA.
The complete generative semantics of LDA allows us to describe the process of sequence
generation where mixing components ? = [?1 , ? ? ? , ?k , ? ? ? , ?K ] are K-dimensional
Dirichlet random variables and so are drawn from the K ? 1 dimensional simplex defined
by the Dirichlet distribution D(?|?) with parameters ?. These are then combined with the
individual state-transition probabilities Tk , which are model parameters to be estimated,
PK
and yield the symbol transition probabilities Tm???0 = k=1 Tm???0,k ?k . The overall probability for a sequence sn under such a mixture, which we shall now refer to as a simplicial
mixture [8], denoted as P (sn |T, ?) is equal to
Z
Z
P (sn |T, ?)D(?|?)d? =
4
d?D(?|?)
4
|S|
Y
sm =1
???
|S|
Y
s0 =1
(
K
X
)rnm???0
Tm???0,k ?k
(1)
k=1
Each sequence will have its own expectation under the Dirichlet mixing coefficients and
so the ability of such a representation to model intra-sequence heterogeneity emerges naturally.
The following subsections briefly present the details of the identification of this model,
which also highlights the close relationship between two existing related models, specifically the probabilistic latent semantic analysis (PLSA) [5] and LDA [1] as being instances
of the same theoretical model and differing only in the estimation procedure adopted [4].
2.1 Parameter Estimation and Inference
Exact inference within the LDA framework is not possible [1], however the likelihood can
be lower-bounded by introducing a sequence specific parameterised variational posterior
Qn (?) whose parameters will depend on n
??
? ?
D(?|?)
(2)
log P (sn |T, ?) ? EQn (?) log P (sn |T, ?)
Qn (?)
Where EQn (?) denotes expectation with respect to Qn (?). The bound can be defined using
AP
the Maximum a Posteriori (MAP) estimator, such that Qn (?) = ?(? ? ?M
), in which
n
M AP
M AP
case (2) is equal to log P (sn |T, ?n
) + log D(?n
|?) + H? where H? denotes the
AP
entropy of the delta function around ?M
(which can be discarded in this setting as it
n
does not depend on the model parameters, although it amounts to minus infinity). Forming
a Lagrangian from the above to enforce the constraint that ?M AP is a sample point from a
AP
Dirichlet variable then taking derivatives with respect to the ?M
, a convergent series of
k
t
updates ?kn is obtained where the superscript denotes the t?th iteration. As in [7], for each
observed sequence in the sample a MAP value for the variable ? is iteratively estimated by
the following multiplicative updates
|S|
X
|S|
X
?
?
P kn
Ln + k (?k ? 1)
sm =1
s0 =1
(3)
P
AP
where Ln = sm ???s0 rnm???0 is the length of the sequence sn . Once the MAP values ?M
n
for each sn are obtained a similar multiplicative iteration for the transition probabilities can
be obtained
N
AP
X
?M
T?m???0,k
t+1
t
kn
rnm???0 PK
(4)
T?m???0,k = Tm???0,k
; Tm???0,k
= P|S|
t
M AP
?m???00 ,k
T
0
n=1
l=1 Tm???0,l ?ln
s =1
? kn = (?k ?1)+?t
?
kn
???
rnm???0 PK
Tm???0,k
t
l=1 Tm???0,l ?ln
;
?t+1
kn =
0
The final parameter is that of the prior Dirichlet distribution, maximum likelihood estimaAP
tion yields the estimated distribution parameters ? given the ?M
[6, 1]. Note that both
n
(3) and (4) require an elementwise matrix multiplication and division so these iterations
will scale linearly with the number of non-zero state-transition counts. It is interesting to
note that the MAP estimator under a uniform Dirichlet distribution exactly recovers the
aspect mixture model of [5] as a special case of the MAP estimated LDA model.
2.1.1 Variational Parameter Estimation and Inference
While being optimal in analyzing an existing data set, MAP estimators are notoriously
prone to overfitting, especially where there is a paucity of available data [10] and so the
variational Bayes (VB) approach detailed in [1] can be adopted by considering Qn (?) =
D(?|? n ), where ? n is a sequence-specific variational free parameter vector. The above (2)
can be further lower-bounded by noting that
?
?
|S|
|S| K
X
X
X
Tm???0,k
m???0
(5)
log P (sn |T, ?) ?
???
rn Qm???0,n (k) log ?k
Qm???0,n (k)
s =1
s =1
m
0
k=1
P
where k Qm???0,n (k) = 1, Qm???0,n (k) ? 0 are additional variational parameters. Alternatively, Qm???0,n (.) can also be understood as a variational distribution on a discrete
hidden variable with K possible outcomes that selects which transition matrix is active at
each time step of the generative process.
P
Replacing (5) in (2), expanding and evaluating ED(?|? ) [log ?k ] = ?(?k ) ? ?( k0 ?k0 ),
n
where ? denotes the digamma function, then solving for Qm???0,n (k) and ?kn and finally
combining yields the following multiplicative iterative update for the sequence specific
variational free parameter ? n
t+1
t
?kn
= ?k + exp{?(?kn
)}
|S|
X
sm =1
???
|S|
X
rnm???0 PK
s0 =1
k0 =1
Tm???0,k
Tm???0,k0 exp{?(?kt 0 n )}
(6)
Solving for the transition probabilities and combining with the fixed point solutions for
each Qm???0,n (k) yields the following
N
X
T?m???0,k
t
t
?
0
s00 Tm???0 ,k
n=1
k0 =1 Tm???0,k0 exp{?(?k0 n )}
(7)
As before the parameters of the prior Dirichlet distribution ? given the variational parameters ? n are estimated using standard methods [6, 1].
t
T?m???0,k = Tm???0,k
rnm???0 PK
t
exp{?(?kn
)}
;
t+1
Tm???0,k
=P
2.2 Prediction with Simplicial Mixtures
The predictive probability of observing symbol snext given a sequence of L symbols
sn = {sLn , ? ? ? , s1 } is given as P (snext |sn ) = EP (?|sn ) {P (snext |sm ? ? ? s1 , ?)} ?
PK
k=1 T (snext |sm ? ? ? s1 , k)EQn (?) {?k }. It should be noted that while m-th order Markov
chains form the basis of the representation, the resulting simplicial mixture is not m-th order Markov with any global transition model. Rather it approximates the individual m-th
order models while keeping the generative parameter set compact. The m-th order information of each individual?s past behaviour is embodied in the individual-specific latent
variable estimate. On the other hand in a mixture model one component is responsible
for sequence generation so within a cluster the representation is still global m-th order.
Employing the MAP approximation for the Dirichlet distribution then EQn (?) {?k } =
AP
M AP
AP
AP ) {?k } = ?
E?(???M
where ?M
is the k-th dimension of ?M
. Employing the
n
kn
kn
n
PK
variational Dirichlet approximation then EQn (?) {?k } = ED(?|? n ) {?k } = ?kn / l=1 ?ln
therefore given a new sequence snew , the symbol snext which is most likely to be predicted
from the model as a suggested continuation of the sequence, is the maximum argument of
P (snext |sn ).
3 Distributed Modelling of Dynamic Profiles
3.1 Datasets
3.1.1 Telephone Usage Modelling
The ability to model the usage of a telephone service is of importance at a number of levels, e.g. to obtain a predictive model of customer specific activity and service usage for
the purposes of service provision planning, resource management of switching capacity,
identification of fraudulent usage of services. A representative description can be based on
the distribution of the destination numbers dialled and connected by the customer, in which
case a multinomial distribution over the dialling codes can be employed. One method of
encoding the destination numbers dialled by a customer is to capture the geographic location of the destination, or the mobile service provider if not a land based call. This is useful
in determining the potential demand placed on telecommunication switches which route
traffic from various geographical regions on the service providers network. Two weeks
of transactions from a UK telecommunications operator were logged during weekdays,
amounting to 36,492,082 and 45,350,654 transactions in each week respectively. All transactions made by commercial customers in the Glasgow region of the UK were considered
in this study. This amounts to 1,172,578 transactions from 12,202 high usage customers
in the first week considered and 1,753,304 transactions being made in the following week.
The mapping from dialling number to geographic region or mobile operator was encoded
with 87 symbols amounting to a possible 7,569 symbol transitions. Each customers activity
is defined by a sequence of symbols defining the sequence of calls made over each period
considered and these are employed to encode activity in a customer specific generative
representation.
3.1.2 Web Page Browsing
The second data set used in this study is a selected subset of the msnbc.com user navigation
collection employed in [2]. Sequences of users who visited at least 9 of the overall 17
page categories (frontpage, news, tech, local, opinion, on-air, misc,weather, msn-news,
health, living, business, msn-sports, sports, summary, bbs, travel) have been retained, this
selection criteria is motivated by the observation that there would be little scope in trying
to model interleaved dynamic behavior in observables which are too short to reveal any
intra-sequence heterogeneity. The resulting data set, referred to as WEB, totals 119,667
page requests corresponding to 1,480 web browsing sessions.
0.62
Predictive Perplexity
11
Fraction Prediction Error
0.61
0.6
0.59
10
9
8
7
0.58
0.4
0.6
0.8
1
1.2
1.4
Number of Factors ( log10 )
1.6
1.8
0.4
0.6
0.8
1
1.2
1.4
1.6
Number of Factors ( log10 )
1.8
Figure 1: Left: percentage of incorrect predictions against the number of model factors;
right: predictive perplexity of each model against model order for the PHONE dataset.
Solid straight line: global first order MC, dash: MAP estimated simplicial mixture, solid
line: VB estimated simplicial mixture, dash-dot: mixture model.
3.2 Results
In each experiment the objective assessment of model performance is evaluated by the
PNtest
predictive perplexity, exp{?1/N m=1
log P (snext |sm )}. In addition, the predictive accuracy of all models is measured under a 0-1 loss. Given a number of previously unobserved truncated sequences, the number of times the model correctly predicts the symbol
which follows in the sequence is then counted. In all mixture models naive random initialization of the parameters was employed and parameter estimation was halted when the
5
0.4
4
? ?k P(?|?n)d?
Entropy Rate ( bits )
0.35
0.3
0.25
3
2
0.2
0.15
0.1
1
0.05
1
Simplicial Model
2
Mixture Model
0
5
10
15
20
(k)
Figure 2: Left: distribution of entropy rates for the transition matrices of a 20-factor mixture
and simplicial mixture models (VB). Right: the expected value of the Dirichlet variable
under the variational approximation for one customer indicating the levels of participation
in factor specific behaviors.
in-sample likelihood did not improve by more than 0.001%, no annealing or early stopping
was utilized, fifteen randomly initialized parameter estimation runs for each model were
performed. The number of mixture components for the models ranged from 2 up to 200. On
the PHONE data set the parameters of a global first-order Markov chain (bigram), mixtures
of Markov chains [2], and simplicial mixtures of Markov chains (using both the MAP and
VB estimation procedures) are estimated using the first week of customer transactions and
the predictive capabilities of the models are assessed on the transactions from the following week. The results are summarized in Figure 1, from the predictive perplexity measures
it is clear that the simplicial representation provides a statistically (tested at the 5% level
using a Wilcoxon Rank Sum test) and practically significant reduction in perplexity over
the global and mixture models. This is also reflected in the levels of prediction error under
each model, however the mixture models tend to perform slightly worse than the global
model. As expected the MAP estimated simplicial model performs slightly worse than that
obtained using VB [1]. This also provides an additional insight as to why LDA models
improve upon PLSA, as they are in fact both the same model using different approximations to the likelihood, refer to [10] for an illustrative discussion on the weaknesses of MAP
estimators. As a comparison to different structural models hidden Markov models with a
range of hidden states were also tested on this data set the best results obtained were for a
ten state model which achieved a predictive perplexity score of (mean?standard-deviation)
11.119 ? 0.624 and fraction prediction error of 0.674 ? 0.959, considerably poorer than
that obtained by the models considered here.
In addition to the predictive capability of a simplicial representation of a customers activity
the cost of encoding such a representation can be assessed by measuring the entropy rate
of each of the constituent transition matrices which act as a basis in the representation
of the individual specific generative process. The left hand plot of Figure (2) shows the
distribution of the entropy rates for the transition probabilities in twenty factor simplicial
and mixture models, the results are obtained from fifty randomly initialized estimation
procedures. The entropy rates for the simplicial mixture are significantly lower than that
of a mixture model indicating that the basis of each representation describes a number of
simpler processes.
The final experiment demonstrated considers the WEB data set. The results of ten-fold
cross-validated predictive perplexities again show statistically significant improvement obtained with the VB-estimated simplicial mixture (again tested using the ranksum Wilcoxon
test at the 5% level). The results are summarized in Figure 3. Five of the estimated transition factors of a twenty-factor model are shown in Figure 4, demonstrating once more
that the proposed model creates a low entropy and an easily interpretable dynamic factorial
representation. The numbers on the axes on these charts correspond to the 17 page cat-
egories enumerated earlier and the average strength of each of these factors amongst the
PN
full set of twenty factors computed as N1 n=1 ED(?|? n ) {?k } is also given above each
chart. We can see that a behavioral feature manifested is a keen interest to visit pages about
?news? along with a quite dynamic transition model (left hand chart) which characterizes
around 12% of the behavioral patterns of the entire user population under consideration
while static state-repetition (second chart) or an almost exclusive interest in viewing the
homepage (last chart) etc represent also relatively strong common characteristics of browsing behavior. The distribution of the entropy rates of the full set of these twenty basistransitions in comparison to those obtained from the mixture model is given on the right
hand plot of Figure 3. Clearly, the coding efficiency of a simplicial mixture representation is significantly (statistically tested) superior. Note also these basis-transitions embody
correlated transitions (transitions which appear in similar dynamical contexts and so have
similar functionality), as can be seen from the multiplicative nature of the equations used
for identifying the model. It is not surprising then that state repetitions or transitions which
express focused interest in one of the topic categories appear together on distinct factors.
We can also see a joint interest in msnnews and msnsport being present together on the 4-th
chart of Figure 4 ? indeed, as the prefix of these page categories also indicates, these are
related page categories.
7.8
3
7.6
2.5
Entropy Rates (bits)
Predictive Perplexity
7.4
7.2
7
6.8
6.6
2
1.5
1
6.4
0.5
6.2
6
0.4
0.6
0.8
1
1.2
1.4
1.6
Nr of Factors (log10)
Mixture Model
Simplicial Model
Figure 3: Left: the predictive perplexity for the WEB data (straight line: global firstorder Markov chain, dash-dot: mixture of Markov chains, dotted line: simplicial mixture
estimated by MAP, solid line: simplicial mixture estimated by VB). Right: the distribution
of entropy rates.
4 Conclusions
This paper has presented a linear time method to model finite-state sequences of discrete
symbols which may arise from user or customer activity traces. The main feature of the
proposed approach has been the assumption that heterogeneous user behavior may be ?explained? by the interleaved action of some structurally simple common generator processes.
0.12
0.02
0.03
0.23
0.07
2
2
2
2
2
4
4
4
4
4
6
6
6
6
8
8
8
8
8
10
10
10
10
10
12
12
12
12
12
14
10
15
5
10
15
14
16
16
16
5
14
14
14
16
6
5
10
15
16
5
10
15
5
10
Figure 4: State transition matrices of selected factors from a 20-factor run on WEB.
15
An empirical study has been conducted on two real-world collections of user activity which
has demonstrated this to be an efficient representation, revealed by both objective measures
of prediction performance, low entropy rates, and interpretable representations of the user
profiles provided.
Acknowledgements
Mark Girolami is part of the DETECTOR project funded by the Department of Trade and
Industry (DTI) Management of Information (LINK) Programme and the Engineering &
Physical Sciences Research Council (EPSRC) grant GR/R55184.
References
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[3] H. Frydman, Maximum likelihood estimation in the mover-stayer model, Journal
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[4] M. Girolami and A. Kab?an, On an equivalence between PLSI and LDA, Proc. 26-th
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[7] D. Lee & H. Sebastian Seung, Algorithms for Non-negative Matrix Factorization,
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correlated:1 |
1,672 | 252 | Can Simple Cells Learn Curves? A Hebbian Model in a Structured Environment
Can Simple Cells Learn Curves? A
Hebbian Model in a Structured
Environment
William R. Softky
Divisions of Biology and Physics
103-33 Caltech
Pasadena, CA 91125
[email protected]
Daniel M. Kammen
Divisions of Biology and Engineering
216-76 Caltech
Pasadena, CA 91125
[email protected]
ABSTRACT
In the mammalian visual cortex, orientation-selective 'simple cells'
which detect straight lines may be adapted to detect curved lines
instead. We test a biologically plausible, Hebbian, single-neuron
model, which learns oriented receptive fields upon exposure to unstructured (noise) input and maintains orientation selectivity upon
exposure to edges or bars of all orientations and positions. This
model can also learn arc-shaped receptive fields upon exposure
to an environment of only circular rings. Thus, new experiments
which try to induce an abnormal (curved) receptive field may provide insight into the plasticity of simple cells. The model suggests
that exposing cells to only a single spatial frequency may induce
more striking spatial frequency and orientation dependent effects
than heretofore observed.
1
Introduction
Although most mathematical theories of cortical function assume plasticity of individual cells, there is a strong debate in the biological community between "instructional" (plastic) and "selectional" (hard-wired) models of orientation-selective cells
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(which we will call "simple cells") in striate visual cortex. Thus, a theory of simple
cell learning which can make experimental predictions is desirable.
1.1
Overview of Plasticity Experiments
The most illuminating experiments addressing the plasticity of visual cortex are
collectively called "stripe-rearing." Such experiments artificially restrict the visual
environment of animals (usually kittens) toa few straight, dark, parallel lines (e.g. 3
vertical stripes.) In the many cases studied, examination of the visual cortex reveals
that animals which viewed such limited visual environments posses more simple cells
tuned to the exposed orientation than tuned to other orientations. (For comparison,
the simple cells of animals with normal visual experience are equally distributed
among all orientations.) But the observed changes in cell populations can be equally
well explained by "instructional" and "selectional" hypotheses (Stryker et al.1978).
Although many variations on stripe-rearing have been tried (different orientations
for each eye, one eye closed, etc.), only environments spanning a very restricted
subset (straight lines) of the natural environment have been studied (Hirsch et al.
1983, Blakemore et al. 1978, and see references therein). Conclusions regarding
plasticity have been based on changes in populations of simple cells, rather than on
changes in individual cells. Statistical arguments based on changes in large groups
of cells are questionable, since the well-documented lateral interactions between
cortical neurons may constrain population ratios, e.g. limit the fraction of neurons
responding to a single orientation.
1.2
New Experimental Approach
We propose several experiments to alter the receptive field (RF) of a single cell (see
also Fregnac et al. 1988). How might that be done? The RF ofa simple cell has only
one characteristic spatial frequency (Jones & Palmer 1987 and ref's therein). To try
altering the shape of that RF, it is necessary to present a pattern which is different
from a simple bar or edge, but is still sufficiently similar in spatial frequency to
activate the same population of retinal cells that detect the bar. An arc-shaped RF
satisfies this condition; to generate an arc-shaped RF, an environment of circular
rings (rather than bent bars) is necesary, since complete circles lack sharp end-effects
which could overexcite spatial opponent cells and thus disturb learning.
This paper proposes a very simple Hebbian model of a neuron, and examines the
resulting plasticity upon exposure to edge, bar, and arc-shaped stimuli.
2
Mathematical Model
The model applies a simple Hebbian learning rule to an array of about 400 synapses.
There are several important features of this model. One is that the stimulus is a
visual environment of structured input (bars, edges, or circles) rather than only
stochastic (noise) input, as was used in the previous Hebb-Iearning models of Linsker
(1986) and Kammen & Yuille (1988). (For a review of Hebbian learning and neural
development see Kammen and Yuille 1990). Second, the input is Laplace filtered
Can Simple Cells Learn Curves? A Hebbian Model in a Structured Environment
to simulate the retinal processing stage; and third, all connections are rectified to
be excitatory, like direct afferent input to simple cells.
2.1
Overview
We model the neuron as an array of non-negative synapses, distributed within a
circular region. To let the neuron "see" a single pattern in the visual environment
(see Figure 1, end of text), the array is overlaid on a much larger positive array (the
filtered image), which represents the environment. Each synapse value is multiplied
by its corresponding input pixel, and the sum of these products forms the neuron's
"output." If the output is above a threshold value, each synapse is changed slightly
to make it more like its corresponding pixel (the synapse is increased for a positive
pixel, and decreased for a zero pixel.) If the output is low, nothing is changed. This
process implements the correlation-based ("Hebbian") learning rule for synapse
modification. To ensure maturation, we presented roughly one million training
images to each neuron. Because there are many filtered images, only one is chosen
at random for each iteration, and the neuron is overlapped at some random spatial
offset.
2.2
Input Filtering Process
The visual environment is a collection of N black-on-white pictures of a single shape
(such as straight lines), at fixed contrast. The environment seen by the neuron is
a set of N filtered images, whose non-negative elements are produced from the
pictures by a rectified, Laplace-like, center-surround process similar to that of the
mammalian retina (Van Essen &, Anderson 1988). To determine the RF of a mature
array of synapses, the combined efficacy of all synapses is calculated for each pixel,
and displayed as a grey scale (white
excitatory, black
inhibitory). See Figure
2, at end of text, for several examples of mature RF's.
=
2.3
=
Plasticity Under Visual Stimulation
The neuron's input synapses cover a circle much smaller than the filtered image. A
single exposure to the environment overlaps the synapse array at a random position
on the input image (chosen randomly from the training set). This overlap pairs
each synapse with an input from a filter whose center has like polarity (on or off),
so that each synapse represents a definite polarity of retinal cell.
A typical run involves perhaps 10 6 exposures. There is no time variable, so that
motion and temporal correlations between images are entirely absent. During each
exposure a Hebb rule (section 2.4) changes synaptic weights based on current cell
output and input values. When the neuron is exposed to filtered stochastic input
("noise-rearing"), synapses are intitialized randomly. When the neuron is exposed
to structured environments, synapses are initialized with the orderly synapse arrays
which result from noise-rearing. (As in animals, synapses may evolve in response to
filtered random input before they are exposed to the external environment.)
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2.4
A Choice of Hebb Rules for Learning Plasticity
Hebb postulated (1949) that neurons modify their synapses according to the following rule: the synapse will increase in efficacy if the post-synaptic and presynaptic excitations are coincident. There are many different formulae which satisfy
Hebb's criterion; this model explores some simple representative ones. During each
exposure to input, the synapses are adjusted according to the following type of
hard-limited Hebb rule:
(1)
out
And if (out - thresh)
>0:
(out - thresht
X
ini
X
growth
if ini > 0 and syni < 10
-(out - thresh)" X decay
=
if ini 0 and syni
o otherwise
(2)
(3)
> 0.5
(4)
The constants growth and decay are positive, and the exponent n is at least one.
Both types of threshold depend on the neuron's recent output history: either the
average of the previous 200 outputs, or one half the maximum previous output
(decaying by .9995 each exposure until a new m;~ exceeds it) . This Hebb Rule
assumes that the cell can detect the current input value before its modification by
a synapse.
2.5
Choice of Parameters
The constants growth and decay are not sensitive parameters. We found that only
three parameter regimes exist: all synapses saturate at maximum, all saturate at
minimum, or some at maximum and some at minimum. Only the latter regime is
of interest, because only it contains structured RF's.
Most simulations used n = 1,2,3 with both thresholds. The threshold based on
maximum output enhances learning selectivity, while the averaged output version
can be derived from a principle of "excess information" (See Appendix) . Because
simple cell RF's have approximately Gaussian envelopes (Jones & Palmer 1987),
some simulations were done with Gaussian envelopes modulating the maximum
synapse values. That modification made no difference in the results observed.
3
Results and Discussion
The production of oriented RFs during exposure to unstructured input confirms
previous results by Linsker (1986) and Yuille et al. (1989), but with some important differences. Like those models, the neurons simulated here learn oriented
stripe-patterns as a kind of lowest-energy configuration under exposure to spatially
Can Simple Cells Learn Curves? A Hebbian Model in a Structured Environment
correlated inputs. But unlike those models, we do not use: inhibitory connections
or synapses; a synaptic-density gradient; a global conservation of synapse strength;
or adjustable free parameters which can yield differently-shaped RFs. (In Linsker
1986 the ratio of "on" to "off" synapses is art adjustable parameter; here, on and
off pixels are represented equally.) Also, unlike previous models, mature RF's could
have more than 3 lobes, depending on the ratio of filter size to RF size (Figure 2).
Under exposure to images of bars at all orientations, the neuron developed a mature
RF matching a single one of them. Under exposure to stripes of nearly a single
orientation, development of a mature RF depended on the stripes' spatial frequency.
In all cases, input patterns were learned much more quickly and strongly when
their spatial frequency corresponded to the frequency of the Laplace filters. For
input frequencies near the filter frequency, the resulting RF had a spatial frequency
intermediate between the two. Otherwise, no learning occured unless the input
frequency was a harmonic of the filter frequency, in which case the filter frequency
was learned. Thus, this model predicts that enhanced learning might take place
in kittens exposed to stripes of a single frequency, if that frequency is typical of
simple-cell RF frequencies.
Under exposure to arcs or circles (with diameter ~ 3 x annular width), the model
consistently developed RF's which matched a portion of the circle. These results
suggest that animals which see only circles of a certain scale during the critical
period may develop curved RFs (Barrow 1987) which differ qualitatively from those
observed by such experiments as Jones &, Palmer's (1987), who report seeing no
curved contours in their point-by-point mappings of the RFs of normally-reared
kittens. As with the stripes, the circles' annular width determines the spatial frequency of the retinal and simple cells which will respond best.
Such predictions must be treated with caution, because this paper does not simulate
any version of the competing "selectional" model. It is possible that some of the
effects predicted here for the "instructional" Hebbian model could also be observed
by a "selectional" system.
To experimentally observe such effects in laboratory animals, many other known
biological influences (eye acuity, interneuron effects, etc.) must be accounted for.
We consider such problems elsewhere (Softky &, Kammen in preparation), because
they are of secondary importance to the striking and robust results of the model.
In summary, we have a single-cell model which contains essential biological features
(such as all-excitatory input and synapses, and no global renormalizations). This
model developes mature, oriented receptive fields under exposure to stochastic input
for a wide variety of Hebb rules and for all non-trivial parameter regimes studied,
with no apparent limitations on the number of lobes learned. Under exposure
to structured input characteristic of normal environments, the model maintains
oriented RF's; under exposure to input of "resonant" spatial frequency, the model
develops RF's which reflect any novel orientation, spatial frequency, or curvature
of the stimuli. This general, rule-independent response to the spatial frequency of
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Softky and Kammen
a stimulus - and the specific mechanism for generating abnormally curved RF's may be useful in deciding experimentally whether simple cortical cells are indeed
modifiable by Hebbian mechanisms.
This model does not attempt to explain curve-detection in a normal visual system.
We already know that normal simple cells are not tuned for curves, and there are
credible theories of normal curve-detection (Dobbins et al. 1987.) Rather, this
model proposes using stimuli tuned to the natural spatial frequency of simple cells
to induce a RF property which is distinctly abnormal, in order to better understand
the rules by which normal visual properties emerge.
4
Appendix - Choice of Thresholds for the Hebb Rule
The choice of the average output as a threshold for a Hebb rule can be interpreted
as follows. Consider a developing neuron whose output is the sum of N inputs,
each of which has independent probability distribution of mean a and standard
deviation u. We can calculate the information content in that sum, whose value
has probability distribution (from the central limit theorem) of
P(out)
exp (
oc
-(out - (out))2)
2u 2
?
(5)
The Shannon information (Shannon & Weaver 1962) carried by the sequence is
H( event)
-In P( event) .
(6)
The excess information above the information carried by the average is thus
oc
H(out) - H( < out?
(out - (out) )2
2u 2
(7)
(8)
Thus, a Hebb rule using n = 2 and thresh = (out) is equivalent to learning based
on the excess information carried in the output of an immature neuron.
The alternate threshold ( tmax) enhances selective learning for the following reason.
If we consider the whole ensemble of patterns and shifts, the output characteristic
which best distinguishes a matched synapse pattern from a random one is not its
average output (the two averages are comparable for the all-excitatory case), but
its maximum output. Thus, if a neuron can only 'remember' one characteristic
number to serve as a threshold, then a number which changes during evolution
(e.g. the maximum output) will refine selectivity more than one which is relatively
constant. In addition, storing a maximum rather than an average removes the
need to compute a running average, allowing unhindered evolution even after long
periods of no input.
Can Simple Cells Learn Curves? A Hebbian Model in a Structured Environment
Acknowledgements
D.K. is a Weizmann Postdoctoral Fellow and acknowledges support from the Weizmann Foundation, the James S. McDonnell Foundation and a NSF Presidential
Young Investigator A ward to Christof Koch.
References
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1)
3) Filter retinal
field, then expose
the filtered image
to the neuron's
synapses.
Calculate neuron's
output, then adjust
synaptic weights
according to Hebb
rule.
2) Place retinal field at a
random location on the image.
Figure 1: Synapses Change Slightly During Each of a Million Iterations
Figure 2: Learned Receptive Fields. Top row: Random pixel input, large (1) and
small (r) filter sizes. Bottom row: Structured input, circular rings (1) and edges at
differen t orientations (r).
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1,673 | 2,520 | Mutual Boosting for
Contextual Inference
Michael Fink
Center for Neural Computation
Hebrew University of Jerusalem
Jerusalem, Israel 91904
[email protected]
Pietro Perona
Electrical Engineering Department
California Institute of Technology
Pasadena, CA 91125
[email protected]
Abstract
Mutual Boosting is a method aimed at incorporating contextual
information to augment object detection. When multiple detectors
of objects and parts are trained in parallel using AdaBoost [1],
object detectors might use the remaining intermediate detectors to
enrich the weak learner set. This method generalizes the efficient
features suggested by Viola and Jones [2] thus enabling
information inference between parts and objects in a compositional
hierarchy. In our experiments eye-, nose-, mouth- and face
detectors are trained using the Mutual Boosting framework. Results
show that the method outperforms applications overlooking
contextual information. We suggest that achieving contextual
integration is a step toward human-like detection capabilities.
1
In trod u ction
Classification of multiple objects in complex scenes is one of the next challenges
facing the machine learning and computer vision communities. Although, real-time
detection of single object classes has been recently demonstrated [2], na?ve
duplication of these detectors to the multiclass case would be unfeasible. Our goal is
to propose an efficient method for detection of multiple objects in natural scenes.
Hand-in-hand with the challenges entailing multiclass detection, some distinct
advantages emerge as well. Knowledge on position of several objects might shed
light on the entire scene (Figure 1). Detection systems that do not exploit the
information provided by objects on the neighboring scene will be suboptimal.
A
B
Figure 1: Contextual spatial relationships assist detection A. in absence of facial
components (whitened blocking box) faces can be detected by context (alignment of
neighboring faces). B. keyboards can be detected when they appear under monitors.
Many human and computer vision models postulate explicitly or implicitly that
vision follows a compositional hierarchy. Grounded features (that are
innate/hardwired and are available prior to learning) are used to detect salient parts,
these parts in turn enable detection of complex objects [3, 4], and finally objects are
used to recognize the semantics of the entire scene. Yet, a more accurate assessment
of human performance reveals that the visual system often violates this strictly
hierarchical structure in two ways. First, part and whole detection are often
evidently interacting [5, 6]. Second, several layers of the hierarchy are occasionally
bypassed to enable swift direct detection. This phenomenon is demonstrated by gist
recognition experiments where the semantic classification of an entire scene is
performed using only minimal low level feature information [7].
The insights emerging from observing human perception were adopted by the object
detection community. Many object detection algorithms bypass stages of a strict
compositional hierarchy. The Viola & Jones (VJ) detector [2] is able to perform
robust online face detection by directly agglomerating very low-level features
(rectangle contrasts), without explicitly referring to facial parts. Gist detection from
low-level spatial frequencies was demonstrated by Oliva and Torralba [8]. Recurrent
optimization of parts and object constellation is also common in modern detection
schemes [9]. Although Latent Semantic Analysis (making use of object cooccurrence information) has been adapted to images [10], the existing state of object
detection methods is still far from unifying all the sources of visual contextual
information integrated by the human perceptual system. Tackling the context
integration problem and achieving robust multiclass object detection is a vital step
for applications like image-content database indexing and autonomous robot
navigation.
We will propose a method termed Mutual Boosting to incorporate contextual
information for object detection. Section 2 will start by posing the multiclass
detection problem from labeled images. In Section 3 we characterize the feature sets
implemented by Mutual Boosting and define an object's contextual neighborhood.
Section 4 presents the Mutual Boosting framework aimed at integrating contextual
information and inspired by the recurrent inferences dominating the human
perceptual system. An application of the Mutual Boosting framework to facial
component detection is presented in Section 5. We conclude with a discussion on
the scope and limitations of the proposed framework.
2
Problem setting and basic notation
Suppose we wish to detect multiple objects in natural scenes, and that these scenes
are characterized by certain mutual positions between the composing objects. Could
we make use of these objects' contextual relations to improve detection? Perceptual
context might include multiple sources of information: information originating from
the presence of existing parts, information derived from other objects in the
perceptual vicinity and finally general visual knowledge on the scene. In order to
incorporate these various sources of visual contextual information Mutual Boosting
will treat parts, objects and scenes identically. We will therefore use the term object
as a general term while referring to any entity in the compositional hierarchy.
Let M denote the cardinality of the object set we wish to detect in natural scenes.
Our goal is to optimize detection by exploiting contextual information while
maintaining detection time comparable to M individual detectors trained without
such information. We define the goal of the multiclass detection algorithm as
generating M intensity maps Hm=1,..,M indicating the likelihood of object m appearing
at different positions in a target image.
We will use the following notation (Figure 2):
?
H0+/H0-: raw image input with/without the trained objects (A1 & A2)
?
Cm[i]: labeled position of instance i of object m in image H0+
?
Hm: intensity map output indicating the likelihood of object m appearing in
different positions in the image H0 (B)
B.
Hm
A2.
H0-
A1.
H0+
Cm[1] Cm[2]
Cm[1] Cm[2]
Figure 2: A1 & A2. Input: position of positive and negative examples of eyes in
natural images. B. Output: Eye intensity (eyeness) detection map of image H0+
3
F e a t u r e se t a n d c o n t e x t u a l wi n d o w g e n e r a l i za t i o n s
The VJ method for real-time object-detection included three basic innovations.
First, they presented the rectangle contrast-features, features that are evaluated
efficiently, using an integral-image. Second, VJ introduced AdaBoost [1] to object
detection using rectangle features as weak learners. Finally a cascade method was
developed to chain a sequence of increasingly complex AdaBoost learners to enable
rapid filtering of non-relevant sections in the target image. The resulting cascade of
AdaBoost face detectors achieves a 15 frame per second detection speed, with 90%
detection rate and 2x10-6 false alarms. This detection speed is currently unmatched.
In order to maintain efficient detection and in order to benchmark the performance
of Mutual Boosting we will adopt the rectangle contrast feature framework
suggested by VJ.
It should be noted that the grayscale rectangle features could be naturally extended
to any image channel that preserves the semantics of summation. A diversified
feature set (including color features, texture features, etc.) might saturate later than
a homogeneous channel feature set. By making use of features that capture the
object regularities well, one can improve performance or reduce detection time.
VJ extract training windows that capture the exact area of the training faces. We
term this the local window approach. A second approach, in line with our attempt to
incorporate information from neighboring parts or objects, would be to make use of
training windows that capture wide regions around the object (Figure 3)1.
A
B
Figure 3: A local window (VJ) and a contextual window that captures relative
position information from objects or parts around and within the detected object.
1
Contextual neighborhoods emerge by downscaling larger regions in the original image
to a PxP resolution window.
The contextual neighborhood approach contributes to detection when the applied
channels require a wide contextual range as will be demonstrated in the Mutual
Boosting scheme presented in the following section2.
4
Mutual Boosting
The AdaBoost algorithm maintains a clear distinction between the boosting level
and the weak-learner training level. The basic insight guiding the Mutual Boosting
method reexamines this distinction, stipulating that when multiple objects and parts
are trained simultaneously using AdaBoost; any object detector might combine the
previously evolving intermediate detectors to generate new weak learners. In order
to elaborate this insight it should first be noted that while training a strong learner
using 100 iterations of AdaBoost (abbreviated AB100) one could calculate an
intermediate strong learner at each step on the way (AB2 - AB99). To apply this
observation for our multiclass detection problem we simultaneously train M object
detectors. At each boosting iteration t the M detectors (ABmt-1) emerging at the
previous stage t-1, are used to filter positive and negative3 training images, thus
producing intermediate m-detection maps Hmt-1 (likelihood of object m in the
images4). Next, the Mutual Boosting stage takes place and all the existing Hmt-1
maps are used as additional channels out of which new contrast features are
selected. This process gradually enriches the initial grounded features with
composite contextual features. The composite features are searched on a PxP wide
contextual neighborhood region rather than the PxP local window (Figure 3).
Following a dynamic programming approach in training and detection, Hm=1,..,M
detection maps are constantly maintained and updated so that the recalculation of
Hmt only requires the last chosen weak learner WLmn*t to be evaluated on channel
Hn*t-1 of the training image (Figure 4). This evaluation produces a binary detection
layer that will be weighted by the AdaBoost weak-learner weighting scheme and
added to the previous stage map5.
Although Mutual Boosting examines a larger feature set during training, an iteration
of Mutual Boosting detection of M objects is as time-consuming as performing an
AdaBoost detection iteration for M individual objects. The advantage of Mutual
Boosting emerges from introducing highly informative feature sets that can enhance
detection or require fewer boosting iterations. While most object detection
applications extract a local window containing the object information and discard
the remaining image (including the object positional information). Mutual Boosting
processes the entire image during training and detection and makes constant use of
the information characterizing objects? relative-position in the training images.
As we have previously stated, the detected objects might be in various levels of a
compositional hierarchy (e.g. complex objects or parts of other objects).
Nevertheless, Mutual Boosting provides a similar treatment to objects, parts and
scenes enabling any compositional structure of the data to naturally emerge. We will
term any contextual reference that is not directly grounded to the basic features, as a
cross referencing of objects6.
2
The most efficient size of the contextual neighborhoods might vary, from the immediate
to the entire image, and therefore should be empirically learned.
3
Images without target objects (see experimental section below)
4
Unlike the weak learners, the intermediate strong learners do not apply a threshold
5
In order to optimize the number of detection map integral image recalculations these
maps might be updated every k (e.g. 50) iterations rather than at each iteration.
6
Scenes can be crossed referenced as well if scene labels are available (office/lab etc.).
H0+/0- positive / negative raw images
Cm[i] position of instance i of object m=1,..,M in image H0+
initialize boosting-weights of instances i of object m to 1
initialize detection maps Hm+0/Hm-0 to 0
Input
Initialization
For t=1,?,T
For m=1,..,M and n=0,..,M
(A) cutout & downscale local (n=0) or contextual (n>0) windows (WINm)
of instances i of object m (at Cm[i]), from all existing images Hnt-1
For m=1,..,M
normalize boosting-weights of object m instances [1]
(B1&2) select map Hn*t-1 and weak learner WLmn* that minimize error on WINm
decrease boosting-weights of instances that WLmn* labeled correctly [1]
(C) DetectionLayermn* ? WLmn*(Hn*t-1)
calculate ? mt the weak learner contribution factor from the empirical error [1]
(D) update m-detection map Hmt ? Hmt-1 + ?mt DetectionLayermn *
Return strong learner ABmT including WLmn*1,..,T and ?m1,..,T (m=1,..,M)
H0? raw image
H1?
. . . Hn*?
(A)
WIN
m0
WL
m0
(B1)
. . . Hm?
(A)
WIN
m1
(B2)
WL
m1
(B1)
(B2)
m
detection
map
(A)
WIN
mn*
WL
(B1)
(D)
(C)
Detection
Layer
mn*
mn*
Figure 4: Mutual Boosting Diagram & Pseudo code. Each raw image H0 is analyzed
by M object detection maps Hm=1,.,M, updated by iterating through four steps: (A)
cutout & downscale from existing maps H n=0,..,M t-1 a local (n=0) or contextual (n>0)
PxP window containing a neighborhood of object m (B1&2) select best performing
map Hn* and weak learner WLmn* that optimize object m detection (C) run WLmn* on
Hn* map to generate a new binary m-detection layer (D) add m-detection layer to
existing detection map Hm. [1] Standard AdaBoost stages are not elaborated
To maintain local and global natural scene statistics, negative training examples are
generated by pairing each image with an image of equal size that does not contain
the target objects and by centering the local and contextual windows of the positive
and negative examples on the object positions in the positive images (see Figure 2).
By using parallel boosting and efficient rectangle contrast features, Mutual Boosting
is capable of incorporating many information inferences (references in Figure 5):
? Features could be used to directly detect parts and objects (A & B)
? Objects could be used to detect other (or identical) objects in the image (C)
? Parts could be used to detect other (or identical) nearby parts (D & E)
? Parts could be used to detect objects (F)
? Objects could be used to detect parts
A. eye
feature
from
raw
image
B. face
feature
from
raw
image
C. face
E. mouth
feature
from eye
feature
from face
detection
image
detection
image
F. face
feature
from mouth
D. eye
feature
from eye
detection
image
detection
image
Figure 5: A-E. Emerging features of eyes, mouths and faces (presented on windows
of raw images for legibility). The windows? scale is defined by the detected object
size and by the map mode (local or contextual). C. faces are detected using face
detection maps HFace, exploiting the fact that faces tend to be horizontally aligned.
5
Experiments
A.
Pd
In order to test the contribution of the Mutual Boosting process we focused on
detection of objects in what we term a face-scene (right eye, left eye, nose, mouth
and face). We chose to perform contextual detection in the face-scene for two main
reasons. First as detailed in Figure 5, face scenes demonstrate a range of potential
part and object cross references. Second, faces have been the focus of object
detection research for many years, thus enabling a systematic result comparison.
Experiment 1 was aimed at comparing the performance of Mutual Boosting to that
of na?ve independently trained object detectors using local windows.
Pfa
Figure 6: A. Two examples of the CMU/MIT face database. B. Mutual Boosting and
AdaBoost ROCs on the CMU/MIT face database.
Face-scene images were downloaded from the web and manually labeled7. Training
relied on 450 positive and negative examples (~4% of the images used by VJ). 400
iterations of local window AdaBoost and contextual window Mutual Boosting were
performed on the same image set. Contextual windows encompassed a region five
times larger in width and height than the local windows8 (see Figure 3).
7
By following CMU database conventions (R-eye, L-eye, Nose & Mouth positions) we
derive both the local window position and the relative position of objects in the image
8
Local windows were created by downscaling objects to 25x25 grids
Test image detection maps emerge from iteratively summing T m-detection layers
(Mutual Boosting stages C&D). ROC performance on the CMU/MIT face database
(see sample images in Figure 6A) was assessed using a threshold on position Cm[i]
that best discriminated the final positive and negative detection maps Hm+/-T. Figure
6B demonstrates the superiority of Mutual Boosting to grounded feature AdaBoost.
A.
COV
0.25
COV
1.00
COV
4.00
Equal error performance
Our second experiment was aimed at assessing the performance of Mutual Boosting
as we change the detected configurations? variance. Assuming normal distribution
of face configurations we estimated (from our existing labeled set) the spatial
covariance between four facial components (noses, mouths and both eyes). We then
modified the covariance matrix, multiplying it by 0.25, 1 or 4 and generated 100
artificial configurations by positioning four contrasting rectangles in the estimated
position of facial components. Although both Mutual Boosting and AdaBoost
performance degraded as the configuration variance increased, the advantage of
Mutual Boosting persists both in rigid and in varying configurations9 (Figure 7).
MB sigma=0.25
MB sigma=1.00
MB sigma=4.00
AB sigma=0.25
AB sigma=1.00
AB sigma=4.00
Boosting iteration
Figure 7: A. Artificial face configurations with increasing covariance B. MB and
AB Equal error rate performance on configurations with varying covariance as a
function of boosting iterations.
6
D i s c u s s i on
While evaluating the performance of Mutual Boosting it should be emphasized that
we did not implement the VJ cascade approach; therefore we only attempt to
demonstrate that the power of a single AdaBoost learner could be augmented by
Mutual Boosting. The VJ detector is rescaled in order to perform efficient detection
of objects in multiple scales. For simplicity, scale of neighboring objects and parts
was assumed to be fixed so that a similar detector-rescaling approach could be
followed. This assumption holds well for face-scenes, but if neighboring objects
may vary in scale a single m-detection map will not suffice. However, by
transforming each m-detection image to an m-detection cube, (having scale as the
third dimension) multi-scale context detection could be achieved10. The dynamic
programming characteristic of Mutual Boosting (simply reusing the multiple
position and scale detections already performed by VJ) will ensure that the running
time of varying scale context will only be doubled. It should be noted that the facescene is highly structured and therefore it is a good candidate for demonstrating
9
In this experiment the resolution of the MB windows (and the number of training
features) was decreased so that information derived from the higher resolution of the
parts would be ruled out as an explaining factor for the Mutual Boosting advantage. This
procedure explains the superior AdaBoost performance in the first boosting iteration.
10
By using an integral cube, calculating the sum of a cube feature (of any size) requires 8
access operations (only double than the 4 operations required in the integral image case).
Mutual Boosting; however as suggested by Figure 7B Mutual Boosting can handle
highly varying configurations and the proposed method needs no modification when
applied to other scenes, like the office scene in Figure 111. Notice that Mutual
Boosting does not require a-priori knowledge of the compositional structure but
rather permits structure to naturally emerge in the cross referencing pattern (see
examples in Figure 5).
Mutual Boosting could be enhanced by unifying the selection of weak-learners
rather than selecting an individual weak learner for each object detector. Unified
selection is aimed at choosing weak learners that maximize the entire object set
detection rate, thus maximizing feature reuse [11]. This approach is optimal when
many objects with common characteristics are trained.
Is Mutual Boosting specific for image object detection? Indeed it requires labeled
input of multiple objects in a scene supplying a local description of the objects as
well as information on their contextual mutual positioning. But these criterions are
shared by other complex "scenes". DNA sequences include multiple objects (Genes)
in mutual positions, and therefore might be handled by a variant of Mutual
Boosting. The remarkable success of the VJ method stems from abandoning the use
of highly custom-tailored complex features in favor of numerous simple ones.
Mutual Boosting combines parallel boosting, with a similar feature approach to
efficiently incorporate contextual information. We suggest that achieving wide
contextual integration is one step towards human-like object detection capabilities.
References
[1] Freund, Y. and Schapire, R. E. (1997) A Decision-Theoretic Generalization of On-Line
Learning and an Application to Boosting. JCSS 55(1): 119-139
[2] Viola, V. P. and Jones M. (2001) Robust real-time object detection. IEEE ICCV
Workshop on Stat. and Comp. Theories of Vision , Vancouver, Canada, July 13 2001
[3] Tanaka, K., Saito, H., Fukada, Y. and Moriya, M. (1991) Coding visual images of objects
in the inferotemporal cortex of the macaque monkey. J. Neurophys. 66:170-189
[4] Biederman, I. (1987). Recognition-by-components: A theory of human image
understanding. Psychological Review, 94, 115?147.
[5] Navon, D. (1977). Forest before trees: The precedence of global features in visual
perception. Cog. Psych. 9, 353-383.
[6] Biederman, I., Mezzanotte, R. J., & Rabinowitz, J. C. (1982). Scene perception:
Detecting the judging objects undergoing relational violations. Cog. Psych. 14, 143?177
[7] Biederman, I. (1981). On the semantics of a glance at a scene. In M. Kubovy, & J. R.
Pomerantz, Perceptual organization (pp. 213?253). Hillsdale, NJ: Erlbaum.
[8] Oliva, A., Torralba, A. B. (2002) Scene-Centered Description from Spatial Envelope
Properties. Biologically Motivated Computer Vision 2002: 263-272
[9] Weber, M., Welling, M., & Perona, P. (2000) Unsupervised Learning of Models for
Recognition. ECCV (1) 2000: 18-32
[10] Barnard K. and Forsyth D. (2001) Learning the semantics of words and pictures. In
IEEE ICCV, volume 2, pages 408--415, Vancouver, Canada, July 2001
[11] Schapire, R. E. and Singer. Y. (2000) Boostexter: A boosting-based system for text
categorization. Machine Learning, 39(2-3):135--168, May/June 2000.
11
MB is currently aimed at detecting objects in office-scenes (Caltech 360? office DB)
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1,674 | 2,521 | Necessary Intransitive Likelihood-Ratio
Classifiers
Gang Ji and Jeff Bilmes
SSLI-Lab, Department of Electrical Engineering
University of Washington
Seattle, WA 98195-2500
{gang,bilmes}@ee.washington.edu
Abstract
In pattern classification tasks, errors are introduced because of differences between the true model and the one obtained via model estimation.
Using likelihood-ratio based classification, it is possible to correct for this
discrepancy by finding class-pair specific terms to adjust the likelihood
ratio directly, and that can make class-pair preference relationships intransitive. In this work, we introduce new methodology that makes necessary corrections to the likelihood ratio, specifically those that are necessary to achieve perfect classification (but not perfect likelihood-ratio
correction which can be overkill). The new corrections, while weaker
than previously reported such adjustments, are analytically challenging
since they involve discontinuous functions, therefore requiring several
approximations. We test a number of these new schemes on an isolatedword speech recognition task as well as on the UCI machine learning
data sets. Results show that by using the bias terms calculated in this
new way, classification accuracy can substantially improve over both the
baseline and over our previous results.
1
Introduction
Statistical pattern recognition is often based on Bayes decision theory [4], which aims to
achieve minimum error rate classification. In previous work [2], we observed that multiclass Bayes classification can be viewed as a tournament style game, where the winner
between players is decided using log likelihood ratios. Supposing the classes (players) are
{c1 , c2 , ? ? ? , cM }, and the observation (game) is x, the winner of each pair of classes is
determined, with the assumption of equal priors, by the sign of the log likelihood ratio
(x|ci )
Lij (x) = ln PP (x|c
, in which case if Lij > 0 class ci wins and otherwise class cj wins.
j)
A practical game strategy can be obtained by fixing a comparison order, {i1 , i2 , ? ? ? , iM },
as a permutation of {1, 2, ? ? ? , M }, where class ci1 plays with class ci2 , the winner plays
with class ci3 , and so on until a final winner is ultimately found. This yields a transitive
game [8] ? assuming no ties, the ultimate winner is identical regardless of the comparison
order.
To perform these procedures optimally, correct likelihood ratios are needed, which requires
correct probabilistic models and sufficient training data. This is never the case given a fi-
nite amount of training data or the wrong model family, typical in practice. In previous
work [2], we introduced a method to correct for the difference between the true and an
approximate log likelihood ratio. In this work, we improve upon the correction method by
using an expression that can still lead to perfect correction, but is weaker than what we used
before. We show that this new condition can achieve a significant improvement over baseline results, both on a medium vocabulary isolated-word automatic speech recognition task
and on the UCI machine learning data sets. The paper is organized as follows: Section 2
describes the general scheme and describes past work. Section 3 discusses the weaker correction condition, and its approximations. Section 4 provides various experimental results
on an isolated-word speech recognition task. Section 5 contains the experimental results
on the UCI data. Finally, Section 6 concludes.
2
Background
A common problem in many probabilistic machine learning settings is the lack of a correct statistical model. In a generative pattern classification setting, this occurs because
only an estimated quantity P? (x|c)1 of a distribution is available, rather than the true classconditional model P (x|c). In the likelihood ratio decision scheme described above, only
? ij (x) = ln(P? (x|ci )/P? (x|cj )), is available for decision
an imperfect log likelihood ratio, L
making rather than the true log likelihood ratio Lij (x).
One approach to correct for this inaccuracy is to use richer class conditional likelihoods,
more complicated parametric forms of Lij (x) itself, and/or more training data. In previous
work [2], we proposed a different approach that requires no change in generative models,
no increase in free parameters, and no additional training data but still yields improved
? ij (x)
accuracy. The key idea is to compensate for the difference between Lij (x) and L
2
using a bias term ?ij (x) computed from test data such that:
? ij (x).
Lij (x) ? ?ij (x) = L
(1)
If it is assumed that a single bias term is used for all data, so that ?ij (x) = ?ij , we found
that the best ?ij is as follows:
1
1?
?
?ij = (D(ikj) ? D(jki)) ?
D(ikj) ? D(jki)
,
(2)
2
2
where D(ikj) = EP (x|ci ) ln Lij (x) is the Kullback-Leibler (KL) divergence [3] between
? ij (x) is its estimation. Under the assumption
?
P (x|ci ) and P (x|cj ) and D(ikj)
= EP (x|ci ) L
(referred to as assumption A in Section 3.1) of symmetric KL-divergence for the true model
(e.g., equal covariance matrices in the Gaussian case), the bias term can be solved explicitly
as
1?
?
?ij = ?
D(ikj) ? D(jki)
.
(3)
2
? ij (x) + ?ij can lead to an inWe saw how the augmented likelihood ratio Sij (x) = L
transitive game [8, 13], since Sij (x) can specify intransitive preferences amongst the set
{1, 2, ? ? ? , M }. We therefore investigated a number of intransitive game playing strategies. Moreover, we observed that if the correction was optimal, the true likelihood ratios
would be obtained which are clearly transitive. We therefore hypothesized and experimentally verified that the existence of intransitivity was a good indicator of the occurrence of a
classification error.
This general approach can be improved upon in several ways. First, better intransitive
strategies can be developed (for detecting, tolerating, and utilizing the intransitivity of a
1
2
In this paper, we use ?hatted? letters to describe estimated quantities.
Note that by bias, we do not mean standard parameter bias in statistical parameter estimation.
classifier); second, the assumption of symmetric KL-divergence could be relaxed; and third,
the above criterion is stricter than required to obtain perfect correction. In this work, we
advance on the latter two of the above three possible avenues for improvement.
3
Necessary Intransitive Scheme
An ?ij (x) that solves Equation 1 is a sufficient condition for a perfect correction of the
estimated likelihood ratio since given such a quantity, the true likelihood ratio would be
attainable. This condition, however, is stricter than required because it is only the sign of
the likelihood ratio that is needed to decide the winning class. We therefore should ask for
a condition that corrects only for the discrepancy in sign between the true and estimated
ratio, i.e., we want to find a function ?ij (x) that minimizes
Z n
o2
? ij (x) ? Pij (x) dx.
J[?ij ] =
sgn [Lij (x) ? ?ij (x)] ? sgnL
Rn
Clearly the ?ij (x) that minimizes J[?ij ] is the one such that
? ij (x),
sgn [Lij (x) ? ?ij (x)] = sgnL
?x ? suppPij = {x : Pij (x) 6= 0}.
(4)
As can be seen, this condition is weaker than Equation 1, weaker in the sense that any
solution to Equation 1 solves Equation 4 but not vice versa. Note also that Equation 4
provides necessary conditions for an additive bias term to achieve perfect correction, since
any such correction must achieve parity in the sign. Therefore, it might make it simpler
to find a better bias term since Equation 4 (and therefore, set of possible ? values) is less
constrained. As will be seen, however, analysis of this weaker condition is more difficult.
In the following sections, therefore we introduce several approximations to this condition.
Note that as in previous work, we henceforth assume ?ij (x) = ?ij is a constant. In this
case, the equation providing the best ?ij values is:
n
o
? ij (x) .
EPij {sgn [Lij (x) ? ?ij ]} = EPij sgnL
(5)
3.1
The difficulty with the sign function
The main problem in trying to solve for ?ij in Equation 5 is the existence of a discontinuous
function. In this section, therefore, we work towards obtaining an analytically tractable
approximation. The {?1, 0, 1}-valued sign function sgn(z) is defined as 2u(z) ? 1, where
u(z) is the Heaviside step function. We obtain an approximation via a Taylor expansion as
follows:
sgn(z + ) = sgn(z) + sgn0 (z) + o() = sgn(z) + 2?(z) + o(),
(6)
where ?(z) is the Dirac delta function [7]. It can be defined as the derivative
of the HeavZ
iside step function u0 (z) = ?(z), and it satisfies the sifting property
f (z0 ). Therefore, it follows that [6, page 263]
Z
Z
f (z)?[g(z)] dz =
Rn
Zg
f (z)?(z ? z0 ) =
R
f (z)
? d?,
|?g(z)|
where ?g is the gradient of g and Zg = {z ? Rn : g(z) = 0} is the zero set of g with
Lebesgue measure ? [12].
Of course, the Taylor expansion is valid only for a differentiable function, otherwise the
error terms can be arbitrarily large. If, however, we find and use a suitable continuous and
differentiable approximation rather than the discrete sign function, the above expansion
becomes more appropriate. There exists a trade-off, however, between the quality of the
sign function approximation (a better sign function should yield a better approximation in
Equation 4) and the error caused by the o() term in Equation 6 (a better sign function
approximation will have a greater error when the higher-order Taylor terms are dropped).
We therefore expect that ideally there will exist an optimal balance between the two. The
shifted sigmoid with free parameter ? (defined and used below) allows us to easily explore
this trade-off simply by varying ?.
Retaining the first-order Taylor term, and applying this to the left side of Equation 5,
EPij sgn [Lij (x) ? ?ij ] ? EPij sgnLij (x) ? 2EPij ?ij ? [Lij (x)] .
The distribution under which the expectation in Equation 5 is taken can also influence our
results. If it is known that the true class of x is always ci , the ci -conditional distribution
(i)
should be used, i.e., Pij (x) = P (x|ci ), yielding a class-conditional correction term ?ij ,
(i)
(i)
? ij (x)+? . The symmetric
and a class-conditional likelihood-ratio correction Sij (x) = L
ij
case arises when x is of class cj . If, on the other hand, neither ci nor cj is the true classes
(i.e., x is sampled from some other class-conditional distribution, say P (x|ck ), k 6= i, j),
it does not matter which distribution for Pij (x) is used since, for a given comparison order
in a game playing strategy, the current winner will ultimately play using the true class
distribution P (x|ck ) of x (when one of i or j will equal k). It is therefore valid to consider
only the case when either x is of class ci (we denote this event by Ci (x)) or when x is of
class cj (event Cj (x)). Note that these two events are disjoint.
In practice, however, we do not know which of the two events is correct. The ideal choice
in either case can be expressed using indicators as follows:
(i)
(j)
Aij (x) = ?ij 1{Ci (x)} + ?ij 1{Cj (x)} .
Taking the expected value of Aij (X) with respect to p(x|Ci (x) ? Cj (x)) yields
(i)
(j)
?ij P (ci ) + ?ij P (cj )
?ij = Ep(x|Ci (x)?Cj (x)) [Aij (X)] =
.
P (ci ) + P (cj )
? ij (x) + ?ij that is obtained simply
This results in a single likelihood correction Sij (x) = L
by integrating in Equation 5 with respect to the average distribution over class ci and cj ,
i.e.,
P (ci )P (x|ci ) + P (cj )P (x|cj )
?
Pij (x) = p(x|Ci (x) ? Cj (x)) =
.
P (ci ) + P (cj )
With these assumptions, and supposing the zero set ZLij = {x ? Rn : P (x|ci ) =
P (x|cj )} of Lij (x) is Lebesgue measurable with measure ?, we get:
Z
Z
{sgnLij (x) ? 2?ij ? [Lij (x)]} Pij (x) dx =
sgnLij (x)Pij (x) dx ? 2?(Pi , Pj )?ij ,
Rn
Rn
where
Z
?(Pi , Pj ) =
Z
Pij (x)? [Lij (x)] dx =
Rn
ZLij
Pij (x)
? d?.
|?Lij (x)|
Therefore,
1
?ij =
?(Pi , Pj )
Z "
Rn
#
? ij (x)
sgnLij (x) ? sgnL
Pij (x) dx.
2
(7)
As can be seen, ?ij is composed of two factors, the integral and the 1/?(Pi , Pj ) factor.
The integral is bounded between -1 and 1 and determines the direction of the correction.
? ij (x) always agree, the integral is zero and there is no correction. The
When Lij (x) and L
? ij is negative
correction favors i when ?ij is positive. This occurs when Lij is positive and L
?
more often than Lij is negative and Lij is positive, a situation improved upon by giving i
?help.? Similarly, when ?ij is negative, the correction biases towards j.
The maximum amount of absolute likelihood correction possible is determined by the (always positive) 1/?(Pi , Pj ) factor. This is affected by two quantities, the mass around and
the log-likelihood ratio gradient at the decision boundary. Low mass at the decision boundary increases the maximum possible correction because any errors in the integral factor
are being de-weighted. High gradient at the decision boundary also increases the maximum possible correction because any decision boundary deviation causes a higher change
in likelihood ratio than if the gradient was low. Since we are correcting the likelihood ratio
directly, this needs to be reflected in ?ij .
When P (x|ci ) and P (x|cj ) are multivariate Gaussians with means ?i and ?j , identical
covariance matrices ?, and equal priors, this becomes:
1
T
?1
e? 8 (?i ??j ) ? (?i ??j )
2?(?i ? ?j )T ??1 (?i ? ?j )
?(Pi , Pj ) = p
As the means diverge from each other, both the mass at the decision boundary decreases
and the likelihood-ratio gradient increases, thereby increasing the maximum amount of
correction.
Unfortunately, it is quite difficult to explicitly evaluate ?(Pi , Pj ) without knowing the true
probability distributions. In this initial work, therefore, our investigations simplify by only
computing the direction and not the magnitude of the correction. As will be seen, this
assumption yields a likelihood-ratio adjustment that is similar in form to our previous KLdivergence based adjustment. More practically, the assumption significantly simplifies the
derivation and still yields reasonable empirical results. Under this assumption, expression
for ?ij becomes:
1
1
? ij (x)].
EP (x) [sgnLij (x)] ? EPij (x) [sgnL
(8)
2 ij
2
The left term on the right of the equality is quite similar to the left difference on the right
of the equality in the KL-divergence case (Equation 2). Again, because we have no information about the true class conditional models, we assume the left term in Equation 8 to
be zero (denote this as assumption B). Comparing this with the corresponding assumption
for the KL-divergence case (assumption A, Equations 2 and 3), it can be shown that 1) they
are not identical in general, and 2) in the Gaussian case, A implies B but not vice versa,
meaning B is weaker than A.
?ij =
Under assumption B, an expression for the resulting ?ij can be derived using the weak law
of large numbers yielding:
?
?
X
X
?
?
P (x|cj )
P (x|ci ) ?
1
?
sgn ln
?
sgn ln
,
(9)
?ij ?
2(Ni + Nj )
P? (x|ci )
P? (x|cj )
x?Ci
x?Cj
where x ? Ci and x ? Cj correspond to the samples as they are classified in a previous
recognition pass; Ni and Nj are number of samples from model ci and cj respectively. One
can immediately see the similarity between this equation and the one using KLD [2].
Like in [2], since the true classes are unknown, we perform a previous classification pass
(e.g., using the original likelihood ratios) to get estimates and use these in Equation 9.
Note that there are three potential sources of error in the analysis above. The first is the
?(Pi , Pj ) factor that we neglected. The second is assumption B, that (since weaker) can
be less severe than in the corresponding KL-divergence case. The third is the error due to
the discontinuity of the sign function. To address the third problem, rather than using the
sign function in Equation 9, we can approximate it with a continuous differential function
with the goal of balancing the trade-off mentioned above. There are a number of possible
sign-function approximations, including hyperbolic and arc tangent, and shifted sigmoid
function, the latter of which is the most flexible because of its free parameter ?.3
Specifically, the sigmoid function has the form f (z) = 1+e1??z , where the free parameter ?
(an inverse temperature) determines how well the curve will approximate the discontinuous
function. Using the sigmoid function, we can approximate the sign function as sgnz ?
2
? 1. Note that the approximation improves as ? increases. Hence,
1+e??z
?
?
X
X
1
2
2
? . (10)
?
?ij ?
?
1?
1?
? ij (x)
?L
2(Ni + Nj ) x?c
1 + e? L? ji (x)
1
+
e
x?c
j
i
4
Speech Recognition Evaluation
As in previous work [2], we implemented this technique on NYNEX PHONEBOOK [10,
1], a medium vocabulary isolated-word speech corpus. Gaussian mixture hidden Markov
models (HMMs) produced probability scores P? (x|ci ) where here x is a matrix of feature
values (one dimension as MFCC features and the other as time frames), and ci is a word
identity. The HMMs use four hidden states per phone, and 12 Gaussian mixtures per state
(standard for this task [10]). This yields approximately 200k free model parameters in total.
In our experiments, the steps are: 1) calculate P? (x|ci ) using full inference (no Viterbi
approximation) for each test case and for each word; 2) classify the test examples using
? ij = ln P? (x|ci )/P? (x|cj ); 3) using the hypothesized (and
just the log likelihood ratios L
error-full) class labels, calculate the test-set bias term using one of the techniques described
? ij + ?ij . Since
above; and 4) classify again using the augmented likelihood ratio Sij = L
the procedure is no longer transitive, we run 1000 random tournament-style games (as in
[2]) and choose the most frequent winner as the ultimate winner.
Table 1: Word error rates % on speech data with various sign approximations.
SIZE
ORIG
SIGN
TANH
ATAN
SIG (.1)
SIG (1)
SIG (10)
SIG (100)
SIG (200)
SIG (400)
75
150
300
600
2.34
3.31
5.23
7.39
1.76
2.83
4.75
6.64
1.76
2.84
4.75
6.61
1.76
2.83
4.70
6.60
1.82
2.65
4.74
6.66
1.76
2.83
4.75
6.64
1.56
2.65
4.29
6.04
1.57
2.47
3.95
5.70
1.33
2.68
4.34
6.74
1.34
2.43
4.34
6.74
KLD[2]
1.91
2.72
4.29
5.91
The results are shown in Table 1, where the first column gives the test-set vocabulary size
(number of different classes). The second column shows the baseline word error rates
? ij . The remaining columns are the bias-corrected results with various
(WERs) using only L
sign approximations, namely sign (Equation 9), hyperbolic and arc tangent, and the shifted
sigmoid with various ? values (thus allowing us to investigate the trade-off mentioned in
Section 3.1). From the results we can see that larger-? sigmoid is usually better, with
overall performance increasing with ?. This is because with large ?, the shifted sigmoid
curve better approximates the sign function. For ? = 100, the results are even better than
our previous KL-divergence (KLD) results reported in [2] (right-most column in the table).
It can also been seen that when ? is greater than 100, the WERs are not consistently better.
This indicates that the inaccuracies due to the Taylor error term start adversely affecting
the results at around ? = 100.
3
Note that the other soft sign functions can also be defined to utilize a ? smoothness parameter.
5
UCI Dataset Evaluation
Table 2: Error rates in % (and std where applicable) on the UCI data.
data
australian
breast
chess
cleve
corral
crx
diabetes
flare
german
glass
glass2
heart
hepatitis
iris
letter
lymphography
mofn-3-7-10
pima
satimage
segment
shuttle-small
soybean-large
vehicle
vote
waveform-21
NN baseline
16.75(3.51)
2.94(1.16)
0.56
25.67(3.40)
2.44(1.26)
17.41(3.18)
28.04(3.08)
20.98(2.26)
29.96(3.49)
42.16(2.06)
28.82(2.57)
21.83(3.77)
19.46(7.10)
8.13(1.60)
38.66
24.46(4.86)
0
25.96(2.01)
15.80
7.53
0.87
8.47(1.31)
28.39(4.68)
7.40(2.22)
26.21
KLD
16.33(3.66)
2.62(1.15)
0.46
24.35(2.82)
1.82(1.16)
17.25(2.67)
26.88(3.56)
19.37(2.16)
28.54(3.45)
39.63(1.76)
26.23(2.61)
21.48(4.26)
16.10(6.13)
6.84(1.44)
34.66
23.81(4.57)
0
25.22(2.95)
14.25
7.40
0.77
8.29(1.39)
28.15(4.62)
6.94(1.77)
26.17
sign
16.17(3.63)
2.63(1.15)
0.47
24.01(2.27)
1.19(1.16)
17.11(2.91)
27.41(4.13)
18.29(2.25)
28.82(2.53)
41.92(1.92)
26.95(2.65)
21.19(4.52)
17.16(6.92)
6.26(1.47)
37.10
23.29(4.52)
0
24.82(2.87)
14.40
7.27
0.87
7.18(1.08)
27.70(4.44)
6.94(1.77)
26.12
sig(10)
16.32(3.75)
2.65(1.15)
0.37
24.01(3.94)
1.19(1.16)
17.26(3.00)
27.18(1.98)
18.46(1.85)
28.25(3.71)
40.95(2.00)
26.23(2.57)
21.09(4.23)
15.82(6.94)
6.84(1.44)
37.00
23.29(4.86)
0
25.96(2.19)
14.25
7.53
0.77
8.47(1.31)
28.39(4.75)
7.17(2.05)
26.14
NB baseline
14.89(1.97)
2.45(1.93)
12.66
17.91(2.37)
12.77(3.66)
15.05(3.67)
25.71(2.13)
20.24(2.31)
24.58(2.57)
44.12(7.96)
22.36(9.01)
15.50(6.01)
16.18(5.92)
6.99(1.78)
30.68
16.62(8.64)
8.59
25.71(2.13)
19.15
12.21
1.40
8.71(2.70)
38.92(4.47)
9.91(1.72)
21.45
KLD
14.29(2.45)
2.29(2.02)
12.76
15.55(1.81)
9.57(2.12)
14.02(3.91)
24.79(2.68)
19.55(2.63)
26.55(1.88)
42.24(8.64)
21.15(9.25)
15.11(5.34)
18.29(5.96)
6.99(1.78)
30.88
18.27(9.25)
4.57
24.79(2.68)
19.35
11.73
1.41
9.13(2.60)
38.59(5.05)
9.68(2.49)
21.11
sign
14.76(2.45)
2.13(2.07)
13.04
15.22(1.82)
9.57(2.62)
13.06(3.67)
24.24(3.49)
18.70(1.87)
24.79(2.30)
42.06(9.22)
21.77(9.25)
15.11(5.72)
18.04(5.92)
6.99(1.78)
30.48
17.34(8.91)
1.56
24.24(3.49)
19.25
11.82
1.50
8.35(2.65)
38.79(4.46)
9.68(1.72)
20.15
sig(10)
14.76(2.37)
1.86(2.07)
12.85
16.22(2.61)
12.05(4.80)
15.05(3.67)
24.66(2.59)
16.64(2.34)
24.25(2.50)
42.28(7.93)
22.36(9.01)
15.11(6.01)
15.45(4.56)
6.99(1.78)
30.64
15.31(8.91)
3.42
24.66(2.59)
18.70
12.21
1.50
8.37(2.70)
37.84(4.43)
9.68(1.72)
21.40
In order to show that our methodology is general beyond isolated-word speech recognition,
we also evaluated this technique on the entire UCI machine learning repository [9]. In
our experiments, baseline classifiers are built using one of: 1) the Matlab neural network
(NN) toolbox with feed-forward 3-layer perceptrons having different number of hidden
units and training epochs (optimized over a large set to achieve the best possible baseline
for each test case), and trained using the Levenberg-Marquardt algorithm [11], or 2) the
MLC++ toolbox to produce na??ve Bayes (NB) classifiers that have been smoothed using
Dirichlet priors. In each case (i.e., NN or NB), we augmented the resulting likelihood ratios
with bias correction terms thereby evaluating our technique using quite different forms of
baseline classifiers. Unlike the above, with these data sets we have only tried one random
tournament game to decide the winner so far.
For the NN results, hidden units use logistic sigmoid, and output units use a soft-max
function, making the network outputs interpretable as posterior probabilities P (c|x), where
x is the sample and c is the class. While our bias correction described above is in terms
of likelihoods ratios Lij (x), posteriors can be used as well if the posteriors are divided by
the priors giving the relation P (c|x)/P (c) = P (x|c)/p(x) (i.e., scaled likelihoods) which
produces the standard Lij (x) values when used in a likelihood ratio .
As was done in [5], for the small data sets the experimental results use 5-fold crossvalidation using randomly selected chunks ? results show mean and standard deviation
(std) in parentheses. For the larger data sets, we use the same held out training/test sets
as in [5] (so std is not shown). The experimental procedure is similar to that described in
Section 4, except that scaled likelihoods are used for the NN baselines. Again, first-pass
error-full test-set hypothesized answers are used to compute the bias corrections.
Table 5 shows our results for both the NN (columns 2?5) and NB (columns 6?9) baseline classifiers. Within each baseline group, the first column shows the baseline accuracy
(with the 5-fold standard derivations when the data set is small). The second column shows
results using KL-divergence based bias corrections ? these are the first published KLD
results on the UCI data. The third column shows results with sign-based correction (Equation 9), and the forth column shows the sigmoid (? = 10) case (Equation 10).
While not the point of this paper, one immediately sees that the NB baseline results are
often better than the NN baseline results (15 out of 25 times). Using the NN as a baseline,
the table shows that the KLD results are almost always better than the baseline 24 times
(out of 25). Also, the sign correction is better than the baseline 23 out of 25 times, and
the sigmoid(10) results are better 20 times. Also (not shown in the table), we found that
? = 10 is slightly better than ? = 1 but there is no advantage using ? = 100. These results
therefore show that the NN KLD correction typically beats the sign and sigmoid correction,
possibly owing to the error in the Taylor approximation. Using the NB classifier as the
baseline, however, shows not only improved baseline results in general but also that the
sigmoid(10) improves more often. Specifically, the KLD results are better than the baseline
16 times, sign is better than the baseline 18 times, and sigmoid(10) beats the baseline 19
times, suggesting that sigmoid(10) typically wins over the KLD case.
6
Discussion
We have introduced a new necessary intransitive likelihood ratio classifier. This was done
by using sign-based corrections to likelihood ratios and by using continuous differentiable
approximations of the sign function in order to be able to vary the inherent trade-off between sign-function approximation accuracy and Taylor error. We have applied these techniques to both a speech recognition corpus and the UCI data sets, as well as applying
previous KL-divergence based corrections to the latter data. Results on the UCI data sets
confirm that our techniques reasonably generalize to data sets other than speech recognition. This suggests that the framework could be applied to other machine learning tasks.
This work was supported in part by NSF grant IIS-0093430 and IIS-0121396.
References
[1] Jeff Bilmes. Burried Markov models for speech recognition. In IEEE Intl. Conf. on Acoustics,
Speech, and Signal Processing, March 1999.
[2] Jeff Bilmes, Gang Ji, and M. Meil?a. Intransitive likeilhood-ratio classifiers. In Neural Information Processing Systems: Natural and Synthetic, December 2001.
[3] T. M. Cover and J. A. Thomas. Elements of Information Theory. John Wiley and Sons, Inc.,
1991.
[4] Richard O. Duda, Peter E. Hart, and David G. Stork. Pattern Classification. John Wiley and
Sons, second edition, 2001.
[5] Nir Friedman, Dan Geiger, and Moises Goldszmidt. Bayesian network classifiers. Machine
Learning, 29(2-3):131?163, 1997.
[6] D. S. Jones. Generalised Functions. McCraw-Hill Publishing Company Limited, 1966.
[7] J. Kevorkian. Partial Differential Equations: Analytical Solution Techniques. New York:
Springer, 2000.
[8] R. Duncan Luce and Howard Raiffa. Games and Decisions: Introduction and Critical Survey.
Dover, 1957.
[9] P. M. Murphy and D. W. Aha. UCI Repository of Machine Learning Database, 1995.
[10] J. Pitrelli, C. Fong, S. H. Wong, J. R. Spitz, and H. C. Lueng. PhoneBook: a phonetically-rich
isolated-word telephone-speech database. In IEEE Intl. Conf. on Acoustics, Speech, and Signal
Processing, 1995.
[11] W. H. Press, B. P. Flannery, S. A. Teukolsky, and W. T. Vetterling. Numerical Recipes in C: The
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[12] M. M. Rao. Measure Theory and Integration. John Wiley and Sons, 1987.
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1,675 | 2,522 | The Diffusion Mediated Biochemical Signal
Relay Channel
Peter J. Thomas?, Donald J. Spencer?
Computational Neurobiology Laboratory
(Terrence J. Sejnowski, Director)
Salk Institute for Biological Studies
La Jolla, CA 92037
Sierra K. Hampton, Peter Park, Joseph P. Zurkus
Department of Electrical and Computer Engineering
University of California San Diego
La Jolla, CA 92093
Abstract
Biochemical signal-transduction networks are the biological
information-processing systems by which individual cells, from neurons
to amoebae, perceive and respond to their chemical environments. We
introduce a simplified model of a single biochemical relay and analyse
its capacity as a communications channel. A diffusible ligand is released
by a sending cell and received by binding to a transmembrane receptor
protein on a receiving cell. This receptor-ligand interaction creates a
nonlinear communications channel with non-Gaussian noise. We model
this channel numerically and study its response to input signals of
different frequencies in order to estimate its channel capacity. Stochastic
effects introduced in both the diffusion process and the receptor-ligand
interaction give the channel low-pass characteristics. We estimate the
channel capacity using a water-filling formula adapted from the additive
white-noise Gaussian channel.
1
Introduction: The Diffusion-Limited Biochemical Signal-Relay
Channel
The term signal-transduction network refers to the web of biochemical interactions by
which single cells process sensory information about their environment. Just as neural
networks underly the interaction of many multicellular organisms with their environments,
these biochemical networks allow cells to perceive, evaluate and react to chemical stimuli [1]. Examples include chemical signaling across the synaptic cleft, calcium signaling within the postsynaptic dendritic spine, pathogen localization by the immune system,
?
?
Corresponding author: [email protected]
[email protected]
growth-cone guidance during neuronal development, phototransduction in the retina, rhythmic chemotactic signaling in social amoebae, and many others. The introduction of quantitative measurements of the distribution and activation of chemical reactants within living
cells [2] has prepared the way for detailed quantitative analysis of their properties, aided
by numerical simulations. One of the key questions that can now be addressed is the fundamental limits to cell-to-cell communication using chemical signaling.
To communicate via chemical signaling cells must contend with the unreliability inherent
in chemical diffusion and in the interactions of limited numbers of signaling molecules
and receptors [3]. We study a simplified situation in which one cell secretes a signaling
molecule, or ligand, which can be detected by a receptor on another cell. Limiting ourselves
to one ligand-receptor interaction allows a treatment of this communications system using
elementary concepts from information theory.
The information capacity of this fundamental signaling system is the maximum of the
mutual information between the ensemble of input signals, the time-varying rate of ligand
secretion s(t), and the output signal r(t), a piecewise continuous function taking the values
one or zero as the receptor is bound to ligand or unbound. Using numerical simulation we
can estimate the channel capacity via a standard ?water-filling? information measure [4],
as described below.
2
Methods: Numerical Simulation of the Biochemical Relay
We simulate a biochemical relay system as follows: in a two-dimensional rectangular volume V measuring 5 micrometers by 10 micrometers, we locate two cells spaced 5 micrometers apart. Cell A emits ligand molecules from location xs = [2.5?, 2.5?] with rate
s(t) ? 0; they diffuse with a given diffusion constant D and decay at a rate ?. Both secretion and decay occur as random Poisson processes, and diffusion is realized as a discrete
random walk with Gaussian-distributed displacements. The boundaries of V are taken to
be reflecting. We track the positions of each of N particles {xi , i = 1, ? ? ? , N } at intervals
of ?t = 1msec. The local concentration in a neighborhood of size ? around a location x
is given by the convolution
c?(x, t) =
Z X
N
?(x0 ? xi )g(x ? x0 , ?) dx0
(1)
V i=1
where g(?, ?) is a normalized Gaussian distribution in the plane, with mean 0 and variance
? 2 . The motions of the individual particles cause c?(x, t) to fluctuate about the mean concentration, causing the local concentration at cell B, c?(xr , t) to be a noisy, low-pass filtered
version of the original signal s(t) (see Figure 1).
Cell B, located at xr = [7.5?, 2.5?], registers the presence of ligand through binding and
unbinding transitions, which form a two-state Markov process with time-varying transition
rates. Given an unbound receptor, the binding transition happens at a rate that depends on
the ligand concentration around the receptor: k+ c?(xr , t). The size of the neighborhood ?
reflects the range of the receptor, with binding most likely in a small region close to xr .
Once the receptor is bound to a ligand molecule, no more binding events occur until the
receptor releases the ligand. The receiver is insensitive to fluctuations in c?(xr , t) while it is
in the bound state (see Figure 1). The unbinding transition occurs with a fixed rate k? .
For concreteness, we take values for D, ?, k? , k+ , and ? appropriate for cyclic AMP
signaling between Dictyostelium amoebae, a model organism for chemical communica1
?1
tion: D = 0.25?2 msec?1 , ? = 1 sec?1 , ? = 0.1?, k? = 1 sec?1 , k+ = 2??
.
2 sec
Kd = k? /k+ is the dissociation constant, the concentration at which the receptor on average is bound half the time. For the chosen values of the reaction constants k? , we have
Figure 1: Biochemical Signaling Simulation.
Top: Cell A secretes a signaling molecule (red dots) with a time-varying rate r(t).
Molecules diffuse throughout the two-dimensional volume, leading to locally fluctuating
concentrations that carry a corrupted version of the signal. Molecules within a neighborhood of cell B can bind to a receptor molecule, giving a received signal s(t) ? {0, 1}.
Bottom Left: Input signal. Mean instantaneous rate of molecule release (thousands of
molecules per second). Molecule release is a Poisson process with time-varying rate.
Bottom Center: Local concentration fluctuations, as seen by cell B, indicated by the number of molecules within 0.2 microns of the receptor. The receptor is sensitive to fluctuations
in local concentrations only while it is unbound. While the receptor is bound, it does not
register changes in the local concentration (indicated by constant plateaus corresponding to
intervals when r(t) = 1 in bottom right panel.
Bottom Right: Output signal r(t). At each moment the receptor is either bound (1) or
unbound (0). The receiver output is a piecewise constant function with a finite number of
transitions.
Kd ? 15.9 molecules
? 26.4nMol, comparable to the most sensitive values reported for
?2
the cyclic AMP receptor [2]. At this concentration the volume V = 50?2 contains about
800 signaling molecules, assuming a nominal depth of 1?.
3
Results: Estimating Information Capacity via Frequency Response
Communications channels mediated by diffusion and ligand receptor interaction are nonlinear with non-Gaussian noise. The expected value of the output signal, 0 ? E[r] < 1, is
a sigmoidal function of the log concentration for a constant concentration c:
E[r] =
1
c
=
c + Kd
1 + e?(y?y0 )
(2)
where y = ln(c), y0 = ln(Kd ). The mean response saturates for high concentrations,
c Kd , and the noise statistics become pronouncedly Poissonian (rather than Gaussian)
for low concentrations.
Several different kinds of stimuli can be used to characterize such a channel. The steadystate response to constant input reflects the static (equilibrium) transfer function. Concentrations ranging from 100Kd to 0.01Kd occupy 98% of the steady-state operating range,
0.99 > E[r] > 0.01 [5]. For a finite observation time T the actual fraction of time spent
bound, r?T , is distributed about E[r] with a variance that depends on T . The biochemical relay may be used as a binary symmetric channel randomly selecting a ?high? or ?low?
secretion rate, and ?decoding? by setting a suitable threshold for r?T . As T increases, the
variance of r?T and the probability of error decrease.
The binary symmetric channel makes only crude use of this signaling mechanism. Other
possible communication schemes include sending all-or-none bursts of signaling molecule,
as in synaptic transmission, or detecting discrete stepped responses. Here we use the frequency response of the channel as a way of estimating the information capacity of the
biochemical channel.
For an idealized linear channel with additive white Gaussian noise (AWNG channel) the
channel capacity under a mean input power constraint P is given by the so-called ?waterfilling formula? [4],
Z
1 ?max
(? ? N (?))+
log2 1 +
d?
(3)
C=
2 ?=?min
N (?)
given the constraining condition
Z ?max
(? ? N (?))+ d? ? P
(4)
?=?min
where the constant ? is the sum of the noise and the signal power in the usable frequency
range, N (?) is the power of the additive noise at frequency ? and (X)+ indicates the
positive part of X. The formula applies when each frequency band (?, ? +d?) is subject to
noise of power N (?) independently of all other frequency bands, and reflects the optimal
allocation of signal power S(?) = (? ? N (?))+ , with greater signal power invested in
frequencies at which the noise power is smallest. The capacity C is in bits/second.
For an input signal of finite duration T = 100 sec, we can independently specify the amplitudes and phases of its frequency components at ? = [0.01 Hz, 0.02 Hz, ? ? ? , 500 Hz],
where 500 Hz is the Nyquist frequency given a 1 msec simulation timestep. Because the
population of secreted signaling molecules decays exponentially with a time constant of
1/? = 1 sec, the concentration signal is unable to pass frequencies ? ? 1Hz (see Figure 2) providing a natural high-frequency cutoff. For the AWGN channel the input and
Figure 2: Frequency Response of Biochemical Relay Channel. The sending cell secreted
signaling molecules at a mean rate of 1000 + 1000 sin(2??t) molecules per second. From
top to bottom, the input frequencies were 1.0, 0.5, 0.2, 0.1, 0.05, 0.02 and 0.01 Hz. The
total signal duration was T = 100 seconds.
Left Column: Total number of molecules in the volume. Attenuation of the original signal
results from exponential decay of the signaling molecule population.
Right Column: A one-second moving average of the output signal r(t), which takes the
value one when the receptor molecule is bound to ligand, and zero when the receptor is
unbound.
Figure 3: Frequency Transmission Spectrum Noise power N (?), calculated as the total
power in r(t)??
r in all frequency components save the input frequency ?. Frequencies were
binned in intervals of 0.01 Hz = 1/T . The maximum possible power in r(t) over all frequencies is 0.25; the power successfully transmitted by the channel is given by 0.25/N (?).
The lower curve is N (?) for input signals of the form s(t) = 1000 + 1000 sin 2??t, which
uses the full dynamic range of the receptor. Decreasing the dynamic range used reduces the
amount of power transmitted at the sending frequency: the upper curve is N (?) for signals
of the form s(t) = 1000 + 500 sin 2??t.
output signals share the same units (e.g. rms voltage); for the biological relay the input s(t) is in molecules/second while the output r(t) is a function with binary range
{r = 0, r = 1}. The maximum of the mean output power for a binary function r(t)
R
2
T
is T1 t=0 |r(t) ? r?| dt ? 14 . This total possible output power will be distributed between different frequencies depending on the frequency of the input. We wish to estimate
the channel capacity by comparing the portion of the output power present in the sending
frequency ? to the limiting output power 0.25. Therefore we set the total output power
constant to ? = 0.25. Given a pure sinusoidal input signal s(t) = a0 + a1 sin(2??t), we
consider the power in the output spectrum at ? Hz to be the residual power from the input
and the rest of the power in the spectrum of r(t) to be analogous to the additive noise power
spectrum N (?) in the AWNG channel. We calculate N (?) to be the total power of r(t) ? r?
in all frequency bands except ?. For signals of length T = 100 sec, the possible frequencies are discretized at intervals ?? = 0.01 Hz. Because the noise power N (?) ? 0.25, the
water-filling formula (3) for the capacity reduces to
Z
1 1Hz
0.25
Cest =
log2
d?.
(5)
2 0.01Hz
N (?)
As mentioned above frequencies ? ? 1 Hz do not transmit any information about the
signal (see Figure 2) and do not contribute to the capacity. We approximate this integral using linear interpolation of log2 (N (?)) between the measured values at ? =
[0.01, 0.02, 0.05, 0.1, 0.2, 0.5, 1.0] Hz. (See Figure 3.) This procedure gives an estimate
of the channel capacity, Cest = 0.087 bits/second.
4
Discussion & Conclusions
Diffusion and the Markov switching between bound and unbound states create a low-pass
filter that removes high-frequency information in the biochemical relay channel. A general
Poisson-type communications channel, such as commonly encountered in optical communications engineering, can achieve an arbitrarily large capacity by transmitting high frequencies and high amplitudes, unless bounded by a max or mean amplitude constraint [6].
In the biochemical channel, the effective input amplitude is naturally constrained by the
saturation of the receptor at concentrations above the Kd . And the high frequency transmission is limited by the inherent dynamics of the Markov process. Therefore this channel
has a finite capacity.
The channel capacity estimate we derived, Cest = 0.087 bits/second, seems quite low
compared to signaling rates in the nervous system, requiring long signaling times to transfer
information successfully. However temporal dynamics in cellular systems can be quite
deliberate; cell-cell communication in the social amoeba Dictyostelium, for example, is
achieved by means of a carrier wave with a period of seven minutes. In addition, cells
typically possess thousands of copies of the receptors for important signaling molecules,
allowing for more complex detection schemes than those investigated here.
Our simplified treatment suggests several avenues for further work. For example, signal
transducing receptors often form Markov chains with more complicated dynamics reflecting many more than two states [7]. Also, the nonlinear nature of the channel is probably not
well served by our additive noise approximation, and might be better suited to a treatment
via multiplicative noise [8].
Whether cells engage in complicated temporal coding/decoding schemes, as has been proposed for neural information processing, or whether instead they achieve efficient communication by evolutionary matching of the noise characteristics of sender and receiver,
remain to be investigated. We note that the dependence of the channel capacity C on such
parameters as the system geometry, the diffusion and decay constants, the binding constants
and the range of the receptor may shed light on evolutionary mechanisms and constraints
on communication within cellular biological systems.
Acknowledgments
This work would not have been possible without the generous support of the Howard
Hughes Medical Institute and the resources of the Computational Neurobiology Laboratory, Terrence J. Sejnowski, Director.
References
[1] Rappel, W.M., Thomas, P.J., Levine, H. & Loomis, W.F. (2002) Establishing Direction during
Chemotaxis in Eukaryotic Cells. Biophysical Journal 83:1361-1367.
[2] Ueda, M., Sako, Y., Tanaka, T., Devreotes, P. & Yanagida, T. (2001) Single Molecule Analysis of
Chemotactic Signaling in Dictyostelium Cells. Science 294:864-867.
[3] Detwiler, P.B., Ramanathan, S., Sengupta, A. & Shraiman, B.I. (2000) Engineering Aspects of
Enzymatic Signal Transduction: Photoreceptors in the Retina. Biophysical Journal79:2801-2817.
[4] Cover, T.M. & Thomas, J.A. (1991) Elements of Information Theory, New York: Wiley.
[5] Getz, W.M. & Lansky, P. (2001) Receptor Dissociation Constants and the Information Entropy of
Membranes Coding Ligand Concentration. Chem. Senses 26:95-104.
[6] Frey, R.M. (1991) Information Capacity of the Poisson Channel. IEEE Transactions on Information Theory 37(2):244-256.
[7] Uteshev, V.V. & Pennefather, P.S. (1997) Analytical Description of the Activation of Multi-State
Receptors by Continuous Neurotransmitter Signals at Brain Synapses. Biophysical Journal72:11271134.
[8] Mitra, P.P. & Stark, J.B. (2001) Nonlinear limits to the information capacity of optical fibre
communications. Nature411:1027-1030.
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1,676 | 2,523 | Phonetic Speaker Recognition with Support
Vector Machines
W. M. Campbell, J. P. Campbell, D. A. Reynolds, D. A. Jones, and T. R. Leek
MIT Lincoln Laboratory
Lexington, MA 02420
wcampbell,jpc,dar,daj,[email protected]
Abstract
A recent area of significant progress in speaker recognition is the use
of high level features?idiolect, phonetic relations, prosody, discourse
structure, etc. A speaker not only has a distinctive acoustic sound but
uses language in a characteristic manner. Large corpora of speech data
available in recent years allow experimentation with long term statistics
of phone patterns, word patterns, etc. of an individual. We propose the
use of support vector machines and term frequency analysis of phone sequences to model a given speaker. To this end, we explore techniques
for text categorization applied to the problem. We derive a new kernel
based upon a linearization of likelihood ratio scoring. We introduce a
new phone-based SVM speaker recognition approach that halves the error rate of conventional phone-based approaches.
1 Introduction
We consider the problem of text-independent speaker verification. That is, given a claim of
identity and a voice sample (whose text content is a priori unknown), determine if the claim
is correct or incorrect. Traditional speaker recognition systems use features based upon the
spectral content (e.g., cepstral coefficients) of the speech. Implicitly, these systems model
the vocal tract and its associated dynamics over a short time period. These approaches have
been quite successful, see [1, 2].
Traditional systems have several drawbacks. First, robustness is an issue because channel effects can dramatically change the measured acoustics of a particular individual. For
instance, a system relying only on acoustics might have difficulty confirming that an individual speaking on a land-line telephone is the the same as an individual speaking on
a cell phone [3]. Second, traditional systems also rely upon seemingly different methods
than human listeners [4]. Human listeners are aware of prosody, word choice, pronunciation, accent, and other speech habits (laughs, etc.) when recognizing speakers. Potentially
because of this use of higher level cues, human listeners are less affected by variation in
channel and noise types than automatic algorithms.
An exciting area of recent development pioneered by Doddington [5] is the use of ?high
level? features for speaker recognition. In Doddington?s idiolect work, word N -grams from
conversations were used to characterize a particular speaker. More recent systems have
used a variety of approaches involving phone sequences [6], pronunciation modeling [7],
and prosody [8]. For this paper, we concentrate on the use of phone sequences [6]. The
processing for this type of system uses acoustic information to obtain sequences of phones
for a given conversation and then discards the acoustic waveform. Thus, processing is done
at the level of terms (symbols) consisting of, for example, phones or phone N -grams.
This paper is organized as follows. In Section 2, we discuss the NIST extended data speaker
recognition task. In Section 3.1, we present a method for obtaining a phone stream. Section 3.2 shows the structure of the SVM phonetic speaker recognition system. Section 4
discusses how we construct a kernel for speaker recognition using term weighting techniques for document classification. We derive a new kernel based upon a linearization of
a likelihood ratio. Finally, Section 5 shows the applications of our methods and illustrates
the dramatic improvement in performance possible over standard phone-based N -gram
speaker recognition methods.
2 The NIST extended data task
Experiments for the phone-based speaker recognition experiments were performed based
upon the NIST 2003 extended data task [9]. The corpus used was a combination of phases
II and III of the Switchboard-2 corpora [10].
Each potential training utterance in the NIST extended data task consisted of a conversation
side that was nominally of length 5 minutes recorded over a land-line telephone. Each
conversation side consisted of a speaker having a conversation on a topic selected by an
automatic operator; conversations were typically between unfamiliar individuals.
For training and testing, a jacknife approach was used to increase the number of tests.
The data was divided into 10 splits. For training, a given split contains speakers to be
recognized (target speakers) and impostor speakers; the remaining splits could be used
to construct models describing the statistics of the general population?a ?background?
model. For example, when conducting tests on split 1, splits 2-10 could be used to construct
a background.
Training a speaker model was performed by using statistics from 1, 2, 4, 8, or 16 conversation sides. This simulated a situation where the system could use longer term statistics
and become ?familiar? with the individual; this longer term training allows one to explore
techniques which might mimic more what human listeners perceive about an individual?s
speech habits. A large number of speakers and tests were available; for instance, for 8 conversation training, 739 distinct target speakers were used and 11, 171 true trials and 17, 736
false trials were performed. For additional information on the training/testing structure we
refer to the NIST extended data task description [9].
3 Speaker Recognition with Phone Sequences
3.1 Phone Sequence Extraction
Phone sequence extraction for the speaker recognition process is performed using the
phone recognition system (PPRLM) designed by Zissman [11] for language identification.
PPRLM uses a mel-frequency cepstral coefficient front end with delta coefficients. Each
phone is modeled in a gender-dependent context-independent (monophone) manner using
a three-state hidden Markov model (HMM). Phone recognition is performed with a Viterbi
search using a fully connected null-grammar network of monophones; note that no explicit
language model is used in the decoding process.
The phone recognition system encompassed multiple languages?English (EG), German
(GE), Japanese (JA), Mandarin (MA), and Spanish (SP). In earlier phonetic speaker recog-
nition work [6], it was found that these multiple streams were useful for improving accuracy. The phone recognizers were trained using the OGI multilanguage corpus which had
been hand labeled by native speakers.
After a ?raw? phone stream was obtained from PPRLM, additional processing was performed to increase robustness. First, speech activity detection marks were used to eliminate
phone segments where no speech was present. Second, silence labels of duration greater
than 0.5 seconds were replaced by ?end start? pairs. The idea in this case is to capture some
of the ways in which a speaker interacts with others?does the speaker pause frequently,
etc. Third, extraneous silence was removed at the beginning and end of the resulting segments. Finally, all phones with short duration were removed (less than 3 frames).
3.2 Phonetic SVM System
Our system for speaker recognition using phone sequences is shown in Figure 1. The scenario for its usage is as follows. An individual makes a claim of identity. The system then
retrieves the SVM models of the claimed identity for each of the languages in the system.
Speech from the individual is then collected (a test utterance). A phone sequence is derived
using each of the language phone recognizers and then post-processing is performed on the
sequence as discussed in Section 3.1. After this step, the phone sequence is vectorized by
computing frequencies of N -grams?this process will be discussed in Section 4. We call
this term calculation since we compute term types (unigram, bigram, etc.), term probabilities and weightings in this step [12]. This vector is then introduced into a SVM using the
speaker?s model in the appropriate language and a score per language is produced. Note
that we do not threshold the output of the SVM. These scores are then fused using a linear
weighting to produce a final score for the test utterance. The final score is compared to a
threshold and a reject or accept decision is made based upon whether the score was below
or above the threshold, respectively.
An interesting aspect of the system in Figure 1 is that it uses multiple streams of phones
in different languages. There are several reasons for this strategy. First, the system can be
used without modification for speakers in multiple languages. Second, although not obvious, from experimentation we show that phone streams different from the language being
spoken provide complimentary information for speaker recognition. That is, accuracy improves with these additional systems. A third point is that the system may also work in other
languages not represented in the phone streams. It is known that in the case of language
identification, language characterization can be performed even if a phone recognizer is not
available in that particular language [11].
EG phone
recognizer
Phone PostProcessing
Term
Calculation
EG Speaker
Model SVM
GE phone
recognizer
Phone PostProcessing
Term
Calculation
GE Speaker
Model SVM
?
O
speech
O
O
SP phone
recognizer
Phone PostProcessing
Term
Calculation
SP Speaker
Model SVM
Figure 1: Phonetic speaker recognition using support vector machines
score
Training for the system in Figure 1 is based upon the structure of the NIST extended data
corpus (see Section 2). We treat each conversation side in the corpus as a ?document.? From
each of these conversation sides we derive a single (sparse) vector of weighted probabilities. To train a model for a given speaker, we use a one-versus-all strategy. The speaker?s
conversations are trained to a SVM target value of +1. The conversations sides not in the
current split (see Section 2) are used as a background. That is, all conversation sides not in
the current split are used as the class for SVM target value of ?1. Note that this strategy
ensures that speakers that are used as impostors are ?unseen? in the training data.
4 Kernel Construction
Possibly the most important aspect of the process of phonetic speaker recognition is the
selection of the kernel for the SVM. Of particular interest is a kernel which will preserve
the identity cues a particular individual might present in their phone sequence. We describe
two steps for kernel construction.
Our first step of kernel construction is the selection of probabilities to describe the phone
stream. We follow the work of [5, 6]. The basic idea is to use a ?bag of N -grams? approach.
For a phone sequence, we produce N -grams by the standard transformation of the stream;
e.g., for bigrams (2-grams) the sequence of phones, t1 , t2 , ..., tn , is transformed to the
sequence of bigrams of phones t1 t2 , t2 t3 , ..., tn?1 tn . We then find probabilities of N grams with N fixed. That is, suppose we are considering unigrams and bigrams of phones,
and the unique unigrams and bigrams are designated d1 , ..., dM and d1 d1 , ... dM dM ,
respectively; then we calculate probabilities and joint probabilities
#(tk = di )
p(di ) = P
k #(tk = dk )
#(tk tl = di dj )
p(di dj ) = P
k,l #(ti tj = dk dl )
(1)
where #(tk = di ) indicates the number of phones in the conversation side equal to d i , and
an analogous definition is used for bigrams. These probabilities then become entries in a
vector v describing the conversation side
v = [p(d1 ) . . .
p(dM ) p(d1 d1 ) . . .
t
p(dM dM )] .
(2)
In general, the vector v will be sparse since the conversation side will not contain all potential unigrams, bigrams, etc.
A second step of kernel construction is the selection of the ?document component? of
term weighting for the entries of the vector v in (2) and the normalization of the resulting
vector. By term weighting we mean that for each entry, vi , of the vector v, we multiply
by a ?collection? (or background) component, wi , for that entry. We tried two distinct
approaches for term weighting.
TFIDF weighting. The first is based upon the standard TFIDF approach [12, 13]. From
the background section of the corpus we compute the frequency of a particular N -gram
using conversation sides as the item analogous to a document. I.e., if we let DF (t i ) be the
number of conversation sides where a particular N -gram, ti , is observed, then our resulting
term-weighted vector has entries
# of conversation sides in background
vi log
.
(3)
DF (ti )
We follow the weighting in (3) by a normalization of the vector to unit length x 7? x/kxk 2 .
Log-likelihood ratio weighting. An alternate method of term weighting may be derived
using the following strategy. Suppose that we have two conversation sides from speakers,
spk1 and spk2 . Further suppose that the sequence of N -grams (for fixed N ) in each conversation side is t1 ,t2 , ..., tn and u1 , u2 , ..., um respectively. We denote the unique set
of N -grams as d1 , ..., dM . We can build a ?model? based upon the conversation sides for
each speaker consisting of the probability of N -grams, p(di |spkj ). We then compute the
likelihood ratio of the first conversation side as is standard in verification problems [1]; a
linearization of the likelihood ratio computation will serve as the kernel. Proceeding,
n
Y
p(t1 , t2 , . . . , tn |spk2 )
p(ti |spk2 )
=
p(t1 , . . . , tn |background) i=1 p(ti |background)
(4)
where we have made the assumption that the probabilities are independent. We then consider the log of the likelihood ratio normalized by the number of observations,
n
1X
p(ti |spk2 )
score =
log
n i=1
p(ti |background)
M
X
#(ti = dj )
p(dj |spk2 )
log
=
(5)
n
p(dj |background)
j=1
=
M
X
p(dj |spk1 ) log
j=1
p(dj |spk2 )
p(dj |background)
.
If we now ?linearize? the log function in (5) by using log(x) ? x ? 1, we get
score ?
M
X
p(dj |spk1 )
X
p(dj |spk2 )
?
p(dj |spk1 )
p(dj |background) j=1
M
M
X
p(dj |spk1 )
p(dj |spk2 )
?1
p(dj |background)
j=1
=
j=1
(6)
M
X
p(dj |spk1 )
p(dj |spk2 )
p
p
?1
p(dj |background) p(dj |background)
j=1
p
Thus, (6) suggests we use a term weighting given by 1/ p(dj |background). Note that the
strategy used for constructing a kernel is part of a general process of finding kernels based
upon training on one instance and testing upon another instance [2].
=
5 Experiments
Experiments were performed using the NIST extended data task ?v1? lists (which encompass the entire Switchboard 2 phase II and III corpora). Tests were performed for 1, 2, 4, 8,
and 16 training conversations. Scoring was performed using the SVM system shown in Figure 1. Five language phone recognizers were used?English (EG), German (GE), Japanese
(JA), Mandarin (MA), and Spanish (SP). The resulting phone sequences were vectorized as
unigram and bigram probabilities (2). Both the standard TFIDF term weighting (3) and the
log-likelihood ratio (TFLLR) term weighting (6) methods were used. We note that when
a term did not appear in the background, it was ignored in training and scoring. A linear
kernel was used x ? y + 1 to compare the vectors of term weights. Training was performed
using the SVMTorch package [14] with c = 1. Comparisons of performance for different
strategies were typically done with 8 conversation training and English phone streams since
these were representative of overall performance.
Table 1: Comparison of different term weighting strategies, English only scores, 8 conversation training
Term Weighting Method EER
TFIDF
7.4%
TFLLR
5.2%
Results were compared via equal error rates (EERs)?the error at the threshold which produces equal miss and false alarm probabilities, Pmiss = Pfa . Table 1 shows the results
for two different weightings, TFIDF (3) and TFLLR (6), using English phones only and 8
training conversations. The table illustrates that the new TFLLR weighting method is more
effective. This may be due to the fact the IDF is too ?smooth?; e.g., for unigrams, the IDF is
approximately 1 since a unigram almost always appears in a given 5 minute conversation.
Also, alternate methods of calculating the TF component of TFIDF have not been explored
and may yield gains compared to our formulation.
We next considered the effect on performance of the language of the phone stream for
the 8 conversation training case. Figure 2 shows a DET plot (a ROC plot with a special
scale [15]) with results corresponding to the 5 language phone streams. The best performing system in the figure is an equal fusion of all scores from the SVM outputs for each language and has an EER of 3.5%; other fusion weightings were not explored in detail. Note
that the best performing language is English, as expected. Note, though, as we indicated in
Section 3.1 that other languages do provide significant speaker recognition information.
SP
40
JA
GE
MA
Miss probability (in %)
20
EG
10
5
Fused Scores
2
1
0.5
0.2
0.1
0.1 0.2
0.5
1
2
5
10
20
False Alarm probability (in %)
40
Figure 2: DET plot for the 8 conversation training case with varying languages and TFLLR
weighting. The plot shows in order of increasing EER?fused scores, EG, MA, GE, JA, SP
Table 2: Comparison of equal error rates (EERs) for different conversation training lengths
using the TFLLR phonetic SVM and the standard log likelihood ratio (LLR) method
# Training
SVM EER LLR EER SVM EER
Conversations
Reduction
1
13.4%
21.8%
38%
2
8.6%
14.9%
42%
4
5.3%
10.3%
49%
8
3.5%
8.8%
60%
16
2.5%
8.3%
70%
Miss probability (in %)
40
20
Standard
Trigram
Standard
Bigram
10
5
2
SVM
1
0.5
0.2
0.1
0.10.2 0.5 1 2
5
10
20
False Alarm probability (in %)
40
Figure 3: DET plot for 8 conversation training showing a comparison of the SVM approach
(solid line) to the standard log likelihood ratio approach using bigrams (dash-dot line) and
the standard log likelihood ratio approach using trigrams (dashed line)
Table 2 shows the effect of different training conversation lengths on the EER. As expected,
more training data leads to lower error rates. We also see that even for 1 training conversation, the SVM system provides significant speaker characterization ability. Figure 3 shows
DET plots comparing the performance of the standard log likelihood ratio method [6] to
our new SVM method using the TFLLR weighting. We show log likelihood results based
on both bigrams and trigrams; in addition, a slightly more complex model involving discounting of probabilities is used. One can see the dramatic reduction in error, especially
apparent for low false alarm probabilities. The EERs of the standard system are 8.8% (trigrams, see Table 2) and 10.4% (bigrams), whereas our new SVM system produces an EER
of 3.5%; thus, we have reduced the error rate by 60%.
6 Conclusions and future work
An exciting new application of SVMs to speaker recognition was shown. By computing
frequencies of phones in conversations, speaker characterization was performed. A new
kernel was introduced based on the standard method of log likelihood ratio scoring. The
resulting SVM method reduced error rates dramatically over standard techniques.
Acknowledgements
This work was sponsored by the United States Government Technical Support Working
Group under Air Force Contract F19628-00-C-0002. Opinions, interpretations, conclusions, and recommendations are those of the authors and are not necessarily endorsed by
the United States Government.
References
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1,677 | 2,524 | Sensory Modality Segregation
Virginia R. de Sa
Department of Cognitive Science
University of California, San Diego
La Jolla, CA 92093-0515
[email protected]
Abstract
Why are sensory modalities segregated the way they are? In this paper
we show that sensory modalities are well designed for self-supervised
cross-modal learning. Using the Minimizing-Disagreement algorithm on
an unsupervised speech categorization task with visual (moving lips) and
auditory (sound signal) inputs, we show that very informative auditory
dimensions actually harm performance when moved to the visual side of
the network. It is better to throw them away than to consider them part
of the ?visual input?. We explain this finding in terms of the statistical
structure in sensory inputs.
1
Introduction
In previous work [1, 2] we developed a simple neural network algorithm that learned categories from co-occurences of patterns to different sensory modalities. Using only the
co-occuring patterns of lip motion and acoustic signal, the network learned separate visual
and auditory networks (subnets) to distinguish 5 consonant vowel utterances. It performed
almost as well as the corresponding supervised algorithm, where the utterance label is
given, on the same data and significantly better than a strategy of separate unsupervised
clustering in each modality followed by clustering of these clusters (This strategy is used
to initialize our algorithm).
In this paper we show that the success of this biologically motivated algorithm depends
crucially on the statistics of features derived from different sensory modalities. We do this
by examining the performance when the two ?network-modalities? or pseudo-modalities
are made up of inputs from the different sensory modalities.
The Minimizing-Disagreement Algorithm
The Minimizing-Disagreement (M-D) algorithm is designed to allow two (or more) modalities (or subnets) to simultaneously train each other by finding a local minimum of the
number of times the individual modalities disagree on their classification decision (see Figure 1). The modalities are essentially trained by running Kohonen?s LVQ2.1 algorithm[3]
but with the target class set by the output of the subnet of the other modality (receiving a
co-occuring pattern) not a supervisory external signal. The steps of the algorithm are as
follows.
feedback of class picked
by auditory input
Multi-sensory object area
"Class" Units
Hidden Units
Modality/Network 1
(Visual)
Modality/Network 2
(Auditory)
visual input
Figure 1: The network for Minimizing-Disagreement algorithm. The weights from the
hidden units to the output units determine the ?labels? of the hidden units. These weights
are updated throughout training to allow hidden units to change classes if needed. During
training each modality creates an output label for the other as shown on the right side of
the figure. After training, each modality subnet is tested separately.
1. Initialize hidden unit weight vectors in each modality (unsupervised clustering)
2. Initialize hidden unit labels using unsupervised clustering of the activity patterns
across the hidden units from both modalities
3. Repeat for each presentation of input patterns X1 (n) and X2 (n) to their respective
modalities
? For each modality, find the two nearest hidden unit weight vectors to the
respective input pattern
? Find the hypothesized output class in each modality (as given by the label of
the hidden unit with closest weight vector). The label of a hidden unit is the
output unit to which it projects most strongly.
? For each modality update the hidden unit weight vectors according to the
LVQ2.1 rule (Only the rules for modality 1 are given below)
Updates are performed only if the current pattern X1 (n) falls within c(n) of
the border between two hidden units of different classes (one of them agreeing with the output from the other modality). In this case
~w1i? (n) = ~w1i? (n ? 1) + ?(n)
~w1 j? (n) = ~w1 j? (n ? 1) ? ?(n)
(X1 (n) ? ~w1i? (n ? 1))
||X1 (n) ? ~w1i? (n ? 1)||
(X1 (n) ? ~w1 j? (n ? 1))
||X1 (n) ? ~w1 j? (n ? 1)||
where ~w1i? is the weight vector of the hidden unit with the same label, and
~w1 j? is the weight vector of the hidden unit with another label.
? Update the labeling weights using Hebbian learning between the winning
hidden unit and the output of the other modality.
In order to discourage runaway to one of the trivial global minima of disagreement, where
both modalities only ever output one class, weights to the output class neurons are renormalized at each step. This normalization means that the algorithm is not modifying the
output weights to minimize the disagreement but instead clustering the hidden unit representation using the output class given by the other modality. This objective is better
for these weights as it balances the goal of agreement with the desire to avoid the trivial
solution of all hidden units having the same label.
Vx
time
Vy
motion
vectors
image areas
Ax
frequency
vectors
time
Ay
frequency channels
Figure 2: An example Auditory and Visual pattern vector. The figure shows which
dimensions went into each of Ax, Ay, Vx, and Vy.
2
Experiments
2.1 Creation of Sub-Modalities
The original auditory and visual data were collected using an 8mm camcorder and directional microphone. The speaker spoke 118 repetitions of /ba/, /va/, /da/, /ga/, and /wa/. The
first 98 samples of each utterance class formed the training set and the remaining 20 the
test set. The auditory feature vector was encoded using a 24 channel mel code1 over 20
msec windows overlapped by 10 msec. This is a coarse short time frequency encoding,
which crudely approximates peripheral auditory processing. Each feature vector was linearly scaled so that all dimensions lie in the range [-1,1]. The final auditory code is a (24 ?
9) 216 dimension vector for each utterance. An example auditory feature vector is shown
in Figure 2 (bottom).
The visual data were processed using software designed and written by Ramprasad Polana
[4]. Visual frames were digitized as 64 ? 64 8 bit gray-level images using the Datacube
MaxVideo system. Segments were taken as 6 frames before the acoustically determined
utterance offset and 4 after. The normal flow was computed using differential techniques
between successive frames. Each pair of frames was then averaged and then these averaged
frames were divided into 25 equal areas (5 ? 5) and the motion magnitudes within each
frame were averaged within each area. The final visual feature vector of dimension (5
frames ? 25 areas) 125 was linearly normalized as for the auditory vectors. An example
visual feature vector is shown in Figure 2 (top).
The original auditory and visual feature vectors were divided into two parts (called Ax, Ay
and Vx,Vy as shown in Figure 2). The partition was arbitrarily determined as a compromise
between wanting a similar number of dimensions and similar information content in each
part. (We did not search over partitions; the experiments below were performed only for
this partition). Our goal is to combine them in different ways and observe the performance
of the minimizing-disagreement algorithm.
We first benchmarked the divided ?sub-modalities? to see how useful they were for the task.
For this, we ran a supervised algorithm on each subset. The performance measurements
are shown in Table 1.
1 linear
spacing below 1000 Hz and logarithmic above 1000 Hz.
Sub-Modality
Ax
Ay
Vx
Vy
Supervised Performance
89 ? 2
91 ? 2
83 ? 2
77 ? 3
Table 1: Supervised performance of each of the sub-modalities. All numbers give percent correct classifications on independent test sets ? standard deviations.
2.2
Creation of Pseudo-Modalities
Pseudo-modalities were created by combining all combinations (of 3 or less) of Ax, Ay, Vx
and Vy; thus Ax+Vx+Vy (Ax+V) would be a pseudo-modality. The idea is to test all possible combinations of pseudo-modalities and compare the resulting performance of the final
individual subnets with what a supervised algorithm could do with the same dimensions.
2.3
Pseudo-Modality Experiments
In order to allow fair comparison, appropriate parameters were found for each modality
division. The data were divided into 75% Training, and 25% Test data. Optimal parameters
were selected by observing performance on the training data, and performance is reported
on the test data.
The results for all possible divisions are presented in Figure 3. Each network has the
following key. The light gray bar and number represents the test-set performance of the
pseudo-modality consisting of the sub-modalities listed below it. The darker bar and number represents the test-set performance of the other pseudo-modality. The black outlines
(and numbers above the outline) give the performance of the corresponding supervised algorithm (LVQ2.1) with the same data. Thus, the empty area between the shaded area and
black outline represents the loss from lack of supervision.
Looking at the figure, one can make several comparisons. For each submodality, we can
ask: To get the best performance of a subnet using those dimensions, where should one
put the other sub-modalities in a M-D network? For instance, to answer that question
of Ax, one would compare the performance of the Ax subnet in Ax/Ay+V network with
that of the Ax+Ay subnet in the Ax+Ay/Vx+Vy network, with that of the Ax+Vx+Vy
subnet in the Ax+Vx+Vy/Ay network etc. The subnet containing Ax that performs the
best is the Ax+Ay subnet (trained with co-modality Vx+Vy). In fact, it turns out that
for each submodality, the architecture for optimal post-training performance of the subnet
containing that submodality, is to put the dimensions from the same ?real? modality on the
same side and those from the other modality on the other side.
This raises the question: Is performance better for the Ax+Ay/Vx+Vy network than the
Ax/Ay+Vx+Vy network because the benefit of having Ay with Ax is greater than that of
having Ay with Vx and Vy (in other words, are there some higher order relationships between dimensions in Ax and those in Ay that require both dimensions to be learned by
the same subnet) OR is it actually harmful to have Ay on the opposite side from Ax? We
can answer this question by comparing the performance of the Ax/Ay+Vx+Vy network
with that of the Ax/Vx+Vy network as shown in Figure 4. For that particular division,
the results are not significantly different (even though we have removed the most useful
dimensions), but for all the other divisions, performance is improved when dimensions are
removed so that only dimensions from one ?real? sensory modality are on one side. For
example, the two graphs in the second column show that it is actually harmful to include
the very useful Ax dimensions on the visual side of the network ? we do better when we
97
91
88
69
73
97
98
97
83
77
72
73
77
68
67
58
Ax
Ay,Vx,Vy
Ay
97
90
93
86
79
71
Ax,Vx,Vy Ax,Ay,Vy Vx
91
91
75
73
Ax,Ay,Vx Vy
91
67
Pseudo?
Modality1
Ax,Ay
Vx,Vy
Ax,Vx
Ay,Vy
Ay,Vx
Pseudo?
Modality2
Ax,Vy
Figure 3: Self-supervised and Supervised performances for the various pseudomodality divisions. Standard errors for the self-supervised performance means are ?1.
Those for the supervised performances are ?.5.
throw them away. Note that this is true even though a supervised network with Ax+Vx+Vy
does much better than a supervised network with Vx+Vy ? this is not a simple feature
selection result.
2.4 Correlational structure is important
Why do we get these results? The answer is that the results are very dependent on the
statistical structure between dimensions within and between different sensory modalities.
Consider a simpler system of two 1-Dimensional modalities and two classes of objects.
Assume that the sensation detected by each modality has a probability density given by a
Gaussian of different mean for each class. The densities seen by each modality are shown
in Figure 5. In part A) of the Figure, the joint density for the stimuli to both modalities
is shown for the case of conditionally uncorrelated stimuli (within each class, the inputs
are uncorrelated). Parts C) and D) show the changing joint density as the sensations to the
two modalities become more correlated within each class. Notice that the density changes
from a ?two blob? structure to more of a ?ridge? structure. As it does this the projection
of the joint density gives less indication of the underlying bi-modal structure and the local
minimum of the Minimizing-Disagreement Energy function gets shallower and narrower.
This means that the M-D algorithm would be less likely to find the correct boundary.
A more intuitive explanation is shown in Figure 6. In the figure imagine that there are two
classes of objects, with densities given by the thick curve and the thin curve and that this
marginal density is the same in each one-dimensional modality. The line drawing below
the densities, shows two possible scenarios for how the ?modalities? may be related. In
the top case, the modalities are conditionally independent. Given that a ?thick? object
is present, the particular pattern to each modality is independent. The lines represent a
possible sampling of data (where points are joined if they co-occured). The minimizing
disagreement algorithm wants to find a line from top to bottom that crosses the fewest lines
? within the pattern space, disagreement is minimized for the dashed line shown.
97
98
97
83
77
73
69
97
91
88
72
73
77
67
68
58
Ax
88
Ay,Vx,Vy
Ax,Ay,Vx Vy
97
91
86
97
86
84
72
71
Ax,Vx,Vy Ax,Ay,Vy Vx
Ay
83
75
84
77
P?M 1 P?M 2
75
70
60
Ax
Ay,Vx,Vy
Ay
Ax,Vx,Vy Ax,Ay,Vy Vx
Ax,Ay,Vx Vy
Figure 4: This figure shows the benefits of having a pseudo-modality composed of
dimensions from only ONE real modality (even if this means throwing away useful
dimensions). Standard errors for the self-supervised performance means are ?1. Those
for the supervised performances are ?.5.
Example Densities for
A
Joint Density with
two classes in one modality
B
? =0
0.4
0.3
0.25
0.2
0.15
0.1
0.05
0
0.2
0.1
4
2
0
-4
-2
-2
0
-4
-2
4
2
2
-4
4
Joint Density with
C
Joint Density with
D
? = 0.2
0.25
0.2
0.15
0.1
0.05
0
4
2
0
-4
-2
-2
0
2
-4
4
? = 0.5
0.25
0.2
0.15
0.1
0.05
0
4
2
0
-4
-2
-2
0
2
-4
4
Figure 5: Different joint densities with the same marginal densities
Modality 1
Conditionally Independent
Modality 2
Modality 1
Highly Correlated
Modality 2
Figure 6: Lines are joined between co-occuring patterns in two imaginary 1-D modalities
(as shown at top). The M-D algorithm wants to find a partition that crosses the fewest lines.
Conditional Information (I(X;Y |Class))
(with diagonal zeroed)
Within-Class Correlation Coefficients
(averaged over each class)
Figure 7: Statistical Structure of our data
In the bottom case, the modalities are strongly dependent. In this case there are many local
minima for minimum disagreement, that are not closely related to the class boundary. It is
easy for the networks to minimize the disagreement between the outputs of the modalities,
without paying attention to the class. Having two very strongly dependent variables, one
on each side of the network, means that the network can minimize disagreement by simply
listening to those units.
To verify that our auditory-visual results were due to statistical differences between the
dimensions, we examined the statistical structure of our data. It turns out that, within a
class, the correlation coefficient between most pairs of dimensions is fairly low. However,
for related auditory features (similar time and frequency band) correlations are high and
also for related visual features. This is shown in Figure 7. We also computed the conditional
mutual information between each pair of features given the class I(x; y|Class). This is
also shown in Figure 7. This value is 0 if and only if the two features are conditionally
independent given the class. The graphs show that many of the auditory dimensions are
highly dependent on each other (even given the class), as are many of the visual dimensions.
This makes them unsuitable for serving on the other side of a M-D network.
2.5
Discussion
The minimizing-disagreement algorithm was initially developed as a model of selfsupervised cortical learning and the importance of conditionally uncorrelated structure was
mentioned in [5]. Since then people have used similar partly-supervised algorithms to deal
with limited labeled data in machine learning problems [6, 7]. They have also emphasized
the importance of conditional independence between the two sides of the input. However
in the co-training style algorithms, inputs that are conditionally dependent are not helpful,
but they are also not as harmful. Because the self-supervised algorithm is dependent on the
class structure being evident in the joint space as its only source of supervision, it is very
sensitive to conditionally dependent relationships between the modalities.
We have shown that different sensory modalities are ideally suited for teaching each other.
Sensory modalities are also composed of submodalities (e.g. color and motion for the
visual modality) which are also likely to be conditionally independent (and indeed may be
actively kept so [8, 9, 10]). We suggest that brain connectivity may be constrained not only
due to volume limits, but because limiting connectivity may be beneficial for learning.
Acknowledgements
A preliminary version of this work appeared in a book chapter [5] in the book, Psychology
of Learning and Motivation. This work is supported by NSF CAREER grant 0133996.
References
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G. Tesauro, and J. Alspector, editors, Advances in Neural Information Processing
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[2] Virginia R. de Sa and Dana H. Ballard. Category learning through multimodality
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[3] Teuvo Kohonen. Improved versions of learning vector quantization. In IJCNN International Joint Conference on Neural Networks, volume 1, pages I?545?I?550, 1990.
[4] Ramprasad Polana. Temporal Texture and Activity Recognition. PhD thesis, Department of Computer Science, University of Rochester, 1994.
[5] Virginia R. de Sa and Dana H. Ballard. Perceptual learning from cross-modal feedback. In R. L. Goldstone, P.G. Schyns, and D. L. Medin, editors, Psychology of
Learning and Motivation, volume 36, pages 309?351. Academic Press, San Diego,
CA, 1997.
[6] A. Blum and T. Mitchell. Combining labeled and unlabeled data with co-training. In
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(COLT-98), pages 92?100, 1998.
[7] Ion Muslea, Steve Minton, and Craig Knoblock. Active + semi-supervised learning =
robust multi-view learning. In Proceedings of the 19th International Conference on
Machine Learning (ICML 2002), pages 435?442, 2002.
[8] C. McCollough. Color adaptation of edge-detectors in the human visual system. Science, 149:1115?1116, 1965.
[9] P.C. Dodwell and G.K. Humphrey. A functional theory of the mccollough effect.
Psychological Review, 1990.
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1,678 | 2,525 | Fast Algorithms for Large-State-Space HMMs with
Applications to Web Usage Analysis
Pedro F. Felzenszwalb1 , Daniel P. Huttenlocher2 , Jon M. Kleinberg2
2
1
AI Lab, MIT, Cambridge MA 02139
Computer Science Dept., Cornell University, Ithaca NY 14853
Abstract
In applying Hidden Markov Models to the analysis of massive data
streams, it is often necessary to use an artificially reduced set of
states; this is due in large part to the fact that the basic HMM
estimation algorithms have a quadratic dependence on the size of
the state set. We present algorithms that reduce this computational
bottleneck to linear or near-linear time, when the states can be
embedded in an underlying grid of parameters. This type of state
representation arises in many domains; in particular, we show an
application to traffic analysis at a high-volume Web site.
1
Introduction
Hidden Markov Models (HMMs) are used in a wide variety of applications where
a sequence of observable events is correlated with or caused by a sequence of unobservable underlying states (e.g., [8]). Despite their broad applicability, HMMs
are in practice limited to problems where the number of hidden states is relatively
small. The most natural such problems are those where some abstract categorization provides a small set of discrete states, such as phonemes in the case of speech
recognition or coding and structure in the case of genomics. Recently, however,
issues arising in massive data streams, such as the analysis of usage logs at hightraffic Web sites, have led to problems that call naturally for HMMs with large state
sets over very long input sequences.
A major obstacle in scaling HMMs up to larger state spaces is the computational
cost of implementing the basic primitives associated with them: given an n-state
HMM and a sequence of T observations, determining the probability of the observations, or the state sequence of maximum probability, takes O(T n2 ) time using the
forward-backward and Viterbi algorithms. The quadratic dependence on the number of states is a long-standing bottleneck that necessitates a small (often artificially
coarsened) state set, particularly when the length T of the input is large.
In this paper, we present algorithms that overcome this obstacle for a broad class
of HMMs. We improve the running times of the basic estimation and inference
primitives to have a linear or near-linear dependence on the number of states, for
a family of models in which the states are embedded as discrete grid points in an
underlying parameter space, and the state transition costs (the negative logs of
the state transition probabilities) correspond to a possibly non-metric distance on
this space. This kind of embedded-state model arises in many domains, including
object tracking, de-noising one-dimensional signals, and event detection in time
series. Thus the algorithms can be seen as extending the applicability of HMMs to
problems that are traditionally solved with more restricted linear Gaussian statespace models such as Kalman filtering. Non-Gaussian state-space techniques are
a research focus in their own right (e.g., [6]) and our methods could be used to
improve their efficiency.
Given a structured embedding of states in an underlying d-dimensional space, our
approach is to reduce the amount of work in the dynamic programming iterations
of the Viterbi and forward-backward algorithms. For the Viterbi algorithm, we
make use of distance transform (also known as Voronoi surface) techniques, which
are widely used in computer vision, image processing, and discrete computational
geometry [2]. For a broad class of distance functions on the embedding space
(including functions that are far from obeying the triangle inequality), we are able
to run each dynamic programming step of the Viterbi algorithm in O(n) time,
yielding an overall running time of O(T n). In the case of the forward-backward
algorithm, we are able to achieve O(T n) time for any transition probabilities that
can be decomposed into a constant number of box filters [10]. Box filters are discrete
convolution kernels that can be computed in linear time; many functions, including
the Gaussian, can be expressed or approximated as the composition of a few box
filters. Moreover, in the case of the forward-backward algorithm, we are able to
obtain a running time of O(T n log n) for arbitrary state transition probabilities, as
long as they are based only on differences in the embedded positions of the states.
A motivating application for our work comes from the analysis of Web usage data
[1]. We focus on the Internet Archive site (www.archive.org) as a prototypical
example of a high-traffic site (millions of page-visits per month) offering an array
of digital items for download. An important question at such a site is to determine
variations in user interest in the items being offered. We use a coin-tossing HMM
model in which the discrete states correspond to the current probability of a user
downloading a given item; this state set has a natural embedding in the interval
[0, 1]. We study the effect of increasing the number of states, and find that a fairly
large state set (of size roughly a hundred or more) is needed in order to detect
brief but significant events that affect the download rate. With tens of millions of
observations and a state set of this size, practical analysis would be computationally
prohibitive without the faster HMM algorithms described here.
It should be noted that our methods can also be used in belief revision and belief
propagation algorithms for Bayesian networks (e.g., [7]), as these algorithms are
essentially variants of the Viterbi and forward-backward algorithms for HMMs.
The methods are also applicable to continuous Markov models, which have recently
been employed for Web user modeling based on duration of page views [9].
2
Hidden Markov Models
We briefly review HMMs; however we assume that the reader is familiar both with
HMMs and with the Viterbi and forward-backward estimation algorithms. Rabiner
[8] provides a good introduction to HMMs; we use notation similar to his. An HMM
can be represented by a 5-tuple ? = (S, V, A, B, ?) where S = {s1 , . . . , sn } is a finite
set of (hidden) states, V = {v1 , . . . , vm } is a finite set of observable symbols, A is an
n ? n matrix with entries aij corresponding to the probability of going from state i
to state j, B = {bi (k)} where bi (k) specifies the probability of observing symbol vk
in state si , and ? is an n-vector with each entry ?i corresponding to the probability
Function
aij = p if |i ? j| ? d,
aij = 0 otherwise
aij ? exp(?|i ? j|2 /2? 2 )
aij ? exp(?k|i ? j|)
aij = p if |i ? j| ? d,
aij = q otherwise
aij ? exp(?|i ? j|2 /2? 2 )
if |i ? j| ? d,
aij ? exp(?k|i ? j|) otherwise
Viterbi
Forward-Backward
Min-filter
Box sum
L22
dist. trans.
L1 dist. trans.
Gaussian approx.
FFT
Combin. min-filter
Combin. box sum
Combin. dist. trans.
FFT
Table 1: Some transition probabilities that can be handled efficiently using our
techniques (see text for an explanation). All running times are O(T n) except those
using the FFT which are O(T n log n).
that the initial state of the system is si .
Let qt denote the state of the system at time t, while ot denotes the observed
symbol at time t. Given a sequence of observations O = (o1 , . . . , oT ) there are three
standard estimation (or inference) problems that have wide applications:
1. Find a state sequence Q = (q1 , . . . , qT ) maximizing P (Q|O, ?).
2. Compute P (O|?), the probability of an observation sequence being generated by ?.
3. Compute the posterior probabilities of each state, P (qt = si |O, ?).
As is well known these problems can be solved in O(T n2 ) time using the Viterbi
algorithm for the first task and the forward-backward algorithm for the others. We
show how to solve them more efficiently for a wide range of transition probabilities based on differences between states that are embedded in an underlying grid.
This grid can be multi-dimensional, however in this paper we consider only the
one-dimensional case. Table 1 lists some widely applicable transition probability
distributions that can be handled by our methods. The algorithms for each distribution differ slightly and are explained in the subsequent sections. The distributions
given in the bottom part of the table can be computed as combinations of the basic distributions in the top part. Other distributions can be obtained using these
same combination techniques, as long as only a constant number of distributions
are being combined.
An additional problem, which we do not explicitly consider here, is that of determining the best model ? given some set of observed sequences {O1 , . . . , Ol }. However
the most widely used technique for solving this problem, expectation maximization (EM), requires repeatedly running the forward-backward algorithm. Thus our
algorithms also indirectly make the model learning problem more efficient.
2.1
Viterbi Algorithm
The Viterbi algorithm is used to find a maximum posterior probability state sequence, that is a sequence Q = (q1 , . . . , qT ) maximizing P (Q|O, ?). The main
computation is to determine the highest probability along a path, accounting for
the observations and ending in a given state. While there are an exponential number of possible paths, the Viterbi algorithm uses a dynamic programming approach
Figure 1: An example of the L1 distance transform for a grid with n = 9 points
containing the point set P = {1, 3, 7}. The distance transform value at each point
is given by the height of the lower envelope, depicted as a dashed contour.
(see e.g., [8]), employing the recursive equation
?t+1 (j) = bj (ot+1 ) max (?t (i)aij ) ,
i
where ?t (i), for i = 1, 2, . . . , n, encodes the highest probability along a path which
accounts for the first t observations and ends in state si . The maximization term
takes O(n2 ) time, resulting in an overall time of O(T n2 ) for a sequence of length
T . Computing ?t for each time step is only the first pass of the Viterbi algorithm.
In a subsequent backward pass, a minimizing path is found. This takes only O(T n)
time, so the forward computation is the dominant part of the running time.
In general a variant of the Viterbi algorithm is employed that uses negative
log probabilities rather than probabilities, such that the computation becomes
0
?t+1
(j) = b0j (ot+1 ) + mini (?t0 (i) + a0ij ), where 0 is used to denote a negative log
probability. We now turn to the computation of ? 0 for restricted forms of the transition costs a0ij , where there is an underlying parameter space such that the costs
can be expressed in terms of a distance between parameter values corresponding to
the states. Let us denote such cost functions by ?(i ? j). Then,
0
?t+1
(j) = b0j (ot+1 ) + min (?t0 (i) + ?(i ? j)) .
i
(1)
We now show how the minimization in the second term can be computed in O(n)
time rather than O(n2 ). The approach is based on a generalization of the distance
transform, which is defined for sets of points on a grid. Consider a grid with N
locations and a point set P on that grid. The distance transform of P specifies for
each grid location, the distance to the closest point in the set P ,
DP (j) = min ?(i ? j).
i?P
Clearly the distance transform can be computed in O(N 2 ) time by considering all
pairs of grid locations. However, it can also be computed in linear time for many
distance functions using simple algorithms (e.g., [2, 5]). These algorithms have small
constants and are fast in practice. The algorithms work for distance transforms of
d-dimensional grids, not just for the one-dimensional case that we illustrate here.
In order to compute the distance transform efficiently it is commonly expressed as,
DP (j) = min (?(i ? j) + 1(i)) ,
i
where 1(i) is an indicator function for the set P such that 1(i) = 0 when i ? P and
1(i) = ? otherwise. Intuitively one can think of a collection of upward facing cones,
one rooted at each grid location that is in the set P . The transform is then obtained
by taking the lower envelope (or minimum) of these cones. For concreteness consider
the one-dimensional case with the L1 distance between grid locations. In this case
the ?cones? are v-shapes of slope 1 rising from the value y = 0 at each grid location
that corresponds to a point of the set P , as illustrated in Figure 1.
It is straightforward to verify that a simple two-pass algorithm correctly computes
this one-dimensional distance transform. First the vector D(j) is initialized to 1(j).
Then in the forward pass, each successive element of D(j) is set to the minimum
of its own value and one plus the value of the previous element (this is done ?in
place? so that updates affect one another).
j = 1, ..., n ? 1 : D(j) = min(D(j), D(j ? 1) + 1).
The backward pass is analogous,
j = n ? 2, ..., 0 : D(j) = min(D(j), D(j + 1) + 1).
Consider the example in Figure 1. After the initialization step the value of D is
(?, 0, ?, 0, ?, ?, ?, 0, ?), after the forward pass it is (?, 0, 1, 0, 1, 2, 3, 0, 1) and
after the backward pass the final answer of (1, 0, 1, 0, 1, 2, 1, 0, 1).
This computation of the distance transform does not depend on the form of the
function 1(i). This suggests a generalization of distance transforms where the indicator function 1(i) is replaced with an arbitrary function,
Df (j) = min (?(i ? j) + f (i)) .
i
The same observation was used in [4] to efficiently compute certain tree-based cost
functions for visual recognition of multi-part objects. Intuitively, the upward-facing
cones are now rooted at height f (i) rather than at zero, and are positioned at every
grid location. The function Df is as above the lower envelope of these cones.
This generalized distance transform Df is precisely the form of the minimization
term in the computation of the Viterbi recursion ? 0 in equation (1), where each
state corresponds to a grid point. The algorithm above can be used to compute
each step of the Viterbi minimization in O(n) time when ? is the L1 norm, giving
an O(T n) algorithm overall. This corresponds to the problem in the third row of
Table 1. The computation for the second row of the table is similar, except that
computing the distance transform for the L2 distance squared is a bit more involved
(see [5]). The distribution in the first row of the table can be handled using a linear
time algorithm for the min-filter [3].
Combinations of transforms can be formed by computing each function separately
and then taking the minimum of the results. The entries in the bottom part of
Table 1 show two such combinations. The function in the fourth row is often of
practical interest, where the probability is p of staying near the current state and
q of transitioning to any other state. The function in the last row is a so-called
?truncated quadratic?, arising commonly in robust statistics. In the experimental
section we use a similar function that is the combination of two linear components
with different slopes.
2.2
Forward-Backward Algorithm
The forward-backward algorithm is used to find the probability of the observed
sequence given the the model, P (O|?). The computation also determines the posterior probability of the states at each time, P (qt = si |O, ?). Most of the work
in the forward-backward algorithm is spent in determining the so-called forward
and backward probabilities at each step (again see [8] or any other introduction to
HMMs). The forward probabilities at a given time can be expressed as the n-vector
?t (i) = P (o1 , o2 , . . . , ot , qt = si |?),
i.e., the probability of the partial observation sequence up until time t and the
state at time t, given the model. The backward probabilities ?t can be expressed
analogously and are not considered here. The standard computation is to express
the vector ?t recursively as
?t+1 (j) = bj (ot+1 )
n
X
(?t (i)aij ) .
i=1
In this form it is readily apparent that computing ?t+1 from ?t involves O(n2 )
operations, as each of the n entries in the vector involves a sum of n terms.
When the transition probabilities are based just on the differences between the
underlying coordinates corresponding to the states, aij = h(j ? i), the recursive
computation of ? becomes
?t+1 (j) = bj (ot+1 )
n
X
(?t (i)h(j ? i)) .
i=1
The summation term is simply the convolution of ?t with h. In general, this discrete
convolution can be computed in O(n log n) time using the FFT. While this is a
simple observation, it enables efficient calculation of the forward and backward
probabilities for problems where the states are embedded in a grid.
In certain specific cases convolution can be computed in linear time. One case
of particular interest is the so-called box sum, in which the convolution kernel is a
constant function within a region. That is, h(j) = k over some interval and h(j) = 0
outside that interval. A Gaussian can be well approximated by convolution of just
a few such box filters [10], and thus it is possible to approximately compute the
functions in the first and second rows of Figure 1 in O(T n) time. Similarly to the
Viterbi case, functions can be created from combinations of box-sums. In this case
a weighted sum of the individual functions is used rather than their minimum.
3
Coin-Tossing Models and Web Usage Analysis
We now turn to the application mentioned in the introduction: using a coin-tossing
model with a one-dimensional embedding of states to estimate the download probability of items at a Web site. Our data comes from the Internet Archive site
(www.archive.org), which offers digital text, movie, and audio files. Each item on
the site has a separate description page, which contains the option to download it;
this is similar to the paper description pages on CiteSeer or the e-print arXiv and
to the item description pages at online retailers (with the option to purchase). On a
site of this type, the probability that a user chooses to acquire an item, conditioned
on having visited the description page, can be viewed as a measure of interest [1].
This ratio of acquisitions to visits is particularly useful as a way of tracking the
changes in user interest in an item. Suppose the item is featured prominently on
the site; or an active off-site link to the item description drives a new sub-population
of users to it; or a technical problem makes it impossible to obtain the item ? these
are all discrete events that can have a sudden, significant effect on the fraction of
users who download the item. By identifying such discrete changes, we can discover
the most significant events, both on the site and on the Web at large, that have
affected user interest in each item. Such a history of events can be useful to site
administrators, as feedback to the users of the site, and for researchers.
This type of change-detection fits naturally into the framework of HMMs. For a
fixed item, each observation corresponds to a user?s visit to the item description,
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Step Size .01
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State for Corresponding Visit
State for Corresponding Visit
Step Size .1
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Time of Visit
1.045e+09
1.05e+09
1.03e+09
1.035e+09
1.04e+09
Time of Visit
1.045e+09
1.05e+09
Figure 2: Estimate of underlying download bias; best state sequence for models
with step sizes of .1 (9 states) on the left and .01 (81 states) on the right.
and there are two observable symbols V = {1, 0}, corresponding to the decision
to download or not. We assume a model in which there is a hidden coin of some
unknown bias that is flipped when the user visits the description and whose outcome
determines the download decision. Thus, each state si corresponds to a discretized
value pi of the underlying bias parameter. The natural observation cost function
b0i (k) is simply the negative log of the probability p for a head and (1 ? p) for a tail.
The points at which state transitions occur in the optimal state sequence thus
become candidates for discrete changes in user interest. The form of the state transition costs is based on our assumptions about the nature of these changes. As
indicated above, they often result from the introduction of a new sub-population
with different interests or expectations; thus, it is natural to expect that the transition cost should rise monotonically as the change in bias increases, but that even
large changes should happen with some regularity.
We quantize the underlying bias parameter values equally, such that |pi ?pj | ? |i?j|
and use a cost function of the form
a0ij = min (k1 |i ? j|, k2 |i ? j| + k3 ) ,
where the ki are positive constants and k1 > k2 . This two-slope linear model is
monotone increasing but once the change in bias becomes large enough the rate of
increase is small. The model prefers constant or small changes in bias but allows
for arbitrarily large changes, similarly to the ?truncated model? common in robust
statistics.
Figure 2 shows the best state sequence obtained with the Viterbi algorithm under
this model, using two different discretizations of the parameter space, for an input
sequence of 11, 159 visits from August 2002 to April 2003 to the description page
for a particular video in the Internet Archive. On the left is a 9-state model with
probabilities ranging from .1 to .9 in steps of size .1. On the right is an 81-state
model with the same range of .1 to .9 but where the steps are of size .01. The x-axis
shows the visit time (UTC in billions of seconds since the epoch) and the y-axis
shows the bias associated with the state in the optimal sequence at that time.
We begin by observing that both models capture a number of discrete changes
in download behavior. These changes correspond to genuine external events. In
particular, both models capture the long-term drop and rebound in bias which corresponds to the time period where the item was highlighted on a top-level page,
as well as the two rightmost short downward spikes which correspond to technical
problems that made downloads temporarily impossible. Even though these latter
failures were relatively short-lived, lasting a few hours out of the several-month
range, they are detected easily by the stochastic model; in contrast, temporal windowing techniques miss such short events.
The two plots, however, exhibit some subtle but important differences that illustrate
the qualitatively greater power we obtain from a larger state set. In particular,
the 81-state model has four short downward spikes rather than three in the time
interval from 1.045 to 1.05. The latter two are the technical failures identified by
both models, but the first two correspond to two distinct off-site referring pages each
of which drove a significant amount of low-interest user traffic to the item. While
the 81-state model was able to resolve these as separate events, the 9-state model
blurs them into an artificial period of medium bias, followed by a downward spike
to the lowest possible state (i.e. the same state it used for the technical failures).
Finally, the 81-state model discovers a gradual decline in the download rate near
the end of the plot that is not visible when there are fewer states.
We see that a model with a larger state set is able to pick up the effects of different
types of events ? both on-site and off-site highlighting of the item, as well as technical problems ? and that these events often result in sudden, discrete changes.
Moreover, it appears that beyond a certain point, the set of significant events remains roughly fixed even as the resolution in the state set increases. While we do
not show the result here, an 801-state model with step size .001 produces a plot
that is qualitatively indistinguishable from the 81 state model with step size .01 ?
only the y-values provide more detail with the smaller step size.
References
[1] J. Aizen, D. Huttenlocher, J. Kleinberg, A. Novak, ?Traffic-Based Feedback on
the Web,? To appear in Proceedings of the National Academy of Sciences.
[2] G. Borgefors, ?Distance Transformations in Digital Images?, Computer Vision,
Graphics and Image Processing, Vol. 34, pp. 344-371, 1986.
[3] Y. Gil and M. Werman, ?Computing 2D Min, Max and Median Filters? IEEE
Trans. PAMI, Vol. 15, 504-507, 1993.
[4] P. Felzenszwalb, D. Huttenlocher, ?Efficient Matching of Pictorial Structures,?
Proc. IEEE Computer Vision and Pattern Recognition Conf., 2000, pp. 66-73.
[5] A. Karzanov, ?Quick algorithm for determining the distances from the points of
the given subset of an integer lattice to the points of its complement?, Cybernetics
and System Analysis, 1992. (Translation from the Russian by Julia Komissarchik.)
[6] G. Kitagawa, ?Non-Gaussian State Space Modeling of Nonstationary Time Series?, Journal of the American Statistical Association, 82, pp. 1032-1063, 1987.
[7] J. Pearl, Probabilistic Reasoning in Intelligent Systems: Networks of Plausible
Inference, Morgan Kaufmann, 1988.
[8] L. Rabiner, ?A tutorial on hidden Markov models and selected applications in
speech recognition,? Proceedings of the IEEE Vol. 77(2), pp. 257-286, 1989.
[9] S.L. Scott, P. Smyth, ?The Markov Modulated Poisson Process and Markov
Poisson Cascade with Applications to Web Traffic Data,? Bayesian Statistics
7(2003), to appear.
[10] W.M. Wells, ?Efficient synthesis of Gaussian filters by cascaded uniform filters?, IEEE Trans. PAMI, Vol. 8(2), pp. 234-239, 1986.
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1,679 | 2,526 | Learning the k in k-means
Greg Hamerly, Charles Elkan
{ghamerly,elkan}@cs.ucsd.edu
Department of Computer Science and Engineering
University of California, San Diego
La Jolla, California 92093-0114
Abstract
When clustering a dataset, the right number k of clusters to use is often
not obvious, and choosing k automatically is a hard algorithmic problem. In this paper we present an improved algorithm for learning k while
clustering. The G-means algorithm is based on a statistical test for the
hypothesis that a subset of data follows a Gaussian distribution. G-means
runs k-means with increasing k in a hierarchical fashion until the test accepts the hypothesis that the data assigned to each k-means center are
Gaussian. Two key advantages are that the hypothesis test does not limit
the covariance of the data and does not compute a full covariance matrix.
Additionally, G-means only requires one intuitive parameter, the standard statistical significance level ?. We present results from experiments
showing that the algorithm works well, and better than a recent method
based on the BIC penalty for model complexity. In these experiments,
we show that the BIC is ineffective as a scoring function, since it does
not penalize strongly enough the model?s complexity.
1
Introduction and related work
Clustering algorithms are useful tools for data mining, compression, probability density estimation, and many other important tasks. However, most clustering algorithms require the
user to specify the number of clusters (called k), and it is not always clear what is the best
value for k. Figure 1 shows examples where k has been improperly chosen. Choosing k is
often an ad hoc decision based on prior knowledge, assumptions, and practical experience.
Choosing k is made more difficult when the data has many dimensions, even when clusters
are well-separated.
Center-based clustering algorithms (in particular k-means and Gaussian expectationmaximization) usually assume that each cluster adheres to a unimodal distribution, such
as Gaussian. With these methods, only one center should be used to model each subset
of data that follows a unimodal distribution. If multiple centers are used to describe data
drawn from one mode, the centers are a needlessly complex description of the data, and in
fact the multiple centers capture the truth about the subset less well than one center.
In this paper we present a simple algorithm called G-means that discovers an appropriate
k using a statistical test for deciding whether to split a k-means center into two centers.
We describe examples and present experimental results that show that the new algorithm
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Figure 1: Two clusterings where k was improperly chosen. Dark crosses are k-means
centers. On the left, there are too few centers; five should be used. On the right, too many
centers are used; one center is sufficient for representing the data. In general, one center
should be used to represent one Gaussian cluster.
is successful. This technique is useful and applicable for many clustering algorithms other
than k-means, but here we consider only the k-means algorithm for simplicity.
Several algorithms have been proposed previously to determine k automatically. Like our
method, most previous methods are wrappers around k-means or some other clustering
algorithm for fixed k. Wrapper methods use splitting and/or merging rules for centers to
increase or decrease k as the algorithm proceeds.
Pelleg and Moore [14] proposed a regularization framework for learning k, which they call
X-means. The algorithm searches over many values of k and scores each clustering model
using the so-called Bayesian Information Criterion [10]: BIC(C|X) = L(X|C) ? p2 log n
where L(X|C) is the log-likelihood of the dataset X according to model C, p = k(d + 1)
is the number of parameters in the model C with dimensionality d and k cluster centers,
and n is the number of points in the dataset. X-means chooses the model with the best BIC
score on the data. Aside from the BIC, other scoring functions are also available.
Bischof et al. [1] use a minimum description length (MDL) framework, where the description length is a measure of how well the data are fit by the model. Their algorithm starts
with a large value for k and removes centers (reduces k) whenever that choice reduces
the description length. Between steps of reducing k, they use the k-means algorithm to
optimize the model fit to the data.
With hierarchical clustering algorithms, other methods may be employed to determine the
best number of clusters. One is to build a merging tree (?dendrogram?) of the data based
on a cluster distance metric, and search for areas of the tree that are stable with respect
to inter- and intra-cluster distances [9, Section 5.1]. This method of estimating k is best
applied with domain-specific knowledge and human intuition.
2
The Gaussian-means (G-means) algorithm
The G-means algorithm starts with a small number of k-means centers, and grows the
number of centers. Each iteration of the algorithm splits into two those centers whose data
appear not to come from a Gaussian distribution. Between each round of splitting, we run
k-means on the entire dataset and all the centers to refine the current solution. We can
initialize with just k = 1, or we can choose some larger value of k if we have some prior
knowledge about the range of k.
G-means repeatedly makes decisions based on a statistical test for the data assigned to each
center. If the data currently assigned to a k-means center appear to be Gaussian, then we
want to represent that data with only one center. However, if the same data do not appear
Algorithm 1 G-means(X, ?)
1: Let C be the initial set of centers (usually C ? {?
x}).
2: C ? kmeans(C, X).
3: Let {xi |class(xi ) = j} be the set of datapoints assigned to center cj .
4: Use a statistical test to detect if each {xi |class(xi ) = j} follow a Gaussian distribution
(at confidence level ?).
5: If the data look Gaussian, keep cj . Otherwise replace cj with two centers.
6: Repeat from step 2 until no more centers are added.
to be Gaussian, then we want to use multiple centers to model the data properly. The
algorithm will run k-means multiple times (up to k times when finding k centers), so the
time complexity is at most O(k) times that of k-means.
The k-means algorithm implicitly assumes that the datapoints in each cluster are spherically
distributed around the center. Less restrictively, the Gaussian expectation-maximization
algorithm assumes that the datapoints in each cluster have a multidimensional Gaussian
distribution with a covariance matrix that may or may not be fixed, or shared. The Gaussian
distribution test that we present below are valid for either covariance matrix assumption.
The test also accounts for the number of datapoints n tested by incorporating n in the
calculation of the critical value of the test (see Equation 2). This prevents the G-means
algorithm from making bad decisions about clusters with few datapoints.
2.1
Testing clusters for Gaussian fit
To specify the G-means algorithm fully we need a test to detect whether the data assigned
to a center are sampled from a Gaussian. The alternative hypotheses are
? H0 : The data around the center are sampled from a Gaussian.
? H1 : The data around the center are not sampled from a Gaussian.
If we accept the null hypothesis H0 , then we believe that the one center is sufficient to
model its data, and we should not split the cluster into two sub-clusters. If we reject H0
and accept H1 , then we want to split the cluster.
The test we use is based on the Anderson-Darling statistic. This one-dimensional test has
been shown empirically to be the most powerful normality test that is based on the empirical
cumulative distribution function (ECDF). Given a list of values xi that have been converted
to mean 0 and variance 1, let x(i) be the ith ordered value. Let zi = F (x(i) ), where F is
the N (0, 1) cumulative distribution function. Then the statistic is
n
A2 (Z)
= ?
1X
(2i ? 1) [log(zi ) + log(1 ? zn+1?i )] ? n
n i=1
(1)
Stephens [17] showed that for the case where ? and ? are estimated from the data (as in
clustering), we must correct the statistic according to
A2? (Z)
= A2 (Z)(1 + 4/n ? 25/(n2 ))
(2)
Given a subset of data X in d dimensions that belongs to center c, the hypothesis test
proceeds as follows:
1. Choose a significance level ? for the test.
2. Initialize two centers, called ?children? of c. See the text for good ways to do this.
3. Run k-means on these two centers in X. This can be run to completion, or to some
early stopping point if desired. Let c1 , c2 be the child centers chosen by k-means.
4. Let v = c1 ? c2 be a d-dimensional vector that connects the two centers. This is
the direction that k-means believes to be important for clustering. Then project
X onto v: x0i = hxi , vi/||v||2 . X 0 is a 1-dimensional representation of the data
projected onto v. Transform X 0 so that it has mean 0 and variance 1.
5. Let zi = F (x0(i) ). If A2? (Z) is in the range of non-critical values at confidence
level ?, then accept H0 , keep the original center, and discard {c1 , c2 }. Otherwise,
reject H0 and keep {c1 , c2 } in place of the original center.
A primary contribution of this work is simplifying the test for Gaussian fit by projecting
the data to one dimension where the test is simple to apply. The authors of [5] also use
this approach for online dimensionality reduction during clustering. The one-dimensional
representation of the data allows us to consider only the data along the direction that kmeans has found to be important for separating the data. This is related to the problem
of projection pursuit [7], where here k-means searches for a direction in which the data
appears non-Gaussian.
We must choose the significance level of the test, ?, which is the desired probability of
making a Type I error (i.e. incorrectly rejecting H0 ). It is appropriate to use a Bonferroni
adjustment to reduce the chance of making Type I errors over multiple tests. For example, if
we want a 0.01 chance of making a Type I error in 100 tests, we should apply a Bonferroni
adjustment to make each test use ? = 0.01/100 = 0.0001. To find k final centers the
G-means algorithm makes k statistical tests, so the Bonferroni correction does not need to
be extreme. In our tests, we always use ? = 0.0001.
We consider two ways to initialize the two child centers. Both approaches initialize with
c ? m, where c is a center and m is chosen. The first method chooses m as a random
d-dimensional vector such that ||m|| is small compared to the distortion of the data. A
second method finds p
the main principal component s of the data (having eigenvalue ?),
and chooses m = s 2?/?. This deterministic method places the two centers in their
expected locations under H0 . The principal component calculations require O(nd2 + d3 )
time and O(d2 ) space, but since we only want the main principal component, we can use
fast methods like the power method, which takes time that is at most linear in the ratio of
the two largest eigenvalues [4]. In this paper we use principal-component-based splitting.
2.2
An example
Figure 2 shows a run of the G-means algorithm on a synthetic dataset with two true clusters
and 1000 points, using ? = 0.0001. The critical value for the Anderson-Darling test is
1.8692 for this confidence level. Starting with one center, after one iteration of G-means,
we have 2 centers and the A2? statistic is 38.103. This is much larger than the critical value,
so we reject H0 and accept this split. On the next iteration, we split each new center and
repeat the statistical test. The A2? values for the two splits are 0.386 and 0.496, both of
which are well below the critical value. Therefore we accept H0 for both tests, and discard
these splits. Thus G-means gives a final answer of k = 2.
2.3
Statistical power
Figure 3 shows the power of the Anderson-Darling test, as compared to the BIC. Lower is
better for both plots. We run 1000 tests for each data point plotted for both plots. In the left
14
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Figure 2: An example of running G-means for three iterations on a 2-dimensional dataset
with two true clusters and 1000 points. Starting with one center (left plot), G-means splits
into two centers (middle). The test for normality is significant, so G-means rejects H0 and
keeps the split. After splitting each center again (right), the test values are not significant,
so G-means accepts H0 for both tests and does not accept these splits. The middle plot is
the G-means answer. See the text for further details.
1
1
G-means
X-means
0.8
P(Type II error)
P(Type I error)
0.8
G-means
X-means
0.6
0.4
0.2
0.6
0.4
0.2
0
0
0
30
60
90
120 150
number of datapoints
180
210
0
30
60
90
120 150
number of datapoints
180
210
Figure 3: A comparison of the power of the Anderson-Darling test versus the BIC. For
the AD test we fix the significance level (? = 0.0001), while the BIC?s significance level
depends on n. The left plot shows the probability of incorrectly splitting (Type I error) one
true 2-d cluster that is 5% elliptical. The right plot shows the probability of incorrectly not
splitting two true clusters separated by 5? (Type II error). Both plots are functions of n.
Both plots show that the BIC overfits (splits clusters) when n is small.
plot, for each test we generate n datapoints from a single true Gaussian distribution, and
then plot the frequency with which BIC and G-means will choose k = 2 rather than k = 1
(i.e. commit a Type I error). BIC tends to overfit by choosing too many centers when the
data is not strictly spherical, while G-means does not. This is consistent with the tests of
real-world data in the next section. While G-means commits more Type II errors when n is
small, this prevents it from overfitting the data.
The BIC can be considered a likelihood ratio test, but with a significance level that cannot
be fixed. The significance level instead varies depending on n and ?k (the change in the
number of model parameters between two models). As n or ?k decrease, the significance
level increases (the BIC becomes weaker as a statistical test) [10]. Figure 3 shows this
effect for varying n. In [11] the authors show that penalty-based methods require problemspecific tuning and don?t generalize as well as other methods, such as cross validation.
3
Experiments
Table 1 shows the results from running G-means and X-means on many large synthetic. On
synthetic datasets with spherically distributed clusters, G-means and X-means do equally
Table 1: Results for many synthetic datasets. We report distortion relative to the optimum
distortion for the correct clustering (closer to one is better), and time is reported relative to
k-means run with the correct k. For BIC, larger values are better, but it is clear that finding
the correct clustering does not always coincide with finding a larger BIC. Items with a star
are where X-means always chose the largest number of centers we allowed.
dataset
synthetic
k=5
synthetic
k=20
synthetic
k=80
synthetic
k=5
synthetic
k=20
synthetic
k=80
synthetic
k=5
synthetic
k=20
synthetic
k=80
d
2
k found
9.1? 9.9
18.1? 3.2
20.1? 0.6
70.5?11.6
80.0? 0.2
171.7?23.7
5.0? 0.0
*20.0? 0.0
20.0? 0.1
*80.0? 0.0
80.2? 0.5
229.2?36.8
5.0? 0.0
*20.0? 0.0
20.0? 0.0
*80.0? 0.0
80.0? 0.0
171.5?10.9
method
G-means
X-means
G-means
X-means
G-means
X-means
G-means
X-means
G-means
X-means
G-means
X-means
G-means
X-means
G-means
X-means
G-means
X-means
2
2
8
8
8
32
32
32
7
7
6
6
5
5
4
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4
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8
BIC(?104 )
-0.19?2.70
0.70?0.93
0.21?0.18
14.83?3.50
1.84?0.12
40.16?6.59
-0.74?0.16
-2.28?0.20
-0.18?0.17
14.36?0.21
1.45?0.20
52.28?9.26
-3.36?0.21
-27.92?0.22
-2.73?0.22
-11.13?0.23
-1.10?0.16
11.78?2.74
distortion(? optimal)
0.89? 0.23
0.37? 0.12
0.99? 0.01
9.45?28.02
1.00? 0.01
48.49?70.04
1.00? 0.00
0.47? 0.03
0.99? 0.00
0.47? 0.01
0.99? 0.00
0.57? 0.06
1.00? 0.00
0.76? 0.00
1.00? 0.00
0.76? 0.01
1.00? 0.00
0.84? 0.01
10
12
0
0
2
4
6
8
10
time(? k-means)
13.2
2.8
2.1
1.2
2.2
1.8
4.6
11.0
2.6
4.0
2.9
6.5
4.4
29.9
2.3
21.2
2.8
53.3
12
Figure 4: 2-d synthetic dataset with 5 true clusters. On the left, G-means correctly chooses
5 centers and deals well with non-spherical data. On the right, the BIC causes X-means to
overfit the data, choosing 20 unevenly distributed clusters.
well at finding the correct k and maximizing the BIC statistic, so we don?t show these
results here. Most real-world data is not spherical, however.
The synthetic datasets used here each have 5000 datapoints in d = 2/8/32 dimensions.
The true ks are 5, 20, and 80. For each synthetic dataset type, we generate 30 datasets with
the true center means chosen uniformly randomly from the unit hypercube, and choosing ?
so that no two clusters are closer than 3? apart. Each cluster is also given a transformation
to make it non-spherical, by multiplying the data by a randomly chosen scaling and rotation
matrix. We run G-means starting with one center. We allow X-means to search between 2
and 4k centers (where here k is the true number of clusters).
The G-means algorithm clearly does better at finding the correct k on non-spherical data. Its
results are closer to the true distortions and the correct ks. The BIC statistic that X-means
uses has been formulated to maximize the likelihood for spherically-distributed data. Thus
it overestimates the number of true clusters in non-spherical data. This is especially evident
when the number of points per cluster is small, as in datasets with 80 true clusters.
0
1
1
2
2
3
3
4
4
Digit
Digit
0
5
5
6
6
7
7
8
8
9
9
5
10
15
20
25
30
Cluster
10
20
30
40
50
60
Cluster
Figure 5: NIST and Pendigits datasets: correspondence between each digit (row) and each
cluster (column) found by G-means. G-means did not have the labels, yet it found meaningful clusters corresponding with the labels.
Because of this overestimation, X-means often hits our limit of 4k centers. Figure 4 shows
an example of overfitting on a dataset with 5 true clusters. X-means chooses k = 20 while
G-means finds all 5 true cluster centers. Also of note is that X-means does not distribute
centers evenly among clusters; some clusters receive one center, but others receive many.
G-means runs faster than X-means for 8 and 32 dimensions, which we expect, since the
kd-tree structures which make X-means fast in low dimensions take time exponential in
d, making them slow for more than 8 to 12 dimensions. All our code is written in Matlab;
X-means is written in C.
3.1
Discovering true clusters in labeled data
We tested these algorithms on two real-world datasets for handwritten digit recognition:
the NIST dataset [12] and the Pendigits dataset [2]. The goal is to cluster the data without
knowledge of the labels and measure how well the clustering captures the true labels. Both
datasets have 10 true classes (digits 0-9). NIST has 60000 training examples and 784
dimensions (28?28 pixels). We use 6000 randomly chosen examples and we reduce the
dimension to 50 by random projection (following [3]). The Pendigits dataset has 7984
examples and 16 dimensions; we did not change the data in any way.
We cluster each dataset with G-means and X-means, and measure performance by comparing the cluster labels Lc.with the true labels Lt . We define the partition quality (PQ) as
Pkt Pkc
Pkt
2
2
pq =
i=1
j=1 p(i, j)
i=1 p(i) where kt is the true number of classes, and kc is
the number of clusters found by the algorithm. This metric is maximized when Lc induces
the same partition of the data as Lt ; in other words, when all points in each cluster have the
same true label, and the estimated k is the true k. The p(i, j) term is the frequency-based
probability that a datapoint will be labeled i by Lt and j by Lc . This quality is normalized
by the sum of true probabilities, squared. This statistic is related to the Rand statistic for
comparing partitions [8].
For the NIST dataset, G-means finds 31 clusters in 30 seconds with a PQ score of 0.177.
X-means finds 715 clusters in 4149 seconds, and 369 of these clusters contain only one
point, indicating an overestimation problem with the BIC. X-means receives a PQ score
of 0.024. For the Pendigits dataset, G-means finds 69 clusters in 30 seconds, with a PQ
score of 0.196; X-means finds 235 clusters in 287 seconds, with a PQ score of 0.057.
Figure 5 shows Hinton diagrams of the G-means clusterings of both datasets, showing that
G-means succeeds at identifying the true clusters concisely, without aid of the labels. The
confusions between different digits in the NIST dataset (seen in the off-diagonal elements)
are common for other researchers using more sophisticated techniques, see [3].
4
Discussion and conclusions
We have introduced the new G-means algorithm for learning k based on a statistical test
for determining whether datapoints are a random sample from a Gaussian distribution with
arbitrary dimension and covariance matrix. The splitting uses dimension reduction and a
powerful test for Gaussian fitness. G-means uses this statistical test as a wrapper around
k-means to discover the number of clusters automatically. The only parameter supplied
to the algorithm is the significance level of the statistical test, which can easily be set in
a standard way. The G-means algorithm takes linear time and space (plus the cost of the
splitting heuristic and test) in the number of datapoints and dimension, since k-means is
itself linear in time and space. Empirically, the G-means algorithm works well at finding
the correct number of clusters and the locations of genuine cluster centers, and we have
shown it works well in moderately high dimensions.
Clustering in high dimensions has been an open problem for many years. Recent research
has shown that it may be preferable to use dimensionality reduction techniques before clustering, and then use a low-dimensional clustering algorithm such as k-means, rather than
clustering in the high dimension directly. In [3] the author shows that using a simple,
inexpensive linear projection preserves many of the properties of data (such as cluster distances), while making it easier to find the clusters. Thus there is a need for good-quality,
fast clustering algorithms for low-dimensional data. Our work is a step in this direction.
Additionally, recent image segmentation algorithms such as normalized cut [16, 13] are
based on eigenvector computations on distance matrices. These ?spectral? clustering algorithms still use k-means as a post-processing step to find the actual segmentation and
they require k to be specified. Thus we expect G-means will be useful in combination with
spectral clustering.
References
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[2] C.L. Blake and C.J. Merz. UCI repository of machine learning databases, 1998. http://www.ics.uci.edu/?mlearn/MLRepository.html.
[3] Sanjoy Dasgupta. Experiments with random projection. In Uncertainty in Artificial Intelligence: Proceedings of the Sixteenth Conference
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[4] Gianna M. Del Corso. Estimating an eigenvector by the power method with a random start. SIAM Journal on Matrix Analysis and Applications,
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[5] Chris Ding, Xiaofeng He, Hongyuan Zha, and Horst Simon. Adaptive dimension reduction for clustering high dimensional data. In Proceedings
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[6] Fredrik Farnstrom, James Lewis, and Charles Elkan. Scalability for clustering algorithms revisited. SIGKDD Explorations, 2(1):51?57, 2000.
[7] Peter J. Huber. Projection pursuit. Annals of Statistics, 13(2):435?475, June 1985.
[8] L. Hubert and P. Arabie. Comparing partitions. Journal of Classification, 2:193?218, 1985.
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[10] Robert E. Kass and Larry Wasserman. A reference Bayesian test for nested hypotheses and its relationship to the Schwarz criterion. Journal of
the American Statistical Association, 90(431):928?934, 1995.
[11] Michael J. Kearns, Yishay Mansour, Andrew Y. Ng, and Dana Ron. An experimental and theoretical comparison of model selection methods. In
Computational Learing Theory (COLT), pages 21?30, 1995.
[12] Yann LeCun, L?eon Bottou, Yoshua Bengio, and Patrick Haffner. Gradient-based learning applied to document recognition. Proceedings of the
IEEE, 86(11):2278?2324, 1998.
[13] Andrew Ng, Michael Jordan, and Yair Weiss. On spectral clustering: Analysis and an algorithm. Neural Information Processing Systems, 14,
2002.
[14] Dan Pelleg and Andrew Moore. X-means: Extending K-means with efficient estimation of the number of clusters. In Proceedings of the 17th
International Conf. on Machine Learning, pages 727?734. Morgan Kaufmann, San Francisco, CA, 2000.
[15] Peter Sand and Andrew Moore. Repairing faulty mixture models using density estimation. In Proceedings of the 18th International Conf. on
Machine Learning. Morgan Kaufmann, San Francisco, CA, 2001.
[16] Jianbo Shi and Jitendra Malik. Normalized cuts and image segmentation. IEEE Transactions on Pattern Analysis and Machine Intelligence,
22(8):888?905, 2000.
[17] M. A. Stephens. EDF statistics for goodness of fit and some comparisons. American Statistical Association, 69(347):730?737, September 1974.
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1,680 | 2,527 | Fast Feature Selection from Microarray
Expression Data via Multiplicative
Large Margin Algorithms
Claudio Gentile
DICOM, Universit`a dell?Insubria
Via Mazzini, 5, 21100 Varese, Italy
[email protected]
Abstract
New feature selection algorithms for linear threshold functions are described which combine backward elimination with an adaptive regularization method. This makes them particularly suitable to the classification of microarray expression data, where the goal is to obtain accurate
rules depending on few genes only. Our algorithms are fast and easy to
implement, since they center on an incremental (large margin) algorithm
which allows us to avoid linear, quadratic or higher-order programming
methods. We report on preliminary experiments with five known DNA
microarray datasets. These experiments suggest that multiplicative large
margin algorithms tend to outperform additive algorithms (such as SVM)
on feature selection tasks.
1
Introduction
Microarray technology allows researchers to simultaneously measure expression levels associated with thousands or ten thousands of genes in a single experiment (e.g., [7]). However, the number of replicates in these experiments is often seriously limited (tipically a
few dozen). This gives rise to datasets having a large number of gene expression values
(numerical components) and a relatively small number of samples. As a popular example,
in the ?Leukemia? dataset from [10] we have only 72 observations of the expression level
of 7129 genes. It is clear that in this extreme scenario machine learning methods related to
feature selection play a fundamental role for increasing efficiency and enhancing the comprehensibility of the results. Besides, in biological and medical research finding accurate
class prediction rules which depend on the level of expression of few genes is important for
a number of activities, ranging from medical diagnostics to drug discovery.
Within the classification framework, a regularization method (also called penalty-based
or feature weighting method) is an indirect route to feature selection. Whereas a (direct)
feature selection method searches in the combinatorial space of feature subsets, a regularization method constrains the magnitudes of the parameters assigning them a ?degree
of relevance? during learning, thereby performing feature selection as a by-product of its
learning mechanism (see, e.g., [16, 19, 17, 14, 4, 20]). Feature selection is a wide and
active field of research; the reader is referred to [15] for a valuable survey. See also, e.g.,
[3, 6] (and references therein) for specific work on gene expression data.
In this paper, we introduce novel feature selection algorithms for linear threshold functions,
whose core learning procedure is an incremental large margin algorithm called 1 ALMAp
(Approximate Large Margin Algorithm w.r.t. norm p) [8]. Our ALMA p -based feature selection algorithms lie between a direct feature selection method and a regularization method.
These algorithms might be considered as a refinement on a recently proposed method,
specifically tested on microarray expression data, called Recursive Feature Elimination
(RFE) [13]. RFE uses Support Vector Machines (SVM) as the core learning algorithm,
and performs backward selection to greedily remove the feature whose associated weight
is smallest in absolute value until only the desired number of features remain. Our algorithms operate in a similar fashion, but they allow us to eliminate many features at once by
exploiting margin information about the current training set. The degree of dimensionality
reduction is ruled by the norm p in ALMA p . The algorithms start by being aggressive (simulating a multiplicative algorithm when the number of current features is large) and end by
being gentle (simulating an additive algorithm such as SVM when few features are left).
From a computational standpoint, our algorithms lie somewhere between a 1-norm and a
2-norm penalization method. However, unlike other regularization approaches specifically
tailored to feature selection, such as those in [4, 20], we do avoid computationally intensive linear (or nonlinear) programming methods. This is because we not only solve the
optimization problem associated to regularization in an approximate way, but also use an
incremental algorithm having the additional capability to smoothly interpolate between the
two kinds of penalizations.
Our algorithms are simple to implement and turn out to be quite fast. We made preliminary
experiments on five known DNA microarray datasets. In these experiments, we compared
the margin-based feature selection performed by our multiplicative algorithms to a standard correlation-based feature selection method applied to both additive (SVM-like) and
multiplicative (Winnow-like) core learning procedures. When possible, we tried to follow
previous experimental settings, such as those in [13, 22, 20]. The conclusion of our preliminary study is that a multiplicative (large margin) algorithm is often better that an SVM-like
algorithm when the goal is to compute linear threshold rules that are both accurate and
depend on the value of few components (as is often the case in gene expression datasets).
2
Preliminaries and notation
An example is a pair (x, y), where x is an instance vector lying in Rf and y ? {?1, +1}
is the binary label associated with x. A training set S is a sequence of examples S =
((x1 , y1 ), ..., (xm , ym )) ? (Rf ? {?1, +1})m . When F ? {1, ..., f } is a set of features
and v ? Rf , we denote by v |F the subvector of v where the features/dimensions not in F
are eliminated. Also, S|F denotes the training set S|F = ((x1|F , y1 ), ..., (xm|F , ym )). A
weight vector w = (w1 , ..., wf ) ? Rf represents a hyperplane passing through the origin.
As usual, we associate with w the (zero threshold) linear threshold function w : x ?
sign(w ? x) = 1 if w ? x ? 0 and = ?1 otherwise. When p ? 1 we denote by ||w||p
Pf
Pf
the p-norm of w, i.e., ||w||p = ( i=1 |wi |p )1/p (also, ||w||? = limp?? ( i=1 |wi |p )1/p
p
= maxi |wi |). We say that norm q is dual to norm p if q = p?1
. In this paper we assume
that p and q are some pair of dual values, with p ? 2. We use p-norms for instance
vectors and q-norms for weight vectors. For notational brevity, throughout this paper we
? = x/||x||p , where p will be clear from the surrounding context.
use normalized instances x
The (normalized) p-norm margin (or just the margin) of a hyperplane w with ||w|| q ? 1
? . If this margin is positive then w classifies (x, y)
on example (x, y) is defined as y w ? x
correctly. Notice that ||x||p ? f 1/p ||x||? for any x ? Rf . Hence if p is logarithmic in
the number of features/dimensions of x, i.e., p = ln f , we obtain ||x||(ln f ) ? e ||x||? .
1
Broadly speaking, as the norm parameter p is varied, ALMA p is able to (approximately) interpolate between Support Vector Machines [5] and (large margin versions of) multiplicative classification
algorithms, such as Winnow [16]. Compared to Winnow, ALMA p is more flexible (since we can
adjust the norm parameter p) and requires less tuning. See Section 3 for details.
ALGORITHM ALMA p (S, ?)
Input: Training set S = ((x1 , y1 ), ..., (xm , ym )); norm parameter p ? 2; approximation parameter ? ? (0, 1].
Initialization: w 1 = 0; k = 1.
For t = 1, 2, ... do:
Get example
weights as follows:
?(xt , yt ) and updateq
8 (p?1) 1
2 ?1
? ; ?k =
Set: ?k =
.
?
p?1
k
k
? t ? (1 ? ?) ?k
If yt wk ? x
then: w0 = T?1 (T(w ) + ? y x
k
k t ? t ),
k
p
wk+1 = w0k /||w 0k ||q , where q = p?1
,
k ? k + 1.
Output: Final weight vector w k = (wk,1 , ..., wk,f ); final margin ? = ?k .
Figure 1: The approximate large margin algorithm ALMA p .
Also, ||w||1 ? 1 implies ||w||q ? 1 for any q > 1. Thus if ||w||1 ? 1 the (ln f )-norm
y w ?x
y w ?x
margin ||x
||(ln f ) is actually bounded from below by the ?-norm margin ||x||? divided by
some constant. Arguing about the ?-norm margin is convenient when dealing with sparse
hyperplanes, i.e., with hyperplanes having only a small number of relevant features (e.g.,
[14]). We say that a training set S = ((x1 , y1 ), ..., (xm , ym )) is linearly separable with
?t ? ?
margin ? > 0 when there exists a hyperplane w with ||w||q ? 1 such that yt w ? x
for t = 1, ..., m. Given ? ? (0, 1], we say that hyperplane w 0 is an ?-approximation to w
? t ? (1 ? ?)? holds for t = 1, ..., m. In
(w.r.t. training set S) if ||w 0 ||q ? 1 and yt w0 ? x
particular, if the underlying margin is an ?-norm margin (and ? is not close to 1) then w 0
tends to share the sparsity properties of w. See also Section 3.
3
The large margin algorithm ALMA p
is a large margin variant of the p-norm Perceptron algorithm2 introduced by [11]
(see also [9]). The version of the algorithm we have used in our experiments is described in
Figure 1, where the one-one mapping T = (T1 , ..., Tf ) : Rf ? Rf is the gradient of the
scalar function 21 || ? ||2q and its inverse T?1 = (T1?1 , ..., Tf?1 ) : Rf ? Rf is the gradient
of the (Legendre dual) function 12 || ? ||2p . The mapping T depends on the chosen norm p,
which we omit for notational brevity. One can immediately see that p = q = 2 gives T =
T?1 = identity. See [9] for further discussion about the properties of T. The algorithm in
Figure 1 takes in input a training set S = ((x1 , y1 ), ..., (xm , ym )) ? (Rf ? {?1, +1})m ,
a norm value p ? 2 and a parameter ? ? (0, 1], measuring the degree of approximation
to the optimal margin hyperplane. Learning proceeds in a sequence of trials. ALMA p
maintains a normalized vector w k of f weights. It starts from w 1 = 0 and in the generic
trial t it processes example (xt , yt ). If the current weight vector w k classifies (xt , yt ) with
(normalized) margin not larger than (1 ? ?) ?k then the algorithm updates its internal state.
The update rule consists of the following: First, the algorithm computes w 0k via a (p-norm)
perceptron-like update rule. Second, w 0k is normalized w.r.t. the chosen norm q (recall
that q is dual to p). The normalized vector w k+1 will then be used in the next trial. After
sweeping (typically more than once) through the training set, the algorithm outputs an f dimensional vector w k which represents the linear model the algorithm has learned from
the data. The output also includes the final margin ? = ?k , where k is the total number of
updates (plus one) the algorithm took to compute w k . This margin is a valuable indication
of the level of ?noise? in the data. In particular, when the training set S is linearly separable,
ALMA p
2
The p-norm Perceptron algorithm is a generalization of the classical Perceptron algorithm, obtained by setting p = 2.
we can use ? to estimate from above the true margin ? ? of S (see Theorem 1). In turn, ? ?
helps us in setting up a reliable feature selection process (see Section 4). Theorem 1 is a
convergence result stating two things [8]: 1. ALMA p (S, ?) computes an ?-approximation
to the maximal p-norm margin hyperplane after a finite number of updates; 2. the margin
? output by ALMA p (S, ?) is an upper bound on the true margin of S.3
? t > 0. Then the
Theorem 1 [8] Let ? ? = maxw?Rf : ||w||q =1 mint=1,...,m yt w ? x
number of updates made by the algorithm in Figure 1 (i.e., the number of trialst such that
2 8
(p?1) 2
? t ? (1 ? ?) ?k ) is upper bounded by 2(?
yt w k ? x
+ ? ? 4 = O ?2p?1
? )2
? ?1
(? ? )2 .
Furthermore, throughout the run of the algorithm we have ?k ? ? ? ? ? , for k = 1, 2, ...
(recall that ? is the last ?k produced by ALMA p ). Hence the previous bound is also an
? t ? (1 ? ?) ?.
upper bound on the number of trials t such that yt wk ? x
Recalling Section 2, we notice that setting p = O(ln f ) makes ALMA p useful when
learning sparse hyperplanes. In particular, the above theorem gives us the following
?-norm margin upper bound on the number of updates: O ln f / (?2 (? ? )2 ) , where
? ? = maxw?Rf : ||w||1 =1 mint=1,...,m yt w ? xt / ||xt ||? . This is similar to the behavior exhibited by classifiers based on linear programming (e.g., [17, 19, 4] and references
therein), as well as to the performance achieved by multiplicative algorithms, such as the
zero-threshold Winnow algorithm [11].
4
The multiplicative feature selection algorithms
We now describe two feature selection algorithms based on ALMA p . The algorithms differ
in the way features are eliminated. The first algorithm, called ALMA - FS (ALMA-based
Feature Selection), is strongly influenced by its training behavior: If ALMA p has made
many updates during training then arguably this corresponds to a high level of noise in
the data (w.r.t. a linear model). In this case the feature selection mechanism tends to be
prudent in eliminating features. On the other hand, if the number of updates is small we
can think of the linear model computed by ALMA p as an accurate one for the training
data at hand, so that one can reliably perform a more aggressive feature removal. The
second algorithm, called ALMA ln -RFE, performs Recursive Feature Elimination (RFE) on
the linear model computed by ALMA p , and might be seen as a simplified version of the
first one, where the rate of feature removal is constant and the final number of features
is fixed ahead of time. ALMA - FS is described in Figure 2. It takes in input a training
set S = ((x1 , y1 ), ..., (xm , ym )) ? (Rn ? {?1, +1})m and a parameter ? (which is the
same as ALMA p ?s). Then the algorithm repeatedly invokes ALMA p on the same training
set but progressively reducing the set F of current features. The algorithm starts with
F = {1, ..., n}, being n the dimension of the input space. Then, on each repeat-until
iteration, the algorithm: sets the norm p to the logarithm4 of the number f of current
features, runs ALMA p for the given values of ? and p, gets in output w and ?, and computes
the new (smaller) F to be used in the next iteration. Computing the new F amounts to
sorting the components of w according to decreasing absolute value and then keeping,
among the f features, only the largest ones (thereby eliminating features which are likely to
be irrelevant). Here c(?) ? [0, 1] is a suitable function whose value will be specified later.
We call a repeat-until iteration of this kind a feature selection stage. ALMA - FS terminates
when it reaches a local minimum F , where the algorithm is unable to drop any further
features.
ALMA - FS uses the output produced by ALMA p in the most natural way, retaining only the
features corresponding to (supposedly) relevant components of w. We point out that here
the discrimination between relevant and irrelevant components is based on the margin ?
3
A more general statement holds for the nonseparable case (see [8] for details). In this case, the
? parameter in ALMA p (.,?) is similar to the C parameter in SVM [5].
4
In order to prevent p < 2, we actually set p = 2 when ln f < 2.
ALGORITHM ALMA - FS(S, ?)
Input: Training set S = ((x1 , y1 ), ..., (xm , ym )); approx. param. ? ? (0, 1].
Initialization: F = {1, 2, ..., n}; f := |F | = n.
Repeat
? Set p := max{2, ln f } and run ALMA p (S|F , ?), getting in output w =
(w1 , ..., wf ) ? Rf and ? > 0;
? Sort w1 , ..., wf according to decreasing |wi | and let wi1 , ..., wif be the
p
sorted sequence; set q = p?1
and compute the smallest f ? ? f s.t.
Pf ?
q
q
(1)
j=1 |wij | ? 1 ? (c(?) ?) ;
? Set F = {i1 , i2 , ..., if ? }; f := |F | = f ? ;
Until F does not shrink any more.
Output: Final weight vector w = (w1 , ..., wf ).
Figure 2:
ALMA - FS :
Feature selection using ALMA p where p is logarithmic in f .
output by ALMA p . In turn, ? depends on the number of training updates made by ALMA p ,
i.e., on the ?amount of noise? in the data. This criterion can be viewed as a margin-based
criterion according to the following fact: If in any given stage ALMA p has computed an
?-approximation to the maximal margin hyperplane for a (linearly separable) training sequence S, then the (smaller) vector computed at the end of that stage will be an (? + c(?))approximation to the maximal margin hyperplane for the new (linearly separable) sequence
where some features have been eliminated. This statement follows directly from (1) and
Theorem 1. We omit the details due to space limitations. From this point of view, a reasonable choice of c(?) is one which insures ? + c(?) ? 1 for ? ? [0, 1] and the two limiting
conditions lim??0 ? + c(?) = 0 and lim??1 ? + c(?) = 1. The simplest function satisfying the conditions above (the one we used in the experiments) is c(?) = ? (1 ? ?).
ALMA - FS starts with a relatively large value of the norm parameter p (making it fairly
aggressive at the beginning), and then progressively reduces this parameter so that the algorithm can focus in later stages on the remaining features. This heuristic approach allows
us to keep a good approximation capability (as measured by the margin) while dropping a
lot of irrelevant components from the weight vectors computed by ALMA p .
ALMA ln - RFE is a simplified version of ALMA - FS that halves the number of features in
each stage, and uses again a norm p logarithmic in the number of current features. The
? parameter is replaced by nf , the desired number of features. ALMA ln -RFE(S, nf ) is
obtained from the algorithm in Figure 2 upon replacing the definition of f ? in (1) by f ? =
max{bf /2c, nf }, so that the number of training stages is always logarithmic in n/nf .
5
Experiments
We tested ALMA - FS and ALMA ln -RFE on a few well-known microarray datasets (see below). For the sake of comparison, we tended to follow previous experimental settings, such
as those described in [13, 22, 20]. Our results are summarized in Table 1. For each dataset,
we first generated a number of random training/test splits. Since we used on-line algorithms, the output depends on the order of the training sequence. Therefore our random
splits also included random permutations of the training set. The results shown in Table 1
are averaged over these random splits.
Five datasets have been used in our experiments.
1. The ALL-AML dataset [10] contains 72 samples, each with expression profiles about
7129 genes. The task is to distinguish between the two variants of leukemia ALL and
AML. We call this dataset the ?Leukemia? dataset. We used the first 38 examples as training set and the remaining 34 as test set. This seems to be a standard training/test split (e.g.,
[10, 21, 13, 22]). The results have been averaged over 1000 random permutations of the
training set.
2. The ?Colon Cancer? dataset [2] contains 62 expression profiles for tumor and normal
samples concerning 2000 genes. Following [20], we randomly split the dataset into a training set of 50 examples and a test set of 12. The random split was performed 1000 times.
3. In the ER+/ER? dataset from [12] the task is to analyze expression profiles of breast
cancer and classify breast tumors according to ER (Estrogen Receptor) status. This dataset
(which we call the ?Breast? dataset) contains 58 expression profiles concerning 3389 genes.
We randomly split 1000 times into a training set of size 47 and a test set of size 11.
4. The ?Prostate? cancer dataset from [18] contains 102 samples with expression profiles
concerning 12600 genes. The task is to separate tumor from normal samples. As in [18],
we estimated the test error through a Leave-One Out Cross Validation (LOOCV)-like estimator. In particular, for this dataset we randomly split 1000 times into a training set of
101 examples and a test set of 1 example, and then averaged the results. (This is roughly
equivalent to LOOCV with 10 random permutations of the training set.)
5. In the ?Lymphoma? dataset [1] the goal is to separate cancerous and normal tissues in
a large B-Cell lymphoma problem. The dataset contains 96 expression profiles concerning
4026 genes, 62 samples are in the classes ?DLCL?, ?FL? and ?CLL? (malignant) and the
remaining 34 are labelled ?otherwise?. As in [20], we randomly split the dataset into a
training set of size 60 and a test set of size 36. The random split was performed 1000 times.
We made no preprocessing on the data. All our experiments have been run on a PC with
a single AMD Athlon processor running at 1300 Mhz. The running times we will be giving are measured on this machine. We compared on these datasets ALMA - FS (?FS? in
Table 1) and ALMA ln -RFE (?ln-RFE?) to three more feature selection algorithms: a fast
approximation to Recursive Feature Elimination applied to SVM (called ALMA 2 -RFE, abbreviated as ?2-RFE? in Table 1), and a standard feature selection method based on correlation coefficients (e.g., [10]) applied to both (an approximation to) SVM and ALMA ln f ,
being f the number of features selected by the correlation method. We call the last two
methods ALMA 2 -CORR (?2-CORR? in Table 1) and ALMA ln -CORR (?ln-CORR? in Table 1), respectively. In all cases our base learning algorithm was ALMA p (.,?), where
? ? {0.5, 0.6, 0.7, 0.8, 0.9}, and p was either 2 (to approximate SVM) or logarithmic
in the number of features the algorithm was operating on (to simulate a multiplicative large
margin algorithm). For each combination (algorithm, number of genes), only the best
accuracy results (w.r.t. ?) are shown. On the ?Colon cancer?, the ?Breast? and the ?Lymphoma? datasets we run ALMA p by cycling 50 times over the current training set. On the
?Leukemia? and the ?Prostate? datasets (which are larger) we cycled 100 times. In Table
1 we give, for each dataset, the average error and the number of features (?# GENES?) selected by the algorithms.5 The only algorithm which tries to determine the final number of
features as a part of its inference mechanism is ALMA - FS: all the others take this number
as an explicit input parameter.
The main goal of this experimental study was to carry out a direct comparison between different feature selection methods combined with different core learning algorithms. Feature
selection performed by ALMA - FS, ALMA ln -RFE and ALMA 2 -RFE is margin-based, while
feature selection performed by ALMA 2 -CORR and ALMA ln -CORR is correlation-based.
According to [15], the former falls within the category of wrapper methods, while the latter is an example of filter methods. The two core learning algorithms we employed are the
SVM-like algorithm ALMA 2 and the (large margin) Winnow-like algorithm ALMA p , with
logarithmic p. The first has been used with ALMA 2 -RFE and ALMA 2 -CORR, the second
has been used with ALMA - FS, ALMA ln -RFE and ALMA ln -CORR.
The accuracy results we have obtained are often superior to those reported in the litera5
Observe that, due to the on-line nature of the algorithms, different sets of genes get selected on
different runs. Therefore one could also collect statistics about the gene selection frequency over the
runs. Details will be given in the full paper.
Table 1: Experimental results on five microarray datasets. The percentages denote the average fraction of misclassified patterns in the test set, while ?# GENES? denotes the average
number of genes (features) selected. The results refer to the same training/test splits. Notice
that ALMA - FS (?FS?) determines automatically the number of genes to select. According
to Wilcoxon signed rank test, ? 0.5% accuracy difference might be considered significant.
L EUKEMIA
# GENES
20
26.5
40
60
100
200
ALL
C OLON
C ANCER
20
22.6
40
60
100
200
ALL
B REAST
20
38.5
40
60
100
200
ALL
P ROSTATE
20
30.8
40
60
100
200
ALL
LYMPHOMA
20
30.8
40
60
100
200
ALL
FS
?
3.0%
?
?
?
?
?
?
12.7%
?
?
?
?
?
?
9.5%
?
?
?
?
?
?
9.5%
?
?
?
?
?
?
8.1%
?
?
?
?
?
2- RFE
5.8%
?
6.7%
8.9%
9.0%
7.2%
3.5%
17.0%
?
15.4%
14.8%
14.3%
13.2%
13.0%
11.5%
?
10.7%
10.1%
10.4%
11.9%
15.8%
8.4%
?
8.1%
8.1%
9.3%
9.8%
10.0%
10.1%
?
7.9%
7.4%
6.6%
6.3%
7.2%
ln- RFE
3.3%
?
3.0%
3.2%
2.5%
3.1%
3.3%
13.1%
?
12.1%
12.0%
12.6%
12.4%
13.3%
10.3%
?
9.9%
9.9%
9.8%
9.6%
10.0%
7.8%
?
9.4%
10.3%
10.2%
9.9%
10.4%
9.9%
?
7.4%
6.8%
6.0%
5.6%
5.5%
2- CORR
5.9%
?
5.0%
4.3%
4.0%
3.0%
3.5%
15.4%
?
14.4%
14.2%
13.7%
13.9%
13.0%
6.1%
?
6.5%
7.5%
13.1%
14.6%
15.8%
11.5%
?
10.2%
8.5%
6.9%
8.4%
10.0%
12.6%
?
10.5%
9.5%
8.2%
7.4%
7.2%
ln- CORR
3.7%
?
3.6%
2.9%
2.9%
4.5%
3.3%
14.8%
?
14.0%
13.6%
13.1%
13.2%
13.3%
5.5%
?
6.5%
8.5%
10.4%
14.5%
10.0%
10.4%
?
8.0%
7.7%
6.5%
7.2%
10.4%
12.3%
?
10.2%
9.2%
8.3%
7.7%
5.5%
ture, though this should not be considered very significant.6 From our direct comparison,
however, a few (more reliable) conclusions can be drawn. First, on these gene expression
6
In fact, the results on feature selection applied to microarray datasets are not readily comparable
across different papers, due to the randomness in the training/test splits (which is a relevant source
of variance) and the different preprocessing of the data. That said, we briefly mention a few results
reported by other researchers on the same datasets. On the ?Leukemia? dataset, [22] report 0% test
error for a logistic regression algorithm that chooses the number of features to extract by LOOCV.
The same error rate is reported by [21] for a linear SVM using 20 genes. [20] use linear SVM as the
underlying learning algorithm. On the ?Colon Cancer? dataset, the authors report an average accuracy
of 16.4% without feature selection and an accuracy ranging between 15.0% and 16.9% (depending
on the number of genes selected) for the RFE and the AROM (Approximation of the Zero-Norm
Minimization) methods. On the ?Lymphoma? dataset the same authors report 7.1% average error for
linear SVM and 5.9% to 6.8% average error (again depending on the number of genes selected) for
the RFE and the AROM methods. On the ?Prostate? dataset, [18] use a k-NN classifier and report a
LOOCV accuracy comparable to ALMA 2 -RFE?s (but worse than ALMA ln -CORR?s).
datasets a large margin Winnow-like algorithm generally outperforms an SVM-like algorithm. Second, despite the common wisdom [15] according to which wrapper methods tend
to be more accurate than filter methods, it is hard to tell here how the two methods compare
(see [22] for similar results). Third, knowing the ?optimal? number of genes beforehand
is a valuable side information. Notice that, unlike many of the methods proposed in the
literature, ALMA - FS tries to determine in an automatic way a ?good? number of features to
select.7 In fact, due to the scarcity of examples and the large number of vector components,
the repeated use of cross-validation on the same validation set might lead to overfitting.
ALMA - FS seems to do a fine job of it on three out of five datasets (on the ?Breast? dataset
?FS? should only be compared to ?2-RFE? and ?ln-RFE?). Finally, we would like to stress
that our feature selection algorithms are quite fast. To give an idea, on the ?Colon Cancer?
and the ?Breast? datasets our algorithms take on average just a few seconds, while on the
?Prostate? dataset they take just a few minutes.
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7
The reader might object that the number of selected features can depend on the value of parameter
? in ALMAp . In practice, however, we observed that ? does not have a big influence on this number.
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1,681 | 2,528 | Online Learning via Global Feedback
for Phrase Recognition
Xavier Carreras
Llu??s M`arquez
TALP Research Center, LSI Department
Technical University of Catalonia (UPC)
Campus Nord UPC, E?08034 Barcelona
{carreras,lluism}@lsi.upc.es
Abstract
This work presents an architecture based on perceptrons to recognize
phrase structures, and an online learning algorithm to train the perceptrons together and dependently. The recognition strategy applies learning
in two layers: a filtering layer, which reduces the search space by identifying plausible phrase candidates, and a ranking layer, which recursively
builds the optimal phrase structure. We provide a recognition-based feedback rule which reflects to each local function its committed errors from
a global point of view, and allows to train them together online as perceptrons. Experimentation on a syntactic parsing problem, the recognition
of clause hierarchies, improves state-of-the-art results and evinces the
advantages of our global training method over optimizing each function
locally and independently.
1
Introduction
Over the past few years, many machine learning methods have been successfully applied
to Natural Language tasks in which phrases of some type have to be recognized. Generally,
given an input sentence ?as a sequence of words? the task is to predict a bracketing
for the sentence representing a structure of phrases, either sequential or hierarchical. For
instance, syntactic analysis of Natural Language provides several problems of this type,
such as partial parsing tasks [1, 2], or even full parsing [3].
The general approach consists of decomposing the global phrase recognition problem into
a number of local learnable subproblems, and infer the global solution from the outcomes
of the local subproblems. For chunking problems ?in which phrases are sequentially
structured? the approach is typically to perform a tagging. In this case, local subproblems
include learning whether a word opens, closes, or is inside a phrase of some type (noun
phrase, verb phrase, . . . ), and the inference process consists of sequentially computing
the optimal tag sequence which encodes the phrases, by means of dynamic programming
[1, 4, 5]. When hierarchical structure has to be recognized, additional local decisions are
required to determine the embedding of phrases, resulting in a more complex inference
process which recursively builds the global solution [3, 2, 6, 7]. In general, a learning
system for these tasks makes use of several learned functions which interact in some way
to determine the structure.
A usual methodology for solving the local subproblems is to use a discriminative learning
algorithm to learn a classifier for each local decision [1, 2]. Each individual classifier is
trained separately from the others, maximizing some local measure such as the accuracy of
the local decision. However, when performing the phrase recognition task, the classifiers
are used together and dependently, in the sense that one classifier predictions? may affect
the prediction of another. Indeed, the global performance of a system is measured in terms
of precision and recall of the recognized phrases, which, although related, is not the local
classification accuracy measure for which the local classifiers are usually trained.
In this direction, recent works in the area provide alternative strategies in which the learning
process is driven from the global level. The general idea consists of moving the learning
strategy from the binary classification setting to a general ranking context into which the
global problem can be casted. Crammer and Singer [8] present a label-ranking algorithm,
in which several perceptrons receive feedback from the ranking they produce over a training
instance. Har-Peled et al. [9] study a general learning framework in which the constraints
between a number of linear functions and an output prediction allow to effectively learn a
desired label-ranking function. For structured outputs, and motivating this work, Collins
[10] introduces a variant of the perceptron for tagging tasks, in which the learning feedback
is globally given from the output of the Viterbi decoding algorithm.
In this paper we present a global learning strategy for the general task of recognizing
phrases in a sentence. We adopt the general phrase recognition strategy of our previous
work [6]. Given a sentence, learning is first applied at word level to identify phrase candidates of the solution. Then, learning is applied at a higher-order level in which phrase
candidates are scored to discriminate among competing ones. The overall strategy infers
the global solution by exploring with learning components a number of plausible solutions.
As a main contribution, we propose a recognition-based feedback rule which allows to learn
the decisions in the system as perceptrons, all in one go. The learning strategy works online
at sentence level. When visiting a sentence, the perceptrons are first used to recognize the
set of phrases, and then updated according to the correctness of the global solution. As a result, each local function is automatically adapted to the recognition strategy. Furthermore,
following [11] the final model incorporates voted prediction methods for the perceptrons
and the use of kernel functions. Experimenting on the Clause Identification problem [2] we
show the effectiveness of our method, evincing the benefits over local learning strategies
and improving the best results for the particular task.
2
2.1
Phrase Recognition
Formalization
Let x be a sentence formed by n words xi , with i ranging from 0 to n ? 1, belonging
to the sentence space X . Let K be a predefined set of phrase categories. For instance,
in syntactic parsing K may include noun phrases, verb phrases, prepositional phrases and
clauses, among others. A phrase, denoted as (s, e)k , is the sequence of consecutive words
spanning from word xs to word xe , having s ? e, with category k ? K.
Let ph1 = (s1 , e1 )k1 and ph2 = (s2 , e2 )k2 be two different phrases. We define that ph1 and
ph2 overlap iff s1 < s2 ? e1 < e2 or s2 < s1 ? e2 < e1 , and we note it as ph1 ? ph2 . Also,
we define that ph1 is embedded in ph2 iff s2 ? s1 ? e1 ? e2 , and we note it as ph1 ? ph2 .
Let P be the set of all possible phrases, expressed as P = {(s, e)k | 0 ? s ? e, k ? K}.
A solution for a phrase recognition problem is a set y of phrases which is coherent with
respect to some constraints. We consider two types of constraints: overlapping and embedding. For the problem of recognizing sequentially organized phrases, often referred to
as chunking, phrases are not allowed to overlap or embed. Thus, the solution space can
be formally expressed as Y = {y ? P | ? ph1 , ph2 ? y ph1 6? ph2 ? ph1 6? ph2 } . More
generally, for the problem of recognizing phrases organized hierarchically, a solution is a
set of phrases which do not overlap but may be embedded. Formally, the solution space is
Y = {y ? P | ? ph1 , ph2 ? y ph16?ph2 } .
In order to evaluate a phrase recognition system we use the standard measures for recognition tasks: precision (p) ?the ratio of recognized phrases that are correct?, recall (r)
2pr
.
?the ratio of correct phrases that are recognized? and their harmonic mean F1 = p+r
2.2
Recognizing Phrases
The mechanism to recognize phrases is described here as a function which, given a sentence
x, identifies the set of phrases y of x: R : X ? Y. We assume two components within this
function, both being learning components of the recognizer. First, we assume a function P
which, given a sentence x, identifies a set of candidate phrases, not necessarily coherent,
for the sentence, P(x) ? P. Second, we assume a score function which, given a phrase,
produces a real-valued prediction indicating the plausability of the phrase for the sentence.
The phrase recognizer is a function which searches a coherent phrase set for a sentence x
according to the following optimality criterion:
X
R(x) = arg
max
score((s, e)k , x, y)
(1)
y?P(x) | y?Y
(s,e)k ?y
That is, among all the coherent subsets of candidate phrases, the optimal solution is defined
as the one whose phrases maximize the summation of phrase scores.
The function P is only used to reduce the search space of the R function. Note that the
R function constructs the optimal phrase set by evaluating scores of phrase candidates,
and, regarding the length of the sentence, there is a quadratic number of possible phrases,
that is, the set P. Thus, considering straightforwardly all phrases in P would result in a
very expensive exploration. The function P is intended to filter out phrase candidates from
P by applying decisions at word level. A simple setting for this function is a start-end
classification for each phrase type: each word of the sentence is tested as k-start ?if it
is likely to start phrases of type k? and as k-end ?if it is likely to end phrases type k.
Each k-start word xs with each k-end word xe , having s ? e, forms the phrase candidates
(s, e)k . Assuming start and end binary classification functions, hkS and hkE , for each type
k ? K, the filtering function is expressed as:
P(x) = { (s, e)k ? P | hkS (xs ) = +1 ? hkE (xe ) = +1 }
Alternatives to this setting may be to consider a single pair of start-end classifiers, independent of phrase types, or to perform a different tagging for identifying phrases, such as the
well-known begin-inside classification. In general, each classifier will be applied to each
word in the sentence, and deciding the best strategy for identifying phrase candidates will
depend on the sparseness of phrases in a sentence, the length of phrases and the number of
categories.
Once the phrase candidates are identified, the optimal coherent phrase set is selected according to (1). Due to its nature, there is no need to explicitly enumerate each possible
coherent phrase set, which would result in an exponential exploration. Instead, by guiding
the exploration through the problem constraints and using dynamic programming the optimal coherent phrase set can be found in polynomial time over the sentence length. For
chunking problems, the solution can be found in quadratic time by performing a Viterbistyle exploration from left to right [4]. When embedding of phrases is allowed, a cubic-time
bottom-up exploration is required [6]. As noted above, in either cases there will be the additional cost of applying a quadratic number of decisions for scoring phrases.
Summarizing, the phrase recognition system is performed in two layers: the identification
layer, which filters out phrase candidates in linear time, and the scoring layer, which selects
the optimal phrase chunking in quadratic or cubic time.
3
Additive Online Learning via Recognition Feedback
In this section we describe an online learning strategy for training the learning components
of the Phrase Recognizer, namely the start-end classifiers in P and the score function. The
learning challenge consists in approximating the functions so as to maximize the global F1
measure on the problem, taking into account that the functions interact. In particular, the
start-end functions define the actual input space of the score function.
Each function is implemented using a linear separator, hw : Rn ? R, operating in a
feature space defined by a feature representation function, ? : X ? Rn , for some instance
space X . The function P consists of two classifiers per phrase type: the start classifier (hkS )
and the end classifier (hkE ). Thus, the P function is formed by a prediction vector for each
classifier, noted as wkS or wkE , and a unique shared representation function ?w which maps a
word in context into a feature vector. A prediction is computed as hkS (x) = wkS ? ?w (x), and
similarly for the hkE , and the sign is taken as the binary classification. The score function
computes a real-valued score for a phrase candidate (s, e)k . We implement this function
with a prediction vector wk for each type k ? K, and also a shared representation function
?p which maps a phrase into a feature vector. The score prediction is then given by the
expression: score((s, e)k , x, y) = wk ? ?p ((s, e)k , x, y).
3.1
The FR-Perceptron Learning Algorithm
We propose a mistake-driven online learning algorithm for training the parameter vectors
all together. We give the algorithm the name FR-Perceptron since it is a Perceptron-based
learning algorithm that approximates the prediction vectors in P as Filters of words, and the
score vectors as Rankers of phrases. The algorithm starts with all vectors initialized to 0,
and then runs repeatedly in a number of epochs T through all the sentences in the training
set. Given a sentence, it predicts its optimal phrase solution as specified in (1) using the
current vectors. As in the traditional Perceptron algorithm, if the predicted phrase set is
not perfect the vectors responsible of the incorrect prediction are updated additively. The
algorithm is as follows:
? Input: {(x1 , y 1 ), . . . , (xm , y m )}, xi are sentences, y i are solutions in Y
? Define: W = {wkS , wkE , wk |k ? K}.
? Initialize: ?w ? W w = 0;
? for t = 1 . . . T , for i = 1 . . . m :
1. y? = RW (xi )
2. recognition learning feedback(W, xi , y i , y?)
? Output: the vectors in W .
We now describe the recognition-based learning feedback. By analyzing the dependencies
between each function and a solution, we derive a feedback rule which naturally fits the
phrase recognition setting. Let y ? be the gold set of phrases for a sentence x, and y? the set
predicted by the R function. Let goldS(xi , k) and goldE(xi , k) be, respectively, the perfect
indicator functions for start and end boundaries of phrases of type k. That is, they return 1
if word xi starts/ends some k-phrase in y ? and -1 otherwise. We differentiate three kinds
of phrases in order to give feedback to the functions being learned:
? Phrases correctly identified: ?(s, e)k ? y ? ? y?:
? Do nothing, since they are correct.
? Missed phrases: ?(s, e)k ? y ? \ y?:
1. Update misclassified boundary words:
if (wkS ? ?w (xs ) ? 0) then wkS = wkS + ?w (xs )
if (wkE ? ?w (xe ) ? 0) then wkE = wkE + ?w (xe )
2. Update score function, if applied:
if (wkS ? ?w (xs ) > 0 ? wkE ? ?w (xe ) > 0) then wk = wk + ?p ((s, e)k , x, y)
? Over-predicted phrases: ?(s, e)k ? y?\y ? :
1. Update score function: wk = wk ? ?p ((s, e)k , x, y)
2. Update words misclassified as S or E:
if (goldS(xs , k) = ?1) then wkS = wkS ? ?w (xs )
if (goldE(xe , k) = ?1) then wkE = wkE ? ?w (xe )
This feedback models the interaction between the two layers of the recognition process.
The start-end layer filters out phrase candidates for the scoring layer. Thus, misclassifying
the boundary words of a correct phrase blocks the generation of the candidate and produces a missed phrase. Therefore, we move the start or end prediction vectors toward the
misclassified boundary words of a missed phrase. When an incorrect phrase is predicted,
we move away the prediction vectors from the start or end words, provided that they are
not boundary words of a phrase in the gold solution. Note that we deliberately do not care
about false positives start or end words which do not finally over-produce a phrase.
Regarding the scoring layer, each category prediction vector is moved toward missed
phrases and moved away from over-predicted phrases. It is important to note that this
feedback operates only on the basis of the predicted solution y?, avoiding to make updates
for every prediction the function has made. Thus, the learning strategy is taking advantage
of the recognition process, and concentrates on (i) assigning high scores for the correct
phrases and (ii) making the incorrect competing phrases to score lower than the correct
ones. As a consequence, this feedback rule tends to approximate the desired behavior of
the global R function, that is, to make the summation of the scores of the correct phrase
set maximal with respect to other phrase set candidates. This learning strategy is closely
related to other recent works on learning ranking functions [10, 8, 9].
A Note on the Convergence Assuming linear separability for each start, end and score
function, it can be shown that (i) the mistakes of the start-end filters are bounded (applying
Novikoff?s proof); (ii) between two consecutive updates in the start-end layer, there is room
only for a finite number of updates of the score function; and (iii) once the start-end filters
have converged, the correct solution is always considered in the score layer as candidate,
and in this state the overall learning process converges (applying the proof of Collins for a
perceptron tagger [10]).
4
Experiments on Clause Identification
Clause Identification is the problem of recognizing the clauses of a sentence. A clause can
be roughly defined as a phrase with a subject, possibly implicit, and a predicate. Clauses in
a sentence form a hierarchical structure which constitutes the skeleton of the full syntactic
tree. In the following example, the clauses are annotated with brackets:
( (When (you don?t have any other option)), it is easy (to fight) .)
We followed the setting of the CoNLL-2001 competition 1 . The problem consists of recognizing the set of clauses on the basis of words, part-of-speech tags (PoS), and syntactic
base phrases (or chunks). There is only one category of phrases to be considered, namely
the clauses. The data consists of a training set (8,936 sentences, 24,841 clauses), a development set (2,012 sentences, 5,418 clauses) and a test set (1,671 sentences, 5,225 clauses).
Representation Functions We now describe the representation functions ?w and ?p ,
which respectively map a word or a phrase and their local context into a feature vector in
{0, 1}n . Their design is inspired in our previous work [6]. For the function ?w (xi ) we
capture the form, PoS and chunk tags of words in a window around xi , that is, words xi+l
with l ? [?Lw , +Lw ]. Each attribute type, together with each relative position l and each
returned value forms a final binary indicator feature (for instance, ?the word at position -2
is that? is a binary feature). Also, we consider the word decisions of the words to the left
of xi , that is, binary flags indicating whether the [?Lw , ?1] words in the window are starts
and/or ends of a phrase. For the function ?p (s, e) we represent the context of the phrase
by capturing a [?Lp , 0] window of forms, PoS and chunks at the s word, and a separate
[0, +Lp ] window at the e word. Furthermore, we represent the (s, e) phrase by evaluating
a pattern from s to e which captures the relevant elements in the sentence fragment from
word s to word e 2 . We experimentally set both Lw and Lp to 3.
On this problem we were interested in comparing the FR-Perceptron algorithm versus other
alternative learning methods. The system to train was composed by the start and end functions which identify clause candidates, and a score function for clauses. As alternatives, we
first considered a batch classification setting, in which each function is trained separately
with binary classification loss. To do so, we generated three data sets from training examples, one for each function. For the start-end sets, we considered an example for each word
in the data. To train the score classifier, we generated only the phrase candidates formed
with all pairs of correct phrase boundaries. This latter generation greatly reduces the real
instance space in which the scoring function operates. The alternative of generating all possible phrases as examples would be more realistic, but infeasible for the learning algorithm
since it would produce 1,377,843 examples, with a 98.2% of negatives. As a secondary
intermediate approach, we considered a simple model which learns all the functions online
via binary classification loss. That is, the training sentences are visited online as in the
FR-Perceptron: first, the start-end functions are applied to each word, and according to
their positive decisions, phrase examples are generated to train the score function. In this
way, the input of the score function is dynamically adapted to the start-end behavior, but a
classification feedback is given to each function for each decision taken.
The functions of the system were actually modeled as Voted Perceptrons [11], which compute a prediction as an average of all vectors generated during training. For the batch
classification setting, we modeled the functions as Voted Perceptrons and also as SVMs3 .
In all cases, a function can be expressed in dual form as a combination of training instances,
which allows the use of kernel functions. We work with polynomial kernels of degree 2. 4
We trained the perceptron models for up to 20 epochs via the FR-Perceptron algorithm and
via classification feedback, either online (CO-VP) or batch (CB-VP). We also trained SVM
classifiers (Cl-SVM), adjusting the soft margin C parameter on the development set.
1
Data and details at the CoNLL-2001 website: http://cnts.uia.ac.be/conll2001 .
The following elements are considered in a pattern: a) Punctuation marks and coordinate conjunctions; b) The word that; c) Relative pronouns; d) Verb phrase chunks; and e) The top clauses
within the s to e fragment, already recognized through the bottom up search (a clause in a pattern
reduces all the elements within it into an atomic element).
3
We used the SVMlight package available at http://svmlight.joachims.org .
4
Initial tests revealed poor performance for the linear case and no improvements for degrees > 2.
2
Figure 1: Performance on the development set with respect to the number of epochs. Top:
global F1 (left) and precision/recall on starts (right). Bottom: given the start-end filters,
upper bound on the global F1 (left) and number of proposed phrase candidates (right).
100
Precision/Recall on Start Words
90
global F Measure
88
86
84
82
80
FR-Perceptron
CO-VP
CB-VP
SVM
78
76
0
5
10
15
90
85
80
75
70
65
precision FR-Perceptron
recall FR-Perceptron
precision CO-VP
recall CO-VP
60
55
50
20
0
97
5
10
45000
P - number of phrase candidates
96
global F upper bound
95
95
94
93
92
91
90
FR-Perceptron
CO-VP
CB-VP
SVM
89
88
0
5
10
number of epochs
15
15
20
FR-Perceptron
CO-VP
CB-VP
SVM
40000
35000
30000
25000
20000
15000
10000
5000
20
0
5
10
15
20
number of epochs
Figure 1 (top, left) shows the performance curves in terms of the F1 measure with respect
to the number of training epochs. Clearly, the FR-Perceptron model exhibits a much better
curve than classification models, being at any epoch more than 2 points higher than the
online model, and far from the batch models. To get an idea of how the learning strategy
behaves, it is interesting to look at the other plots of Figure 1. The top right plot shows the
performance of the start function. The FR-Perceptron model exhibits the desirable filtering
behavior for this local decision, which consists in maintaining a very high recall (so that no
correct candidates are blocked) while increasing the precision during epochs. In contrast,
the CO-VP model concentrates mainly on the precision. The same behavior is observed for
the other classification models, and also for the end local decision. The start-end behavior
is also shown from a global point of view at the bottom plots. The left plot shows the
maximum achievable global F1 , assuming a perfect scorer, given the phrases proposed by
the start-end functions. Additionally, the right plot depicts the filtering capabilities in terms
of the number of phrase candidates produced, out of a total number of 300,511 possible
phrases. The FR-Perceptron behavior in the filtering layer is clear: while it maintains a
high recall on identifying correct phrases (above 95%), it substantially reduces the number
of phrase candidates to explore in the scoring layer, and thus, it progressively simplifies the
input to the score function. Far from this behavior, the classification-based models are not
sensitive to the global performance in the filtering layer and, although they aggressively
reduce the search space, provide only a moderate upper bound on the global F1 .
Table 4 shows the performance of each model, together with the results of our previous
system [6], which held the best results on the problem. There, the same decisions were
learned by AdaBoost classifiers working in a richer feature space. Also, the score function
was a robust combination of several classifiers. These were trained taking into account the
errors of the start-end classifiers, which required a tuning procedure to select the amount
of introduced errors. Our new approach is much simpler to learn, since the interaction
between functions is naturally ruled by the recognition feedback. Looking at results, we
substantially improve the global F1 .
CB-VP
SVM
CO-VP
FR-Perceptron
AdaBoost [6]
T
8
19
20
?
development
prec. recall F?=1
83.84 80.55 82.16
84.31 82.83 83.57
91.06 80.62 85.52
90.56 85.73 88.08
92.53 82.48 87.22
prec.
82.22
83.19
89.25
88.17
90.18
test
recall
78.09
80.00
77.62
82.10
78.11
F?=1
80.10
81.57
83.03
85.03
83.71
Table 1: Results of Clause Identification on the CoNLL-2001 development and test sets.
The T column shows the optimal number of epochs on the development set.
5
Conclusion
We have presented a global learning strategy for the general problem of recognizing structures of phrases, in which, typically, several different learning functions interact to explore
and recognize the structure. The effectiveness of our method has been empirically proved
in the problem of clause identification, where we have shown that a considerable improvement can be obtained by exploiting high-order global dependencies in learning, in contrast
to concentrating only on the local subproblems. These results suggest to scale up global
learning strategies to more complex problems found in the natural language area (such as
full parsing or machine translation), or other structured domains.
Acknowledgements
Research partially funded by the European Commission (Meaning, IST-2001-34460) and
the Spanish Research Department (Hermes, TIC2000-0335-C03-02; Petra, TIC2000-1735C02-02). Xavier Carreras is supported by a grant from the Catalan Research Department.
References
[1] E. F. Tjong Kim Sang and S. Buchholz. Introduction to the CoNLL-2000 Shared Task: Chunking. In Proc. of CoNLL-2000 and LLL-2000, 2000.
[2] Erik F. Tjong Kim Sang and Herv?e D?ejean. Introduction to the CoNLL-2001 Shared Task:
Clause Identification. In Proc. of CoNLL-2001, 2001.
[3] A. Ratnaparkhi. Learning to Parse Natural Language with Maximum-Entropy Models. Machine
Learning, 34(1):151?175, 1999.
[4] V. Punyakanok and D. Roth. The Use of Classifiers in Sequential Inference. In Advances in
Neural Information Processing Systems 13 (NIPS?00), 2001.
[5] T. Kudo and Y. Matsumoto. Chunking with Support Vector Machines . In Proc. of 2nd Conference of the North American Chapter of the Association for Computational Linguistics, 2001.
[6] X. Carreras, L. M`arquez, V. Punyakanok, and D. Roth. Learning and Inference for Clause
Identification. In Proceedings of the 14th ECML, Helsinki, Finland, 2002.
[7] T. Kudo and Y. Matsumoto. Japanese Dependency Analyisis using Cascaded Chunking . In
Proc. of CoNLL-2002, 2002.
[8] K. Crammer and Y. Singer. A Family of Additive Online Algorithms for Category Ranking.
Journal of Machine Learning Research, 3:1025?1058, 2003.
[9] S. Har-Peled, D. Roth, and D. Zimak. Constraint Classification for Multiclass Classification
and Ranking. In Advances in Neural Information Processing Systems 15 (NIPS?02), 2003.
[10] M. Collins. Discriminative Training Methods for Hidden Markov Models: Theory and Experiments with Perceptron Algorithms. In Proceedings of the EMNLP?02, 2002.
[11] Y. Freund and R. E. Schapire. Large Margin Classification Using the Perceptron Algorithm.
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1,682 | 2,529 | A Fast Multi-Resolution Method for Detection of
Significant Spatial Disease Clusters
Daniel B. Neill
Department of Computer Science
Carnegie Mellon University
Pittsburgh, PA 15213
[email protected]
Andrew W. Moore
Department of Computer Science
Carnegie Mellon University
Pittsburgh, PA 15213
[email protected]
Abstract
Given an N ?N grid of squares, where each square has a count and an underlying population, our goal is to find the square region with the highest
density, and to calculate its significance by randomization. Any density
measure D, dependent on the total count and total population of a region, can be used. For example, if each count represents the number
of disease cases occurring in that square, we can use Kulldorff?s spatial
scan statistic DK to find the most significant spatial disease cluster. A
naive approach to finding the maximum density region requires O(N 3 )
time, and is generally computationally infeasible. We present a novel
algorithm which partitions the grid into overlapping regions, bounds the
maximum score of subregions contained in each region, and prunes regions which cannot contain the maximum density region. For sufficiently
dense regions, this method finds the maximum density region in optimal
O(N 2 ) time, in practice resulting in significant (10-200x) speedups.
1
Introduction
This paper develops fast methods for detection of spatial overdensities: discovery of spatial regions with high scores according to some density measure, and statistical significance
testing in order to determine whether these high-density regions can reasonably have occurred by chance. A major application is in identifying clusters of disease cases, for purposes ranging from detection of bioterrorism (ex. anthrax) to environmental risk factors
for diseases such as childhood leukemia ([1]-[3]). [4] discusses many other applications,
including astronomy (identifying star clusters), reconnaissance, and medical imaging.
Consider the case in which counts are aggregated to a uniform 2-d grid. Assume an N ? N
grid of squares G, where each square si j ? G is associated with a count ci j and an underlying
population pi j . For example, a square?s count may be the number of disease cases in that
geographical region in a given time period, while its population may be the total number of
people ?at-risk? for the disease. Our goal is to find the square region S ? ? G with the highest
density according to a density measure D: S? = arg maxS D(S). We use the abbreviations
mdr for the ?maximum density region? S? , and mrd for the ?maximum region density?
D(S? ), throughout. The density measure D must be an increasing function of the total
count of the region, C(S) = ?S ci j , and a decreasing function of the total population of the
region, P(S) = ?S pi j . In the case of a uniform underlying population, P(S) ? k 2 , where k
is the size of region S. But we focus on the more interesting case: non-uniform populations.
The problem of finding significant spatial overdensities is distinct from that solved by gridbased hierarchical methods such as CLIQUE [5], MAFIA [6], and STING [7], which also
look for ?dense clusters.? There are three main differences:
1. Our method is applicable to any density measure D, while the other algorithms are
specific to the ?standard? density measure D1 (S) = C(S)
P(S) . The D1 measure is the
number of points per unit population, for example this corresponds to the region
with the highest observed disease rate. Unlike many other density measures, D 1 is
monotonic: if a region S with density d is partitioned into any set of disjoint subregions, at least one subregion will have density d 0 ? d. Thus it is not particularly
useful to find the ?region? with maximum D1 , since this will be the single square
c
with highest pii jj . Instead, the other algorithms search for maximally sized regions
with D1 greater than some threshold, relying on the monotonicity of D1 by first
finding dense units (1 ? 1 squares), then merging adjacent units in bottom-up fashion. For a non-monotonic measure such as Kulldorff?s, it is possible to have a large
dense region where none of its subregions are themselves dense, so bottom-up can
fail. Here, we will optimize with respect to arbitrary non-monotonic density measures, and thus use a different approach from CLIQUE, MAFIA, or STING.
2. Our method deals with non-uniform underlying populations: this is particularly
important for real-world epidemiological applications, in which an overdensity of
disease cases is more significant if the underlying population is large.
3. Our goal is not only to find the highest scoring region, but also to test whether that
region is a true cluster or if it is likely to have occurred by chance.
1.1 The spatial scan statistic
A non-monotonic density measure which is of great interest to epidemiologists is Kulldorff?s spatial scan statistic [8], which we denote by DK . This assumes that counts ci j are
generated by an inhomogeneous Poisson process with mean qpi j , where q is the underlying
?disease rate? (or expected value of the D1 density). We then calculate the log of the likelihood ratio of two possibilities: that the disease rate q is higher in the region than outside the
region, and that the disease rate is identical inside and outside the region. For a region with
count C and population P, in a grid with total count Ctot and population Ptot , we can calCtot
Ctot
C
tot ?C
culate DK = C log CP + (Ctot ? C) log CPtot
?P ? Ctot log Ptot , if P > Ptot , and 0 otherwise. [8]
proved that the spatial scan statistic is individually most powerful for finding a significant
region of elevated disease rate: it is more likely to detect the overdensity than any other test
statistic. Note, however, that our algorithm is general enough to use any density measure,
and in some cases we may wish to use measures other than Kulldorff?s. For instance, if
we have some idea of the size of the maximum density region, we can use the D r measure,
C(S)
Dr (S) = P(S)
r , 0 < r < 1, with larger r corresponding to tests for smaller clusters.
Once we have found the maximum density region (mdr) of grid G according to our density
measure, we must still determine the statistical significance of this region. Since the exact
distribution of the test statistic is only known in special cases (such as D1 density with a
uniform underlying population), in general we must perform Monte Carlo simulation for
our hypothesis test. To do so, we run a large number R of random replications, where
a replica has the same underlying populations pi j as G, but assumes a uniform disease
tot (G)
0
rate qrep = CPtot
(G) for all squares. For each replica G , we first generate all counts ci j
randomly from an inhomogeneous Poisson distribution with mean qrep pi j , then compute
the maximum region density (mrd) of G0 and compare this to mrd(G). The number of
replicas G0 with mrd(G0 ) ? mrd(G), divided by the total number of replications R, gives
us the p-value for our maximum density region. If this p-value is less than .05, we can
conclude that the discovered region is statistically significant (unlikely to have occurred
by chance) and is thus a ?spatial overdensity.? If the test fails, we have still discovered the
maximum density region of G, but there is not sufficient evidence that this is an overdensity.
1.2 The naive approach
The simplest method of finding the maximum density region is to compute the density of
all square regions of sizes k = kmin . . . N.1 Since there are (N ? k + 1)2 regions of size k,
there are a total of O(N 3 ) regions to examine. We can compute the density of any region
S in O(1), by first finding the count C(S) and population P(S), then applying our density
measure D(C, P).2 This allows us to compute the mdr of an N ? N grid G in O(N 3 ) time.
However, significance testing by Monte Carlo replication also requires us to find the mrd for
each replica G0 , and compare this to mrd(G). Since calculation of the mrd takes O(N 3 ) time
for each replica, the total complexity is O(RN 3 ), and R is typically large (we assume R =
1000). Several simple tricks may be used to speed up this procedure for cases where there
is no significant spatial overdensity. First, we can stop examining a replica G 0 immediately
if we find a region with density greater than mrd(G). Second, we can use the Central Limit
Theorem to halt our Monte Carlo testing early if, after a number of replications R 0 < R, we
can conclude with high confidence that the region is not significant. For cases where there
is a significant spatial overdensity, the naive approach is still extremely computationally
expensive, and this motivates our search for a faster algorithm.
2
Overlap-multires partitioning
Since the problem of detection of spatial overdensities is closely related to problems such
as kernel density estimation and kernel regression, this suggests that multi-resolution partitioning techniques such as kd-trees [9] and mrkd-trees [10] may be useful in speeding up
our search. The main difference of our problem from kernel density estimation, however,
is that we are only interested in the maximum density region; thus, we do not necessarily
need to build a space-partitioning tree at all resolutions. Also, the assumption that counts
are aggregated to a uniform grid simplifies and speeds up partitioning, eliminating the need
for a computationally expensive instance-based approach. These observations suggest a
top-down multi-resolution partitioning approach, in which we search first at coarse resolutions (large regions), then at successively finer resolutions as necessary. One option would
be to use a ?quadtree? [11], a hierarchical data structure in which each region is recursively
partitioned into its top left, top right, bottom left, and bottom right quarters. However, a
simple partitioning approach fails because of the non-monotonicity of our density measure:
a dense region may be split into two or more separate subregions, none of which is as dense
as the original region. This problem can be prevented by a partitioning approach in which
adjacent regions partially overlap, an approach we call ?overlap-multires partitioning.?
To explain how this method works, we first define some notation. We denote a region S by
an ordered triple (x, y, k), where (x, y) is the upper left corner of the region and k is its size.
Next, we define the ?-children of a region S = (x, y, k) as the four overlapping subregions
of size k ? ? corresponding to the top left, top right, bottom left, and bottom right corners
of S: (x, y, k ? ?), (x + ?, y, k ? ?), (x, y + ?, k ? ?), and (x + ?, y + ?, k ? ?). Next, we
define a region as ?even? if its size is 2k for some k ? 2, and ?odd? if its size is 3 ? 2k for
some k ? 0. We define the ?gridded children? (g-children) of an even region S = (x, y, k)
as its ?-children for ? = 4k . Thus the four g-children of an even region are odd, and each
overlaps 32 with the directly adjacent child regions. Similarly, we define the g-children of
an odd region S = (x, y, k) as its ?-children for ? = 3k . Thus the four g-children of an odd
region are even, and each overlaps 12 with the directly adjacent child regions. Note that
1 We
assume that a region must have size at least kmin to be significant: here kmin = 3.
2 An old trick allows us to compute the count of any k ? k region in O(1): we first form a matrix of
the cumulative counts, then compute each region?s count by adding at most four cumulative counts.
even though a region has four g-children, and each of its g-children has four g-children,
it has only nine (not 16) distinct grandchildren, several of which are the child of multiple
regions. Figure 1 shows the first two levels of such a tree.
Figure 1: The first two levels of the overlapmutires tree. Each node represents a gridded region (denoted by a thick square) of the entire
dataset (thin square and dots).
Next, we assume that the size of the entire grid is a power of two: thus the entire grid
G = (0, 0, N) is an even region. We define the set of ?gridded? regions of G as G and all of
its ?gridded descendents? (its g-children, g-grandchildren, etc.). Our algorithm focuses its
search on the set of gridded regions, only searching non-gridded regions when necessary.
This technique is useful because the total number of gridded regions is O(N 2 ), as in the
simple quadtree partitioning method. This implies that, if only gridded regions need to be
searched, our total time to find the mdr of a grid is O(N 2 ). Since it takes ?(N 2 ) time to
generate the grid, this time bound is optimal.
2.1 Top-down pruning
So when can we search only gridded regions, or alternatively, when does a given nongridded region need to be searched? Our basic method is branch-and-bound: we perform
a top-down search, and speed up this search by pruning regions which cannot possibly
contain the mdr. Our first step is to derive an upper bound Dmax (S, k) on the density of
subregions of minimum size k contained in a given region S (Section 2.2). Then we can
compare Dmax (S, k) to the density D(S? ) of the best region found so far: if Dmax (S, k) <
D(S? ), we know that no subregion of S with size k or more can be the mdr.
We can use this information for two types of pruning. First, if Dmax (S, kmin ) < D(S? ), we
know that no subregion of S can be optimal; we can prune the region completely, and not
search its (gridded or non-gridded) children. Second, we can show that (for 0 < k < n) any
region of size 2k + 1 or less is contained entirely in an odd gridded region of size 23 ? 2k .
Thus, if Dmax (G, 2n?1 + 2) < D(S? ) for the entire grid G, any optimal non-gridded region
must be contained in an odd gridded region. Similarly, if Dmax (S, 2k + 2) < D(S? ) for an
odd gridded region S of size 3 ? 2k , any optimal non-gridded subregion of S must be within
an odd gridded subregion of S. Thus we can search only gridded regions if two conditions
hold: 1) no subregion of G of size 2n?1 + 2 or more can be optimal, and 2) for each odd
gridded region of size 3 ? 2k , no subregion of size 2k + 2 or more can be optimal.
2.2 Bounding subregion density
To bound the maximum subregion density Dmax (S, k), we must find the highest possible
score D(S0 ) of a subregion S0 ? S of size k or more. Let C = C(S), P = P(S), and K =
size(S). We assume that these are known, as well as lower and upper bounds [d min , dmax ]
on the D1 density of subregions of S. Let c = C(S0 ) and p = P(S0 ); these are presently
unknown. We can prove that, if D(S0 ) > D(S), the maximum value of D(S0 ) occurs when
S has the maximum allowable D1 density dmax , and S ? S0 has the minimum allowable D1
min
density dmin : this gives us pdmax + (P ? p)dmin = C. Thus p = dC?Pd
and c = dmax p =
max ?dmin
C?Pdmin
1?dmin /dmax .
Then computing D(c, p) gives us a guaranteed upper bound on Dmax (S, k).
We can place tighter bounds on Dmax (S, k) if we also have a lower bound pmin (S, k) on the
population of a size k subregion S0 ? S: in this case, if the value calculated for p in the
equation above is less than pmin , we know that D(c0 , pmin ), where c0 = C ? (P ? pmin )dmin ,
is a tighter upper bound for Dmax . We can bound pmin in several ways. First, if we know
the minimum population ps,min of a single square s ? S, then pmin ? k2 ps,min . Second,
if we know the maximum population ps,max of a single square s ? S, then pmin ? P ?
(K 2 ? k2 )ps,max . At the beginning of our algorithm, we calculate ps,max (S) = max pi j and
ps,min (S) = min pi j (where si j ? S) for each gridded region S. This calculation can be
done recursively (bottom-up) in O(N 2 ). The resulting population statistics are used for the
original grid and for all replicas. For non-gridded regions, we use the population statistics
of the region?s gridded parent (either an odd gridded region or the entire grid G); these
bounds will be looser for the child region than for the parent, but are still correct. We
also initially calculate dmax and dmin . This is done simply by finding the global maximum
0)
0
0
and minimum values of the D1 density: dmax = max C(S
P(S0 ) (where S ? G and size(S ) =
c
kmin ), and dmin = min pii jj (where si j ? G).3 Alternatively, we could compute dmax and dmin
recursively (bottom-up) for each gridded region S, but in practice we find that the global
values are sufficient for good performance on most test cases.
2.3 The algorithm
Our algorithm, based on the overlap-multires partitioning scheme above, is a top-down,
best-first search of the set of gridded regions, followed by a top-down, best-first search
of any non-gridded regions as necessary. We use priority queues (q1,q2) for our search:
each step of the algorithm takes the ?best? (i.e. highest density) region from a queue,
examines it, and (if necessary) adds its children to queues. The ?-children and g-children
of a region S are defined above; note that the 1-children of S are its ?-children with ? = 1.
We also assume that regions are ?marked? once added to a queue, so that a region will
not be searched more than once. Finally, we use the rules and density bounds derived
above to speed up our search, by pruning subregions when Dmax (S, k) ? D(S? ). The basic
pseudocode outline of our method is as follows:
Add G to q1.
If D_max(G,N/2+2)>mrd, add 1-children(G) to q2 with k1=N/2+2.
While q1 not empty:
Get best region S from q1.
If D(S)>mrd, set mdr=S and mrd=D(S).
If D_max(S,k_min)>mrd, add g-children(S) to q1.
If size(S)=3(2?k) and D_max(S,2?k+2)>mrd, add 1-children(S) to q2 with k1=2?k+2.
While q2 not empty:
Get best region S and value k1(S) from q2.
If D(S)>mrd, set mdr=S and mrd=D(S).
If D_max(S,k1(S))>mrd, add 1-children(S) to q2 with same k1.
These steps are first performed for the original grid, allowing us to calculate its mdr and
mrd. We then perform these steps to calculate the mrd of each replica; however, several
techniques allow us to reduce the amount of computation necessary for a replica. First, we
can stop examining a replica G0 immediately if we find a region with density greater than
mrd(G). This is especially useful in cases where there is no significant spatial overdensity
in G. Second, we can use mrd(G) for pruning our search on a replica G0 : if Dmax (S, k) <
mrd(G) for some S ? G0 , we know that no subregion of S of size k or more can have a
greater density than the mdr of the original grid, and thus we do not need to examine any of
those subregions. This is especially useful where there is a significant spatial overdensity
in G: a high mrd will allow large amounts of pruning on the replica grids.
3
Improving the algorithm
The exact version of the algorithm uses conservative estimates of the D1 densities of S0 and
S ? S0 (dmax and dmin respectively), and a loose lower bound on the population of S 0 , to
3 We can use the tighter bound for d
max since we are using it to bound the density of a square
region S0 of size at least kmin ; we cannot use the tighter bound for dmin since S ? S0 is not square.
calculate Dmax (S, k). This results in a loose upper bound on Dmax which is guaranteed to be
correct, but allows little pruning to be done. We can derive tighter bounds on D max in two
ways: by using a closer approximation to the D1 density of S ? S0 , and by using a tighter
lower bound on the population of S0 . These improvements are discussed below.
3.1 The outer density approximation
To derive tighter bounds on the maximum density of a subregion S 0 contained in a given
region S, we first note that (under both the null hypothesis and the alternative hypothesis)
we assume that at most one disease cluster Sdc exists, and that the disease rate q is expected
to be uniform outside Sdc (or uniform everywhere, if no disease cluster exists). Thus, if
Sdc is contained entirely in the region under consideration S, we would expect that the
0
0
maximum density subregion
h S iof S is Sdc , and that the disease rate of S ? S is equal to the
Ctot ?C
C?c
disease rate outside S: E P?p = Ptot ?P = dout . Assuming that the D1 density of S ? S0 is
equal to its expected value dout , we obtain the equation pdmax + (P ? p)dout = C. Solving
out
for p, we find p = dC?Pd
. Then Dmax (S, k) = D(c, p), where c = dmax p.
max ?dout
The problem with this approach is that we have not compensated for the variance in densities: our calculated value of Dmax is an upper bound for the maximum subregion density
D(S0 ) only in the most approximate probabilistic sense. We would expect the D1 density of
S ? S0 to be less than its expected value half the time, and thus we would expect D(S 0 ) to be
less than Dmax at least half the time; in practice, our bound will be correct more often, since
we are still using a conservative approximation of the D1 density of S0 . Note also that we
expect to underestimate Dmax if the disease cluster Sdc is not contained entirely in S: this is
acceptable (and desirable) since a region not containing Sdc does not need to be expanded.
We can improve the correctness of our probabilistic bound by also considering the variC?c
tot ?C
ance of P?p
? CPtot
by a inho?P . Assuming that all counts outside Sdc are generated
i
h
C?c
tot ?C
mogeneous Poisson distribution with parameter qpi j , we obtain: ?2 P?p ? CPtot
?P =
h
i
?2 Po(q(P?p)) ? Po(q(Ptot ?P)) = q + q = q(Ptot ?p) . Since the actual value of
P?p
Ptot ?P
P?p
Ptot ?P
(P?p)(Ptot ?P)
Ctot
the parameter q is not known, we use a conservative empirical estimate: q = Ptot
?p . From
h
i q
Ctot
C?c
tot ?C
this, we obtain ? P?p
? CPtot
?P =
(P?p)(Ptot ?P) . Then we can compute p by solving
pdmax + (P ? p)(dout ? b?) = C, and obtain c = dmax p and Dmax = D(c, p) as before.
By adjusting our approximation of the minimum density in this manner, we compute a
higher score Dmax , reducing the likelihood that we will underestimate the maximum subregion density and prune a region that should not necessarily be pruned. Given a constant b,
the D1 density of S ? S0 will be greater than dout ? b? with probability P(Z < b), where Z
is chosen randomly from the unit normal. For b = 2, there is an 98% chance that we will
underestimate D1 (S ?S0 ), giving a guaranteed correct upper bound for the maximum subregion density. In practice, the maximum subregion density will be lower than our computed
value of Dmax more often, since our estimates for dmax and q are conservative. Thus, though
our algorithm is approximate, it is very likely to converge to the globally optimal mdr. In
fact, our experiments demonstrate that b = 1 is sufficient to obtain the correct region with
over 90% probability, approaching 100% for sufficiently dense regions.
3.2 Cached population statistics
A final step in making the algorithm tractable is to cache certain statistics about the minimum populations of subsquares of gridded regions. This is only performed once: it need
not be repeated for each replica (since populations need not be randomized). Although
there is no room to describe it, we have empirically shown it to give an important acceleration if populations are highly non-uniform. The results below make use of this.
4
Results
We first describe results with artificially generated grids and then real-world case data. An
artificial grid is generated from a set of parameters (N, k, ?, ?, q0 , q00 ). The grid generator
first creates an N ? N grid, and randomly selects a k ? k ?test region.? Then the population
of each square is chosen randomly from a normal distribution with mean ? and standard
deviation ? (populations less than zero are set to zero). Finally, the count of each square is
chosen randomly from a Poisson distribution with parameter qpi j , where q = q0 inside the
test region and q = q00 outside the test region.
We tested three different adjustments for density variance (b = 0, 1, 2). The approximate
algorithm was tested for grids of size N = 512; test region sizes of k = 16 and k = 4 were
used, and the disease rate q was set to .002 inside the test region and .001 outside the
test region. We used three different population distributions for testing: the ?standard?
distribution (? = 104 , ? = 103 ), and two types of ?highly varying? populations. For the
?city? distribution, we randomly selected a ?city region? with size 16: square populations
were generated with ? = 107 and ? = 106 inside the city, and ? = 104 and ? = 103 outside
the city. For the ?high-?? distribution, we generated all square populations with ? = 10 4
and ? = 5 ? 103 . We first compared the performance of each variant of the algorithm to
the naive approach for the three test cases; see Table 1 for results. For large test regions
(k = 16), all variants of the algorithm had runtimes of ?20 minutes, as compared to 44
hours for the naive approach, a speedup of 122-155x. For small test regions (k = 4), we
observed that performance generally decreased with increasing b: the algorithm achieved
average speedups of 133x for b = 0, 61x for b = 1, and 18x for b = 2.
Next, we tested accuracy by generating 50 artificial grids for each population distribution,
and computing the percentage of test grids on which the algorithm was able to find the
correct mdr (see Table 2). For the large test region (k = 16), all variants were able to find the
correct mdr with high (97-100%) accuracy. For the small test region, accuracy improved
significantly with increasing b: the non-variance adjusted version (b = 0) achieved only
45% accuracy, while the variance adjusted versions (b = 1 and b = 2) achieved 89% and
99% accuracy respectively. These results demonstrate that the approximate algorithm (with
variance adjustment and cached population statistics) is able to achieve high performance
and accuracy even for very small test regions and highly non-uniform populations.
Finally, we measured the performance of the approximate algorithm on a grid generated
from real-world data. We used a database of (anonymized) Emergency Department data
collected from Western Pennsylvania hospitals in the period 1999-2002. This dataset contained a total of 630,000 records, each representing a single ED visit and giving the latitude
and longitude of the patient?s home location to the nearest .005 degrees (? 13 mile, a sufficiently low resolution to ensure anonymity). For each record, the latitude L and longitude
l?lmin
min
l were converted to a grid square si j by i = L?L
.005 and j = .005 ; this created a 512 ? 512
grid. We tested for spatial clustering of ?recent? disease cases: the ?count? of each square
was the number of ED visits in that square in the last two months, and the ?population? of
that square was the total number of ED visits in that square. See Figure 2 for a picture of
this dataset, including the highest scoring region. We tested six variants of the approximate
algorithm on the ED dataset; the presence/absence of cached population statistics did not
significantly affect the performance or accuracy for this test, so we focus on the variation
in b. All three variants (b = 0, 1, 2), as well as the naive algorithm, found the maximum
density region (of size 101) and found it statistically significant (p-value 0/1000). The major difference, of course, was in runtime and number of regions searched (see Table 3).
The naive algorithm took 2.7 days to find the mdr and perform 1000 Monte Carlo replications, while each of the variants of the approximate algorithm performed the same task
in ?2 hours or less. The approximate algorithm took 19 minutes (a speedup of 209x) for
b = 0, 47 minutes (a speedup of 85x) for b = 1, and 126 minutes (a speedup of 31x) for
b = 2. Thus we can see that all three variants find the correct region in much less time than
Figure 2: The left picture shows the
?population? distribution within Western PA and the right picture shows the
?counts? distribution. The winning region is shown as a square.
Table 1: Performance of algorithm, N = 512
method
naive
test
all
time (orig+1000 reps)
2 : 37 + 43 : 36 : 40
speedup
x1
b=0
b=1
b=2
std, k = 16
std, k = 16
std, k = 16
0 : 42 + 16 : 40
0 : 43 + 16 : 20
0 : 41 + 17 : 00
x151
x154
x148
b=0
b=1
b=2
std, k = 4
std, k = 4
std, k = 4
0 : 41 + 17 : 00
0 : 41 + 29 : 10
0 : 42 + 1 : 13 : 00
x148
x88
x36
b=0
b=1
b=2
city, k = 16
city, k = 16
city, k = 16
0 : 42 + 16 : 30
0 : 46 + 20 : 40
0 : 41 + 18 : 40
x153
x122
x135
b=0
b=1
b=2
city, k = 4
city, k = 4
city, k = 4
0 : 43 + 24 : 30
0 : 44 + 2 : 11 : 00
0 : 47 + 7 : 06 : 50
x104
x20
x6.1
b=0
b=1
b=2
high-?, k = 16
high-?, k = 16
high-?, k = 16
0 : 41 + 17 : 00
0 : 41 + 16 : 40
0 : 41 + 17 : 00
x148
x151
x148
b=0
b=1
b=2
high-?, k = 4
high-?, k = 4
high-?, k = 4
0 : 44 + 17 : 15
0 : 45 + 34 : 10
1 : 08 + 3 : 20 : 00
x146
x75
x13
Table 2: Accuracy of algorithm
method
test
standard
city
high-?
accuracy
(k = 16)
96%
98%
98%
accuracy
(k = 4)
52%
36%
46%
b=0
b=0
b=0
b=1
b=1
b=1
standard
city
high-?
100%
100%
100%
90%
88%
90%
b=2
b=2
b=2
standard
city
high-?
100%
100%
100%
98%
98%
100%
Table 3: Emergency Dept. dataset
method
naive
b=0
b=1
b=2
time (orig+1000 reps)
4 : 05 + 65 : 50 : 00
4 : 20 + 14 : 36
4 : 22 + 42 : 20
4 : 36 + 2 : 01 : 12
speedup
x1
x209
x85
x31
the naive approach. This is very important for applications such as real-time detection of
disease outbreaks: if a system is able to detect an outbreak in minutes rather than days,
preventive measures or treatments can be administered earlier, possibly saving many lives.
Thus we have presented a fast overlap-multires partitioning algorithm for detection of
spatial overdensities, and demonstrated that this method results in significant (10-200x)
speedups on real and artificially generated datasets. We are currently applying this algorithm to national-level hospital and pharmacy data, attempting to detect statistically significant indications of a disease outbreak based on changes in the spatial clustering of disease
cases. Application of a fast partitioning method using the techniques presented here may
allow us to achieve the difficult goal of automatic real-time detection of disease outbreaks.
References
[1] S. Openshaw, et al. 1988. Investigation of leukemia clusters by use of a geographical analysis machine. Lancet 1, 272-273.
[2] L. A. Waller, et al. 1994. Spatial analysis to detect disease clusters. In N. Lange, ed. Case Studies in Biometry. Wiley, 3-23.
[3] M. Kulldorff and N. Nagarwalla. 1995. Spatial disease clusters: detection and inference. Statistics in Medicine 14, 799-810.
[4] M. Kulldorff. 1999. Spatial scan statistics: models, calculations, and applications. In Glaz and Balakrishnan, eds. Scan
Statistics and Applications. Birkhauser: Boston, 303-322.
[5] R. Agrawal, et al. 1998. Automatic subspace clustering of high dimensional data for data mining applications. Proc. ACMSIGMOD Intl. Conference on Management of Data, 94-105.
[6] S. Goil, et al. 1999. MAFIA: efficient and scalable subspace clustering for very large data sets. Northwestern University,
Technical Report No. CPDC-TR-9906-010.
[7] W. Wang, et al. 1997. STING: a statistical information grid approach to spatial data mining. Proc. 23rd Conference on Very
Large Databases, 186-195.
[8] M. Kulldorff. 1997. A spatial scan statistic. Communications in Statistics: Theory and Methods 26(6), 1481-1496.
[9] F. P. Preparata and M. I. Shamos. 1985. Computational Geometry: An Introduction. Springer-Verlag: New York.
[10] K. Deng and A. W. Moore. 1995. Multiresolution instance-based learning. Proc. 12th Intl. Joint Conference on Artificial
Intelligence, 1233-1239.
[11] H. Samet. 1990. The Design and Analysis of Spatial Data Structures. Addison-Wesley: Reading.
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1,683 | 253 | 638
Zipser
Subgrouping Reduces Complexity and Speeds Up
Learning in Recurrent Networks
David Zipser
Department of Cognitive Science
University of California, San Diego
La Jolla, CA 92093
1 INTRODUCTION
Recurrent nets are more powerful than feedforward nets because they allow simulation of
dynamical systems. Everything from sine wave generators through computers to the brain are
potential candidates, but to use recurrent nets to emulate dynamical systems we need learning
algorithms to program them. Here I describe a new twist on an old algorithm for recurrent nets
and compare it to its predecessors.
2 BPTT
In the beginning there was BACKPROPAGATION THROUGH TUvffi (BPTT) which was
described by Rumelhart, Williams, and Hinton (1986). The idea is to add a copy of the whole
recurrent net to the top of a growing feedforward network on each update cycle. Backpropagating through this stack corrects for past mistakes by adding up all the weight changes from
past times. A difficulty with this method is that the feedforward net gets very big. The obvious
solution is to truncate it at a fixed number of copies by killing an old copy every time a new
copy is added. The truncated-BPTT algorithm is illustrated in Figure 1. It works well, more
about this later.
3RTRL
It turns out that it is not necessary to keep an ever growing stack of copies of the recurrent
net as BPTT does. A fixed number of parameters can record all of past time. This is done in
the REAL TI!\.1E RECURRENT LEARNING (RTRL) algorithm of Williams and Zipser
(1989). The derivation is given elsewhere (Rumelhart, Hinton, & Williams, 1986), but a
Sub grouping Reduces Complexity
IN
t-l
IN
IN
t -k +2
t -k +1
i~
I
i<
~r
Figure 1: BPTT.
-!::::~
l
i;f~
639
640
Zipser
simple rational comes from the fact that error backpropagation is linear, which makes it
possible to collapse the whole feedforward stack ofBPTT into a few fixed size data structures.
The biggest and most time consuming to update of these is the matrix of p values whose
update rule is
P it <t + 1) = f
'(Sk
<t? [
L
W kl
leU
i~
P <t) + c5 ik
Zj
<t) ]
ieU,jeUuI,keU
where z,,(t) represents the value of a signal, either an input or recurrent; the sets of subscriptss
are defined so that if z" is an input then k E I and if z"is a signal from a recurrently connected
unit then k E U, s" are net values; d,,, is the Kronecker delta; and wk.l is the recurrent weight
matrix. For a network with n units and w weights there are nw of these p values, and it takes
O(wn 2) operations to update them. As n gets big this gets very big and is computationally unpleasant. This unpleasantness is cured to some degree by the new variant ofRTRL described
below.
4 SUBGROUPED RTRL
The value of n in the factor wn 2 , which causes all the trouble for RTRL, can be reduced by
viewing a recurrent network as consisting of a set of subnetworks all connected together. A
full y recurrent network wi th n units and m inpu ts can be divided into g full y recurren t su bnets,
each with n/g units (assuming g is a factor of n). Each unit in a subnet will receive as input
the original m inputs and the activities of the n - n/g units in the other subnets. The effect of
subgrouping is to reduce the number of p values per weight to n/g and the number of
operations to update the pto O(wn 2/g 2 ). If g is increased in proportion to n, which keeps the
size of the sub-nets constant, n2/g 2 is a constant and the complexity is reduced to O(w). If all
this is confusing try Figure 2.
5 TESTING THESE ALGORITHMS
To see if the subgrouped algorithm works, I compared its performance to RTRL and BPTT
on the problem of training a Turing machine to balance parentheses. The network "sees" the
same tape as the Turing machine, and is trained to produce the same outputs. A fully recurrent
network with 12 units was the smallest that learned this task. All three algorithms learned the
task in about the same number oflearning cycles. RTRL and subgrouped RTRL succeeded
50%, and BPTT succeeded 80% of the time. Subgrouped RTRL was 10 times faster than
RTRL, whereas BPTT was 28 times faster.
References
Rumelhart, D. E., Hinton, G. E., & Williams, R. J. (1986). Learning internal representations
by error propagation. In D. E. Rumelhart, J. L. McClelland, & the PDP Research Group
(Eds.), Parallel distributed processing: Explorations in the microstructure of cognition. Vol.
1. Foundationa. Cambridge, MA: MIT Press.
Williams, R. J., & Zipser, D. (1989). A learning algorithm for continually running fully
recurrent neural networks. Neural Computation, 1, 270-280.
Subgrouping Reduces Complexity
Fully Recurrent
---t.~
...........::::'., Activity only
ALtivity and Error
..
..
SubgrOlJ'ped
Figure 2: Suhgroupcd-RTRL
641
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1,684 | 2,530 | Model Uncertainty in Classical Conditioning
A. C. Courville*1,3 , N. D. Daw2,3 , G. J. Gordon4 , and D. S. Touretzky2,3
1
Robotics Institute, 2 Computer Science Department,
3
Center for the Neural Basis of Cognition,
4
Center for Automated Learning and Discovery
Carnegie Mellon University, Pittsburgh, PA 15213
{aaronc,daw,ggordon,dst}@cs.cmu.edu
Abstract
We develop a framework based on Bayesian model averaging to explain
how animals cope with uncertainty about contingencies in classical conditioning experiments. Traditional accounts of conditioning fit parameters within a fixed generative model of reinforcer delivery; uncertainty
over the model structure is not considered. We apply the theory to explain the puzzling relationship between second-order conditioning and
conditioned inhibition, two similar conditioning regimes that nonetheless result in strongly divergent behavioral outcomes. According to the
theory, second-order conditioning results when limited experience leads
animals to prefer a simpler world model that produces spurious correlations; conditioned inhibition results when a more complex model is
justified by additional experience.
1
Introduction
Most theories of classical conditioning, exemplified by the classic model of Rescorla and
Wagner [7], are wholly concerned with parameter learning. They assume a fixed (often
implicit) generative model m of reinforcer delivery and treat conditioning as a process
of estimating values for the parameters wm of that model. Typically, these parameters
represent the rates of reinforcers delivered in the presence of various stimuli. Using the
model and the parameters, the probability of reinforcer delivery can be estimated; such
estimates are assumed to give rise to conditioned responses in behavioral experiments.
More overtly statistical theories have treated uncertainty in the parameter estimates, which
can influence predictions and learning [4].
In realistic situations, the underlying contingencies of the environment are complex and
unobservable, and it can thus make sense to view the model m as itself uncertain and subject
to learning, though (to our knowledge) no explicitly statistical theories of conditioning
have yet done so. Under the standard Bayesian approach, such uncertainty can be treated
analogously to parameter uncertainty, by representing knowledge about m as a distribution
over a set of possible models, conditioned on evidence. Here we advance this idea as a highlevel computational framework for the role of model learning in classical conditioning. We
do not concentrate on how the brain might implement these processes, but rather explore
the behavior that a system approximating Bayesian reasoning should exhibit. This work
establishes a relationship between theories of animal learning and a recent line of theory
by Tenenbaum and collaborators, which uses similar ideas about Bayesian model learning
to explain human causal reasoning [9].
We have applied our theory to a variety of standard results in animal conditioning, including acquisition, negative and positive patterning, and forward and backward blocking.
Here we present one of the most interesting and novel applications, an explanation of a
rather mysterious classical conditioning phenomenon in which opposite predictions about
the likelihood of reinforcement can arise from different amounts of otherwise identical
experience [11]. The opposing effects, both well known, are called second-order conditioning and conditioned inhibition. The theory explains the phenomenon as resulting from
a tradeoff between evidence and model complexity.
2
A Model of Classical Conditioning
In a conditioning trial, a set of conditioned stimuli CS ? {A, B, . . . } is presented, potentially accompanied by an unconditioned stimulus or reinforcement signal, US . We represent the jth stimulus with a binary random variable yj such that yj = 1 when the stimulus
is present. Here the index j, 1 ? j ? s, ranges over both the (s ? 1) conditioned stimuli
and the unconditioned stimulus. The collection of trials within an experimental protocol
constitutes a training data set, D = {yjt }, indexed by stimulus j and trial t, 1 ? t ? T .
We take the perspective that animals are attempting to recover the generative process underlying the observed stimuli. We claim they assert the existence of latent causes, represented
by the binary variables xi ? {0, 1}, responsible for evoking the observed stimuli. The relationship between the latent causes and observed stimuli is encoded with a sigmoid belief
network. This particular class of models is not essential to our conclusions; many model
classes should result in similar behavior.
Sigmoid Belief Networks In sigmoid belief networks, local conditional probabilities are
defined as functions of weighted sums of parent nodes. Using our notation,
X
wij xi ? wyj ))?1 ,
P (yj = 1 | x1 , . . . , xc , wm , m) = (1 + exp(?
(1)
i
and P (yj = 0 | x1 , . . . , xc , wm , m) = 1 ? P (yj = 1 | x1 , . . . , xc , wm , m). The weight,
wij , represents the influence of the parent node xi on the child node yj . The bias term wyj
encodes the probability of yj in the absence of all parent nodes. The parameter vector wm
contains all model parameters for model structure m.
The form of the sigmoid belief networks we consider is represented as a directed graphical
model in Figure 1a, with the latent causes as parents of the observed stimuli. The latent
causes encode the intratrial correlations between stimuli ? we do not model the temporal
structure of events within a trial. Conditioned on the latent causes, the stimuli are mutually
independent.
We can express the conditional joint probability of the observed stimuli as
Qs
P
(y
|
x
j
1 , . . . , xc , wm , m).
j=1
Similarly, we assume that trials are drawn from a stationary process. We do not consider
trial order effects, and we assume all trials are mutually independent. (Because of these
simplifying assumptions, the present model cannot address a number of phenomena such
as the difference between latent inhibition, partial reinforcement, and extinction.) The
resulting likelihood function of the training data, with latent causes marginalized, is:
P (D | wm , m) =
T XY
s
Y
t=1 x j=1
P (yjt | x, wm , m)P (x | wm , m),
(2)
w x1
x1
w11 w w22
12
A
B
w y1
w y2
p(D | m) simple model
w x2
x2
w1s
???
???
w2s
complicated model
US
w ys
simple
wins
complicated
wins
D
(b) Marginal likelihood
(a) Sigmoid belief network
Figure 1: (a) An example from the proposed set of models. Conditional dependencies
are depicted as links between the latent causes (x1 , x2 ) and the observed stimuli (A, B,
U S) during a trial. (b) Marginal likelihood of the data, D, for a simple model and a more
complicated model (after MacKay [5]).
where
all combinations of values of x = [x1 , . . . , xc ] and P (x | wm , m) =
Qc the sum is over
xi
?1
(1
+
exp(?1
w
.
xi ))
i=1
Sigmoid belief networks have a number of appealing properties for modeling conditioning.
First, the sigmoid belief network is capable of compactly representing correlations between
groups of observable stimuli. Without a latent cause, the number of parameters required to
represent these correlations would scale exponentially with the number of stimuli. Second,
the parent nodes, interacting additively, constitute a factored representation of state. This is
advantageous as it permits generalization to novel combinations of factors. Such additivity
has frequently been observed in conditioning experiments [7].
2.1
Prediction under Parameter Uncertainty
Consider a particular network structure, m, with parameters wm . Given m and a set of
trials, D, the uncertainty associated with the choice of parameters is represented in a posterior distribution over wm . This posterior is given by Bayes? rule, p(wm | D, m) ?
P (D | wm , m)p(wm | m), where P (D | m) is from Equation 2 and p(wm | m) is the
prior distribution over the parameters
the model parameters are a priQ of m. We
Q assumeQ
ori independent. p(wm | m) = ij p(wij ) i p(wxi ) j p(wyj ), with Gaussian priors
for weights p(wij ) = N (0, 3), latent cause biases p(wxi ) = N (0, 3), and stimulus biases
p(wyj ) = N (?15, 1), the latter reflecting an assumption that stimuli are rare in the absence
of causes.
In conditioning, the test trial measures the conditioned response (CR). This is taken to
be a measure of the animal?s estimate of the probability of reinforcement conditioned on
the present conditioned stimuli CS . This probability is also conditioned on the absence
of the remaining stimuli; however, in the interest of clarity, our notation suppresses these
absent stimuli. In the Bayesian framework, given m, this probability, P (US | CS , m, D)
is determined by integrating over all values of the parameters weighted by their posterior
probability density,
P (US | CS , m, D) =
Z
P (US | CS , wm , m, D)p(wm | m, D) dwm
(3)
2.2
Prediction under Model Uncertainty
In the face of uncertainty about which is the correct model of contingencies in the world ?
for instance, whether a reinforcer is independent of a tone stimulus ? a standard Bayesian
approach is to marginalize out the influence of the model choice,
X
P (US | CS , D) =
P (US | CS , m, D)P (m | D)
(4)
m
=
XZ
m
P (US | CS , wm , m, D)p(wm | m, D)P (m | D) dwm
The posterior over models, p(m | D), is given by:
P (m | D) = P
P (D | m)P (m)
,
0
0
m0 P (D | m )P (m )
P (D | m) =
Z
P (D | wm , m)p(wm | m) dwm
The marginal likelihood P (D | m) is the probability of the data under model m, marginalizing out the model parameters. The marginal likelihood famously confers an automatic
Occam?s razor effect on the average of Equation 4. Under complex models, parameters
can be found to boost the probability of particular data sets that would be unlikely under
simpler models, but any particular parameter choice is also less likely in more complex
models. Thus there is a tradeoff between model fidelity and complexity (Figure 1b).
We also encode a further preference for Q
simpler models through the prior over model strucc
ture, which we factor as P (m) = P (c) i=1 P (li ), where c is the number of latent causes
and li is the number of directed links emanating from xi . The priors over c and li are in
turn given by,
(
(
?3c
10?3li
if 0 ? li ? 4
Ps
0
P5 10
if
0
?
c
?
5
0
?3c
10?3li
10
0
0
P (c) =
and
P
(l
)
=
l
=0
c =0
i
i
0
otherwise
0
otherwise
In the Bayesian model average, we consider the set of sigmoid belief networks with a
maximum of 4 stimuli and 5 latent causes.
This strong prior over model structures is required in addition to the automatic Occam?s
razor effect in order to explain the animal behaviors we consider. This probably is due to
the extreme abstraction of our setting. With generative models that included, e.g., temporal
ordering effects and multiple perceptual dimensions, model shifts equivalent to the addition
of a single latent variable in our setting would introduce a great deal of additional model
complexity and require proportionally more evidential justification.
2.3
Monte Carlo Integration
In order to determine the predictive probability of reinforcement, Bayesian model averaging requires that we evaluate Equation 4. Unfortunately, the integral is not amenable
to analytic solution. Hence we approximate the integral with a sum over samples from
the posterior p(wm , m | D). Acquiring samples is complicated by the need to sample
over parameter spaces of different dimensions. In the simulations reported here, we solved
this problem and obtained samples using a reversible jump Markov chain Monte Carlo
(MCMC) method [2]. A new sample in the chain is obtained by proposing perturbations
to the current sample?s model structure or parameters.1 Jumps include the addition or removal of links or latent causes, or updates to the stimulus biases or weights. To improve
mixing over the different modes of the target distribution, we used exchange MCMC, which
enables fast mixing between modes through the coupling of parallel Markov chains [3].
1
The proposal acceptance probability satisfies detailed balance for each type of jump.
Group
No-X
Few-X
Many-X
A-US
96
96
96
A-X
0
4
48
B-US
8
8
8
Test ;Result
X ;?
X ;CR
X ;?
Test ;Result
XB ;CR
XB ;CR
XB ;?
Table 1: A summary of some of the experiments of Yin et al. [11]. The US was a footshock;
A = white noise or buzzer sound; X = tone; B = click train.
3
Second-Order Conditioning and Conditioned Inhibition
We use the model to shed light on the relationship between two classical conditioning
phenomena, second-order conditioning and conditioned inhibition. The procedures for establishing a second-order excitor and a conditioned inhibitor are similar, yet the results are
drastically different. Both procedures involve two kinds of trials: a conditioned stimulus A
is presented with the US (A-US ); and A is also presented with a target conditioned stimulus X in unreinforced trials (A-X). In second order conditioning, X becomes an excitor ?
it is associated with increased probability of reinforcement, demonstrated by conditioned
responding. But in conditioned inhibition, X becomes an inhibitor, i.e. associated with
decreased probability of reinforcement. Inhibition is probed with two tests: a transfer test,
in which the inhibitor is paired with a second excitor B and shown to reduce conditioned
responding, and a retardation test, in which the time course of response development under
subsequent excitatory X-US training is retarded relative to naive animals.
Yin et al. [11] explored the dimensions of these two procedures in an effort to distill the
essential requirements for each. Under previous theories [8], it might have seemed that the
crucial distinction between second order conditioning and conditioned inhibition had to do
with either blocked versus interspersed trials, or with sequential versus simultaneous presentation of the CS es. However, they found that using only interspersed trials and simultaneous presentation of the conditioned stimuli, they were able to shift from second-order
conditioning to conditioned inhibition simply by increasing the number of A-X pairings. 2
Table 1 summarizes the relevant details of the experiment.
From a theoretical perspective, these results present a challenge for models of conditioning.
Why do animals so drastically change their behavior regarding X given only more of the
same kind of A-X experience? Bayesian model averaging offers some insight.
We simulated the experiments of Yin et al., matching their numbers for each type of trial,
as shown in Table 1. Results of the MCMC approximation of the Bayesian model average
integration are shown in Figure 2. All MCMC runs were at least 5 ? 106 iterations long
excluding a burn-in of 1 ? 106 iterations. The sequences were subsampled to 2.5 ? 104 .
In Figure 2a, we see that P (US | X, D) reveals significant second order conditioning
with few A-X trials. With more trials the predicted probability of reinforcement quickly
decreases. These results are consistent with the findings of Yin et al., as shown in Table 1. With few A-X trials there are insufficient data to justify a complicated model that
accurately fits the data. Due to the automatic Occam?s razor and the prior preference for
simple models, high posterior density is inferred for the simple model of Figure 3a. This
model combines the stimuli from all trial types and attributes them to a single latent cause.
When X is tested alone, its connection to the US through the latent cause results in a large
P (US | X, D).
With more training trials, the preference for simpler models is more successfully offset and
more complicated models ? capable of describing the data more accurately ? are given
2
In other conditions, trial ordering was shown to have an additional effect; this is outside the scope
of the present theory due to our stationarity assumptions.
1
0.8
0.8
P(US | A, D )
P(US | X, D )
0.6
0.4
0.2
0.2
10
20
30
40
50
Number of A?X trials
(a) Second-order Cond.
0.8
P(US | B, D )
P(US | X, B, D )
0.6
0.4
0
0
1
60
0
0
P(US|X,D )
1
0.6
0.4
0.2
10
20
30
40
Number of A?X trials
(b) Summation test
50
60
0
0
4
48
Number of A?X trials
(c) Retardation test
Figure 2: A summary of the simulation results. Error bars indicate the 3? margin in the
standard error of the estimate (we omit very small error bars). (a) P (US | X, D) and
P (US | A, D) as a function of A-X trials. For few trials (2 to 8), P (US | X, D) is high,
indicative of second-order conditioning. (b) P (US | X, B, D) and P (US | B, D) as a
function of number of A-X trials. After 10 trials, X is able to significantly reduce the
predicted probability of reinforcement generated by the presentation of B. (c) Results of
a retardation test. With many A-X trials, acquisition of an excitatory association to X is
retarded.
greater posterior density (Figure 3c). An example of such a model is shown in Figure 3b.
In the model, X is made a conditioned inhibitor by a negative valued weight between x 2
and X. In testing X with a transfer excitor B, as shown in Figure 2, this weight acts to
cancel a positive correlation between B and the US . Note that the shift from excitation
to inhibition is due to inclusion of uncertainty over models; inferring the parameters with
the more complex model fixed would result in immediate inhibition. In their experiment,
Yin et al. also conducted a retardation test of conditioned inhibition for X. We follow
their procedure and include in D 3 X-US trials. Our retardation test results are shown in
Figure 2 and are in agreement with the findings of Yin et al.
A further mystery about conditioned inhibitors, from the perspective of the benchmark
theory of Rescorla and Wagner [7], is the nonextinction effect: repeated presentations of a
conditioned inhibitor X alone and unreinforced do not extinguish its inhibitory properties.
An experiment by Williams and Overmier [10] demonstrated that unpaired presentations of
a conditioned inhibitor can actually enhance its ability to suppress responding in a transfer
test. Our model shows the same effect, as illustrated with a dramatic test in Figure 4. Here
we used the previous dataset with only 8 A-X pairings and added a number of unpaired
presentations of X. The additional unpaired presentations shift the model from a secondorder conditioning regime to a conditioned inhibition regime. The extinction trials suppress
posterior density over simple models that exhibit a positive correlation between X and US ,
shifting density to more complex models and unmasking the inhibitor.
4
Discussion
We have demonstrated our ideas in the context of a very abstract set of candidate models,
ignoring the temporal arrangement of trials and of the events within them. Obviously, both
of these issues have important effects, and the present framework can be straightforwardly
generalized to account for them, with the addition of temporal dependencies to the latent
variables [1] and the removal of the stationarity assumption [4].
An odd but key concept in early models of classical conditioning is the ?configural unit,?
a detector for a conjunction of co-active stimuli. ?Configural learning? theories (e.g. [6])
?2.5
x1
15 10
11
x1
16
16
A
X
B
?13
?14
?14
US
?13
(a) Few A-X trials
16 ?8
0.8
x2
11
11
A
X
B
?14
?14
?14
8
US
?14
(b) Many A-X trials
Average number of latent causes
3
?2.5
2.5
2
1.5
1
0
10
20
30
40
50
60
Number of A?X trials
(c) Model size over trials
Figure 3: Sigmoid belief networks with high probability density under the posterior. (a)
After a few A-X pairings: this model exhibits second-order conditioning. (b) After many
A-X pairings: this model exhibits conditioned inhibition. (c) The average number of latent
causes as a function of A-X pairings.
rely on heuristics for creating such units in response to observations, a rough-and-ready
sort of model structure learning. With a stimulus configuration represented through a latent
cause, our theory provides a clearer prescription for how to reason about model structure.
Our framework can be applied to a reservoir of configural learning experiments, including
negative and positive patterning and a host of others. Another body of data on which our
work may shed light is acquisition of a conditioned response. Recent theories of acquisition
(e.g. [4]) propose that animals respond to a conditioned stimulus (CS ) when the difference
in the reinforcement rate between the presence and absence of the CS satisfies some test of
significance. From the perspective of our model, this test looks like a heuristic for choosing
between generative models of stimulus delivery that differ as to whether the CS and US
are correlated through a shared hidden cause.
To our knowledge, the relationship between second-order conditioning and conditioned inhibition has never been explicitly studied using previous theories. This is in part because the
majority of classical conditioning theories do not account for second-order conditioning at
all, since they typically consider learning only about CS -US but not CS -CS correlations.
Models based on temporal difference learning [8] predict second-order conditioning, but
only if the two CS es are presented sequentially (not true of the experiment considered
here). Second-order conditioning can also be predicted if the A-X pairings cause some
sort of representational change so that A?s excitatory associations generalize to X. Yin et
al. [11] suggest that if this representational learning is fast (as in [6], though that theory
would need to be modified to include any second-order effects) and if conditioned inhibition accrues only gradually by error-driven learning [7], then second-order conditioning
will dominate initially. The details of such an account seem never to have been worked out,
and even if they were, such a mechanistic theory would be considerably less illuminating
than our theory as to the normative reasons why the animals should predict as they do.
Acknowledgments
This work was supported by National Science Foundation grants IIS-9978403 and DGE9987588, and by AFRL contract F30602?01?C?0219, DARPA?s MICA program. We thank
Peter Dayan and Maneesh Sahani for helpful discussions.
10
p(wm,m | D )
8
1
1 X? trial
2 X? trials
3 X? trials
0.8
6
0.6
4
0.4
2
0.2
0
0
0.2
0.4
0.6
P(US | X,B,w ,m,D )
m
(a) Posterior PDF
0.8
1
0
0
P(US | B,D )
P(US | X,B,D )
2
4
6
8
10
Number of X? trials
(b) Summation test
Figure 4: Effect of adding unpaired presentations of X on the strength of X as an inhibitor.
(a) Posterior probability of models which predict different values of P (US | X, B). With
only 1 unpaired presentation of X, most models predict a high probability of US (secondorder conditioning). With 2 or 3 unpaired presentations of X, models which predict a low
P (US | X, B) get more posterior weight (conditioned inhibition). (b) A plot contrasting
P (US | B, D) and P (US | X, B, D) as a function of unpaired X trials. The reduction in
the probability of reinforcement indicates an enhancement of the inhibitory strength of X.
Error bars indicate the 3? margin in the standard error in the estimate (omitting small error
bars).
References
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[2] P. J. Green. Reversible jump Markov chain Monte Carlo computation and Bayesian model
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[3] Y. Iba. Extended ensemble Monte Carlo. International Journal of Modern Physics C,
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[4] S. Kakade and P. Dayan. Acquisition and extinction in autoshaping. Psychological Review,
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[5] D. J. C. MacKay. Bayesian model comparison and backprop nets. In Advances in Neural
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1,685 | 2,531 | Log-Linear Models for Label Ranking
Ofer Dekel
Computer Science & Eng.
Hebrew University
Christopher D. Manning
Computer Science Dept.
Stanford University
Yoram Singer
Computer Science & Eng.
Hebrew University
[email protected] [email protected] [email protected]
Abstract
Label ranking is the task of inferring a total order over a predefined set of
labels for each given instance. We present a general framework for batch
learning of label ranking functions from supervised data. We assume that
each instance in the training data is associated with a list of preferences
over the label-set, however we do not assume that this list is either complete or consistent. This enables us to accommodate a variety of ranking
problems. In contrast to the general form of the supervision, our goal is
to learn a ranking function that induces a total order over the entire set
of labels. Special cases of our setting are multilabel categorization and
hierarchical classification. We present a general boosting-based learning
algorithm for the label ranking problem and prove a lower bound on the
progress of each boosting iteration. The applicability of our approach is
demonstrated with a set of experiments on a large-scale text corpus.
1
Introduction
This paper discusses supervised learning of label rankings ? the task of associating instances with a total order over a predefined set of labels. The ordering should be performed
in accordance with some notion of relevance of the labels. That is, a label deemed relevant
to an instance should be ranked higher than a label which is considered less relevant. With
each training instance we receive supervision given as a set of preferences over the labels.
Concretely, the supervision we receive with each instance is given in the form of a preference graph: a simple directed graph for which the labels are the graph vertices. A directed
edge from a label y to another label y 0 denotes that according to the supervision, y is more
relevant to the instance than y 0 . We do not impose any further constraints on the structure
of the preference graph.
The approach we employ distills and generalizes several learning settings. The simplest
setting is multiclass categorization in which each instance is associated with a single label
out of k possible labels. Such a setting was discussed for instance in [10] where a boosting algorithm called AdaBoost.MR (MR stands for Multiclass Ranking) for solving this
problem was described and analyzed. Using the graph representation for multiclass problems, the preference graph induced by the supervision has k vertices and k ? 1 edges. A
directed edge points from the (single) relevant label to each of the k ? 1 irrelevant labels
(Fig. 1a). An interesting and practical generalization of multiclass problems is multilabel
problems [10, 6, 4], in which a set of relevant labels (rather than a single label) is associated with each instance. In this case the supervision is represented by a directed bipartite
1
3
1
1
2
2
3
4
5
2
1
5
4
2
3
4
(a)
5
3
4
(b)
5
(c)
(d)
Figure 1: The supervision provided to the algorithm associates every training instance with
a preference graph. Different graph topologies define different learning problems. Examples that fit naturally in our generalized setting: (a) multiclass single-label categorization
where 1 is the correct label. (b) multiclass multilabel categorization where {1, 2} is the set
of correct labels. (c) A multi-layer graph that encodes three levels of label ?goodness?, useful for instance in hierarchical multiclass settings. (d) a general (possibly cyclic) preference
graph with no predefined structure.
graph where the relevant labels constitute one side of the graph and the irrelevant labels
the other side and there is a directed edge from each relevant label to each irrelevant label. (Fig. 1b). Similar settings are also encountered in information retrieval and language
processing tasks. In these settings the set of labels contains linguistic structures such as
tags and parses [1, 12] and the goal is to produce a total order over, for instance, candidate
parses. The supervision might consist of information that distinguishes three goodness levels (Fig. 1c); for instance, the Penn Treebank [13] has notations to mark not only the most
likely correct parse implicitly opposed to incorrect parses, but also to mark other possibly
correct parses involving different phrasal attachments (additional information that almost
all previous work in parsing has ignored). Additionally, one can more fully rank the quality
of the many candidate parses generated for a sentence based on how many constituents
or dependencies each shares with the correct parse ? much more directly and effectively
approaching the metrics on which parser quality is usually assessed. For concreteness, we
use the term label ranking for all of these problems.
Our learning framework decomposes each preference graph into subgraphs, where the
graph decomposition procedure may take a general form and can change as a function
of the instances. Ranking algorithms, especially in multilabel categorization problems, often reduce the ranking task into multiple binary decision problems by enumerating over all
pairs of labels [7, 6, 4]. Such a reduction can easily be accommodated within our framework by decomposing the preference graph into elementary subgraphs, each consisting of a
single edge. Another approach is to compare a highly preferred label (such as the correct or
best parse of a sentence) with less preferred labels. Such approaches can be analyzed within
our framework by defining a graph decomposition procedure that generates a subgraph for
each relevant label and the neighboring labels that it is preferred over. Returning to multilabel settings, this decomposition amounts to a loss that counts the number of relevant
labels which are wrongly ranked below irrelevant ones.
The algorithmic core of this paper is based on boosting-style algorithms for exponential
models [2, 8]. Specifically, the boosting-style updates we employ build upon the construction used in [2] for solving multiclass problems. Our framework employing graph decomposition can also be used in other settings such as element ranking via projections [3, 11].
Furthermore, settings in which a semi-metric is defined over the label-set can also be reduced to the problem of label ranking, such as the parse ordering case mentioned above or
when the labels are arranged in a hierarchical structure. We employ such a reduction in the
category ranking experiments described in Sec. 4.
The paper is organized as follows: a formal description of our setting is given in Sec. 2. In
Sec. 3 we present an algorithm for learning label ranking functions. We demonstrate the
merits of our approach on the task of category-ranking in Sec. 4 and conclude in Sec. 5.
2
Problem Setting
Let X be an instance domain and let Y be a set of labels, possibly of infinite cardinality. A
label ranking for an instance x ? X is a total order over Y, where y y 0 implies that y
is preferred over y 0 as a label for x. A label ranking function f : X ? Y ? R induces a
label ranking for x ? X by y y 0 ?? f (x, y) > f (x, y 0 ). Overloading our notation,
we denote the label ranking induced by f for x by f (x).
We assume that we are provided with a set of base label-ranking
Pn functions, h1 , . . . , hn ,
and aim to learn a linear combination of the form f (x, y) =
j=1 ?j hj (x, y). We are
also provided with a training set S = {(xi , Gi )}m
where
every
example is comprised
i=1
of an instance xi ? X and a preference graph Gi . As defined in the previous section,
a preference graph is a directed graph G = (V, E), for which the set of vertices V is
defined to be the set of labels Y and E is some finite set of directed edges. Every edge
in a directed graph e ? E is associated with an initial vertex, init(e) ? V , and a terminal
vertex, term(e) ? V . The existence of a directed edge between two labels in a preference
graph indicates that init(e) is preferred over term(e) and should be ranked higher. We
require preference graphs to be simple, namely to have no more than a single edge between
any pair of vertices and to not contain any self-loops. However, we impose no additional
constraints on the supervision, namely, the set of edges in a preference graph may be sparse
and may even include cycles. This form of supervision was chosen for its generality and
flexibility. If Y is very large (possibly infinite), it would be unreasonable to require that the
training data contain a complete total order over Y for every instance.
Informally, our goal is for the label ranking induced by f to be as consistent as possible with
all of the preference graphs given in S. We say that f (xi ) disagrees with
a preference graph
Gi = (Vi , Ei ) if there exists an edge e ? Ei for which f xi , init(e) ? f xi , term(e) .
Formally, we define a function ? that indicates when such a disagreement occurs
1 if ?e ? E s.t. f x, init(e) ? f x, term(e)
?(f (x), G) =
0 otherwise .
A simple measure of empirical ranking accuracy immediately follows from the definition
of ?: We define the 0 ? 1 error attained by a ranking function f on a training set S to be
the number of training examples for which f (xi ) disagrees with Gi , namely,
?0?1 (f, S) =
m
X
?(f (xi ), Gi ) .
i=1
The 0 ? 1 error may be natural for certain ranking problems, however in general it is a
rather crude measure of ranking inaccuracy, as it is invariant to the exact number of edges
in Gi with which f (xi ) disagrees. Many ranking problems require a more refined notion of
ranking accuracy. Thus, we define the disagreement error attained by f (xi ) with respect
to Gi to be the fraction of edges in Ei with which f (xi ) disagrees. The disagreement
error attained on the entire training set is the sum of disagreement errors over all training
examples. Formally, we define the disagreement error attained on S as
m
X
e ? Ei s.t. f x, init(e) ? f x, term(e)
?dis (f, S) =
.
Ei
i=1
Both the 0 ? 1 error and the disagreement error are reasonable measures of ranking inaccuracy. It turns out that both are instances of a more general notion of ranking error of which
additional meaningful instances exist. The definition of this generalized error is slightly
more involved but enables us to present a unified account of different measures of error.
The missing ingredient needed to define the generalized error is a graph decomposition
procedure A that we assume is given together with the training data. A takes as its input
3
2
1
5
A1
7??
4
3
5
A2
7??
4
3
1
5
A3
7??
4
1
4
5
1
2
2
1
2
3
3
5
5
?dis =
2
2
1
1
3
1
3
1
2
4
5
1
5
2
3
2
3
5
2
3
1
1
4
3
8
4
5
5
2
4
5
1
3
?Dom =
2
4
?dom =
3
5
5
Figure 2: Applying different graph decomposition procedures induces different error functions: A1 induces ?dis , A2 induces ?Dom and A3 induces ?dom . The errors above are with
respect to the order 1 2 3 4 5. Dashed edges without arrowheads disagree with
this total order, and the errors are the fraction of subgraphs that contain disagreeing edges.
a preference graph Gi and returns a set of si subgraphs of Gi , denoted {Gi,1 , . . . , Gi,si },
where Gi,k = (Vi , Ei,k ). Each subgraph Gi,k is itself a preference graph and therefore
?(f (xi ), Gi,k ) is well defined. We now define the generalized error attained by f (xi ) with
respect to Gi as the fraction of subgraphs in A(Gi ) with which f (xi ) disagrees. The
generalized error attained on S is the sum of generalized errors over all training instances.
Formally, the generalized ranking error is defined as
?gen (f, S, A) =
si
m
X
1 X
?(f (xi ), Gi,k ) where {Gi,1 , . . . , Gi,si } = A(Gi ) .
s
i=1 i
(1)
k=1
Previously used losses for label ranking are special cases of the generalized error and are
derived by choosing an appropriate decomposition procedure A. For instance, when A is
defined to be the identity transformation on graphs (A(G) = {G}), then the generalized
ranking error is reduced to the 0 ? 1 error. Alternatively, for a graph G with s edges, we
can define A to return s different subgraphs of G, each consisting of a single edge from G
(Fig. 2 top) and the generalized ranking error reduces to the disagreement error.
An additional meaningful measure of error is the domination error. A vertex is said to
dominate the set of neighboring vertices that are connected to its outgoing edges. We
would like every vertex in the preference graph to be ranked above all of its dominated
neighbors. The domination error attained by f (xi ) with respect to Gi is the fraction of
vertices with outgoing edges which are not ranked above all of their dominated neighbors.
Formally, let A be the procedure that takes a preference graph G = (V, E) and returns a
subgraph for each vertex with outgoing edges, each such subgraph consisting of a dominating vertex, its dominated neighbors and edges between them (Fig. 2 middle). Now define
?Dom (f, S) = ?gen (f, S, A) . Minimizing the domination error is useful for solving multilabel classification problems. In these problems Y is of finite cardinality and every instance
xi is associated with a set of correct labels Yi ? Y. In order to reduce this problem to a
ranking problem, we construct preference graphs Gi = (Y, Ei ), where Ei contains edges
from every vertex in Yi to every vertex in Y \ Yi . In this case, the domination loss simply
counts the number of labels in Yi that are not ranked above all of the labels in Y \ Yi .
A final interesting measure of error is the dominated error, denoted ?dom . The dominated
error is proportional to the number of labels with incoming edges that are not ranked below
all of the labels that dominate them. Its graph decomposition procedure is depicted at the
bottom of Fig. 2. Additional instances of the generalized ranking error exist, and can be
tailored to fit most ranking problems. In the next section we set aside the specifics of the
decomposition procedure and derive a minimization procedure for the generalized error.
I NPUT: training data S = {(xi , Gi )}m
i=1 s.t. xi ? X and Gi is a preference graph,
a decomposition procedure A and a set of base ranking functions {h1 , . . . , hn }.
I NITIALIZE : ?1 = (0, 0, . . . , 0)
?i,e,j = hj xi , term(e) ? hj xi , init(e)
[1 ? i ? m, e ? Ei , 1 ? j ? n]
P
? = maxi,e j |?i,e,j |
I TERATE : For t = 1, 2, . . .
X
exp (?t ? ? i,e )
P
[1 ? i ? m, e ? Ei ]
qt,i,e =
1 + e0 ?Ei,k exp (?t ? ? i,e0 )
k:e?Ei,k
X qt,i,e ?i,e,j
X ?qt,i,e ?i,e,j
+
?
[1 ? j ? n]
Wt,j
=
Wt,j
=
s
s
i
i
i,e:?i,e,j >0
i,e:?i,e,j <0
!
+
Wt,j
1
[1 ? j ? n]
?t,j = ln
?
2
Wt,j
?t+1 = ?t ?
?t
?
Figure 3: A boosting based algorithm for generalized label ranking.
3
Minimizing the Generalized Ranking Error
Our goal is to minimize the generalized error for a given training set S and graph decomposition procedure A. This task generalizes standard classification problems which are known
to be NP-complete. Hence we do not attempt to minimize the error directly but rather minimize a smooth, strictly convex, upper bound on ?gen . The disagreement of f (xi ) and a
preference graph Gi,k = (Vi,k , Ei,k ) can be upper bounded by
?
?
X
?(f, xi , Gi,k ) ? log2 ?1 +
exp f xi , term(e) ? f xi , init(e) ?
e?Ei,k
Denoting the right hand side of the above as L(f (xi ), Gi,k ), we define the loss attained by
f on the entire training set S to be
si
m
X
1 X
L(f (xi ), Gi,k ) where Gi,1 , . . . , Gi,si = A(Gi ) .
L(f, S, A) =
s
i=1 i
k=1
From the definition of the generalized error in Eq. (1), we conclude the upper bound
?gen (f, S, A) ? L(f, S, A) . A boosting-based algorithm that globally minimizes the
loss is given in Fig. 3. On every iteration, a weight qt,i,e is calculated for every edge in
the training data, and the algorithm focuses on satisfying each edge with proportion to its
weight. This set of weights plays the role of the distribution vector common in boosting
algorithms for classification. The following theorem bounds the decrease in loss on every
iteration of the algorithm by a non-negative auxiliary function.
Theorem 1 Let S = {(xi , Gi )}m
i=1 be a training set such that every xi ? X and every
Gi is a preference graph. Let A be a graph decomposition procedure that defines for each
preference graph Gi a set of subgraphs {Gi,1 , . . . , Gi,si } = A(Gi ). Denote
P by ft the
ranking function obtained at iteration t of the algorithm given in Fig. 3 (ft = j ?t,j hj ).
Using the notation defined in Fig. 3, the decrease in loss on iteration t is bounded by
2
n q
q
1X
+
?
L(ft , S, A) ? L(ft+1 , S, A) ?
Wt,j ? Wt,j
.
? j=1
Proof Define ?t,i,k to be the difference between the loss attained by ft and the loss attained
by ft+1 on
P (xi , Gi,k ), that is ?t,i,k = L(ft (xi ), Gi,k ) ? L(ft+1 (xi ), Gi,k ), and define
?t,i,k = e?Ei,k exp(?t ? ? i,e ). We can now rewrite L(ft (xi ), Gi,k ) as log 1 + ?t,i,k .
Using the inequality ? log(1 ? a) ? a (which holds when log(1 ? a) is defined), we get
?t,i,k ? ?t+1,i,k
?t,i,k = log 1 + ?t,i,k ? log 1 + ?t+1,i,k = ? log 1 ?
1 + ?t,i,k
X exp(?t ? ? i,e ) ? exp(?t+1 ? ? i,e )
?t,i,k ? ?t+1,i,k
P
.
(3)
?
=
0
1 + ?t,i,k
1 +
e0 ?Ei,k exp(?t ? ? i,e )
e?Ei,k
The algorithm sets ?t+1 = ?t ? (1/?)?t and therefore exp(?t+1 ? ? i,e ) in Eq. (3) can be
replaced by exp(?t ? ? i,e ) exp(?(1/?)?t ? ? i,e ), yielding:
!
X
1
exp(?t ? ? i,e )
P
1 ? exp ? ?t ? ? i,e
?t,i,k ?
.
0
1 +
?
e0 ?Ei,k exp(?t ? ? i,e )
e?Ei,k
Summing both sides of the above over the subgraphs in A(Gi ), and plugging in qt,i,e ,
?
?
si
X
X
X
exp(?t ? ? i,e )
?
? 1 ? exp ? 1 ?t ? ? i,e
P
?t,i,k ?
1 +
?
e0 ?Ei,k exp(?t ? ? i,e0 )
e?Ei k:e?Ei,k
k=1
X
1
.
(4)
=
qt,i,e 1 ? exp ? ?t ? ? i,e
?
e?Ei
We now rewrite (1/?)?t ? ? i,e in more convenient form
n
n
X
X
1
1
? ?t ? ? i,e = ?
?t,j ?i,e,j =
(|?i,e,j |/?) (?sign(?i,e,j )?t,j ) .
?
?
j=1
j=1
(5)
The rationale behind this rewriting is that we now think of (|?i,e,1 |/?) , . . . , (|?i,e,n |/?) as
coefficients in a subconvex combination of (?sign(?i,e,1 )?
Pt,1 ) , . . . , (?sign(?i,e,n )?t,n ),
since ?j (|?i,e,j |/?) ? 0 and from the definition of ?, j (|?i,e,1 |/?) ? 1. Plugging
Eq. (5) into Eq. (4) and using the concavity of the function 1 ? exp(?) in Eq. (4), we obtain
?
?
??
si
n
X
X
X
?t,i,k ?
qt,i,e ?1 ? exp ?
(|?i,e,j |/?) (?sign(?i,e,j )?t,j )??
j=1
e?Ei
k=1
?
X
n
X
e?Ei,k j=1
qt,i,e (|?i,e,j |/?) 1 ? exp (?sign(?i,e,j )?t,j ) .
Finally, we sum both sides of the above over all of S and plug in W + , W ? and ? to get
si
n X
X
L(ft , S, A) ? L(ft+1 , S, A) =
?t,i,k
i=1 k=1
?
=
n m
1 X X X qt,i,e |?i,e,j |
1 ? exp (?sign(?i,e,j )?t,j )
? j=1 i=1
si
e?Ei
q
q
?
?
?
?
??
?
+
n
W
W
X
t,j
t,j
1
+ ?
? ?
?Wt,j
? + Wt,j
??
1? q
1? q
+
? j=1
W
W?
t,j
=
2
n q
q
1 X
+
?
Wt,j ? Wt,j
.
? j=1
t,j
Thm. 1 proves that the losses attained on each iteration form a monotonically nonincreasing sequence of positive numbers, that must therefore converge. However, we are
interested in proving a stronger claim, namely that the vector sequence (?t )?
t=1 converges
to a globally optimal weight-vector ?? . Since the loss is a convex function, it suffices to
show that the vector sequence converges to a stationary point of the loss. It is easily verified
that the non-negative auxiliary function which bounds the decrease in loss equals zero only
?
at stationary points of the loss. This fact implies that (?t )?
t=1 indeed converges to ? if
the set of all feasible values for ? is compact and the loss has a unique global minimum.
Compactness of the feasible set and uniqueness of the optimum can be explicitly enforced
by adding a form of natural regularization to the boosting algorithm. The specifics of this
technique exceed the scope of this paper and are discussed in [5]. In all, the boosting
algorithm of Fig. 3 converges to the globally optimal weight-vector ?? .
4
Experiments
To demonstrate our framework, we chose to
learn a category ranking problem on a subset of the Reuters Corpus, Vol. 1 [14]. The
full Reuters corpus is comprised of approximately 800, 000 textual news articles, collected over a period of 12 months in 1996?
1997. Most of the articles are labeled by
one or more categories. For the purpose of
these experiments, we limited ourselves to
the subset of articles collected during January 1997: approximately 66, 000 articles
labeled by 103 different categories.
0?1
dis
Dom
dom
?0?1
0.63
0.73
0.59
0.59
?dis
0.068
0.063
0.049
0.067
?Dom
0.42
0.51
0.35
0.41
?dom
0.12
0.14
0.10
0.10
Figure 4: The test error averaged over 5fold cross validation. The rows correspond
to different optimization problems: minimizing ?0?1 , ?dis , ?Dom and ?dom . Errors
are measured using all 4 error measures.
An interesting aspect of the Reuters corpus is that the categories are arranged in a hierarchy. The set of possible labels contains both general categories and more specific ones,
where the specific categories refine the general categories. This concept is best explained
with an example: three of the categories in the corpus are Economics, Government Finance and Government Borrowing. It would certainly be correct to categorize an article on
government borrowing as either government finance or economics, however these general
categories are less specific and do not describe the article as well. Furthermore, misclassifying such an article as government revenue is by far better than misclassifying it as sports.
In summary, the category hierarchy induces a preference over the set of labels. We exploit
this property to generate supervision for the label ranking problem at hand.
Formally, we view every category as a vertex in a rooted tree, where the tree root corresponds to a general abstract category that is relevant to all of the articles in the corpus and
every category is a specific instance of its parent in the tree. The labels associated with an
article constitute a set of paths from the tree root to a set of leaves. The original corpus is
somewhat inconsistent in that not all paths end in a leaf, but rather end in some inner vertex.
To fix this inconsistency, we added a dummy child vertex to every inner vertex and diverted
all paths that originally end in this inner vertex to its new child. Our learning problem then
becomes the problem of ranking leaves. The severity of wrongly categorizing an article
by a leaf is proportional to the graph distance between this leaf and the closest correct leaf
given in the corpus. The preference graph that encodes this preference is a multi-layer
graph where the top layer contains all of the correct labels, the second layer contains all of
their sibling vertices in the tree and so on. Every vertex in the multi-layer preference graph
has outgoing edges to all vertices in lower layers, but there are no edges between vertices
in the same layer. For practical purposes, we conducted experiments using only 3-layer
preference graphs generated by collapsing all of the layers below 3 to a single layer.
All of the experiments were carried out using 5-fold cross validation. The word counts
for each article were used to construct base ranking functions in the following way: for
every word w and every category y, let w(xi ) denote the number of appearances of w in
the article xi . Then, define
log(w(xi )) + 1 if w(xi ) > 0 and yi = y
hw,y (xi , yi ) =
(6)
0
otherwise .
For each training set, we first applied a heuristic feature selection method common in boosting applications [10] to select some 3200 informative words. These words then define
103 ? 3200 base ranking functions as shown in Eq. (6). Next, we ran our learning algorithm
using each of the 4 graph decomposition procedures discussed above: zero-one, disagreement, domination and dominated. After learning each problem, we calculated all four error
measures on the test data. The results are presented in Fig. 4. Two points are worth noting.
First, these results are not comparable with previous results for multilabel problems using
this corpus, since label ranking is a more difficult task. For instance, an average preference
graph in the test data has 820 edges, and the error for such a graph equals zero only if every
single edge agrees with the ranking function. Second, the experiments clearly indicate that
the results obtained by minimizing the domination loss are better than the other ranking
losses, no matter what error is used for evaluation. In particular, employing the domination
loss yields significantly better results than using the disagreement loss which has been the
commonly used decomposition method in categorization problems [7, 10, 6, 4].
5
Summary
We presented a general framework for label ranking problems by means of preference
graphs and the graph decomposition procedure. This framework was shown to generalize
other decision problems, most notably multilabel categorization. We then described and
analyzed a boosting algorithm that works with any choice of graph decomposition. We
are currently exporting the approach to learning in inner product spaces, where different
graph decomposition procedures result in different bindings of slack variables. Another
interesting question is whether the graph decomposition approach can be combined with
probabilistic models for orderings [9] to achieve algorithmic efficiency.
References
[1] M. Collins and N. Duffy. New ranking algorithms for parsing and tagging: Kernels over discrete
structures, and the voted perceptron. In 30th Annual Meeting of the ACL, 2002.
[2] M. Collins, R.E. Schapire, and Y. Singer. Logistic regression, AdaBoost and Bregman distances. Machine Learning, 47(2/3):253?285, 2002.
[3] K. Crammer and Y. Singer. Pranking with ranking. NIPS 14, 2001.
[4] K. Crammer and Y. Singer. A new family of online algorithms for category ranking. Jornal of
Machine Learning Research, 3:1025?1058, 2003.
[5] O. Dekel, S. Shalev-Shwartz, and Y. Singer. Smooth epsilon-insensitive regression by loss
symmetrization. COLT 16, 2003.
[6] A. Elisseeff and J. Weston. A kernel method for multi-labeled classification. NIPS 14, 2001.
[7] Y. Freund, R. Iyer, R. E.Schapire, and Y. Singer. An efficient boosting algorithm for combining
preferences. In Machine Learning: Proc. of the Fifteenth International Conference, 1998.
[8] G. Lebanon and J. Lafferty. Boosting and ML for exponential models. NIPS 14, 2001.
[9] G. Lebanon and J. Lafferty. Conditional models on the ranking poset. NIPS 15, 2002.
[10] R. E. Schapire and Y. Singer. BoosTexter: A boosting-based system for text categorization.
Machine Learning, 32(2/3), 2000.
[11] A. Shashua and A. Levin. Ranking with large margin principle. NIPS 15, 2002.
[12] K. Toutanova and C. D. Manning. Feature selection for a rich HPSG grammar using decision
trees. In Proceedings of the Sixth Conference on Natural Language Learning (CoNLL), 2002.
[13] The Penn Treebank Project. http://www.cis.upenn.edu/?treebank/.
[14] Reuters Corpus Vol. 1. http://about.reuters.com/researchandstandards/corpus/.
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1,686 | 2,532 | Boosting versus Covering
Kohei Hatano?
Tokyo Institute of Technology
[email protected]
Manfred K. Warmuth
UC Santa Cruz
[email protected]
Abstract
We investigate improvements of AdaBoost that can exploit the fact
that the weak hypotheses are one-sided, i.e. either all its positive
(or negative) predictions are correct. In particular, for any set
of m labeled examples consistent with a disjunction of k literals
(which are one-sided in this case), AdaBoost constructs a consistent
hypothesis by using O(k 2 log m) iterations. On the other hand,
a greedy set covering algorithm ?nds a consistent hypothesis of
size O(k log m). Our primary question is whether there is a simple
boosting algorithm that performs as well as the greedy set covering.
We ?rst show that InfoBoost, a modi?cation of AdaBoost proposed by Aslam for a di?erent purpose, does perform as well as
the greedy set covering algorithm. We then show that AdaBoost
requires ?(k 2 log m) iterations for learning k-literal disjunctions.
We achieve this with an adversary construction and as well as in
simple experiments based on arti?cial data. Further we give a variant called SemiBoost that can handle the degenerate case when the
given examples all have the same label. We conclude by showing
that SemiBoost can be used to produce small conjunctions as well.
1
Introduction
The boosting method has become a powerful paradigm of machine learning. In this
method a highly accurate hypothesis is built by combining many ?weak? hypotheses.
AdaBoost [FS97, SS99] is the most common boosting algorithm. The protocol is as
follows. We start with m labeled examples labeled with ?1. AdaBoost maintains
a distribution over the examples. At each iteration t, the algorithm receives a ?1
valued weak hypothesis ht whose error (weighted by the current distribution on the
examples) is slightly smaller than 12 . It then updates its distribution so that after
the update, the hypothesis ht has weighted error exactly 12 . The ?nal hypothesis is
a linear combination of the received weak hypotheses and it stops when this ?nal
hypothesis is consistent with all examples.
It is well known [SS99] that if each weak hypothesis has weighted error
at most
?
1
?
,
then
the
upper
bound
on
the
training
error
reduces
by
a
factor
of
1 ? ?2
2
2
?
This research was done while K. Hatano was visiting UC Santa Cruz under the EAP
exchange program.
and after O( ?12 log m) iterations, the ?nal hypothesis is consistent with all examples.
Also, it has been shown that if the ?nal hypotheses are restricted to (unweighted)
majority votes of weak hypotheses [Fre95], then this upper bound on the number
of iterations cannot be improved by more than a constant factor.
However, if there always is a positively one-sided weak hypothesis (i.e. its positive
predictions are always correct) that has error1 at most 12 ? ?2 , then a set cover algorithm can be used to reduce the training error by a factor2 of 1 ? ? and O( ?1 log m)
weak hypotheses su?ce to form a consistent hypothesis [Nat91]. In this paper
we show that the improved factor is also achieved by InfoBoost, a modi?cation of
AdaBoost developed by Aslam [Asl00] based on a di?erent motivation.
In particular, consider the problem of ?nding a consistent hypothesis for m examples
labeled by a k literal disjunction. Assume we use the literals as the pool of weak
hypotheses and always choose as the weak hypothesis a literal that is consistent
with all negative examples. Then it can be shown that, for any distribution D on
the examples, there exists a literal (or a constant hypothesis) h with weighted error
1
at most 12 ? 4k
(See e.g. [MG92]). Therefore, the upper bound on the training error
of AdaBoost reduces by a factor of 1 ? 4k12 and O(k 2 log m) iterations su?ce.
However, a trivial greedy set covering algorithm, that follows a strikingly similar
protocol as the boosting algorithms, ?nds a consistent disjunction with O(k log m)
literals. We show that InfoBoost mimics the set cover algorithm in this case (and
attains the improved factor of 1 ? k1 ).
We ?rst explain the InfoBoost algorithm in terms of constraints on the updated
distribution. We then show that ?(k 2 log m) iterations are really required by AdaBoost using both an explicit construction (which requires some assumptions) and
arti?cial experiments. The di?erences are quite large: For m = 10, 000 random
examples and a disjunction of size k = 60, AdaBoost requires 2400 iterations (on
the average), whereas Covering and InfoBoost require 60 iterations. We then show
that InfoBoost has the improved reduction factor if the weak hypotheses happen
to be one-sided. Finally we give a modi?ed version of AdaBoost that exploits the
one-sidedness of the weak hypotheses and avoids some technical problems that can
occur with InfoBoost. We also discuss how this algorithm can be used to construct
small conjunctions.
2
Minimizing relative entropy subject to constraints
Assume we are given a set of m examples (x1 , y1 ), . . . , (xm , ym ). The instances xi
are in some domain X and the labels yi are in {?1, 1}. The boosting algorithms
maintain a distribution Dt over the examples. The initial distribution is D1 and is
typically uniform. At the t-th iteration, the algorithm chooses a weak3 hypothesis
ht : X ? {?1, 1} and then updates its distribution. The most popular boosting
algorithm does this as follows:
AdaBoost: Dt+1 (i) =
1
Dt (i) exp{?yi ht (xi )?t }
,
Zt
This assumes equal weight on both types of examples.
Wipe out the weights of positive examples that are correctly classified and re-balance
both types of examples.
3
For the sake of simplicity we focus on the case when the range of the labels and the
weak hypotheses is ?1 valued. Many parts of this paper generalize to the range [?1, 1]
[SS99, Asl00].
2
Here Zt is a normalization constant and the coe?cient ?t depends on the error t
t
at iteration t: ?t = 12 ln 1?
and t = PrDt [ht (xi ) = yi ]. The ?nal hypothesis is
t
given by the sign of the following linear combination of the chosen weak hypotheses:
T
H(x) = t=1 ?t ht (x). Following [KW99, Laf99], we motivate the updates on the
distributions of boosting algorithms as a constraint minimization of the relative
entropy between the new and old distributions:
AdaBoost: Dt+1 = argminD?[0,1]m , i D(i)=1 ?(D, Dt ), s.t. Pr[ht (xi ) = yi ] =
Here the relative entropy is de?ned as ?(D, D ) =
the updated distribution is constraint to half.
i
D
1
.
2
D(i)
D(i) ln D
(i) and error w.r.t.
y i \ ht
The constraint can be easily understood using the table of
+1
Figure 1. There are two types of misclassi?ed examples:
?1
false positive (weight c) and false negative (weight b). The
AdaBoost constraint means b + c = 12 w.r.t. the updated Figure 1:
of examples.
distribution Dt+1 .
?1
b
d
+1
a
c
Four types
The second boosting algorithm we discuss in this paper has the following update:
InfoBoost: Dt+1 (i) =
Dt (i) exp{?yi ht (xi )?t [ht (xi )]}
,
Zt
where ?t [?1] = 12 ln 1?[?1]
t [?1] , t [?1] = PrDt [ht (xi ) = yi |ht (xi ) = ?1] and Zt is
the normalization factor. The ?nal hypothesis is given by the sign of H(x) =
T
t=1 ?t [ht (x)] ht (x).
In the original paper [Asl00], the InfoBoost update was motivated by seeking a
distribution Dt+1 for which the error of ht is half and yi and ht (xi ) have mutual
information zero. Here we motivate InfoBoost as a minimization of the same relative
entropy subject to the AdaBoost constraint b + c = 12 and a second simultaneously
enforced constraint a + b = 12 . Note that the second constraint is the AdaBoost
constraint w.r.t. the constant hypothesis 1. A natural question is why not just do
two steps of AdaBoost at each iteration t: One for ht and and then, sequentially,
one for 1. We call the latter algorithm AdaBoost with Bias, since the constant
hypothesis introduces a bias into the ?nal hypothesis. See Figure 2 for an example
of the di?erent updates.
Dt :
Dt+1 : InfoB.
y i \ ht
+1
?1
+1
yi \ ht
+1
?1
+1
0
0
2
5
0
?1
2
5
1
5
Dt+1 : AdaB.
?1
1
2
1
2
Dt+1 : AdaB.w.Bias
yi \ ht
+1
?1
+1
yi \ ht
+1
?1
+1
1
3
0
1
5
0
?1
1
2
1
6
?1
3
10
1
2
Figure 2: Updating based on a positively one-sided hypothesis ht (weight c is 0):
The updated distributions on the four types of examples are quite di?erent.
We will show in the next section that in the case of learning disjunctions, AdaBoost
with Bias (and plain AdaBoost) can require many more iterations than InfoBoost
and the trivial covering algorithm. This is surprising because the AdaBoost with
Bias and InfoBoost seem so similar to each other (simultaneous versus sequential
enforcement of the same constraints). A natural extension would be to constrain
the errors of all past hypotheses to half which is the Totally Corrective Algorithm
of [KW99]. However this can lead to subtle convergence problems (See discussion
in [RW02]).
3
Lower bounds of AdaBoost for Learning k disjunctions
So far we did not specify how the weak hypothesis ht is chosen at iteration t. We
assume there is a pool H of weak hypotheses and distinguish two methods:
Greedy: Choose a ht ? H for which the normalization factor Zt in the update of
the algorithm is minimized.
1
Minimal: Choose ht with error smaller than a given
threshold 2 ? ?.
The greedy method is motivated by the fact that t Zt upper bounds the training
error of the ?nal hypothesis ([SS99, Asl00]) and this method greedily minimizes this
upper bound. Note that the Zt factors are di?erent for AdaBoost and InfoBoost.
In our lower bounds on the number of iterations the example set is always consistent with a k-literal monotone disjunction over N variables. More precisely the
instances xi are in {?1}N and the label yi is xi,1 ? xi,2 ? . . . ? xi,k . The pool of weak
learners consists of the N literals Xj , where Xj (xi ) = xij . For the greedy method
we show that on random data sets InfoBoost and the covering algorithm use drastically fewer iterations than AdaBoost with Bias.
We chose 10, 000 examples as follows: The ?rst
k bits of each example are chosen independently
at random so that the probability of label +1 is
half (i.e. the probability of +1 for each of the
?rst k bits is 1 ? 2?1/k ); the remaining N ? k irrelevant bits of each example are chosen +1 with
probability half. Figure 3 shows the number of
iterations as function of the size of disjunction k
(averaged over 20 runs) of AdaBoost with Bias
until consistency is reached on all 10, 000 exam- Figure 3: Average # of steps
ples. The number of iteration in this very simple of AdaBoost with Bias for k =
setting grows quadratically with k. If the num- 10, 20, 30, 40, 50, 60.
ber of iterations is divided by k 2 then the resulting curve is larger than a constant.
In contrast the number of iterations of the greedy covering algorithm and InfoBoost
is provably linear in k: For k = 60 and m = 10, 000, the former require 60 iterations
on the average, whereas AdaBoost with Bias with the greedy choice of the weak
hypothesis requires 1200 even though it never chooses irrelevant variables as weak
learners (Plain AdaBoost requires twice as many iterations).
The above construction is not theoretical. However we now give an explicit construction for the minimal method of choosing the weak hypothesis for which the
number of iterations of greedy covering and InfoBoost grow linearly in k and the
number of iterations of AdaBoost with Bias is quadratic in k.
For any dimension N we de?ne an example set which is the rows of the following
(N + 1) ? N dimensional matrix x: All entries on the main diagonal and above are
+1 and the remaining entries ?1. In particular, the last row is all ?1 (See Figure
4). The i-th instance xi is the i-th row of this matrix and the ?rst N examples
(rows) are labeled +1 and the label of the last row yN +1 is ?1.
Clearly the literal XN is consistent with the labels and thus always has error 0
w.r.t. any distribution on the examples. But note that the disjunction of the last k
literals is also consistent (for any k). We will construct a distribution on the rows
that gives high probability to the early rows (See Figure 4) and allows the following
?minimal? choice of weak hypotheses: At iteration t, AdaBoost with Bias is given
1
1
the weak hypothesis Xt . This weak hypothesis will have error 12 ? 2k
(? = 2k
) w.r.t.
current distribution of the examples.
Contrary to our construction, the initial distribution for boosting applications is
typically uniform. However, using padding this can be avoided but makes the construction more complicated. For any precision parameter ? (0, 1) and disjunction
size k, we de?ne the dimension
?
?
1
1
?
ln
?
ln
1
?
1
k
k2
2?
k ?
N := ?
ln
?
+
1
?
?
?
2
2? 2
? ln 1 ? 2
?
k(k+1)
The initial distribution D1 is de?ned as
? 1
,
?
? 2k
?
?
1
1
1
?
1?
k(k+1)
k
D1 (xt ) :=
? 1 ? N ?1 D(x ),
?
t
?
t=1
? 21
,
2
2
k(k+1)
t?2
for t = 1
,
for 2 ? t ? N ? 1
.
for t = N
for t=N + 1.
The example xN has the lowest probability w.r.t. D1 (See Figure 4). However one
can show that its probability is at least ?.
D1 (xi )
? +
? ?
? ?
? ?
?
? ?
?
?
?
xi,j
+
+
?
?
?
?
?
+
+
+
?
?
?
?
+
+
+
+
+
?
?
+
+
+
+
+
+
?
y
? ? +i
? ? +
? ? +
? ? +
? ?
? ? +
? ?
+
?
?
?
?
?
?
?
?
Figure 4: The examples (rows of the matrix), the labels, and the distribution D1 .
Also
probability D1 (x?t ) of the ?rst t examples is
for t ? N ? 1, the
t?1
1
1
2
1 ? k(k+1)
. AdaBoost with Bias does two update steps at
2 1? 1? k
iteration t (constrain the error of ht to half and then sequentially the error of 1 to
half.)
t }
t (i) = Dt (i) exp{?yi ht (xi )?t } and Dt+1 = Dt (i) exp{?yi ?
.
D
Zt
Zt
1
2
t
ln 1??
t =
?t and ?
1
2
(x
D
)
ln D t(x ?N ) . The ?nal
t
N +1
T
hypothesis is the sign of the following linear combination: H(x) = t=1 ?t ht (x) +
T
t .
t=1 ?
The Z?s are normalization factors, ?t =
Proposition 1. For AdaBoost with Bias and t ? N , PrDt [Xt (xi ) = yi ] =
1
2
?
1
2k .
Proof. (Outline) Since each literal Xt is one-sided, Xt classi?es the negative example
xN +1 correctly. Since PrDt [Xt (xi ) = yi ] = Dt (xN +1 )+Dt (x?t ) and Dt (xN +1 ) = 12
1
it su?ces to show that Dt (x?t ) = 2k
for t ? N . The proof is by induction on t.
For t = 1, the statement follows from the de?nition of D1 . Now assume that the
statement holds for any t < t. Then we have
Dt (x?t ) = Dt (x?t?1 ) + Dt (xt ) = Dt?1 (x?t?1 )
e??t?1 e? t?1
+ Dt (xt ).
t?1
Zt?1 Z
(1)
Note that the example xt is not covered by any previous hypotheses X1 , . . . , Xt?1 ,
and thus we have
Dt (xt ) = D1 (xt )
t?1
e?j e? j
.
j
Zj Z
j=1
(2)
1
Using the inductive assumption that PrDt [Xt (xi ) = yi ] = 12 ? 2k
, for t < t, one
1
k+1
1
1
can show that ?t = 2 ln k?1 , Zt = k (k ? 1)(k + 1), Dt (x?N ) = 12 + 2(k+1)
,
1
1
1
k+2
1
Dt (xN +1 ) = ?
, ?
t = ln
, and Zt =
k(k + 2). Substituting
2
2(k+1)
2
k
k+1
these values into the formulae (1) and (2), completes the proof.
Theorem 2. For the described examples set, initial distribution D1 , and minimal choice of weak hypotheses, AdaBoost with Bias needs at least N iterations to
construct a final hypothesis whose error with respect to D1 is below ?.
Proof. Let t be any integer smaller than N . At the end of the iteration t, the
examples xt+1 , . . . , xN are not correctly classi?ed by the past weak hypotheses
X1 , . . . , Xt . In particular, the ?nal linear combination evaluated at xN is
H(xN ) =
t
t
t
t
t k+1 t k+2
+ ln
< 0.
?j Xj (xN )+
?
j = ?
?j +
?
j = ? ln
2
k?1 2
k
j=1
j=1
j=1
j=1
Thus sign(H(xN )) = ?1 and the ?nal hypothesis has error at least D1 (xN ) ? ?
with respect to D1 .
To show a similar lower bound for plain AdaBoost we use the same example set and
the following sequence of weak hypotheses X1 , 1, X2 , 1, . . . XN , 1. For odd iteration
1
numbers t the above proposition shows the error of the weak hypothesis is 12 ? 2k
and
1
for even iteration numbers one can show that the hypothesis 1 has error 12 ? 2(k+1)
.
4
InfoBoost and SemiBoost for one-sided weak hypotheses
Aslam proved the following upper bound on the training error[Asl00] of InfoBoost:
Theorem 3. The training error of the final hypothesis
produced by InfoBoost is
T
Z
,
where
Z
=
Pr
[h
(x
)
=
+1]
1 ? ?t [+1]2 + PrDt [ht (xi ) =
bounded
by
t
Dt t i
t=1 t
?1] 1 ? ?t [?1]2 and edge4 ?t [?1] = 1 ? 2?t [?1].
Let ?t = 1 ? 2?t . If ?t [+1] = ?t [?1] = ?t , then Zt = 1 ? ?t2 , as?for AdaBoost.
However, if ht is one-sided, InfoBoost gives the improved factor of 1 ? ?t :
?
Corollary 4. For t ? 2, if ht is one-sided w.r.t. Dt , then Zt = 1 ? ?t .
4
The edge ? and error are related as follows: ? = 1?2 and =
1
1
? 2?
;
2
=
1
2
? ? = 0.
Proof. Wlog. assume ht is always correct when it predicts +1. Then ?t [+1] = 1 and
the ?rst summand in the expression for Zt given in the above theorem disappears.
Recall that InfoBoost maintains the distribution Dt over examples so that PrDt [yi =
+1] = 12 for t ? 2. So the second summand becomes
2 Pr[ht (xi ) = ?1, yi = +1] Pr[ht (xi ) = ?1, yi = ?1]
Dt
Dt
= 2 Pr[yi = +1] Pr[yi = ?1] Pr[ht (xi ) = ?1|yi = +1] Pr[ht (xi ) = ?1|yi = ?1]
Dt
Dt
Dt
Dt
= Pr[ht (xi ) = ?1|yi = +1].
Dt
By the de?nition of ?t , we have
1 ? ?t = 2PrDt [ht (xi ) = yi ]
= 2PrDt [ht (xi ) = ?1, yi = +1] (because of one-sidedness of ht )
= 2PrDt [yi = +1] Pr[ht (xi ) = ?1|yi = +1]
Dt
2
= PrDt [ht (xi ) = ?1|yi = +1] (because PrDt [yi = +1] = 12 )
This corollary implies that if a one-sided hypothesis is chosen at each iteration,
then InfoBoost constructs a ?nal hypothesis consistent with all m examples within
2
? ln m iterations. When the considered weak hypotheses are positively one-sided,
then the trivial greedy covering algorithm (which simply chooses the set that covers
the most uncovered positive examples), achieves the improved factor of 1 ? ?, which
means at most ?1 ln m iterations. By a careful analysis (not included), one can show
that the factor for InfoBoost can be improved to 1 ? ?, if all weak hypotheses are
one-sided. So in this case InfoBoost indeed matches the 1 ? ? factor of the greedy
covering algorithm.
A technical problem arises when InfoBoost is given a set of examples that are
all labeled +1. Then we have ?1 [+1] = ? and ?1 [?1] = ??. This implies
H(x) = ?1 [h1 (xi )]ht (xi ) = ? for any instance xi . Thus InfoBoost terminates in
a single iteration and outputs a hypothesis that predicts +1 for any instance and
InfoBoost cannot be used for constructing a cover.
We propose a natural way to cope with this subtlety. Recall that the ?nal hyT
pothesis of InfoBoost is given by H(x) =
t=1 ?t [ht (x)] ht (x). This doesn?t
seem to be a linear combination of hypotheses from H since the coe?cients vary
with the prediction of weak hypotheses. However observe that ?t [ht (x)] ht (x) =
?
?
?t [+1] h+
= h(x) if h(x) = ?1 and 0 otherwise.
t (x) + ?t [?1] ht (x), where h
+
?
We call h and h the semi hypotheses of h. Note that h+ (x) = h(x)+1
and
2
h(x)?1
?
h (x) =
. So the ?nal hypothesis of InfoBoost and the new algorithm we
2
will de?ne in a moment is a bias plus a linear combination of the the original weak
learners in H.
We propose the following variant of AdaBoost (called Semi-Boost): In each iteration
execute one step of AdaBoost but the chosen weak hypothesis must be a semi
hypothesis of one of the original hypothesis h ? H which has a positive edge.
SemiBoost avoids the outlined technical problem and can handle equally labeled
example sets. Also if all the chosen hypotheses are of the h+ type then the ?nal
hypothesis is a disjunction. If hypotheses are chosen by smallest error (largest
edge), then the greedy covering algorithm is simulated. Analogously, if all the
chosen hypotheses are of the h? type then one can show that the ?nal hypothesis of
SemiBoost is a conjunction. Furthermore, two steps of SemiBoost (with hypothesis
h+ in the ?rst step followed by the sibling hypothesis h? in the second step) are
equivalent to one step of InfoBoost with hypothesis h.
Finally we note that the ?nal hypothesis of InfoBoost (or SemiBoost) is not wellde?ned when it includes both types of one-sided hypotheses, i.e. positive and negative in?nite coe?cients may con?ict each other. We propose two solutions. First,
following [SS99] one can use the modi?ed coe?cients ?[?1] = 12 ln 1??[?1]+?
for
??[?1]+?
small ? > 0. It can be shown that the new Z increases by at most 2?([SS99]).
Second, we allow in?nite coe?cients but interpret the ?nal hypothesis as a version
of a decision list [Riv87]: Whenever more than one semi hypotheses with in?nite
coe?cients are non-zero on the current instance, then the semi hypothesis with the
lowest iteration number determines the label. Once such a consistent decision list
over some set of hypothesis ht and 1 has been found, it is easy the ?nd an alternate
linear combination of the same set of hypotheses (using linear programming) that
maximizes the margin or minimizes the one-norm of the coe?cient vector subject
to consistency.
Conclusion: We showed that AdaBoost can require signi?cantly more iterations
than the simple greedy cover algorithm when the weak hypotheses are one-sided
and gave a variant of AdaBoost that can readily exploit one-sidedness. The open
question is whether the new SemiBoost algorithm gives improved performance on
natural data and can be used for feature selection.
Acknowledgment: This research bene?ted from many discussions with Gunnar
R?
atsch. He encouraged us to analyze AdaBoost with Bias and suggested to write
the ?nal hypothesis of InfoBoost as a linear combination of semi hypotheses. We
also thank anonymous referees for helpful comments.
References
[Asl00] J. A. Aslam. Improving algorithms for boosting. In Proc. 13th Annu.
Conference on Comput. Learning Theory, pages 200?207, 2000.
[Fre95] Y. Freund. Boosting a weak learning algorithm by majority. Inform.
Comput., 121(2):256?285, September 1995. Also appeared in COLT90.
[FS97] Y. Freund and R. E. Schapire:. A decision-theoretic generalization of
on-line learning and an application to boosting. J. Comput. Syst. Sci.,
55(1):119?139, 1997.
[KW99] Jyrki Kivinen and Manfred K. Warmuth. Boosting as entropy projection.
In Proc. 12th Annu. Conf. on Comput. Learning Theory, pages 134?144.
ACM Press, New York, NY, 1999.
[Laf99] J. La?erty. Additive models, boosting, and inference for generalized divergences. In Proc. 12th Annu. Conf. on Comput. Learning Theory, pages
125?133. ACM, 1999.
[MG92] A. A. Razborov M. Goldmann, J. Hastad. Majority gates vs. general
weighted threshold gates. Journal of Computation Complexity, 1(4):277?
300, 1992.
[Nat91] B. K. Natarajan. Machine Learning: A Theoretical Approach. Morgan
Kaufmann, San Mateo, CA, 1991.
[Riv87] R. L. Rivest. Learning decision lists. Machine Learning, 2:229?246, 1987.
[RW02] G. R?
atsch and M. K. Warmuth. Maximizing the margin with boosting.
In Proceedings of the 15th Annual Conference on Computational Learning
Theory, pages 334?350. Springer, July 2002.
[SS99] Robert E. Schapire and Yoram Singer. Improved boosting algorithms using
con?dence-rated predictions. Machine Learning, 37(3):297?336, 1999.
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1,687 | 2,533 | Entrainment of Silicon Central Pattern Generators
for Legged Locomotory Control
Francesco Tenore1, Ralph Etienne-Cummings1,2, M. Anthony Lewis3
Dept. of Electrical & Computer Eng., Johns Hopkins University, Baltimore, MD 21218
2
Institute of Systems Research, University of Maryland, College Park, MD 20742
3
Iguana Robotics, Inc., P.O. Box 625, Urbana, IL 61803
1
{fra, retienne}@jhu.edu, [email protected]
Abstract
We have constructed a second generation CPG chip capable of generating the necessary
timing to control the leg of a walking machine. We demonstrate improvements over a
previous chip by moving toward a significantly more versatile device. This includes a
larger number of silicon neurons, more sophisticated neurons including voltage
dependent charging and relative and absolute refractory periods, and enhanced
programmability of neural networks. This chip builds on the basic results achieved on a
previous chip and expands its versatility to get closer to a self-contained locomotion
controller for walking robots.
1
Introduction
Legged locomotion is a system level behavior that engages most senses and
activates most muscles in the human body. Understanding of biological systems is
exceedingly difficult and usually defies any unifying analysis. Walking behavior is no
exception. Theories of walking are likely incomplete, often in ways that are invisible to
the scientist studying these behavior in animal or human systems. Biological systems
often fill in gaps and details. One way of exposing our incomplete understanding is
through the process of synthesis. In this paper we report on continued progress in
building the basic elements of a motor pattern generator sufficient to control a legged
robot. The focus of this paper is on a 2nd generation chip, that incorporates new features
which we feel important for legged locomotion.
An essential element of most locomotory systems is the Central Patter Generator
(CPG). The CPG is a set of neural circuits found in the spinal cord, arranged to produce
oscillatory periodic waveforms that activate muscles in a coordinated manner. They are
neuron primitives that are used in most periodic biological systems such as the
respiratory, the digestive and the locomotory systems. In this last one, CPGs are
constructed using neurons coupled together to produce phasic relationships required to
achieve coordinated gait-type movements.
The CPG is more than a clock, or even a network of oscillators. Phenomena
such as reflex reversal [7] can only be understood in terms of a system that has at least
one additional state variable over sensory information alone. The CPG or similar circuits
is certainly involved in modulation of sensory information from the periphery [5] and is
of primary importance in providing phase information to the cerebellum. This
information is necessary for coordination of the brain and the spinal cord [6].
Currently, there are two extremes in using CPGs for control of mechanical
devices. The first is to be as faithful to the biological as possible, and then to discover
how biological systems can assist in the control of complex machines. This approach is
similar to that of Rasche et al. [1], based on the Hodgkin-Huxley model [3], and the one
implemented by Simoni and DeWeerth [2], based on the Morris-Lecar model [4]. These
ion-channel based models imply a very large parameter space, making it difficult to work
with in silicon, yet inviting direct comparison with biological counterparts.
Our approach is to start in the other direction. A system of minimal complexity
was built [8,9] and then the question was asked of what additional features should be
added to this minimal system to enable a behavior that is missing in the previous design.
Thus, the two approaches start from different philosophical grounds, but will, hopefully,
converge on similar solutions.
The motivation behind choosing a self-contained silicon system rather than a
software implementation is that the former will use less power and be more compact and
more amenable to the control of a power-autonomous robot.
Previously, a minimal system chip was built using integrate-and-fire neurons
controlling a rudimentary robot [8, 9]. The chip described in this paper is an evolution of
that one. Its main differences with its previous version are the following. The previous
chip contained 2 spiking motoneurons and 2 pacemaker neurons, whereas the current
chip contains 10 neurons of either type. More importantly, all the synapse weights (22 per
neuron) are on-chip and can be used to make the synapse excitatory or inhibitory, while
the previous version weighted the synapse signals outside the chip. The current chip also
has 10 feedback synapses, making all the neurons interconnected. Moreover, the current
chip has the capability of receiving and weighting up to 8 external inputs (instead of 2),
such as sensory feedback signals, to allow better control of the CPG. The possibility of
better tuning the pacemaker and spiking motoneurons created by the chip is achieved
through direct modulation of the pulse width, of the absolute or relative refractory period
and of the discharge strength of each neuron. Finally, the charging and discharging of the
neurons? membrane capacitance is an exponential function of time, as opposed to the
linear function that the previous chip exhibited. This allows for better coupling between
CPGs (unpublished observation).
In this paper, after explaining the architecture of the chip and how simple
networks can be created, a robotic application will be described. The paper will show that
entrainment of multiple CPGs can be achieved by using direct coupling. Analysis and
experiments demonstrating entrainment between multiple CPGs using direct coupling are
presented. Finally, the oscillatory patterns used to control a single-legged robot are
implemented in this chip.
2
Architecture
The CPG emulator chip was fabricated in silicon using a 0.5 ?m CMOS process.
The chip was designed to provide plausible electronic counterparts of biological
elements, such as neurons, synapses, cell membranes, axons, and axon hillocks, for
controlling motor systems. The chip also contains digital memories that can be used with
synapses to modify weights or to modulate the membrane conductance. Through these
components, it is possible to construct non-linear oscillators, which are based on the
central pattern generators of biological organisms.
The chip?s architecture can be seen in figure 1. It is made up of 10 fully interconnected
?neurons? and 22 ?synapses? per neuron. Communication with a particular
neuron/synapse pair occurs through the address register, made up of the neuron/row
register and the synapse/column register. Finally, a weight/data register allows a tunable
amount of current to flow onto or away from the ?neurons? axons.?
Figure 2 shows a detailed view of a single neuron. As can be seen, all neurons
are integrate-and-fire type neurons, in which the current that flows on the axon charges
Neuron/Row Select
Weight Value
Synapse/Column Select
...
Feedback
Neuron 1
Vout1
Neuron 2
Vout 2
Neuron 3
Vout 3
Neuron 4
Vout 4
Neuron 5
Vout5
Neuron 6
Vout6
Neuron 7
Vout 7
Neuron 8
Vout
Neuron 9
Vout9
Neuron 10
Vout10
8
2
Figure 1. Top. Chip micrograph, 3.3x2.1 mm . The 22 synapses per neuron (vertical
lines) are distinguishable. Bottom. System block diagram.
up the membrane capacitor, Cmem. When the voltage across the capacitor reaches a certain
threshold, Vthresh, the hysteretic comparator output goes high. The output of the
comparator does not change if the discharge and refractory period controls are disabled.
Normally, however, the discharge controller is active and its function is to decrease the
voltage on the membrane capacitance until it drops below the hysteretic comparator?s
lower threshold. The comparator output then goes low, the discharge is halted, and the
capacitor can charge up again, thereby making the process start anew.
The i-th neuron can be modeled through the following set of equations:
mem
Ci
mem
dVi
dt
= ? j W + ij I j ? ?k W ? ik I k ?S i I dis ? S i I refrac
?1 if
S i (t + dt ) = ?
?0 if
?
(1)
+
( S i (t ) = 1 ? Vi
> VT ) ? (Vi
> VT )
mem
+
mem
?
( S i (t ) = 0 ? Vi
> VT ) ? (Vi
> VT )
mem
mem
(2)
where Cimem is the membrane capacitance of the i-th neuron, VT+ and VT- are respectively
the high and low thresholds of the hysteretic comparator, Vimem is the voltage on the
capacitor, Si(t) is the state of the hysteretic comparator at time t, W+ij is the excitatory
weight on the j-th excitatory synapse of the i-th neuron and similarly W -ik is the
inhibitory weight on the k-th inhibitory synapse of the i-th neuron. The discharge and
refractory currents, Idis and Irefrac correspond to the discharge and refractory period rates,
respectively.
Neuron 1
An 1
An 4
Dig 1
Dig 4
Vout1
...
...
Vout10
...
Axon Hillock
(Hysteretic Comparator)
Internal bias Analog Inputs Digital Inputs
FB Signals
Weight (Exc)
Weight (Exc)
Weight (Exc)
Weight (Exc)
Vthresh
PW Control
Vout1
Axon
Weight (Inh)
Weight (Inh)
Weight (Inh)
Weight (Inh) Cmem
I
dis
Internal bias Analog Inputs Digital Inputs
FB Signals
Discharge
...
...
An 4
Dig 1
...
An 1
I
Dig 4 Vout1
refrac
Vout10
Refrac Control
Figure 2. Block diagram of a single neuron. The neuron output is fed back to all the
neurons including itself (Vout1 is also a feedback signal).
The speed with which the comparator changes state depends on the amount of
current that the weight, or weights, sets on or remove from the ?axon?. The weights are
set through 8-bit digital-to-analog converters (DACs) and stored in static random access
memory (SRAM) cells. A ninth bit selects the type of weight, either excitatory or
inhibitory. Finally, the three blocks that depend on the comparator output, work as
follows. A weight can be set on any one of these three blocks, just as was done for the
synapses. This allows modulation of the discharge strength, of the refractory period, and
of the pulse width. The refractory period control element prevents current from charging
up the capacitor for as long as it is active. It can be both relative and absolute, depending
on its weight. The pulse-width block allows independent control of the output duty cycle
by modifying the amount of time the output is high. As can be seen in figure 2, the output
from the PW control block is both the neuron output and the feedback signal to all the
neurons, including itself (self-feedback). The chip is thus fully interconnected.
From figure 2, four types of synapses can be identified. The first is the internal
bias synapse, which allows current to flow onto or away from the membrane capacitor,
depending on the type of bias it has, without requiring signals from inside the chip. The
analog and digital synapses require the presence of an external analog or digital voltage
to allow current to flow on the capacitor. The feedback synapses are also internal to the
chip and allow the neurons to influence each other by modulating the charge-up of the
membrane capacitors they are acting upon. This means that one of these synapses is of
self-feedback for a particular neuron. These synapses are considered to be dual mode, in
that they can both excite or inhibit. The 3 final synapses are used to control the discharge
strength, the refractory period, and the pulse width.
It is thus possible to attain two types of waveforms at each neuron output,
depending on the current charging the capacitor. If the current charges up and discharges
the capacitor very quickly, the output is similar to that of a motor neuron. If the current
charges and discharges the capacitor slowly, then the output is that of a pacemaker
envelope neuron, which makes up the CPG.
3
Networks
Two simple networks are described in this section using this chip to understand
the how the chip operates. The first example is shown in figure 3. A pacemaker neuron
feedback
synapse
(exc)
bias
synapse
bias
synapse
Vout
Vthresh
I
PW Control
+
Cmem
feedback
synapse
I
dis
Discharge
I
+
Cmem
Vthresh
PW Control
Vout
I
dis
Discharge
Figure 3. An envelope neuron exciting a motor neuron. The output waveforms are 180?
out-of-phase.
Figure 4. Master slave relationship. When the master spikes, the membrane potential
increases for the duration of the spike.
controls the spiking of a motor neuron such that the spiking only occurs if the envelope is
high. This is done using the internal biasing synapse to charge up the membrane
capacitance of the envelope neuron and the feedback synapse coming from the envelope
neuron to charge up the capacitor of the motor neuron. Similarly, the envelope neuron
can inhibit the spiking which would otherwise occur at a constant rate through the bias
synapse. Note that the bias synapse can either be the internally generated, as the one
shown in figure 3, or it can be the one of the external analog or digital synapse seen in
figure 2.
A second example, shown in figure 4, depicts the effects of a single spike on an
envelope neuron. Depending on where the spike occurs with respect to the slave envelope
neuron, it will either accelerate the charge-up or decelerate the discharge. In this example,
the spike occurred during the membrane potential?s discharge phase. The membrane
potential?s output voltage is shown within the slave output waveform. The two horizontal
lines that delimit it represent the hysteretic comparator?s threshold voltages. Thus, the
slave stays high for a longer period of time, thus decreasing its normal frequency of
oscillation. It is therefore possible to entrain the slave oscillator to the frequency of the
master. This can be done either by increasing the duration of the master spike, increasing
Master oscillator
Spike Entrainer
Slave oscillator
bias
synapse
Master
Spike
Discharger
Spike Entrainer
Spike discharger
Slave
bias
synapse
Figure 5. CPG entrainment.
Figure 6. Phase delay between master envelope and spike entrainer.
the feedback weight with which the master controls the slave, or simply by increasing the
spike frequency. For example, in this latter case, if the master frequency is higher than
the slave?s, then the spike will accelerate the slave such that it reaches the same period.
4
Analysis of pulse coupling
To show that it is possible to entrain two oscillators to have the same frequency
but alter the phase at will, such that any phase between the two waveforms can be
achieved, it is necessary to use a configuration similar to the one described in the
previous section. A master and slave oscillator with different frequencies and both with
approximately 50% duty cycle are set up as shown in figure 5. Another neuron is used to
generate a single spike during the master?s pulse width called the entrainer spike. It is
evoked by the input from the master and has the same frequency, but its phase depends
on the strength of the feedback synapse between these two cells. The spike?s discharge
occurs very slowly, but to ensure that no residual charge is left on the capacitor, a fourth
neuron, 180? out-of-phase with the master, is used. When this neuron is high, it sends a
strong inhibition signal to the spike, thereby resetting it. At this point, the spike can be
used for synchronizing the slave oscillator. As described previously, if the slave
oscillator?s frequency is lower than the master?s (and therefore that of the spike?s), the
spike?s effect is to accelerate until the two are synchronized. This allows for two
pacemaker neurons to be out-of-phase by an arbitrary angle. This is shown in figure 6,
where the coupling weight between master and slave was systematically altered and the
resulting phase variation was recorded. To fine tune the slave oscillator?s desired phase
difference, once the spike master has been set, it is necessary to tune the feedback
strength between the spike and the slave oscillator. A stronger feedback will allow the
Map Function (4.4 ms pulse width)
1
0.9
0.8
Slope = -1
0.7
Phase N+1
0.6
Phase (N+1)
0.5
Slope of |f(x)|< 1
0.4
0.3
0.2
0.1
0
0
0.2
0.4
0.6
0.8
1
PhaseN
Figure 7. Map function illustrating the coupling behavior between two neurons.
two signals to happen virtually at the same time, a weaker weight will cause
some delay between the two. Lewis and Bekey show that adaptation of time is critical to
controlling walking in a robot [10].
Finally, figure 7 shows a map function obtained using a 4.4 ms spike pulse
width. A map function depicts the effect of a spike on a pacemaker neuron at all possible
phases. The curve shows a slope smaller 1 (in absolute value) in the transition region,
which implies that the system is asymptotically stable [9].
5
Experiment
To build on all the results achieved, the oscillatory patterns necessary to control
a single-legged robot were synthesized. Figure 7 shows the waveforms generated to
control a hip?s flexor and extensor muscles and ipsilateral knee?s flexor and extensor.
These waveforms were generated using all 10 available neurons with the procedures
described previously. The hip flexor and extensor are 180? out-of-phase to each other.
The left knee extensor is slightly out-of-phase with its respective hip muscle but the
width of the waveform?s pulse is shorter than that of the hip extensor. As can be seen, the
knee flexor has two bumps, where the purpose of the first bump is to stabilize the knee
when the foot hits the substrate. The waveforms depicted are necessary to drive a robotic
leg with a standard walking gait. Different gaits will have waveforms with different phase
relationships. However, the results shown in the previous sections show that these
waveforms, through simple variations of the timing parameters described, can be
generated with ease.
6
Conclusions
The waveforms needed to control a robotic leg can be generated using a silicon
chip described in this paper. The phase differences between the waveforms, however,
change depending on the type of gait that one wants to implement in a robot. The results
obtained show that any phase difference between two or more waveforms can be
achieved, thus making any gait effectively achievable. Furthermore, the map function that
resulted from on-chip measurements showed that the chip has the capability of
asymptotic coupling stability.
Figure 8. Waveforms generated to control a robotic leg.
References
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[7]. H. Forssberg, S. Grillner, S. Rossignol, ?Phase dependent reflex during walking in
chronic spinal cats,? Brain research, vol. 85, pp. 103-7, 1975.
[8]. M.A. Lewis, R. Etienne-Cummings, A.H. Cohen, M. Hartmann, ?Toward
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conference on robotics and automation, San Francisco, CA, 2000.
[9]. M. A. Lewis, R. Etienne-Cummings, M. J. Hartmann, A. H. Cohen, Z. R. Xu, ?An in
silico central pattern generator: silicon oscillator, coupling, entrainment, and physical
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robot, Autonomous Robots, 12(3) 301-312.
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1,688 | 2,534 | Different Cortico-Basal Ganglia Loops
Specialize in Reward Prediction on
Different Time Scales
Saori Tanaka
Kenji Doya
Nara Institute of Science and Technology
ATR Computational Neuroscience Laboratories
CREST, Japan Science and Technology Corporation
Kyoto, Japan
[email protected]
[email protected]
Go Okada
Kazutaka Ueda
Yasumasa Okamoto
Shigeto Yamawaki
Hiroshima University School of Medicine
CREST, Japan Science and Technology Corporation
Hiroshima, Japan
Abstract
To understand the brain mechanisms involved in reward prediction
on different time scales, we developed a Markov decision task that
requires prediction of both immediate and future rewards, and analyzed subjects? brain activities using functional MRI. We estimated
the time course of reward prediction and reward prediction error on
different time scales from subjects' performance data, and used them
as the explanatory variables for SPM analysis. We found topographic maps of different time scales in medial frontal cortex and
striatum. The result suggests that different cortico-basal ganglia
loops are specialized for reward prediction on different time scales.
1
Intro du ction
In our daily life, we make decisions based on the prediction of rewards on different
time scales; immediate and long-term effects of an action are often in conflict, and
biased evaluation of immediate or future outcome can lead to pathetic behaviors.
Lesions in the central serotonergic system result in impulsive behaviors in humans [1],
and animals [2, 3], which can be attributed to deficits in reward prediction on a long
time scale. Damages in the ventral part of medial frontal cortex (MFC) also cause
deficits in decision-making that requires assessment of future outcomes [4-6].
A possible mechanism underlying these observations is that different brain areas are
specialized for reward prediction on different time scales, and that the ascending
serotonergic system activates those specialized for predictions in longer time scales
[7].
The theoretical framework of temporal difference (TD) learning [8] successfully
explains reward-predictive activities of the midbrain dopaminergic system as well as
those of the cortex and the striatum [9-13]. In TD learning theory, the predicted
amount of future reward starting from a state s(t) is formulated as the ?value function?
V(t) = E[r(t + 1) + ? r(t + 2) + ? 2r(t + 3) + ?]
(1)
and learning is based on the TD error
?(t) = r(t) + ? V(t) ? V(t - 1).
(2)
The ?discount factor? ? controls the time scale of prediction; while only the immediate
reward r(t + 1) is considered with ? = 0, rewards in the longer future are taken into
account with ? closer to 1.
In order to test the above hypothesis [7], we developed a reinforcement learning task
which requires a large value of discount factor for successful performance, and analyzed subjects? brain activities using functional MRI. In addition to conventional
block-design analysis, a novel model-based regression analysis revealed topographic
representation of prediction time scale with in the cortico-basal ganglia loops.
2
2.1
Methods
Markov Decision Task
In the Markov decision task (Fig. 1), markers on the corners of a square present four
states, and the subject selects one of two actions by pressing a button (a1 = left button,
a2 = right button) (Fig. 1A). The action determines both the amount of reward and the
movement of the marker (Fig. 1B). In the REGULAR condition, the next trial is
started from the marker position at the end of the previous trial. Therefore, in order to
maximize the reward acquired in a long run, the subject has to select an action by
taking into account both the immediate reward and the future reward expected from
the subsequent state. The optimal behavior is to receive small negative rewards at
states s 2, s3, and s4 to obtain a large positive reward at state s1 (Fig. 1C). In the
RANDOM condition, next trial is started from a random marker position so that the
subject has to consider only immediate reward. Thus, the optimal behavior is to collect a larger reward at each state (Fig. 1D). In the baseline condition (NO condition),
the reward is always zero.
In order to learn the optimal behaviors, the discount factor ? has to be larger than
0.3425 in REGULAR condition, while it can be arbitrarily small in RANDOM condition.
2.2
fMRI imaging
Eighteen healthy, right-handed volunteers (13 males and 5 females), gave informed
consent to take part in the study, with the approval of the ethics and safety committees
of ATR and Hiroshima University.
A
0
Time
1.0
2.0
2.5
3.0
100
C
B
+r 2
s2
s1
REGULAR condition
s2
-r 1
-r 2
+r 1
s1
100
D
RANDOM condition
+r 2
s2
s1
-r 1
+r 1
-r 2
-r 1
s4
+r 2
4.0 (s)
-r 1
s3
a1
a2
r1 = 20 10 yen
r2 = 100 10 yen
+r 1
-r 1
s4
-r 1
-r 1
s3
s4
-r 1
s3
Fig. 1. (A) Sequence of stimulus and response events in the Markov decision
task. First, one of four squares representing present state turns green (0s). As
the fixation point turns green (1s), the subject presses either the right or left
button within 1 second. After 1s delay, the green square changes its position
(2s), and then a reward for the current action is presented by a number (2.5s)
and a bar graph showing cumulative reward during the block is updated
(3.0s). One trial takes four seconds. Subjects performed five trials in the NO
condition, 32 trials in the RANDOM condition, five trials in the NO condition, and 32 trials in the REGULAR condition in one block. They repeated
four blocks; thus, the entire experiment consisted of 312 trials, taking about
20 minutes. (B) The rule of the reward and marker movement. (C) In the
REGULAR condition, the optimal behavior is to receive small negative rewards ?r 1 (-10, -20, or -30 yen) at states s2, s3, and s4 to obtain a large positive
reward +r2 (90, 100, or 110 yen) at state s1. (D) In the RANDOM condition,
the next trial is started from random state. Thus, the optimal behavior is to
select a larger reward at each state.
A 1.5-Tesla scanner (Marconi, MAGNEX ECLIPSE, Japan) was used to acquire both
structural T1-weighted images (TR = 12 s, TE = 450 ms, flip angle = 20 deg, matrix =
256 ? 256, FoV = 256 mm, thickness = 1 mm, slice gap = 0 mm ) and T2*-weighted
echo planar images (TR = 4 s, TE = 55 msec, flip angle = 90 deg, 38 transverse slices,
matrix = 64 ? 64, FoV = 192 mm, thickness = 4 mm, slice gap = 0 mm, slice gap = 0
mm) with blood oxygen level-dependent (BOLD) contrast.
2.3
Data analysis
The data were preprocessed and analyzed with SPM99 (Friston et al., 1995; Wellcome Department of Cognitive Neurology, London, UK). The first three volumes of
images were discarded to avoid T1 equilibrium effects. The images were realigned to
the first image as a reference, spatially normalized with respect to the Montreal
Neurological Institute EPI template, and spatially smoothed with a Gaussian kernel (8
mm, full-width at half-maximum).
A
RANDOM condition
action
larger
reward
Fig. 2. The selected action
of a representative single
subject (solid line) and the
group average ratio of
selecting optimal action
(dashed line) in (A) RANDOM and (B) REGULAR
conditions.
smaller
reward
1
32
64
96
128
96
128
trial
REGULAR condition
B
action
optimal
nonoptimal
1
32
64
trial
Images of parameter estimates for the contrast of interest were created for each subject.
These were then used for a second-level group analysis using a one-sample t-test
across the subjects (random effects analysis).
We conducted two types of analysis. One was block design analysis using three
boxcar regressors convolved with a hemodynamic response function as the reference
waveform for each condition (RANDOM, REGULAR, and NO). The other was
multivariate regression analysis using explanatory variables, representing the time
course of the reward prediction V(t) and reward prediction error ?(t) estimated from
subjects? performance data (described below), in addition to three regressors representing the condition of the block.
2.4
Estimation of predicted reward V(t) and prediction error ?(t)
The time course of reward prediction V(t) and reward prediction error ?(t) were estimated from each subject?s performance data, i.e. state s(t), action a(t), and reward
r(t), as follows.
If the subject starts from a state s(t) and comes back to the same state after k steps, the
expected cumulative reward V(t) should satisfy the consistency condition
V(t) = r(t + 1) + ? r(t + 2) + ? + ?
k-1
r(t + k) + ? kV(t).
(3)
Thus, for each time t of the data file, we calculated the weighted sum of the rewards
acquired until the subject returned to the same state and estimated the value function
for that episode as
?? r ( t + 1) + ? r ( t + 2 ) + ... + ? k ?1r ( t + k ) ??
?
.
V (t ) =
1? ? k
(4)
The estimate of the value function V(t) at time t was given by the average of all previous episodes from the same state as at time t
V (t ) =
1
L
L
? V? ( t ) ,
l
(5)
l =1
where {t1, ?, tL} are the indices of time visiting the same state as s(t), i.e. s(t1) = ? =
s(tL) = s(t). The TD error was given by the difference between the actual reward r(t)
and the temporal difference of the value function V(t) according to equation (2).
Assuming that different brain areas are involved in reward prediction on different
time scales, we varied the discount factor ? as 0, 0.3, 0.6, 0.8, 0.9, and 0.99.
Fig. 3. (A) In REGULAR vs. RANDOM comparison, significant activation
was observed in DLPFC ((x, y, z) = (46, 45, 9), peak t = 4.06) (p < 0.001
uncorrected). (B) In RANDOM vs. REGULAR comparison, significant activation was observed in lateral OFC ((x, y, z) = (-32, 9, -21), peak t = 4.90) (p <
0.001 uncorrected).
3
3.1
R e sul t s
Behavioral results
Figure 2 summarizes the learning performance of a representative single subject (solid
line) and group average (dashed line) during fMRI measurement. Fourteen subjects
successfully learned to take larger immediate rewards in the RANDOM condition (Fig.
2A) and a large positive reward at s1 after small negative rewards at s2, s3 and s4 in the
REGULAR condition (Fig. 2B).
3.2
Block-design analysis
In REGULAR vs. RANDOM contrast, we observed a significant activation in the
dorsolateral prefrontal cortex (DLPFC) (Fig. 3A) (p < 0.001 uncorrected). In
RANDOM vs. REGULAR contrast, we observed a significant activation in lateral
orbitofrontal cortex (lOFC) (Fig. 3B) (p < 0.001 uncorrected).
The result of block-design analysis suggests differential involvement of neural
pathways in reward prediction on long and short time scales. The result in RANDOM
vs. REGULAR contrast was consistent with previous studies that the OFC is involved
in reward prediction within a short delay and reward outcome [14-20].
3.3
Regression analysis
We observed significant correlation with reward prediction V(t) in the MFC, DLPFC
(all ? ), ventromedial insula (small ? ), dorsal striatum, amygdala, hippocampus, and
parahippocampal gyrus (large ? ) (p < 0.001 uncorrected) (Fig. 4A). We also found
significant correlation with reward prediction error ?(t) in the IPC, PMd, cerebellum
(all ? ), ventral striatum (small ? ), and lateral OFC (large ? ) (p < 0.001 uncorrected)
(Fig. 4B). As we changed the time scale parameter ? of reward prediction, we found
rostro-caudal maps of correlation to V(t) in MFC with increasing ?.
Fig. 4. Voxels with a significant correlation (p < 0.001 uncorrected) with reward
prediction V(t) and prediction error ?(t) are shown in different colors for different settings of the time scale parameter (? = 0 in red, ? = 0.3 in orange, ? =
0.6 in yellow, ? = 0.8 in green, ? = 0.9 in cyan, and ? = 0.99 in blue). Voxels
correlated with two or more regressors are shown by a mosaic of colors. (A)
Significant correlation with reward prediction V(t) was observed in the MFC,
DLPFC, dorsal striatum, insula, and hippocampus. Note the anterior-ventral to
posterior-dorsal gradient with the increase in ? in the MFC. (B) Significant
correlation with reward prediction error ?(t) on ? = 0 was observed in the ventral
striatum.
4
D i s c u ss i o n
In the MFC, anterior and ventral part was involved in reward prediction V(t) on
shorter time scales (0 ? ? ? 0.6), whereas posterior and dorsal part was involved in
reward prediction V(t) on longer time scales (0.6 ? ? ? 0.99). The ventral striatum
involved in reward prediction error ?(t) on shortest time scale (? = 0), while the
dorsolateral striatum correlated with reward prediction V(t) on longer time scales (0.9
? ? ? 0.99). These results are consistent with the topographic organization of
fronto-striatal connection; the rostral part of the MFC project to the ventral striatum,
whereas the dorsal and posterior part of the cingulate cortex project to the dorsolateral
striatum [21].
In the MFC and the striatum, no significant difference in activity was observed in
block-design analysis while we did find graded maps of activities with different
values of ?. A possible reason is that different parts of the MFC and the striatum are
concurrently involved with reward prediction on different time scales, regardless of
the task context. Activities of the DLPFC and lOFC, which show significant differences in block-design analysis (Fig. 3), may be regulated according to the necessity
for the task;
From these results, we propose the following mechanism of reward prediction on
different time scales. The parallel cortico-basal ganglia loops are responsible for
reward prediction on various time scales. The ?limbic loop? via the ventral striatum
specializes in immediate reward prediction, whereas the ?cognitive and motor loop?
via the dorsal striatum specialises in future reward prediction. Each loop learns to
predict rewards on its specific time scale. To perform an optimal action under a given
time scale, the output of the loop with an appropriate time scale is used for actual
action selection.
Previous studies in brain damages and serotonergic functions suggest that the MFC
and the dorsal raphe, which are reciprocally connected [22, 23], play an important role
in future reward prediction. The cortico-cortico projections from the MFC, or the
serotonergic projections from the dorsal raphe to the cortex and the striatum may be
involved in the modulation of these parallel loops.
In present study, using a novel regression analysis based on subjects? performance
data and reinforcement learning model, we revealed the maps of time scales in reward
prediction, which could not be found by conventional block-design analysis. Future
studies using this method under pharmacological manipulation of the serotonergic
system would clarify the role of serotonin in regulating the time scale of reward
prediction.
Acknowledgments
We thank Nicolas Schweighofer, Kazuyuki Samejima, Masahiko Haruno, Hiroshi
Imamizu, Satomi Higuchi, Toshinori Yoshioka, and Mitsuo Kawato for helpful discussions and technical advice.
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| 2534 |@word trial:12 cingulate:1 mri:2 hippocampus:2 specialises:1 tr:2 solid:2 necessity:1 selecting:1 current:1 anterior:2 activation:4 subsequent:1 motor:2 opin:1 medial:5 v:5 half:1 selected:1 short:2 characterization:1 five:2 differential:1 specialize:1 fixation:1 pathway:1 behavioral:1 rostral:1 orbital:2 acquired:2 expected:2 tryptophan:1 behavior:7 brain:9 approval:1 td:4 actual:2 increasing:2 psychopharmacology:3 project:2 underlying:1 circuit:1 mass:1 neurobiol:1 recruit:1 developed:2 informed:1 accumbens:1 corporation:2 temporal:3 uk:1 control:4 positive:3 safety:1 t1:4 consequence:1 striatum:16 sutton:1 receptor:2 modulation:1 fov:2 suggests:2 collect:1 co:2 locked:1 acknowledgment:1 responsible:1 block:11 area:2 drug:1 projection:2 regular:14 anticipation:2 suggest:1 parahippocampal:1 selection:1 context:1 conventional:2 map:4 chronic:1 go:1 regardless:1 starting:1 rule:1 updated:1 pt:1 play:1 substrate:1 mosaic:1 hypothesis:1 observed:8 role:3 connected:1 episode:2 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1,689 | 2,535 | On the Dynamics of Boosting?
Robert E. Schapire
Cynthia Rudin
Ingrid Daubechies
Princeton University
Princeton University
Department of Computer Science
Progr. Appl. & Comp. Math.
35 Olden St.
Fine Hall
Princeton, NJ 08544
Washington Road
[email protected]
Princeton, NJ 08544-1000
{crudin,ingrid}@math.princeton.edu
Abstract
In order to understand AdaBoost?s dynamics, especially its ability to
maximize margins, we derive an associated simplified nonlinear iterated
map and analyze its behavior in low-dimensional cases. We find stable
cycles for these cases, which can explicitly be used to solve for AdaBoost?s output. By considering AdaBoost as a dynamical system, we are
able to prove R?atsch and Warmuth?s conjecture that AdaBoost may fail
to converge to a maximal-margin combined classifier when given a ?nonoptimal? weak learning algorithm. AdaBoost is known to be a coordinate
descent method, but other known algorithms that explicitly aim to maximize the margin (such as AdaBoost? and arc-gv) are not. We consider
a differentiable function for which coordinate ascent will yield a maximum margin solution. We then make a simple approximation to derive a
new boosting algorithm whose updates are slightly more aggressive than
those of arc-gv.
1
Introduction
AdaBoost is an algorithm for constructing a ?strong? classifier using only a training set and
a ?weak? learning algorithm. A ?weak? classifier produced by the weak learning algorithm
has a probability of misclassification that is slightly below 50%. A ?strong? classifier
has a much smaller probability of error on test data. Hence, AdaBoost ?boosts? the weak
learning algorithm to achieve a stronger classifier. AdaBoost was the first practical boosting
algorithm, and due to its success, a number of similar boosting algorithms have since been
introduced (see [1] for an introduction). AdaBoost maintains a distribution (set of weights)
over the training examples, and requests a weak classifier from the weak learning algorithm
at each iteration. Training examples that were misclassified by the weak classifier at the
current iteration then receive higher weights at the following iteration. The end result is a
final combined classifier, given by a thresholded linear combination of the weak classifiers.
Often, AdaBoost does not empirically seem to suffer badly from overfitting, even after
a large number of iterations. This lack of overfitting has been attributed to AdaBoost?s
?
This research was partially supported by NSF Grants IIS-0325500, CCR-0325463, ANI0085984 and AFOSR Grant F49620-01-1-0099.
ability to generate a large margin, leading to a better guarantee on the generalization performance. When it is possible to achieve a positive margin, AdaBoost has been shown to
approximately maximize the margin [2]. In particular, it is known that AdaBoost achieves a
margin of at least 12 ?, where ? is the largest margin that can possibly be attained by a combined classifier (other bounds appear in [3]). Many of the subsequent boosting algorithms
that have emerged (such as AdaBoost? [4], and arc-gv [5]) have the same main outline as
AdaBoost but attempt more explicitly to maximize the margin at the expense of lowering
the convergence rate; the trick seems to be to design an update for the combined classifier
that maximizes the margin, has a fast rate of convergence, and is robust.
For all the extensive theoretical and empirical study of AdaBoost, it is still unknown if
AdaBoost achieves a maximal margin solution, and thus the best upper bound on the probability of error (for margin-based bounds). While the limiting dynamics of the linearly
inseparable case (i.e., ? = 0) are fully understood [6], other basic questions about the dynamics of AdaBoost in the more common case ? > 0 are unknown. For instance, we do
not know, in the limit of a large number of rounds, if AdaBoost eventually cycles among
the base classifiers, or if its behavior is more chaotic.
In this paper, we study the dynamics of AdaBoost. First we simplify the algorithm to reveal a nonlinear iterated map for AdaBoost?s weight vector. This iterated map gives a direct
relation between the weights at time t and the weights at time t + 1, including renormalization, thus providing a much more concise mapping than the original algorithm. We then
provide a specific set of examples in which trajectories of this iterated map converge to a
limit cycle, allowing us to calculate AdaBoost?s output vector directly.
There are two interesting cases governing the dynamics: the case where the optimal weak
classifiers are chosen at each iteration (the ?optimal? case), and the case where permissible
non-optimal weak classifiers may be chosen (the ?non-optimal? case). In the optimal case,
the weak learning algorithm is required to choose a weak classifier which has the largest
edge at every iteration. In the non-optimal case, the weak learning algorithm may choose
any weak classifier as long as its edge exceeds ?, the maximum margin achievable by a
combined classifier. This is a natural notion of non-optimality for boosting; thus it provides
a natural sense in which to measure robustness.
Based on large scale experiments and a gap in theoretical bounds, R?atsch and Warmuth [3]
conjectured that AdaBoost does not necessarily converge to a maximum margin classifier
in the non-optimal case, i.e., that AdaBoost is not robust in this sense. In practice, the weak
classifiers are generated by CART or another heuristic weak learning algorithm, implying
that the choice need not always be optimal. In Section 3, we show this conjecture to be true
using a low-dimensional example. Thus, our low-dimensional study provides insight into
AdaBoost?s large scale dynamical behavior.
AdaBoost, as shown by Breiman [5] and others, is actually a coordinate descent algorithm on a particular exponential loss function. However, minimizing this function in other
ways does not necessarily achieve large margins; the process of coordinate descent must be
somehow responsible. In Section 4, we introduce a differentiable function that can be maximized to achieve maximal margins; performing coordinate ascent on this function yields
a new boosting algorithm that directly maximizes margins. This new algorithm and AdaBoost use the same formula to choose a direction of ascent/descent at each iteration; thus
AdaBoost chooses the optimal direction for this new setting. We approximate the update
rule for coordinate ascent on this function and derive an algorithm with updates that are
slightly more aggressive than those of arc-gv.
We proceed as follows: in Section 2 we introduce some notation and state the AdaBoost
algorithm. Then we decouple the dynamics for AdaBoost in the binary case to reveal a
nonlinear iterated map. In Section 3, we analyze these dynamics for a simple case: the case
where each hypothesis has one misclassified point. In a 3 ? 3 example, we find 2 stable
cycles. We use these cycles to show that AdaBoost produces a maximal margin solution in
the optimal case; this result generalizes to m ? m. Then, we produce the example promised
above to show that AdaBoost does not necessarily converge to a maximal margin solution
in the non-optimal case. In Section 4 we introduce a differentiable function that can be
used to maximize the margin via coordinate ascent, and then approximate the coordinate
ascent update step to derive a new algorithm.
2
Simplified Dynamics of AdaBoost
The training set consists of {(xi , yi )}i=1..m , where each example (xi , yi ) ? X ? {?1, 1}.
Denote by dt ? Rm the distribution (weights) over the training examples at iteration t,
expressed as a column vector. (Denote dTt as its transpose.) Denote by n the total number
of classifiers that can be produced by the weak learning algorithm. Since our classifiers are
binary, n is finite (at most 2m ), but may be very large. The weak classifiers are denoted
h1 , ..., hn , with hj : X ? {1, ?1}; we assume that for every hj on this list, ?hj also
appears. We construct a matrix M so that Mij = yi hj (xi ), i.e., Mij = +1 if training
example i is classified correctly by hypothesis hj , and ?1 otherwise. The (unnormalized)
coefficient of classifier hj for the final
hypothesis is denoted ?j , soP
that the final
Pcombined
n
n
combined hypothesis is fAda (x) = j=1 (?j /k?k1 )hj (x) where k?k1 = j=1 ?j . (In
this paper, either hj or -hj remains unused.) The simplex of n-dimensional vectors with
positive entries that sum to 1 will be denoted ?n . The margin of training example i is
defined by yi fAda (xi ), or equivalently (M?)i /k?k1 , and the edge of hypothesis j with
respect to the training data (weighted by d) is (dT M)j , or 1 ? 2?(probability of error of
? ? ?n that
hj on the training set weighted by d). Our goal is to find a normalized vector ?
? i . We call this minimum margin over training examples the margin
maximizes mini (M?)
of classifier ?. Here is the AdaBoost algorithm and our reduction to an iterated map.
AdaBoost (?optimal? case):
1. Input: Matrix M, Number of iterations tmax
2. Initialize: ?1,j = 0 for j = 1, ..., n
3. Loop for t = 1, ..., tmax
Pm
(a) dt,i = e(?M?t )i / i=1 e(?M?t )i for i = 1, ..., m
(b) jt = argmaxj (dTt M)j
(c) rt = (dTt M)jt
?
?
1+rt
(d) ?t = 12 ln 1?r
t
(e) ?t+1 = ?t + ?t ejt , where ejt is 1 in position jt and 0 elsewhere.
4. Output: ?combined,j = ?tmax +1,j /k?tmax +1 k1
Thus at each iteration, the distribution dt is computed (Step 3a), classifier jt with maximum
edge is selected (Step 3b), and the weight of that classifier is updated (Step 3c, 3d, 3e).
(Note that wlog one can omit from M all the unused columns.)
AdaBoost can be reduced to the following iterated map for the dt ?s. This map gives a
direct relationship between dt and dt+1 , taking the normalization of Step 3a into account
automatically. Initialize d1,i = 1/m for i = 1, ..., m as in the first iteration of AdaBoost.
Reduced Iterated Map:
1. jt = argmaxj (dTt M)j
2. rt = (dTt M)jt
3. dt+1,i =
dt,i
1+Mijt rt
for i = 1, ..., m
To derive this map, consider the iteration defined by AdaBoost and reduce as follows.
?
?
1 + rt
e?(M?t )i e?(Mijt ?t )
1
, so
dt+1,i = Pm
where ?t = ln
?(M?t )i e?(Mijt ?t )
2
1 ? rt
i=1 e
?
?1M
?
?1
1 ? rt 2 ijt
1 ? Mijt rt 2
?(Mijt ?t )
e
=
=
, thus
1 + rt
1 + Mijt rt
dt,i
dt+1,i =
?
?1 .
?1 ?
Pm
1?Mijt rt 2
1+Mijt rt 2
d
i=1 t,i 1+M rt
1?Mij rt
ijt
t
P
Define d+ = {i:M =1} dt,i and d? = 1 ? d+ . Thus, d+ =
ijt
each i such that Mijt = 1, we find:
d
dt,i
? = t,i
?
dt+1,i =
1+rt
1 + rt
d+ + d? 1?rt
1+rt
2
and d? =
1?rt
2 .
For
d
t,i
Likewise, for each i such that Mijt = ?1, we find dt+1,i = 1?r
. Our reduction is
t
Pm
Pm
1
1
complete. To check that i=1 dt+1,i = 1, we see i=1 dt+1,i = 1+rt d+ + 1?r
d? =
t
d?
2d?
d+
2d+
+
3
The Dynamics of Low-Dimensional AdaBoost
= 1.
First we will introduce a simple 3?3 input matrix and analyze the convergence of AdaBoost
in the optimal case. Then we will consider a larger matrix and show that AdaBoost fails to
converge to a maximum margin solution in the non-optimal case.
?
?
?1
1
1
1
?1
1
Consider the following input matrix M =
corresponding to the case of
1
1
?1
three training examples, where each weak classifier misclassifies one example. (We could
add additional hypotheses to M, but these would never be chosen by AdaBoost.) The maximum value of the margin for M is 1/3. How will AdaBoost achieve this result? We are in
the optimal case, where jt = argmaxj (dTt M)j . Consider the dynamical system on the simP3
plex i=1 dt,i = 1, dt,i > 0 ?i defined by our reduced map above. In the triangular region
with vertices (0, 0, 1), (1/3, 1/3, 1/3), (0, 1, 0), jt will be 1. Similarly, we have regions for
jt = 2 and jt = 3 (see Figure 1(a)). Since dt+1 will always satisfy (dTt+1 M )jt = 0, the
dynamics are restricted to the edges of a triangle with vertices (0, 12 , 21 ), ( 12 , 0, 12 ), ( 12 , 12 , 0)
after the first iteration (see Figure 1(b)).
1
second component of d_t
second component of d_t
1
(dT M)2 =0
(dT M)1 =0
1/2
jt =1
jt =3
1/3
jt =2
1/3
first component of d_t
1
(dT M)3 =0
1/2
1
first component of d_t
Figure 1: (a-Left) Regions of dt -space where classifiers jt = 1, 2, 3 will respectively be selected.
(b-Right) All weight vectors d2 , ..., dtmax are restricted to lie on the edges of the inner triangle.
(0,.5,.5)
position along triangle
position along triangle
(0,.5,.5)
(.5,.0,.5)
(.5,0,.5)
(.5,.5,0)
(0,.5,.5)
(0,.5,.5)
(.5,.5,0)
(.5,.5,0)
(.5,.0,.5)
(0,.5,.5)
position along triangle
(.5,.5,0)
(.5,.0,.5)
(0,.5,.5)
position along triangle
0.6
0.5
second component of weight vector
second component of weight vector
0.15
0.15
(0,.5,.5)
(0,.5,.5)
first component of weight vector
0.55
0
0
first component of weight vector
0.5
Figure 2: (a-Upper Left) The iterated map on the unfolded triangle. Both axes give coordinates on
the edges of the inner triangle in Figure 1(b). The plot shows where d t+1 will be, given dt . (b-Upper
Right) The map from (a) iterated twice, showing where dt+3 will be, given dt . There are 6 stable
fixed points, 3 for each cycle. (c-Lower Left) 50 timesteps of AdaBoost showing convergence of
dt ?s to a cycle. Small rings indicate earlier timesteps of AdaBoost, while larger rings indicate later
(3)
(2)
(1)
timesteps. There are many concentric rings at positions dcyc , dcyc , and dcyc . (d-Lower Right) 500
timesteps of AdaBoost on a random 11x21 matrix. The axes are d t,1 vs dt,2 .
On this reduced 1-dimensional phase space, the iterated map has no stable fixed points or
orbits of length
2. ?However, consider the following
periodic orbit of?length 3: ?
?
?
?
(1)T
(2)T
(3)T
3? 5
5?1 1
5?1
1 3? 5
1 3? 5
dcyc = ( 4 , 4 , 2 ), dcyc = ( 2 , 4 , 4 ), dcyc = ( 5?1
4 , 2,
4 ). This
(1)
is clearly a cycle, since starting from dcyc , AdaBoost will choose jt = 1. Then r1 =
?
(1)
(2)
(1)T
(dcyc M)1 = ( 5 ? 1)/2. Now, computing dcyc,i /(1 + Mi,1 r1 ) for each i yields dcyc .
In this way, AdaBoost will cycle between
hypotheses
j = 1, 2, 3,
1, 2, 3, ?
etc. There is
?
?
?
(1)T
(2)T
(3)T
3? 5 1
5?1
5?1 3? 5
1
in fact another 3-cycle, dcyc0 = ( 4 , 2 , 4 ), dcyc0 = ( 2 , 4 , 4 ), dcyc0 =
?
?
3? 5 1
( 5?1
4 ,
4 , 2 ). To find these cycles, we hypothesized only that a cycle of length 3 exists,
visiting each hypothesis in turn, and used the reduced equations from Section 2 to solve for
the cycle coordinates.
We give the following outline of the proof for global stability: This map is a contraction, so any small perturbation from the cycle will diminish, yielding local stability of
the cycles. One only needs to consider the one-dimensional map defined on the unfolded triangle, since within one iteration every trajectory lands on the triangle. This
map and its iterates are piecewise continuous and monotonic in each piece, so one can
find exactly where each interval will be mapped (see Figure 2(a)). Consider the second iteration of this map (Figure 2(b)). One can break the unfolded triangle into intervals and find the region of attraction of each fixed cycle; in fact the whole triangle is the union of both regions of attraction. The convergence to one of these two 3cycles is very fast; Figure 2(b) shows that the absolute slope of the second iterated map
at the fixed points is much less than 1. The combined classifier AdaBoost will output
(1)T
(2)T
(3)T
is: ?combined = ((dcyc M)1 , (dcyc M)2 , (dcyc M)3 )/normaliz. = (1/3, 1/3/1/3), and
since mini (M?combined )i = 1/3 AdaBoost produces a maximal margin solution.
This 3 ? 3 case can be generalized to m classifiers, each having one misclassified training
example; in this case there will be periodic cycles of length m, and the contraction will
also persist (the cycles will be stable). We note that for every low-dimensional case we
tried, periodic cycles of larger lengths seem to exist (such as in Figure 2(d)), but that the
contraction at each iteration does not, so it is harder to show stability.
Now, we give an example to show that non-optimal AdaBoost does not necessarily converge to a maximal margin
input matrix (again, omitting
? solution. Consider the following
!
unused columns): M =
?1
1
1
1
1
?1
1
1
1
1
?1
1
1
1
1
?1
?1
?1
1
1
. For this matrix, the maximal mar-
gin ? is 1/2. In the optimal case, AdaBoost will produce this value by cycling among the
first four columns of M. Recall that in the non-optimal case jt ? {j
: (d?Tt M)j ?? ?}.
?
3?
5
Consider the following initial condition for the dynamics: dT1 = ( 8 , 3?8 5 , 21 , 5?1
4 ).
T
here it is not the optimal
Since (d1 M)5 > ?, we are justified in choosing?j1 = 5, although
?
3? 5
,
),
satisfying
(dT1 M)4 > ? for
choice. Another iteration yields dT2 = ( 14 , 14 , 5?1
4
4
which we choose j2 = 4. At the third iteration, we choose j3 = 3, and at the fourth iteration
we find d4 = d1 . This cycle is the same cycle as in our previous example (although there
is one extra dimension). There is actually a whole manifold of 3-cycles in this non-optimal
?
?
?
T
3? 5
case, since d?1 := (?, 3?4 5 ? ?, 12 , 5?1
4 ) lies on a cycle for any ?, 0 ? ? ?
4 . In any
case, the value of the margin produced by this cycle is 1/3, not 1/2.
We have thus established that AdaBoost is not robust in the sense we described; if the
weak learner is not required to choose the optimal hypothesis at each iteration, but is only
required to choose a sufficiently good weak classifier jt ? {j : (dTt M)j ? ?}, then a
maximum margin solution will not necessarily be attained. In practice, it may be possible
for AdaBoost to converge to a maximum margin solution when hypotheses are chosen to be
only slightly non-optimal; however the notion of non-optimal we are using is a very natural
notion, and we have shown that AdaBoost may not converge to ? here. Note that for some
matrices M, a maximum margin solution may still be attained in the non-optimal case (for
example the simple 3?3 matrix we analyzed above), but it is not attained in general as
shown by our example. We are not saying that the only way for AdaBoost to converge to
a non-optimal solution is to fall into the wrong cycle; there may be many other non-cyclic
ways for the algorithm to fail to converge to a maximum margin solution. Also note that
for the other algorithms mentioned in Section 1 and for the new algorithms in Section 4,
there are fixed points rather than periodic orbits.
4
Coordinate Ascent for Maximum Margins
AdaBoost can be interpreted as an algorithm based on coordinate descent. There are other
algorithms such as AdaBoost? and arc-gv that attempt to maximize the margin explicitly,
but these are not based on coordinate descent. We now suggest a boosting algorithm that
aims to maximize the margin explicitly (like arc-gv and AdaBoost? ) yet is based on coordinate ascent. An important note is that AdaBoost and our new algorithm choose the
direction of descent/ascent (value of jt ) using the same formula, jt = argmaxj (dTt M)j .
This lends further credence to the conjecture that AdaBoost maximizes the margin in the
optimal case, since the direction AdaBoost chooses is the same direction one would choose
to maximize the margin directly via coordinate ascent.
Pm
The function that AdaBoost minimizes via coordinate descent is F (?) = i=1 e?(M?)i .
Consider any ? such that (M?)i > 0 ?i. Then lima?? a? will minimize F , yet the original normalized ? might not yield a maximum margin. So it must be the process of coordinate descent which awards AdaBoost its ability to increase margins, not simply AdaBoost?s
ability to minimize F . Now consider a different function (which bears a resemblance to an
?-Boosting objective in [7]):
?m
!
n
X
X
1
1
?(M?)i
G(?) = ?
ln F (?) = ?
ln
e
where k?k1 :=
?j .
k?k1
k?k1
i=1
j=1
It can be verified that G has many nice properties, e.g., G is a concave function for each
fixed value of k?k1 , whose maximum only occurs in the limit as k?k1 ? ?, and more
importantly, as k?k1 ? ? we have G(?) ? ?(?), where ?(?) = (mini (M?)i )/k?k1 ,
the margin of ?. That is,
Pm ?(M?)i
> e??(?)k?k1
me??(?)k?k1 ?
(1)
i=1 e
?(ln m)/k?k1 + ?(?) ?
G(?)
< ?(?)
(2)
For (1), the first inequality becomes equality only when all m examples achieve the same
minimal margin, and the second inequality holds since we took only one term. Rather than
performing coordinate descent on F as in AdaBoost, let us perform coordinate ascent on
G. The choice of direction jt at iteration t is:
? Pm ?(M?t )i
?
Mij
dG(?t + ?ej ) ??
1
i=1 e
= argmax
+
argmax
?
2 ln(F (?t )).
d?
F
(?
)k?
k
k?
?=0
j
j
t
t 1
t k1
Of these two terms, the second term does not depend on j, and the first term is proportional
to (dTt M)j . Thus the same direction will be chosen here as for AdaBoost.
Now consider the distance to travel along this direction. Ideally, we would like to maximize
G(?t + ?ejt ) with respect to ?, i.e., we would like:
Pm ?(M?t )i ?Mij ?
t M
e
e
dG(?t + ?ejt )
ijt
k?t+1 k1 = i=1
0=
? G(?t + ?ejt )
d?
F (?t + ?ejt )
There is not an analytical solution for ?, but maximization of G(?t +?ejt ) is 1-dimensional
so it can be performed quickly. An approximate coordinate ascent algorithm which avoids
this line search is the following approximation to this maximization problem:
Pm ?(M?t )i ?Mij ?
t M
e
e
ijt
? G(?t ).
0 ? i=1
F (?t + ?ejt )
We can solve for ?t analytically:
?
?
?
?
1 + rt
1
1 + gt
1
? ln
, where gt = max{0, G(?t )}.
(3)
?t = ln
2
1 ? rt
2
1 ? gt
Consider some properties of this iteration scheme. The update for ?t is strictly positive (in
the case of positive margins) due to the Von Neumann min-max theorem and equation (2),
e i ? mini (M?t )i /k?t k1
that is: rt ? ? = mind??m maxj (dT M)j = max???
mini (M?)
e
n
> G(?t ), and thus ?t > 0 ?t. We have preliminary proofs that the value of G increases
at each iteration of our approximate coordinate ascent algorithm, and that our algorithms
converge to a maximum margin solution, even in the non-optimal case.
Our new update (3) is less aggressive than AdaBoost?s, but slightly more aggressive than
arc-gv?s. The other algorithm we mention, AdaBoost? , has a different sort of update. It
converges to a combined classifier attaining a margin inside the interval [? ? ?, ?] within
2(log2 m)/? 2 steps, but does not guarantee asymptotic convergence to ? for a fixed ?.
There are many other boosting algorithms, but some of them require minimization over
non-convex functions; here, we choose to compare with the simple updates of AdaBoost
(due to its fast convergence rate), AdaBoost? , and arc-gv. AdaBoost, arc-gv, and our
algorithm have initially large updates, based on a conservative estimate of the margin.
AdaBoost? ?s updates are initially small based on an estimate of the edge.
0.16
arc?gv, approximate
coord ascent,
and coord ascent
AdaBoost
arc?gv
approximate
coord. ascent
and coord.
ascent
0.5
Margin
Margin
0.65
AdaBoost
0.13
arc?gv
approximate
coordinate
ascent
AdaBoost
AdaBoost*
0.4
0
Iterations 20
150
1100
0.1
90
400
1800
Iterations
10000
Figure 3: (a-Left) Performance of all algorithms in the optimal case on a random 11 ? 21 input
matrix (b-Right) AdaBoost, arc-gv, and approximate coordinate ascent on synthetic data.
Figure 3(a) shows the performance of AdaBoost, arc-gv, AdaBoost? (parameter ? set to
.001), approximate coordinate ascent, and coordinate ascent on G (with a line search for
?t at every iteration) on a reduced randomly generated 11 ? 21 matrix, in the optimal case.
AdaBoost settles into a cycle (as shown in Figure2(d)), so its updates remain consistently
large, causing k?t k1 to grow faster, thus converge faster with respect to G. The values
of rt in the cycle happen to produce an optimal margin solution, so AdaBoost quickly
converges to this solution. The approximate coordinate ascent algorithm has slightly less
aggressive updates than AdaBoost, and is very closely aligned with coordinate ascent; arcgv is slower. AdaBoost? has a more methodical convergence rate; convergence is initially
slower but speeds up later. Artificial test data for Figure 3(b) was designed as follows:
50 example points were constructed randomly such that each xi lies on a corner of the
P11
hypercube {?1, 1}100 . We set yi = sign( k=1 xi (k)), where xi (k) indicates the k th
component of xi . The j th weak learner is hj (x) = x(j), thus Mij = yi xi (j). As expected,
the convergence rate of approximate coordinate ascent falls between AdaBoost and arc-gv.
5
Conclusions
We have used the nonlinear iterated map defined by AdaBoost to understand its update rule
in low-dimensional cases and uncover cyclic dynamics. We produced an example to show
that AdaBoost does not necessarily maximize the margin in the non-optimal case. Then, we
introduced a coordinate ascent algorithm and an approximate coordinate ascent algorithm
that aim to maximize the margin directly. Here, the direction of ascent agrees with the
direction chosen by AdaBoost and other algorithms. It is an open problem to understand
these dynamics in other cases.
References
[1] Robert E. Schapire. A brief introduction to boosting. In Proceedings of the Sixteenth International Joint Conference on Artificial Intelligence, 1999.
[2] Robert E. Schapire, Yoav Freund, Peter Bartlett, and Wee Sun Lee. Boosting the margin: A new
explanation for the effectiveness of voting methods. The Annals of Statistics, 26(5):1651?1686,
October 1998.
[3] Gunnar R?
atsch and Manfred Warmuth. Maximizing the margin with boosting. In Proceedings
of the 15th Annual Conference on Computational Learning Theory, pages 334?350, 2002.
[4] Gunnar R?
atsch and Manfred Warmuth. Efficient margin maximizing with boosting. Journal of
Machine Learning Research, submitted 2002.
[5] Leo Breiman. Prediction games and arcing classifiers. Neural Computation, 11(7):1493?1517,
1999.
[6] Michael Collins, Robert E. Schapire, and Yoram Singer. Logistic regression, AdaBoost and
Bregman distances. Machine Learning, 48(1/2/3), 2002.
[7] Saharon Rosset, Ji Zhu, and Trevor Hastie. Boosting as a regularized path to a maximum margin
classifier. Technical report, Department of Statistics, Stanford University, 2003.
| 2535 |@word achievable:1 stronger:1 seems:1 open:1 d2:1 tried:1 contraction:3 concise:1 mention:1 harder:1 reduction:2 initial:1 cyclic:2 current:1 yet:2 must:2 subsequent:1 happen:1 j1:1 gv:15 plot:1 designed:1 update:14 v:1 implying:1 credence:1 rudin:1 selected:2 intelligence:1 warmuth:4 manfred:2 provides:2 math:2 boosting:15 iterates:1 along:5 ingrid:2 direct:2 constructed:1 prove:1 consists:1 inside:1 introduce:4 expected:1 behavior:3 automatically:1 unfolded:3 considering:1 becomes:1 notation:1 maximizes:4 interpreted:1 minimizes:1 nj:2 guarantee:2 every:5 voting:1 concave:1 exactly:1 classifier:34 rm:1 wrong:1 grant:2 omit:1 appear:1 positive:4 understood:1 local:1 limit:3 path:1 approximately:1 tmax:4 might:1 twice:1 coord:4 appl:1 practical:1 responsible:1 practice:2 union:1 chaotic:1 empirical:1 road:1 suggest:1 map:20 maximizing:2 starting:1 convex:1 insight:1 rule:2 attraction:2 importantly:1 stability:3 notion:3 coordinate:30 limiting:1 updated:1 annals:1 lima:1 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1,690 | 2,536 | Kernels for Structured Natural Language Data
Jun Suzuki, Yutaka Sasaki, and Eisaku Maeda
NTT Communication Science Laboratories, NTT Corp.
2-4 Hikaridai, Seika-cho, Soraku-gun, Kyoto, 619-0237 Japan
{jun, sasaki, maeda}@cslab.kecl.ntt.co.jp
Abstract
This paper devises a novel kernel function for structured natural language
data. In the field of Natural Language Processing, feature extraction
consists of the following two steps: (1) syntactically and semantically
analyzing raw data, i.e., character strings, then representing the results
as discrete structures, such as parse trees and dependency graphs with
part-of-speech tags; (2) creating (possibly high-dimensional) numerical
feature vectors from the discrete structures. The new kernels, called Hierarchical Directed Acyclic Graph (HDAG) kernels, directly accept DAGs
whose nodes can contain DAGs. HDAG data structures are needed to
fully reflect the syntactic and semantic structures that natural language
data inherently have. In this paper, we define the kernel function and
show how it permits efficient calculation. Experiments demonstrate that
the proposed kernels are superior to existing kernel functions, e.g., sequence kernels, tree kernels, and bag-of-words kernels.
1
Introduction
Recent developments in kernel technology enable us to handle discrete structures, such as
sequences, trees, and graphs. Kernel functions suitable for Natural Language Processing
(NLP) have recently been proposed. Convolution Kernels [4, 12] demonstrate how to build
kernels over discrete structures. Since texts can be analyzed as discrete structures, these
discrete kernels have been applied to NLP tasks, such as sequence kernels [8, 9] for text
categorization and tree kernels [1, 2] for (shallow) parsing.
In this paper, we focus on tasks in the application areas of NLP, such as Machine Translation, Text Summarization, Text Categorization and Question Answering. In these tasks,
richer types of information within texts, such as syntactic and semantic information, are required for higher performance. However, syntactic information and semantic information
are formed by very complex structures that cannot be written in simple structures, such as
sequences and trees. The motivation of this paper is to propose kernels specifically suited to
structured natural language data. The proposed kernels can handle several of the structures
found within texts and calculate kernels with regard to these structures at a practical cost
and time. Accordingly, these kernels can be efficiently applied to learning and clustering
problems in NLP applications.
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Figure 1: Examples of structures within texts as determined by basic NLP tools
2
Structured Natural Language Data for Application Tasks in NLP
In general, natural language data contain many kinds of syntactic and semantic structures.
For example, texts have several levels of syntactic and semantic chunks, such as part-ofspeech (POS) chunks, named entities (NEs), noun phrase (NP) chunks, sentences, and discourse segments, and these are bound by relation structures, such as dependency structures,
anaphora, discourse relations and coreference. These syntactic and semantic structures can
provide important information for understanding natural language and, moreover, tackling
real tasks in application areas of NLP. The accuracies of basic NLP tools such as POS taggers, NP chunkers, NE taggers, and dependency structure analyzers have improved to the
point that they can help to develop real applications.
This paper proposes a method to handle these syntactic and semantic structures in a single
framework: We combine the results of basic NLP tools to make one hierarchically structured data set. Figure 1 shows an example of structures within texts analyzed by basic NLP
tools that are currently available and that offer easy use and high performance. As shown
in Figure 1, structures in texts can be hierarchical or recursive ?graphs in graph?. A certain node can be constructed or characterized by other graphs. Nodes usually have several
kinds of attributes, such as words, POS tags, semantic information such as WordNet [3],
and classes of the named entities. Moreover, the relations between nodes are usually directed. Therefore, we should employ a (1) directed, (2) multi-labeled, and (3) hierarchically
structured graph to model structured natural language data.
Let V be a set of vertices (or nodes) and E be a set of edges (or links). Then, a graph
G = (V, E) is called a directed graph if E is a set of directed links E ? V ? V .
Definition 1 (Multi-Labeled Graph) Let ? be a set of labels (or attributes) and M ? V ??
be label allocations. Then, G = (V, E, M ) is called a multi-labeled graph.
Definition 2 (Hierarchically Structured Graph) Let Gi = (Vi , Ei ) be a subgraph in G =
(V, E) where Vi ? V and Ei ? E, and G = {G1 , . . . , Gn } be a set of subgraphs in G.
F ? V ? G represents a set of vertical links from a node v ? V to a subgraph Gi ? G.
Then, G = (V, E, G, F ) is called a hierarchically structured graph if each node has at most
one vertical edge. Intuitively, vertical link fi,Gj ? F from node vi to graph Gj indicates
that node vi contains graph Gj .
Finally, in this paper, we successfully represent structured natural language data by using a
multi-labeled hierarchical directed graph.
Definition 3 (Multi-Labeled Hierarchical Directed Graph) G = (V, E, M, G, F ) is a
multi-labeled hierarchical directed graph.
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letter represents an attribute
Figure 2 shows examples of multi-labeled hierarchical directed graphs. In this paper, we
call a multi-labeled hierarchical directed graph a hierarchical directed graph.
3
Kernels on Hierarchical Directed Acyclic Graph
At first, in order to calculate kernels efficiently, we add one constraint: that the hierarchical
directed graph has no cyclic paths. First, we define a path on a Hierarchical Directed Graph.
If a node has no vertical link, then the node is called a terminal node, which is denoted as
T ? V ; otherwise it is a non-terminal node, which is denoted as T? ? V .
Definition 4 (Hierarchical Path (HiP)) Let p = hvi , ei,j , vj , . . . , vk , ek,l , vl i be a path.
Let ?(v) be a function that returns a subgraph Gi that is linked with v by a vertical link if
v ? T?. Let P(G) be a function that returns the set of all HiPs in G, where links between
v ? G and v ?
/ G are ignored. Then, ph = hh(vi ), ei,j , h(vj ), . . . , h(vk ), ek,l , h(vl )i is
defined as a HiP, where h(v) returns vphx , phx ? P(Gx ) s.t. Gx = ?(v) if v ? T? otherwise
returns v. Intuitively, a HiP is constructed by a path in the path structure, e.g., ph =
hvi , ei,j , vj hvm , em,n , vn i, . . . , vk , ek,l , vl i.
Definition 5 (Hierarchical Directed Acyclic Graph (HDAG)) hierarchical directed graph
G = (V, E, M, G, F ) is an HDAG if there is no HiP from any node v to the same node v.
A primitive feature for defining kernels on HDAGs is a hierarchical attribute subsequence.
Definition 6 (Hierarchical Attribute Subsequence (HiAS)) A HiAS is defined as a list of
attributes with hierarchical information extracted from nodes on HiPs.
For example, let ph = hvi , ei,j , vj hvm , em,n , vn i, . . . , vk , ek,l , vl i be a HiP, then, HiASs in
ph are written as ? (ph ) = hai , aj ham , an i, . . . , ak , al i, which is all combinations for all
ai ? ? (vi ), where ? (v) of node v is a function that returns the set of attributes allocated to
node v, and ? (ph ) of HiP ph is a function that returns all possible HiASs extracted from
HiP ph .
?? denotes all possible HiASs constructed by the attribute in ? and ?i ? ?? denotes the
i?th HiAS. An explicit representation of a feature vector of an HDAG kernel is defined
as ?(G) = (?1 (G), . . . , ?|?? | (G)), where ? represents the explicit feature mapping from
HDAG to the numerical feature space. The value of ?i (G) becomes the weighted number
of occurrences of ?i in G. According to this approach, the HDAG kernel, K(G 1 , G 2 ) =
P|?? |
1
2
i=1 h?i (G ) ? ?i (G )i, calculates the inner product of the weighted common HiASs in
G
v
:
1
:1.0
L5 :1.0
Li :
f1, G :0 .8
: node
1
v
2
:0
.8
L1 :0 .4
L2 :0 .5
G
v
3
:0
.7
L3 :1.2
e2
, 4
:0
e3
, 4
:0
.7
v
4
:0
: s u
.9
L4 :0 .9
g
r a p
er t i c a l
v
i
ei,
j
i
h
G
: di r ec t ed l i nk
: v
.6
b
v
l i nk
j
ei ,j
fi , G
j
f
1
i,
w
e i g
h
t
o
f
La b
w
e i g
h
t
o
f
n
:
:
G : w
j
w
e i g
f r o
e i g
f r o
m
h
m
h
v
t
v
o
i
t
o
i
f
o
d
t o
t o
f
v
v
d
e l
e
Li
v
i
i r e c t e d
l i n
k
j
e r t i c a l
G
l i n
k
j
Figure 3: An Example of Hierarchical Directed Graph ?G? with weight factors
two HDAGs, G 1 and G 2 . In this paper, we use | stand for the meaning of ?such that,? since
it is simple.
X
X
X
KHDAG (G 1 , G 2 ) =
W?i (ph1 )W?i (ph2 ), (1)
h
h
h
1
2
?i ??? ?i ?? (ph
1 )|p1 ?P(G ) ?i ?? (p2 )|p2 ?P(G )
where W?i (ph ) represents the weight value of HiAS ?i in HiP ph . The weight of HiAS ?i
in HiP ph is determined by
Y
Y
Y
Y
W?i (ph ) =
WV (v)
WE (vi , vj )
WF (vi , Gj )
W? (a), (2)
v?V (ph )
ei,j ?E(ph )
fi,Gj ?F (ph )
a?? (?i )
where WV (v), WE (vi , vj ), WF (vi , Gj ), and W? (a) represent the weight of node v, link
from vi to vj , vertical link from vi to subgraph Gj , and attribute a, respectively. An example
of how each weight factor is given is shown in Figure 3. In the case of NL data, for example,
W? (a) might be given by the score of tf ? idf from large scale documents, WV (v) by the
type of chunk such as word, phrase or named entity, WE (vi , vj ) by the type of relation
between vi and vj , and WF (vi , Gj ) by the number of nodes in Gj .
Soft Structural Matching Frameworks
Since HDAG kernels permit not only the exact matching of substructures but also approximate matching, we add the framework of node skip and relaxation of hierarchical information.
First, we discuss the framework of the node skip. We introduce decay function ? V (v)(0 <
?V (v) ? 1), which represents the cost of skipping node v when extracting HiASs from
the HiPs, which is almost the same architecture as [8]. For example, a HiAS under the
node skips is written as h?ha2 , a3 i, ?, ha5 ii from HiP hv1 hv2 , v3 i, v4 , hv5 ii, where ? is the
explicit representation of a node that is skipped.
Next, in the case of the relaxation of hierarchical information, we perform two processes:
(1) we form one hierarchy if there is multiple hierarchy information in the same point, for
example, hhhai , aj ii, ak i becomes hhai , aj i, ak i; and (2) we delete hierarchical information
if there exists only one node, for example, hhai i, aj , ak i becomes hai , aj , ak i.
These two frameworks achieve approximate substructure matching automatically. Table 1
shows an explicit representation of the common HiASs (features) of G 1 and G 2 in Figure 2.
For the sake of simplicity, for all the weights WV (v), WE (vi , vj ), WF (vi , Gj ), and W? (a),
are taken as 1 and for all v, ?V (v) = ? if v has at least one attribute, otherwise ?V (v) = 1.
Efficient Recursive Computation
In general, when the dimension of the feature space |?? | becomes very high, it is computationally infeasible to generate feature vector ?(G) explicitly. We define an efficient
calculation formula between HDAGs G 1 and G 2 , which is written as:
XX
KHDAG (G 1 , G 2 ) =
K(q, r),
(3)
q?Q r?R
Table 1: Common HiASs of G 1 and G 2 in Figure 2: (N.S. represents the node skip, H.R.
represents the relaxation of hierarchical information)
G1
HiAS with ?
HiAS
hP i
hP i
hN i
hN i
hai
hai
hbi
hbi
hci
hci
hdi
hdi
hc, bi
hc, bi
hd, bi
hd, bi
P hai
P hai
P hci
P hci
hhN i, haii
h?hN i, h?i, ?haii
h?hN i, h?i, P i
hhN i, P i
hN, bi
hN, bi
hhN i, hdii
h?hN i, hdii
h?hbi, h?i, ?haii
hhbi, haii
h?hbi, h?i, P i
hhbi, P i
hhbi, hdii
h?hbi, hdii
h?hci, h?i, ?haii
hhci, haii
h?hci, hdii
hhci, hdii
hhdi, haii
hhdi, ?haii
h?hN i, h?i, P haii hhN i, P haii
hhbi, P haii
h?hbi, h?i, P haii
h?hN, bi, h?i, ?haii hhN, bi, haii
h?hN, bi, h?i, P i
hhN, bi, P i
h?hN, bi, hdii
hhN, bi, hdii
h?hN, bi, h?i, P haii hhN, bi, P haii
G2
value
HiAS with ?
2
hP i
1
hN i
2
hai
1
hbi
1
hci
1
hdi
1
hc, bi
1
hhdi, hhbiii
2
P hai
1
P hhcii
3
?
hhN i, ?haii
?2
hhN i, P i
1
hN, bi
? hhN i, ?hhdiii
3
?
hhbi, ?haii
?2
hhbi, P i
?
hhbi, ?hhdiii
?3
hhci, ai
?
hc, di
?
hhdi, ai
2
?
hhN i, P haii
?2
hhbi, P haii
3
?
hhN, bi, ?haii
2
?
hhN, bi, P i
? hhN, bi, ?hhdiii
2
?
hhN, bi, P haii
N.S.
HiAS
value common HiAS
hP i
1
hP i
hN i
1
hN i
hai
1
hai
hbi
1
hbi
hci
1
hci
hdi
1
hdi
hc, bi
1
hc, bi
hhdi, hhbiii
1
P hai
1
P hai
P hhcii
1
hhN i, haii
?
hhN i, haii
hhN i, P i
1
hhN i, P i
hN, bi
1
hN, bi
hhN i, hhdiii
?
hhbi, haii
?
hhbi, haii
hhbi, P i
1
hhbi, P i
hhbi, hhdiii
?
hhci, ai
1
hc, di
1
hhdi, ai
1
hhN i, P haii
1 hhN i, P haii
hhbi, P haii
1
hhbi, P haii
hhN, bi, haii
? hhN, bi, haii
hhN, bi, P i
1
hhN, bi, P i
hhN, bi, hhdiii ?
hhN, bi, P haii 1 hhN, bi, P haii
N.S.+ H.R.
value common HiAS
2
hP i
1
hN i
2
hai
1
hbi
1
hci
1
hdi
1
hc, bi
0
hb, di
2
P hai
0
P hci
4
?
hN, ai
2
?
hN, P i
1
hN, bi
0
hN, di
4
?
hb, ai
?2
hb, P i
0
hb, di
0
hc, ai
0
hc, di
0
hd, ai
2
?
hN, P haii
?2
hb, P haii
4
?
hhN, bi, ai
2
?
hhN, bi, P i
0
hhN, bi, di
2
? hhN, bi, P haii
value
2
1
2
1
1
1
1
1
2
1
?4
?2
1
?2
?4
?2
?2
?3
?
?
?2
?2
?4
?2
?2
?2
where Q = {q1 , . . . , q|Q| } and R = {r1 , . . . , r|R| } represent nodes in G 1 and G 2 , respectively. K(q, r) represents the sum of the weighted common HiASs that are extracted from
the HiPs whose sink nodes are q and r.
? r)I(q, r) + I(q, r)
K(q, r) = JG001 ,G 2 (q, r)H(q, r) + H(q,
(4)
Function I(q, r) returns the weighted number of common attributes of nodes q and r,
X
X
I(q, r) = WV (q)WV (r)
W? (a1 )W? (a2 )?(a1 , a2 ),
(5)
a1 ?? (q) a2 ?? (r)
where ?(a1 , a2 ) = 1 if a1 = a2 , and 0 otherwise. Let H(q, r) be a function that returns the
sum of the weighted common HiASs between q and r including ?(q) and ?(r).
?
?
H(q, r) = I(q, r) + (I(q, r) + ?V (q)?V (r)) H(q, r), if q, r ? T
(6)
I(q, r),
otherwise
X
X
? r) =
(7)
H(q,
WF (q, Gi1 )WF (r, Gj2 )JGi1 ,Gj2 (s, t)
s?Gi1 |Gi1 =?(q) t?Gj2 |Gj2 =?(r)
0
Jx,y
(q, r),
00
(q, r), where x, y are (sub)graphs, be recursive functions
Let Jx,y (q, r),
and Jx,y
to calculate H(q, r) and K(q, r).
0
Jx,y
(q, r) =
00
Jx,y
(q, r) =
?
?
00
Jx,y (q, r) = Jx,y
(q, r)H(q, r) + H(q, r)
X
0
WE (q, t) ?0V (t)Jx,y
(q, t)+Jx,y (q, t) ,
(8)
if ?(r) 6= ?
t?{?(r)?V (y)}
?
0, otherwise
?
X
00
0
?
WE (s, r) ?0V (s)Jx,y
(s, r)+Jx,y
(s, r) , if ?(q) 6= ?
? s?{?(q)?V (x)}
0, otherwise
(9)
(10)
Q
where ?0V (v) = ?V (v) t?Gi |Gi =?(v) ?V (t) if v ? T?, ?0V (v) = ?V (v) otherwise. Function ?(q) returns a set of nodes that have direct links to node q. ?(q) = ? means that no
node has a direct link to s.
Next, we show the formula when using the framework of relaxation of hierarchical information. The functions have the same meanings as in the previous formula. We denote
? r) = H(q, r) + H 0 (q, r).
H(q,
? r) + H 0 (q, r) + H 00 (q, r) I(q, r) + I(q, r)
K(q, r) = JG001 ,G 2 (q, r)H(q,
(11)
H(q, r) = H 0 (q, r) + H 00 (q, r) I(q, r) + H 00 (q, r) + I(q, r)
X
?
? t), if r ? T?
?
WF (r, Gj2 )H(q,
0
H (q, r) = t?Gj2 |Gj2 =?(r)
?
0,
otherwise
X
?
? r), if q, r ? T?
WF (q, Gi1 )H(s, r) + H(q,
?
?
?
?
1 =?(q)
? s?Gi1 |G
i
X
H 00 (q, r) =
WF (q, Gi1 )H(s, r), if q ? T?
?
?
?
1
1
?
? s?Gi |Gi =?(q)
0,
otherwise
00
? r)
Jx,y (q, r) = Jx,y
(q, r)H(q,
0
Jx,y
(q, r) =
?
?
X
(12)
(13)
(14)
(15)
0
? t) , if ?(r) 6= ?
WE (q, t) ?0V (t)Jx,y
(q, t)+Jx,y (q, t)+ H(q,
? t?{?(r)?V (y)}
0, otherwise
(16)
00
? r) are the same as those shown above.
Functions I(q, r), Jx,y
(q, r), and H(q,
According to equation (3), given the recursive definition of KHDAG (q, r), the value between
two HDAGs can be calculated in time O(|Q||R|). In actual use, we may want to evaluate only the subset of all HiASs whose sizes are under n when determining the kernel
value because of the problem discussed in [1]. This can simply realized by not calculating
those HiASs whose size exceeds n when calculating K(q, r); the calculation cost becomes
O(n|Q||R|).
Finally, we normalize the values of the HDAG kernels to remove any bias introduced by
the number of nodes in the graphs. This normalization corresponds to the standard unit
norm normalization of examples in the feature space corresponding to the kernel space
?
K(x,
y) = K(x, y) ? (K(x, x)K(y, y))?1/2 [4].
We will now elucidate an efficient processing algorithm. First, as a pre-process, the nodes
are sorted under two conditions, V (?(v)) ? v and ?(v) ? v, where ?(v) represents all
nodes that have a path to v. The dynamic programming technique can be used to compute
HDAG kernels very efficiently: By following the sorted order, the values that are needed to
calculate K(q, r) have already been calculated in the previous calculation.
4
Experiments
Our aim was to test the efficiency of using the richer syntactic and semantic structures
available within texts, which can be treated now for the first time by our proposed method.
We evaluated the performance of the proposed method in the actual NLP task of Question
Classification, which is similar to the Text Classification task except that it requires many
question:
w
or d
Who i s
W
or d
h
e r
of
W
o
p
r i m
is
P
V
B
e
a t t r i b
p
u
r im
J
m
t e s
i n
i s t e r
( S
e
e q
m
J J
[n
um
b
er ]
- K
of
J a p
a n
?
)
hi e r a r c hi c a l
inister
of
N
I N
N
J a p
N
C
[executive]
[A
a n
N
?
P
.
ountr y
s ia n
C
o
un
W
h
W
tr y ]
P
N
d
e p
W
W
e n
d
h
o
P
e n
c y
s t r u
c t u
V
is
B
J
p
[n
r e s
r im
um
J J
of
b
e
er ]
a t t r i b
m
u
t e s
( D
inister
of
N
I N
N
[executive]
S
- K
,
D
A
J a p
C
[A
N
G
N
o
un
k
s
B
p
J
[n
a n
r im
um
J J
d
e
b
er ]
t he i r
m
r e l a t i on
inister
of
N
I N
N
[executive]
N
s
( H
D
A
J a p
[A
N
s ia n
C
P
G
- K
)
a n
N
C
o
P
un
?
tr y ]
.
ountr y
N P
?
.
P
C
P
n
is
V
)
a n
ountr y
s ia n
- K
o
c hu
tr y ]
Figure 4: Examples of input data of comparison methods
Table 2: Results of question classification by SVM with comparison kernel functions evaluated by F-measure
n
1
HDAG-K DAG-K
DS-K
Seq-K
BOW-K .899
TIME
2
.951
.946
.615
.946
.906
TOP
3
.942
.913
.564
.910
.885
4
.926
.869
.403
.866
.853
1
.748
LOCATION
2
3
.802 .813
.803 .774
.544 .507
.792 .774
.772 .757
4
.784
.729
.466
.733
.745
ORGANIZATION
1
2
3
4
.716 .712 .697
.704 .671 .610
.535 .509 .419
.706 .668 .595
.638 .690 .633 .571
1
.841
NUMEX
2
3
.916 .922
.912 .880
.602 .504
.913 .885
.846 .804
4
.874
.813
.424
.815
.719
more semantic features within texts [7, 10]. We used three different QA data sets written
in Japanese [10].
We compared the performance of the proposed kernel, the HDAG Kernel (HDAG-K), with
DAG kernels (DAG-K), Dependency Structure kernels (DS-K) [2], and sequence kernels
(Seq-K) [9]. Moreover, we evaluated the bag-of-words kernel (BOW-K) [6], that is, the
bag-of-words with polynomial kernels, as the baseline method. The main difference between each method is the ability to treat syntactic and semantic information within texts.
Figure 4 shows the differences of input objects between each method. For better understanding, these examples are shown in English. We used words, named entity tags, and semantic information [5] for attributes. Seq-K only treats word order, DS-K and DAG-K treat
dependency structures, and HDAG-K treats the NP and NE chunks with their dependency
structures. We used the same formula with our proposed method for DAG-K. Comparing
HDAG-K to DAG-K shows the difference in performance between handling the hierarchical structures and not handling them. We extended Seq-K and DS-K to improve the total
performance and to establish a more equal evaluation, with the same conditions, against our
proposed method. Note that though DAG-K and DS-K handle input objects of the same
form, their kernel calculation methods differ as do their return values. We used node skip
parameter ?V (v) = 0.5 for all nodes v in each comparison.
We used SVM [11] as a kernel-based machine learning algorithm. We evaluated the performance of the comparison methods with question type TIME TOP, ORGANIZATION,
LOCATION, and NUMEX, which are defined in the CRL QA-data1 .
Table 2 shows the average F-measure as evaluated by 5-fold cross validation. n in Table 2
indicates the threshold of an attribute?s number, that is, we evaluated only those HiASs that
contain less than n-attributes for each kernel calculation. As shown in this table, HDAGK showed the best performance in the experiments. The experiments in this paper were
designed to investigate how to improve the performance by using the richer syntactic and
semantic structures within texts. In the task of Question Classification, a given question
is classified into Question Type, which reflects the intention of the question. These results
1
http://www.cs.nyu.edu/?sekine/PROJECT/CRLQA/
indicate that our approach, incorporating richer structure features within texts, is well suited
to the tasks in the NLP applications.
The original DS-K requires exact matching of the tree structure, even when it is extended
for more flexible matching. This is why DS-K showed the worst performance in our experiments. The sequence, DAG, and HDAG kernels offer approximate matching by the
framework of node skip, which produces better performance in the tasks that evaluate the
intention of the texts.
The structure of HDAG approaches that of DAG if we do not consider the hierarchical
structure. In addition, the structures of sequences and trees are entirely included in that of
DAG. Thus, the HDAG kernel subsumes some of the discrete kernels, such as sequence,
tree, and graph kernels.
5
Conclusions
This paper proposed HDAG kernels, which can handle more of the rich syntactic and
semantic information present within texts. Our proposed method is a very generalized
framework for handling structured natural language data. We evaluated the performance of
HDAG kernels with the real NLP task of question classification. Our experiments showed
that HDAG kernels offer better performance than sequence kernels, tree kernels, and the
baseline method bag-of-words kernels if the target task requires the use of the richer information within texts.
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1,691 | 2,537 | Auction Mechanism Design for Multi-Robot
Coordination
Curt Bererton, Geoff Gordon, Sebastian Thrun, Pradeep Khosla
{curt,ggordon,thrun,pkk}@cs.cmu.edu
Carnegie Mellon University
5000 Forbes Ave
Pittsburgh, PA 15217
Abstract
The design of cooperative multi-robot systems is a highly active research
area in robotics. Two lines of research in particular have generated interest: the solution of large, weakly coupled MDPs, and the design and implementation of market architectures. We propose a new algorithm which
joins together these two lines of research. For a class of coupled MDPs,
our algorithm automatically designs a market architecture which causes
a decentralized multi-robot system to converge to a consistent policy. We
can show that this policy is the same as the one which would be produced
by a particular centralized planning algorithm. We demonstrate the new
algorithm on three simulation examples: multi-robot towing, multi-robot
path planning with a limited fuel resource, and coordinating behaviors in
a game of paint ball.
1
Introduction
In recent years, the design of cooperative multi-robot systems has become a highly active
research area within robotics [1, 2, 3, 4, 5, 6]. Many planning problems in robotics are best
phrased as MDPs, defined over world states or?in case of partial observability?belief
states [7]. However, existing MDP planning techniques generally scale poorly to multirobot systems because of the curse of dimensionality: in general, it is exponentially harder
to solve an MDP for N agents than it is to solve a single-agent MDP, because the state
and action space for N robots can be exponentially larger than for a single-robot system.
This enormous complexity has confined MDP planning techniques largely to single-robot
systems.
In many cases, robots in a multi-robot system interact only in limited ways. Robots might
seek not to collide with each other [1], coordinate their locations to carry out a joint
task [4, 6], or consume a joint resource with limited availability [8, 9, 10]. While these
problems are not trivially decomposed, they do not necessarily have the worst-case exponential complexity that characterizes the general case. However, so far we lack effective
mechanisms for cooperatively solving such MDPs.
Handling this sort of limited interaction is exactly the strength of market-based planning
algorithms [10, 12]: by focusing their attention on a limited set of important resources and
ignoring all other interactions, these algorithms reduce the problem of cooperating with
other robots to the problem of deciding which resources to produce or consume. Marketbased algorithms are particularly attractive for multi-robot planning because many common
types of interactions can be phrased as constraints on resources such as space (two robots
can?t occupy the same location at once) and time (a robot can only work on a limited
number of tasks at once).
From the point of view of these auction algorithms, the difficult part of the multi-robot
planning problem is to compute the probability distribution of the price of each resource at
every time step: the optimal price for a resource at time t depends on how much each robot
produces or consumes between now and time t, and what each robot?s state is at time t.
The resource usage and state depend on the robots? plans between now and time t, which
in turn depend on the price. Worse yet, future resource usage depends on random events
which can?t be predicted exactly.
In this paper, we bring together resource-allocation tehniques from the auction and MDP
literature. In particular, we propose a general technique for decomposing multi-robot MDP
problems into ?loosely coupled? MDPs which interact only through resource production
and consumption constraints. The decomposition works by turning all interactions into
streams of payments between robots, thereby allowing each robot to learn its own local
value function. Prices can be attached to any function of the visitation frequencies of each
robot?s states and actions. The actual prices for these resources are set by a ?master? agent;
the master agent takes into account the possibility of re-allocating resources at each step,
but it approximates the effect of interactions between robots.
Our approach generalizes a large body of previous literature in multi-robot systems, including prior work by Guestrin and Gordon [11]. Our algorithm can be distributed so that each
robot reasons only about its own local interactions, and it always produces the same answer
as a particular centralized planning algorithm.
2
MDPs, linear programs, and duals
A Markov Decision Process (MDP) is a tuple M = {S, A, T , c, ?, so }. S is a set of N
states. A is a set of M actions. T is the dynamics T (s0 , a, s) = p(s0 | s, a). The reward
function is c : S ? A 7? <. The discount factor is ? ? [0, 1]. Finally, so ? S is the initial
state. For any MDP there is a value function
P which indicates how desirable any state is.
It is defined as V (s) = maxa (c(s, a) + ? s0 p(s0 | s, a)V (s0 )). We can compute V by
solving the Bellman linear program (1). Once we have V , we can compute the optimal
policy by one-step lookahead. Here V ? <N is the vector form of the value function.
ca ? <N is the immediate reward for taking action a and Ta ? <N ?N is the matrix
representation of the transition probabilities for action a. ? is an arbitrary probability
distribution over S which represents the probability of the MDP starting in a particular
state. Typically, ? is a vector in which one entry (the starting state) is set to one and all
other entries are set to zero.
P
maxfa a ca ? fa
minV ? ? V
P
P T
(1)
(2)
a fa ? ?
a Ta fa = ?
?a : V ? ca + ?Ta V
?a : fa ? 0
The dual of the Bellman LP gives us an interesting alternative from which to view the
problem of finding an optimal policy. The dual of the Bellman LP is shown in (2). The
vector fa represents the expected number of times we perform action a from each state.
For the remainder of the paper we will stack all of the fa vectors into one large vector f ,
and collect the equality constraints in (2) into Af = b. Subscripts (e.g., fi or Ai ) will
distinguish the planning problems for different robots.
3
3.1
Algorithm
Loosely coupled MDPs
Our algorithm is designed for multi-robot problems that can be decomposed into separate single-robot MDPs which interact through the production or consumption of fictitious
resources. These resources may be physical goods such as fuel; or they may be logical
resources such as the right to pass over a bridge at a particular time, the right to explore an
area of the environment, or the right to collect reward for achieving a particular subgoal.
Time may be part of the individual robot states, in which case a resource could be the right
to consume a unit of fuel at a particular time (a futures contract).
In more detail, each robot has a vector of state-action visitation frequencies f i which must
satisfy its own local dynamics Ai fi = bi . Its production or consumption of resources is
defined by a matrix Ci : element (j, k) of Ci is the amount of resource j which is produced
or consumed by robot i in state-action pair k. (So, Ci fi is the vector of expected resource
usages for robot i. The sign is arbitrary, so we will assume positive numbers correspond to
consumption.) The robots interact through resource constraints: the instantaneous production and consumption of each resource must balance exactly.
This representation is in many ways related to an undirected dynamic Bayes network: each
node of the network corresponds to the state and action of a single MDP, and a resource
constraint involving a subset of the MDPs plays the role of a clique potential on the corresponding nodes. In this way it is similar to the representation of [11]; but, we do not
assume any particular form for the Ci matrices, while [11] effectively assumes that they
are indicator functions of particular state or action variables.
In the same (trivial) sense as Bayes nets, our representation is completely general: by
collapsing all robots into a single giant agent we can represent an arbitrary MDP. More
importantly, in the more-typical case that some pieces of our model can be written as resource constraints, we can achieve an exponential savings in representation size compared
to the monolithic planning problem.
3.2
Approximation
The resource constraints are what make loosely-coupled MDPs difficult to solve. They
make the value of a joint state depend in a non-linear way on the states of the individual
robots. However, by making a simple approximation we can remove the nonlinearity and
so factor our planning problem: we relax the resource constraints so that they must only
be satisfied in expectation over all time steps, rather than deterministically on each time
step. Under this assumption, knowing the expected resources available to a robot allows
that robot to plan independently: since Ci fi is the vector of expected resource usages for
robot i, adding the constraint Ci fi = k to equation (2) gives us the single-robot resourceconstrained planning problem.
The (approximate) global planning problem then becomes to determine an optimal resource
allocation among robots and corresponding single-robot plans, or equivalently to determine
the optimal resource prices and corresponding single-robot value functions. More formally,
the planning problem is to solve (3):
P
maxfi i ci ? fi
?i : Ai fi = bi
P
(3)
(?)
i C i fi = d
?i : fi ? 0
Without the constraints marked (?), this LP would represent a set of completely uncoupled
robot planning problems. The constraints (?) are the approximated resource constraints:
they say that expected production must equal expected consumption for each resource.
The resource prices are the dual variables for (?), and the local value functions are the dual
variables for the remaining equality constraints.
The quality of our prices and value functions will depend on whether it is valid to assume
a single price for each resource: if the prices stay constant then our approximate plan will
translate perfectly to the physical world. On the other hand, if we are unlucky, we may
find that prices are different than we had planned when we need to buy or sell. In this case
our computed plan will contain overoptimistic, counterintuitive sequences of actions; for
example, in the problem of section 3.4, two robots might each plan to break down at the
same time and be towed by the other. The only way to fix this problem is to make a more
accurate model; in the worst case we will have to combine several robots into one large
MDP so that we can track their joint allocation of resources at all times.
3.3
Action selection
Because the value functions incorporate information about future actions and random
events, the robots only need to look ahead a short time to choose good actions. So, the
robots can run a simple auction to determine their best joint action: each individual robot
estimates its future cost for each action by a single-step backup from its value function.
The difference between these future costs then tells the robot how much it is willing to bid
for the right to execute each action. The optimal joint action is then the feasible action with
the highest sum of bids.
3.4
Example
R3
R2
G
Re
R1
Figure 1: A simple example (left panel): the objective is to have all robots (R1,R2,R3) reach the goal
(G) where they receive a reward. Any action may result in a robot becoming disabled, in which case
it must be towed to the repair area (Re) to continue with the task. The grid shown here is significantly
smaller than the problem solved in our experiments (right panel).
Figure 1 shows a simulator which contains 3 robots. Each robot receives a large reward
upon reaching the goal but incurs a small cost for each step it takes. Robots can break
whenever they take a step, but a functioning robot may tow a failed robot to the repair area
and the repaired robot may then proceed to the goal. Each robot has the action set A = { 8connected move, pickup for towing, request tow}. The state of each robot is its x position,
its y position and its status {towing, going to goal, being towed, doing nothing}. If the grid
is 300 by 300, then the state space size is |S| = 300 ? 300 ? 4 = 360000. The action space
size is |A| = 10. The joint state space of all three robots is |Sjoint | = |S|3 and the joint
action space is |A| = 103 . Clearly, this problem size is such that ordinary MDP solution
methods will be insufficient to determine the optimal value function.
However, this problem lends itself to resource-based decomposition because the robots only
interact through towing. Specifically, we design our Ci matrices to represent the constraint
that the expected number of times a robot executes a pickup action at a position should
be equal to the expected number of times some other robot executes a request-tow action.
Thus, we have a weakly coupled MDP with robot interactions that can be modeled by linear
constraints.
p ? 0 Gi = [ ] ?i = [ ]
repeat
done ? true
for i ? 1 . . . n
send prices p to robot i
f ? frequencies from planning for robot i with costs ci ? CiT p
send expected usage g = Ci f and cost ? = ci ? f to master
if g is not already in Gi
Gi ? [Gi , g] ?i ? [?i , ?] done ? false
end if
end for
p ? new dual variables from solving (4) with current Gi and ?i
until done
Figure 2: The decentralized planning algorithm based on Dantzig-Wolfe decomposition.
3.5
Dantzig-Wolfe decomposition
We have reduced the multi-robot planning problem to the problem of solving the LP (3). So,
one possible planning algorithm is just to pass this LP to a pre-packaged linear-program
solver. This planning algorithm can be fairly efficient, but it is completely centralized:
each agent must communicate its entire dynamics to a central location and wait to receive
its value function in return.
Instead of using this centralized algorithm, we want to produce the same outcome with a
decentralized planner. To do so, we will apply Dantzig-Wolfe decomposition [13, chapter
24]. This decomposition splits our original LP (3) into a master LP (4) and one slave LP (5)
for each robot i. It then solves each slave program repeatedly, generating a new value for
fi each time, and combines these solutions by inserting them into the master LP (Figure 2).
The Dantzig-Wolfe decomposition algorithm is guaranteed to terminate in a finite number
of steps with the correct solution to our original LP and therefore with the correct local
value functions. Each slave LP is the same as the corresponding robot?s MDP except that
it has different state-action costs; so, the robots can run standard MDP planners (which are
often much faster than general LP solvers) to produce their plans. And, instead of sending
whole MDPs and value functions back and forth, the Dantzig-Wolfe decomposition only
needs to send resource prices and expected usages. The master program can be located on
a separate agent, or on an arbitrary robot.
In more detail, the master and slave LPs are:
P
maxqi i cT
i F i qi
P
(?)
C
(F
q
)=d
i
i
i
i
?i : qi ? 0
P
?i :
j qij = 1
(4)
T
maxfi (cT
i ? p Ci )fi
A i fi = b i
fi ? 0
(5)
The master LP is the same as the original problem (3) except that fi has been replaced by
Fi qi . Each column of Fi is one of the solutions fi which we have computed for the ith
slave LP. (For efficiency, instead of storing Fi we keep Gi = Ci Fi and ?i = cT
i Fi .) So,
solving the master LP means finding a convex combination qi of the known solutions for
each slave LP. The slave LP is the same as a single-robot planning problem (2) except that
its costs have been altered by subtracting pT Ci . The vector p is the dual variable for the
constraints (?) from the last time we solved the master LP.
3.6
An economic interpretation
We have described how to use the Dantzig-Wolfe decomposition to derive an efficient distributed planning algorithm for loosely-coupled MDPs. In addition to being efficient and
distributed, our algorithm has an intuitive economic interpretation which leads to interesting links with existing work on market architectures.
It is well known that the dual variables of a linear program have economic significance
[14, 15]. Associated with each row of the constraint matrices Ci in the master program (4)
is a dual variable; that is, there is one dual variable pj for each resource j. We can interpret
this dual variable as a price for resource j. To see why, notice that the slave program
charges robot i a cost of pj [Ci ]j,k each time it visits state-action pair k, and that visiting
state-action pair k consumes an amount [Ci ]j,k of resource j.
The Dantzig-Wolfe algorithm can be interpreted as a search for optimal resource prices.
The master agent repeatedly asks the robots what they would do if the prices were p,
then tries to combine their answers to produce a good plan for all the robots together.
As it combines the single-robot plans, it notices whether it could achieve a higher reward
by increasing or decreasing the supply of each resource; if there is an undersupply of a
resource the master agent assigns it a high price, and if there is an oversupply the master
agent assigns it a low price.
4
Experimental results
3.5
Auction outcome
Performance / Optimal
3
2.5
2
1.5
1
1
2
3
4
5
6
7
8
9
10
Iterations
Figure 3: Auctions for multi-robot path planning with limited fuel usage. Left to right: in an auction
based on the assumption of cheap fuel, all robots go to the globally most tempting goal. If we assume
very expensive fuel, each robot crashes through obstacles and goes to its closest goal. With the
optimal fuel price, the auction trades goal quality against distance to achieve the best possible total
cost. As our algorithm learns better prices, the auction?s outcomes approach the optimal policy.
Our experiments are divided into two groups. First, to investigate the convergence rate
of our algorithm, we collected data from multiple runs on randomly-generated synthetic
problems. Second, to investigate scaling, we applied the algorithm to a large, realistic
problem taken from our ongoing research into robotic laser tag [16].
In our synthetic problem, we randomly place circular obstacles inside a bounded arena to
create a maze. We then place 15 robots in random starting locations and ask them to plan
paths to 10 random goals. Each robot can choose whichever goal it wants, but must pay
a random goal-specific price. The robots are coupled through a constraint on fuel usage:
there is a quadratic penalty on total path length.
In this problem, our algorithm starts from an arbitrary initial guess at the value of a unit
of fuel (which causes the individual robots to make poor policy decisions) and rapidly
improves the estimated value by examining the individual robot plans. We averaged the
performance of our algorithm on 20 random instances; the results are shown in Figure 3.
To demonstrate scaling, we used our learning algorithm to coordinate the robot towing
problem in the simulation shown in figure 4, with a grid size of 300 ? 300 and 9 robots.
Many more robots could be handled, but because we only coordinated towing and not path
Figure 4: Left: an example of the output of the algorithm on a towing problem on a map generated
using the robots on the right. Note that the nearest live robot (R1) tows the damaged robot to the
repair area before heading to the goal. This type of problem was solved for up to 9 robots. Right:
Multi-robot paint ball simulator.
planning in this example, there was a bottleneck at the repair area due to the unmodeled
coordination. The resulting paths executed in a sample problem are shown in figure 4.
Because our algorithm uses an arbitrary MDP planner as a subroutine, very large problems
can be solved by combining our approach with fast planning algorithms.
Figure 4 shows the simulator in which we applied the method to multi-robot paint ball. The
rules of the game are that the last team standing wins and that it takes 4 hits to cause a robot
to fail. There is a repair area to which a tagged teammate may be towed in order to repair it
so that it may continue to play. Robots can only see each other when there are no obstacles
between them.
In this problem, we use our method to select and coordinate predefined policies. Policies
used are: do nothing, attack target i, coordinated attack (with a teammate) target i, tow
teammate i, and be repaired. Currently these policies are hand specified, but in future work
we would like to apply policy search methods to learn these policies. The objective of our
multi-robot planner is to determine at a given time which fixed policy each robot on the
team should be executing so that the team will perform better. Coordination constraints are
that any coordinated attacks or towing/repairing must be consistent: if teammate 1 requests
a tow from teammate 2, then teammate 2 must perform a tow of teammate 1.
To solve the slave problems, we use rollouts of the given policies. This allows us to handle
partial observability as each enemy is tracked with a particle filter, and the particle filter
distribution is used when performing rollouts. Enemy positions are sampled from the particle filters at the beginning of each rollout and each policy is evaluated over several possible
enemy position combinations to determine the performance of a policy. The robots replan
at fixed intervals; the simulation is halted while planning occurs.
We compared our coordination planner to a similar planner without coordination. Each
planner was played against a default behavior of ?attack nearest enemy? over 50 games.
The uncoordinated planner won 42 of 50 games over the default behavior. The coordinated
planner won 48 of 50 games against the default behavior. Thus, the addition of coordination
(via our factored planning algorithm) significantly improved the performance.
5
Conclusions
We have developed a decentralized method for solving large loosely-coupled multi-robot
planning problems. Our algorithm works by finding an optimal solution to an approximate
planning problem in which resource constraints hold only in expectation. It has an intuitive
economic interpretation which facilitates its application to new problems. And, it can be
combined with previous MDP decomposition methods, allowing the user to mix and match
which methods are best suited to their problem. We have applied our algorithm to multirobot towing, optimal use of fuel in a multi-robot path planning problem, and planning for
multi-robot paintball.
Acknowledgements
This project was supported by DARPA?s MICA and MARS programs.
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1,692 | 2,538 | Pairwise Clustering and Graphical Models
Noam Shental
Computer Science & Eng.
Center for Neural Computation
Hebrew University of Jerusalem
Jerusalem, Israel 91904
[email protected]
Assaf Zomet
Computer Science & Eng.
Hebrew University of Jerusalem
Jerusalem, Israel 91904
[email protected]
Tomer Hertz
Computer Science & Eng.
Center for Neural Computation
Hebrew University of Jerusalem
Jerusalem, Israel 91904
[email protected]
Yair Weiss
Computer Science & Eng.
Center for Neural Computation
Hebrew University of Jerusalem
Jerusalem, Israel 91904
[email protected]
Abstract
Significant progress in clustering has been achieved by algorithms that
are based on pairwise affinities between the datapoints. In particular,
spectral clustering methods have the advantage of being able to divide
arbitrarily shaped clusters and are based on efficient eigenvector calculations. However, spectral methods lack a straightforward probabilistic
interpretation which makes it difficult to automatically set parameters using training data.
In this paper we use the previously proposed typical cut framework for
pairwise clustering. We show an equivalence between calculating the
typical cut and inference in an undirected graphical model. We show that
for clustering problems with hundreds of datapoints exact inference may
still be possible. For more complicated datasets, we show that loopy belief propagation (BP) and generalized belief propagation (GBP) can give
excellent results on challenging clustering problems. We also use graphical models to derive a learning algorithm for affinity matrices based on
labeled data.
1
Introduction
Consider the set of points shown in figure 1a. Datasets of this type, where the two clusters are not easily described by a parametric model can be successfully clustered using
pairwise clustering algorithms [4, 6, 3]. These algorithms start by building a graph whose
vertices correspond to datapoints and edges exist between nearby points with a weight that
decreases with distance. Clustering the points is then equivalent to graph partitioning.
a
Figure 1: Clustering as graph partitioning (following [8]). Vertices correspond to datapoints and edges between
adjacent pixels are weighted by the distance. A single isolated datapoint is
marked by an arrow
b
How would we define a good partitioning? One option is the minimal cut criterion. Define:
X
cut(A, B) =
W (i, j)
(1)
i?A,j?B
where W (i, j) is the strength of the weight between node i and j in the graph. The minimal
cut criterion finds clusterings that minimize cut(A, B).
The advantage of using the minimal cut criterion is that the optimal segmentation can be
computed in polynomial time. A disadvantage, pointed out by Shi and Malik [8], is that
it will often produce trivial segmentations. Since the cut value grows linearly with the
number of edges cut, a single datapoint cut from its neighbors will often have a lower cut
value than the desired clustering (e.g, the minimal cut solution separates the full dot in fig1,
instead of the desired ?N? and ?I? clusters).
In order to avoid these trivial clusterings, several graph partitioning criteria have been proposed. Shi and Malik suggested the normalized cut criterion which directly penalizes partitions where one of the groups is small, hence a separation of a single isolated datapoint
is not favored. Minimization of the normalized cut criterion is NP-Complete but it can be
approximated using spectral methods.
Despite the success of spectral methods in a wide range of clustering problems, several
problems remain. Perhaps the most important one is the lack of a straightforward probabilistic interpretation. However, interesting progress in this direction has be made by Meila
and Shi [4] who showed a relation between the top eigenvectors and the equilibrium distribution of a random walk on the graph.
The typical cut criterion, suggested by Blatt et al [1] and later by Gdalyahu et al [2], is
based on a simple probabilistic model. Blatt et al first defines a probability distribution
over possible partitions by:
1
(2)
Pr(A, B) = e?cut(A,B)/T
Z
where Z is a normalizing constant, and the ?temperature? T serves as a free parameter.
Using this probability distribution, the most probable partition is simply the minimal cut.
Thus performing MAP inference under this probability distribution will still lead to trivial
segmentations. However, as Blatt et al pointed out, there is far more information in the
full probability distribution over partitions than solely in the MAP partition. For example,
consider the pairwise correlation p(i, j) defined for any two neighboring nodes in the graph
as the probability that they belong to the same segment:
X
p(i, j) =
Pr(A, B)SAM E(i, j; A, B)
(3)
A,B
with SAM E(i, j; A, B) defined as 1 iff i ? A and j ? A or i ? B and j ? B.
Referring again, to the single isolated datapoint in figure 1, then while that datapoint and its
neighbors do not appear in the same cluster in the most probable partition, they do appear
in the same cluster for the vast majority of partitions. Thus we would expect p(i, j) > 1/2
for that datapoint and its neighbors.
Hence the typical cut algorithm of Blatt et al consists of three stages:
? Preprocessing: Construct the affinity matrix W so that each node will be connected to at most K neighbors. Define the affinities W (i, j) as: W (i, j) =
?d(i,j)2
e ?2 , where di,j is the distance between points i and j, and ? is the mean
distance to the K?th neighbor.
? Estimating pairwise correlations: Use a Markov chain Monte-Carlo (MCMC)
sampling method to estimate p(i, j) at each temperature T .
? Postprocessing: Define the typical cut partition as the connected components of
the graph after removing any links for which p(i, j) < 1/2.
For a given W (i, j) the algorithm has a single free temperature parameter T (see eq. 5).
This parameter implicitly defines the number of clusters. At zero temperature all the datapoints reside in one cluster (this trivially minimizes the cut value), and at high temperatures
every datapoint forms a separate cluster.
In this paper we show that calculating the typical cut is equivalent to performing inference
in an undirected graphical model. We use this equivalence to show that in problems with
hundreds of datapoints, the typical cut may be calculated exactly. We show that when exact
inference is impossible, loopy belief propagation (BP) and generalized belief propagation
(GBP) may give an excellent approximation with very little computational cost. Finally,
we use the standard algorithm for ML estimation in graphical models to derive a learning
algorithm for affinity matrices based on labeled data 1 .
2
The connection between typical cuts and graphical models
An undirected graphical model with pairwise potentials (see [10] for a review) consists of
a graph G and potential functions ?ij (xi , xj ) such that the probability of an assignment x
is given by:
1 Y
?ij (xi , xj )
(4)
Pr(x) =
Z <ij>
where the product is taken over nodes that are connected in the graph G.
To connect this to typical cuts we first define for every partition (A, B) a binary vector x
such that x(i) = 0 if i ? A and x(i) = 1 if i ? B. We then define:
1
e?W (i,j)/T
?ij (xi , xj ) =
(5)
e?W (i,j)/T
1
Observation 1: The typical cut probability distribution (equation 2) is equivalent to that
induced by a pairwise undirected graphical model (equation 4) whose graph G is the same
as the graph used for graph partitioning and whose potentials are given by equation 5.
So far we have focused on partitioning the graph into two segments, but the equivalence holds for any number of segments q. Let (A1 , A2 , ? ? ? , Aq ) be a partitioning of the
graph into q segments (note that these segments need not be connected in G). Define
cut(A1 , A2 , ? ? ? Aq ) in direct analogy to equation 1, and:
1
1
(6)
Pr((A1 , A2 , ? ? ? , Aq ) = e? T cut(A1 ,A2 ,???,Aq )
Z
The implication of observation 1 is that we can use the powerful tools of graphical models
in the context of pairwise clustering. In subsequent sections we provide examples of the
benefits of using graphical models to compute typical cuts.
1
Parts of this work appeared previously in [7].
3
Computing typical cuts using inference in a graphical model
Typical cuts has been successfully used for clustering of datapoints in R n [1] using an
expensive MCMC to calculate pairwise correlations, p(i, j). Using inference algorithms
we provide a deterministic and more efficient estimate of p(i, j). More specifically, we use
inference algorithms to compute the pairwise beliefs
Pq over neighboring nodes b ij (xi , xj ),
and calculate the pairwise correlation as p(i, j) = t=1 bij (t, t).
In cases where the maximal clique size is small enough, we can calculate p(i, j) exactly
using the junction tree algorithm. In all other cases we must resort to approximate inference using the BP and the GBP algorithms. The following subsections discuss exact and
approximate inference for computing typical cuts.
3.1
Exact inference for typical cut clustering
The nature of real life clustering problems seems to suggest that exact inference would be
intractable due to the clique size of the junction tree. Surprisingly, in our empirical studies, we discovered that on many datasets, including benchmark problems from the UCI
repository, we obtain ?thin? junction trees (with maximal clique size less than 20). Figure 2a shows a two dimensional representative result. The temperature parameter T was
automatically chosen to provide two large clusters. As shown previously by Gdalyahu et al
the typical cut criterion does sensible things: it does not favor segmentation of individual
datapoints (as in minimal cut), nor is it fooled by narrow bridges between clusters (as in
simple connected components). However, while previous typical cut algorithms approximate p(i, j) using MCMC, in some cases using the framework of graphical model we can
calculate p(i, j) exactly and efficiently.
300
3.2
200
3
2.8
Figure 2: Clustering ex-
100
amples with clusters indicated by different markers.
In example (a) the pairwise correlations were calculated exactly, while in example (b) we used BP.
2.6
0
2.4
?100
2.2
2
?200
1.8
1.8
2
2.2
2.4
2.6
a.
3.2
2.8
3
3.2
?300
?300
?200
?100
0
100
200
300
b.
Approximate inference for typical cut clustering
Although exact inference is shown to be possible, in the more common case it is infeasible,
and p(i, j) can only be estimated using approximate inference algorithms. In this section
we discuss approximate inference using the BP and the GBP algorithms.
Approximate inference using Belief Propagation In BP the pairwise beliefs over neighboring nodes, bij , are defined using the messages as:
Y
Y
bij (xi , xj ) = ??ij (xi , xj )
mki (xi )
mkj (xj )
(7)
xk ?N (xi )\xj
xk ?N (xj )\xi
Can this be used as an approximation for pairwise clustering?
Observation 2: In case where the messages are initialized uniformly the pairwise beliefs
calculated by BP are only a function of the local potentials, i.e bij (xi , xj ) ? ?ij (xi , xj ).
Proof: Due to the symmetry of the potentials and since the messages are initialized uniformly, all the messages in BP remain uniform. Thus equation 7 will simply give the
normalized local potentials.
A consequence of observation 2 is that we need to break the symmetry of the problem in
order to use BP. We use here the method of conditioning. Due to the symmetry of the
potentials, if exact inference is used then conditioning on a single node xc = 1 and calculating conditional correlations P (xi = xj |xc = 1) should give exactly the same answer
as the unconditional correlations p(i, j) = P (xi = xj ). However, when BP inference
is used, clamping the value of xc causes its outgoing messages to be nonuniform, and as
these messages propagate through the graph they break the symmetry used in the proof of
observation 2. Empirically, this yields much better approximations of the correlations. In
some cases (e.g. when the graph is disconnected) conditioning on a single point does not
break the symmetry throughout the graph and additional points need to be clamped.
In order to evaluate the quality of the approximation provided by BP, we compared BP using conditioning and exact inference over the dataset shown in fig 2a. Figure 3 displays the
results at two different temperatures: ?low? and ?high?. Each row presents the clustering
solution of exact inference and BP, and a scatter plot of the correlations over all of the edges
using the two methods. At the ?low? temperature the approximation almost coincides with
the exact values, but at the ?high? temperature BP over estimates the correlation values.
3.2
Exact
3
2.8
2.8
2.6
2.6
2.4
2.4
2.2
2.2
2
2
1.8
low T
1.8
3.2
3
2
1.8
2.2
2.4
2.6
2.8
3
3.2
3
2.8
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2.6
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2.4
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2.2
2
2
1.8
high T
1.8
2
1.8
2.2
2.4
2.6
2.8
3
3.2
1
low T
1.8
3.2
Exact
BP
Exact correlations
3
2
2.4
2.6
2.8
3
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
0.1
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1
0
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1
3.2
BP
Loopy correlations
1
high T
1.8
0.9
0
2.2
Exact correlations
3.2
2
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
2.2
2.4
2.6
2.8
3
3.2
Loopy correlations
Figure 3: Clustering results at a ?low? temperature (upper row) and a ?high? temperature (lower
row). The left and middle columns present clustering results of exact inference and of BP, respectively. The right column compares the values of the correlations provided by the two methods. Each
dot corresponds to an edge in the graph. At ?low? temperature most of the correlations are close to
1, hence many edges appear as a single dot.
Approximate inference using Generalized Belief Propagation Generalized Belief
Propagation algorithms (GBP) [10] extend the BP algorithm by sending messages that are
functions of clusters of variables, and has been shown to provide a better approximation
than BP in many applications. Can GBP improve the approximation of pairwise correlations in typical cuts?
Our empirical studies show that the performance and convergence of GBP over a general
graph obtained from arbitrary points in Rn , strongly depends on the initial choice of clusters (regions). As also observed by Minka et al [5] a specific choice of clusters may yield
worse results than BP, or may even cause GBP not to converge. However it is far from
obvious how to choose these clusters. In previous uses of GBP [10] the basic clusters used
were chosen by hand. In order to use GBP to approximate p(i, j) in a general graph, one
must obtain a useful automatic procedure for selecting these initial clusters. We have experimented with various heuristics but none of them gave good performance. However, in
the case of ordered graphs such as 2D grids and images, we have found that GBP gives an
excellent approximation when using four neighboring grid points as a region.
Figure 4a shows results of GBP approximations for a 30x30 2D uniform grid. The clique
size in a junction tree is of order 230 hence exact inference is infeasible. We compare the
correlations p(i, j) calculated using an extensive MCMC sampling procedure [9] to those
calculated using GBP with the clusters being four neighboring pixels in the graph. GBP
converges in only 10 iterations and can be seen to provide an excellent approximation.
Figure 4c presents a comparison of the MCMC correlations with those calculated by GBP
on a real 120x80 image shown in fig 4b with affinity based on color similarity. Figure 4d
presents the clustering results, which provides a segmentation of the image.
1
0.9
0.8
0.8
0.7
0.7
0.6
0.6
GBP
GBP
1
0.9
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0
0
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SW?MC
(a)
0.6
0.8
0
0
1
0.2
0.4
0.6
0.8
1
SW?MC
(b)
(c)
(d)
Figure 4: (a) Scatter plot of pairwise correlations in a 30x30 grid, using MCMC [9] and GBP. Each
dot corresponds to the pairwise correlation of one edge at a specific temperature. Notice the excellent
correspondence between GBP and MCMC (c) The same comparison performed over the image in
(b). (d) shows a gray level map of the 15 largest clusters.
4
Learning Affinity Matrices from Labeled Datasets
As noted in the introduction using graphical models to compute typical cuts, can also be
advantageous for other aspects of the clustering problem, apart from computing p(i, j).
One such important advantage is learning the affinity matrix W (i, j) from labeled data.
In many problems, there are multiple ways to define affinities between any two datapoints.
For example, in image segmentation where the nodes are pixels, one can define affinity
based on color similarity, texture similarity or some combination of the two. Our goal is to
use a labeled training set of manually segmented images to learn the ?right? affinities.
More specifically let us assume the ?correct? affinity is a linear combination of a set
of known affinity functions {fk }K
k=1 , each corresponding to different features of the
data. Hence the affinity between neighboring points i and j, is defined by: W (i, j) =
PK
k=1 ?k fk (i, j). In addition assume we are given a labeled training sample, which consists of the following: (i) A graph in which neighboring nodes are connected by edges. (ii)
Affinity values fk (i, j). (iii) A partition of the graph x. Our goal is to estimate the affinity
mixing coefficients ?k .
This problem can be solved using the graphical model defined by the typical cut probability
distribution (Equation 6). Recall that the probability of a partition x is defined as
1 P
1 ? PK
1 ?cut(x)
k=1 ?k fcutk (x)
e
= e? <ij> (1??(xi ?xj ))W (i,j) =
e
(8)
Z
Z
Z(?)
P
Where we have defined: fcutk (x) = <ij> (1 ? ?(xi ? xj ))fk (i, j). fcutk (x) is the cut
value defined by x when only taking into account the affinity function fk , hence it can be
computed using the training sample. Differentiating the log likelihood with respect to ? k
gives the exponential family equation:
P (x) =
? ln P (x)
= ?fcutk (x)+ < fcutk >?
??k
(9)
Equation 9 gives an intuitive definition for the optimal ?: the optimal ? is the one for which
< fcutk >? = fcutk (x), i.e, for optimal ? the expected values of the cuts for each feature
separately, match exactly the values of these cuts in the training set.
Since we are dealing with the exponential family, the likelihood is convex and the ML
solution can be found using gradient ascent. To calculate the gradient explicitly, we use the
linearity of expectation:
X
X
< fcutk >? =
< (1 ? ?(yi ? yj ) >? fk (i, j) =
(1 ? p(i, j)? )fk (i, j)
<ij>
<ij>
Where p(i, j)? are the pairwise correlations for given values of ?.
Equation 9 is visually similar to the learning rule derived by Meila and Shi [4] but the cost
function they are minimizing is actually different, hence the expectations are taken with
respect to completely different distributions.
4.1
Combining learning and GBP approximate inference
We experimented with the learning algorithm on images, with the pixels grid as the graph
and using GBP for approximating p(i, j)? . The three pixel affinity functions, {fk }3k=1 ,
correspond to the intensity differences in the R, G, B color channels. We used a standard
transformation of intensity difference to an affinity function by a Gaussian kernel.
The left pane in Fig 5 shows a synthetic example. There is one training image (fig 5a) but
two different manual segmentations (fig 5b,c). The first and second training segmentations
are based on an illumination-covariant and an illumination-invariant affinities, respectively.
We used gradient ascent as given by equation 9. Figure 5d shows a novel image and figures 5e,f show two different pairwise correlations of this image using the learned ?. Indeed,
the algorithm learns to either ignore or not ignore illumination, based on the training set.
The right pane in figure 5 shows results on real images. For real images, we found that
a preprocessing of the image colors is required in order to learn shadow-invariant linear
transformation. This was done by saturating the image colors. The training segmentation
(figures 5a,b,c) ignores shadows. On the novel image (figure 5d) the most salient edge is a
shadow on the face. Nevertheless, the segmentation based on the learned affinity (figure 5e)
ignores the shadows and segments the facial features from each other. In contrast, a typical
cut segmentation which uses a naive affinity function (combining the three color channels
with uniform weights) segments mostly based on shadows (figure 5f).
5
Discussion
Pairwise clustering algorithms have a wide range of applicability due to their ability to find
clusters with arbitrary shapes. In this paper we have shown how pairwise clustering can be
(a)
(b)
(c)
(a)
(b)
(c)
(d)
(e)
(f)
(d)
(e)
(f)
Figure 5: Left pane: A synthetic example for learning the affinity function. The top row presents the
training set: The input image (a), the clusters of the first (b) and second (c) experiments. The bottom
row presents the result of the learning algorithm: The input image (d), the marginal probabilities
p(i, j) (Eqn. 3) in the first (e) and second (f) experiments. Right pane: Learning a color affinity
function which is invariant to shadows. The top row shows the learning data set: The input image(a),
the pre-processed image (b) and the manual segmentation (invariant to shadows) (c). The bottom row
presents, from left to right, the pre-processed test image (d), an edge map produced by learning the
shadow-invariant affinity (e) and an edge map produced by a naive affinity function, combining the 3
color channels with uniform weights (f). The edge maps were computed by thresholding the pairwise
correlations p(i,j) (Eqn. 3). See text for details. Both illustrations are better viewed in color.
mapped to an inference problem in a graphical model. This equivalence allowed us to use
the standard tools of graphical models: exact and approximate inference and ML learning.
We showed how to combine approximate inference and ML learning in the challenging
problem of learning affinities for images from labeled data. We have only begun to use the
many tools of graphical models. We are currently working on learning from unlabeled sets
and on other approximate inference algorithms.
References
[1] M. Blatt, S. Wiseman, and E. Domany. Data clustering using a mode lgranular magnet. Neural
Computation, 9:1805?1842, 1997.
[2] Y. Gdalyahu, D. Weinshall, and M. Werman. Self organization in vision: Stochastic clustering
for image segmentation, perceptual grouping, and image database organization. IEEE Trans. on
Pattern Analysis and Machine Intelligence, 23(10):1053?1074, 2001.
[3] T. Hofmann and J. M. Buhmann. Pairwise data clustering by deterministic annealing. IEEE
Transactions on Pattern Analysis and Machine Intelligence, 19(1):1?14, 1997.
[4] M. Meila and J. Shi. Learning segmentation by random walks. In Advances in Neural Information Processing Systems 14, 2001.
[5] T. Minka and Y. Qi. Tree-structured approximations by expectation propagation. In Advances in
Neural Information Processing Systems 16, 2003.
[6] A. Ng, M. Jordan, and Y. Weiss. On spectral clustering: Analysis and an algorithm. In Advances
in Neural Information Processing 14, 2001.
[7] N. Shental, A. Zomet, T. Hertz, and Y. Weiss. Learning and inferring image segmentations using
the gbp typical cut. In 9th International Conference on Computer Vision, 2003.
[8] J. Shi and J. Malik. Normalized cuts and image segmentation. In Proc. IEEE Conf. Computer
Vision and Pattern Recognition, pages 731?737, 1997.
[9] J.S. Wang and R.H Swendsen. Cluster monte carlo algorithms. Physica A, 167:565?579, 1990.
[10] J. Yedidia, W. Freeman, and Y. Weiss. Understanding belief propagation and its generalizations.
In G. Lakemeyer and B. Nebel, editors, Exploring Artificial Intelligence in the New Millennium.
Morgan Kaufmann, 2003.
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1,693 | 2,539 | ?
Nonlinear processing in LGN neurons
Vincent Bonin* , Valerio Mante and Matteo Carandini
Smith-Kettlewell Eye Research Institute
2318 Fillmore Street
San Francisco, CA 94115, USA
Institute of Neuroinformatics
University of Zurich and ETH Zurich
Winterthurerstrasse 190
CH-8046 Zurich, Switzerland
{vincent,valerio,matteo}@ski.org
Abstract
According to a widely held view, neurons in lateral geniculate
nucleus (LGN) operate on visual stimuli in a linear fashion. There
is ample evidence, however, that LGN responses are not entirely
linear. To account for nonlinearities we propose a model that
synthesizes more than 30 years of research in the field. Model
neurons have a linear receptive field, and a nonlinear, divisive
suppressive field. The suppressive field computes local root-meansquare contrast. To test this model we recorded responses from
LGN of anesthetized paralyzed cats. We estimate model parameters
from a basic set of measurements and show that the model can
accurately predict responses to novel stimuli. The model might
serve as the new standard model of LGN responses. It specifies
how visual processing in LGN involves both linear filtering and
divisive gain control.
1
In t rod u ct i on
According to a widely held view, neurons in lateral geniculate nucleus (LGN)
operate linearly (Cai et al., 1997; Dan et al., 1996). Their response L(t) is the
convolution of the map of stimulus contrast S(x,t) with a receptive field F(x,t):
L (t ) = [ S? F ] ( 0, t )
The receptive field F(x,t) is typically taken to be a difference of Gaussians in space
(Rodieck, 1965) and a difference of Gamma functions in time (Cai et al., 1997).
This linear model accurately predicts the selectivity of responses for spatiotemporal
frequency as measured with gratings (Cai et al., 1997; Enroth-Cugell and Robson,
1966). It also predicts the main features of responses to complex dynamic video
sequences (Dan et al., 1996).
150 spikes/s
Data
Model
Figure 1. Response of an LGN neuron to a dynamic video sequence along with the
prediction made by the linear model. Stimuli were sequences from Walt Disney?s
?Tarzan?. From Mante et al. (2002).
The linear model, however, suffers from limitations. For example, consider the
response of an LGN neuron to a complex dynamic video sequences (Figure 1). The
response is characterized by long periods of relative silence interspersed with brief
events of high firing rate (Figure 1, thick traces). The linear model (Figure 1, thin
traces) successfully predicts the timing of these firing events but fails to account for
their magnitude (Mante et al., 2002).
The limitations of the linear model are not surprising since there is ample evidence
that LGN responses are nonlinear. For instance, responses to drifting gratings
saturate as contrast is increased (Sclar et al., 1990) and are reduced, or masked, by
superposition of a second grating (Bonin et al., 2002). Moreover, responses are
selective for stimulus size (Cleland et al., 1983; Hubel and Wiesel, 1961; Jones and
Sillito, 1991) in a nonlinear manner (Solomon et al., 2002).
We propose that these and other nonlinearities can be explained by a nonlinear
model incorporating a nonlinear suppressive field. The qualitative notion of a
suppressive field was proposed three decades ago by Levick and collaborators
(1972). We propose that the suppressive field computes local root-mean-square
contrast, and operates divisively on the receptive field output.
Basic elements of this model appeared in studies of contrast gain control in retina
(Shapley and Victor, 1978) and in primary visual cortex (Cavanaugh et al., 2002;
Heeger, 1992; Schwartz and Simoncelli, 2001). Some of these notions have been
applied to LGN (Solomon et al., 2002), to fit responses to a limited set of stimuli
with tailored parameter sets. Here we show that a single model with fixed
parameters predicts responses to a broad range of stimuli.
2
Mod el
In the model (Figure 2), the linear response of the receptive field L(t) is divided by
the output of the suppressive field. The latter is a measure of local root-mean-square
contrast c local. The result of the division is a generator potential
V (t ) = Vmax
L (t )
c50 + clocal
,
where c 50 is a constant.
F(x,t)
Stimulus
S(x,t)
V0
L(t)
Receptive Field
R(t)
Firing
rate
Rectification
H(x,t)
S*(x,t)
c50
clocal
Suppressive Field
Filter
Figure 2. Nonlinear model of LGN responses.
The suppressive field operates on a filtered version of the stimulus, S*=S*H, where
H is a linear filter and * denotes convolution. The squared output of the suppressive
field is the local mean square (the local variance) of the filtered stimulus:
clocal = ?? S ( x, t ) G ( x ) dx dt ,
2
2
*
where G(x) is a 2-dimensional Gaussian.
Firing rate is a rectified version of generator potential, with threshold V thresh:
R(t ) = ??V (t ) ? Vthresh ?? + .
To test the nonlinear model, we recorded responses from neurons in the LGN of
anesthetized paralyzed cats. Methods for these recordings were described elsewhere
(Freeman et al., 2002).
3
Resu l t s
We proceed in two steps: first we estimate model parameters by fitting the model to
a large set of canonical data; second we fix model parameters and evaluate the
model by predicting responses to a novel set of stimuli.
B
60
40
20
0
0.01
0.2
4
Spatial Frequency (cpd)
Response (spikes/s)
Response (spikes/s)
A
80
60
40
20
0
0.5
5
50
Temporal Frequency (Hz)
Figure 3. Estimating the receptive field in an example LGN cell. Stimuli are
gratings varying in spatial (A) and temporal (B) frequency. Responses are the
harmonic component of spike trains at the grating temporal frequency. Error bars
represent standard deviation of responses. Curves indicate model fit.
Response (spikes/s)
0.25 0.50 0.75
Test contrast
100
80
60
40
20
0
1.00
100
D
80
60
40
20
0
0.5
Response (spikes/s)
50
0
C
B
100
Response (spikes/s)
Response (spikes/s)
A
0.25 0.50 0.75 1.00
Mask contrast
100
80
60
40
20
4.0
32.0
Mask diameter (deg)
0
0.01
0.20
4.00
Mask spatial frequency (cpd)
Figure 4. Estimating the suppressive field in the example LGN cell. Stimuli are
sums of a test grating and a mask grating. Responses are the harmonic component of
spike trains at the temporal frequency of test. A: Responses to test alone. B-D:
Responses to test+mask as function of three mask attributes: contrast (B), diameter
(C) and spatial frequency (D). Gray areas indicate baseline response (test alone,
50% contrast). Dashed curves are predictions of linear model. Solid curves indicate
fit of nonlinear model.
3.1
C h a r a c te r i z i n g t he r e c e p ti v e fi e l d
We obtain the parameters of the receptive field F(x,t) from responses to large
drifting gratings (Figure 3). These stimuli elicit approximately constant output in
the suppressive field, so they allow us to characterize the receptive field. Responses
to different spatial frequencies constrain F(x,t) in space (Figure 3A). Responses to
different temporal frequencies constrain F(x,t) in time (Figure 3B).
3.2
C h a r a c te r i z i n g t he s u p p r e s s i v e f i e l d
To characterize the divisive stage, we start by measuring how responses saturate at
high contrast (Figure 4A). A linear model cannot account for this contrast saturation
(Figure 4A, dashed curve). The nonlinear model (Figure 4A, solid curve) captures
saturation because increases in receptive field output are attenuated by increases in
suppressive field output. At low contrast, no saturation is observed because the
output of the suppressive field is dominated by the constant c 50. From these data we
estimate the value of c50.
To obtain the parameters of the suppressive field, we recorded responses to sums of
two drifting gratings (Figure 4B-D): an optimal test grating at 50% contrast, which
elicits a large baseline response, and a mask grating that modulates this response.
Test and mask temporal frequencies are incommensurate so that they temporally
label a test response (at the frequency of the test) and a mask response (at the
frequency of the mask) (Bonds, 1989). We vary mask attributes and study how they
affect the test responses.
Increasing mask contrast progressively suppresses responses (Figure 4B). The linear
model fails to account for this suppression (Figure 4B, dashed curve). The nonlinear
model (Figure 4B, solid curve) captures it because increasing mask contrast
increases the suppressive field output while the receptive field output (at the
temporal frequency of the test) remains constant. With masks of low contrast there
is little suppression because the output of the suppressive field is dominated by the
constant c 50 .
Similar effects are seen if we increase mask diameter. Responses decrease until they
reach a plateau (Figure 4C). A linear model predicts no decrease (Figure 4C, dashed
curve). The nonlinear model (Figure 4C, solid curve) captures it because increasing
mask diameter increases the suppressive field output while it does not affect the
receptive field output. A plateau is reached once masks extend beyond the
suppressive field. From these data we estimate the size of the Gaussian envelope
G(x) of the suppressive field.
Finally, the strength of suppression depends on mask spatial frequency (Figure 4D).
At high frequencies, no suppression is elicited. Reducing spatial frequency increases
suppression. This dependence of suppression on spatial frequency is captured in the
nonlinear model by the filter H(x,t). From these data we estimate the spatial
characteristics of the filter. From similar experiments involving different temporal
frequencies (not shown), we estimate the filter?s selectivity for temporal frequency.
3.3
P r e d i c ti n g r e s p o n s e s t o n o v e l s ti m u l i
We have seen that with a fixed set of parameters the model provides a good fit to a
large set of measurements (Figure 3 and Figure 4). We now test whether the model
predicts responses to a set of novel stimuli: drifting gratings varying in contrast and
diameter.
Responses to high contrast stimuli exhibit size tuning (Figure 5A, squares): they
grow with size for small diameters, reach a maximum value at intermediate diameter
and are reduced for large diameters (Jones and Sillito, 1991). Size tuning , however,
strongly depends on stimulus contrast (Solomon et al., 2002): no size tuning is
observed at low contrast (Figure 5A, circles). The model predicts these effects
(Figure 5A, curves). For large, high contrast stimuli the output of the suppressive
field is dominated by c local, resulting in suppression of responses. At low contrast,
c local is much smaller than c50, and the suppressive field does not affect responses.
Similar considerations can be made by plotting these data as a function of contrast
(Figure 5B). As predicted by the nonlinear model (Figure 5B, curves), the effect of
increasing contrast depends on stimulus size: responses to large stimuli show strong
saturation (Figure 5B, squares), whereas responses to small stimuli grow linearly
(Figure 5B, circles). The model predicts these effects because only large, high
contrast stimuli elicit large enough responses from the suppressive field to cause
suppression. For small, low contrast stimuli, instead, the linear model is a good
approximation.
B
100
Response (spikes/s)
A
80
60
40
20
0
0.50
4.00
32.00
Diameter (deg)
0.00 0.25 0.50 0.75 1.00
Contrast
Figure 5. Predicting responses to novel stimuli in the example LGN cell. Stimuli are
gratings varying in diameter and contrast, and responses are harmonic component of
spike trains at grating temporal frequency. Curves show model predictions based on
parameters as estimated in previous figures, not fitted to these data. A: Responses as
function of diameter for different contrasts. B: Responses as function of contrast for
different diameters.
3.4
M o d e l pe r f or m a nc e
To assess model performance across neurons we calculate the percentage of
variance in the data that is explained by the model (see Freeman et al., 2002 for
methods).
The model provides good fits to the data used to characterize the suppressive field
(Figure 4), explaining more than 90% of the variance in the data for 9/13 cells
(Figure 6A). Model parameters are then held fixed, and the model is used to predict
responses to gratings of different contrast and diameter (Figure 5). The model
performs well, explaining in 10/13 neurons above 90% of the variance in these
novel data (Figure 6B, shaded histogram). The agreement between the quality of the
fits and the quality of the predictions suggests that model parameters are well
constrained and rules out a role of overfitting in determining the quality of the fits.
To further confirm the performance of the model, in an additional 54 cells we ran a
subset of the whole protocol, involving only the experiment for characterizing the
receptive field (Figure 3), and the experiment involving gratings of different
contrast and diameter (Figure 5). For these cells we estimate the suppressive field
by fitting the model directly to the latter measurements. The model explains above
90% of the variance in these data in 20/54 neurons and more than 70% in 39/54
neurons (Figure 6B, white histogram).
Considering the large size of the data set (more than 100 stimuli, requiring several
hours of recordings per neuron) and the small number of free parameters (only 6 for
the purpose of this work), the overall, quality of the model predictions is
remarkable.
Estimating the suppressive field
A
# cells
6
n=13
4
2
0
Size tuning at different contrasts
15 n=54
10
# cells
B
5
0
0
50
100
Explained variance (%)
Figure 6. Percentage of variance in data explained by model. A: Experiments to
estimate the suppressive field. B: Experiments to test the model. Gray histogram
shows quality of predictions. White histogram shows quality of fits.
4
Co n cl u si o n s
The nonlinear model provides a unified description of visual processing in LGN
neurons. Based on a fixed set of parameters, it can predict both linear properties
(Figure 3), as well as nonlinear properties such as contrast saturation (Figure 4A)
and masking (Figure 4B-D). Moreover, once the parameters are fixed, it predicts
responses to novel stimuli (Figure 5).
The model explains why responses are tuned for stimulus size at high contrast but
not at low contrast, and it correctly predicts that only responses to large stimuli
saturate with contrast, while responses to small stimuli grow linearly.
The model implements a form of contrast gain control. A possible purpose for this
gain control is to increase the range of contrast that can be transmitted given the
limited dynamic range of single neurons. Divisive gain control may also play a role
in population coding: a similar model applied to responses of primary visual cortex
was shown to maximize independence of the responses across neurons (Schwartz
and Simoncelli, 2001).
We are working towards improving the model in two ways. First, we are
characterizing the dynamics of the suppressive field, e.g. to predict how it responds
to transient stimuli. Second, we are testing the assumption that the suppressive field
computes root-mean-square contrast, a measure that solely depends on the secondorder moments of the light distribution.
Our ultimate goal is to predict responses to complex stimuli such as those shown in
Figure 1 and quantify to what degree the nonlinear model improves on the
predictions of the linear model. Determining the role of visual nonlinearities under
more natural stimulation conditions is also critical to understanding their function.
The nonlinear model synthesizes more than 30 years of research. It is robust,
tractable and generalizes to arbitrary stimuli. As a result it might serve as the new
standard model of LGN responses. Because the nonlinearities we discussed are
already present in the retina (Shapley and Victor, 1978), and tend to get stronger as
one ascends the visual hierarchy (Sclar et al., 1990), it may also be used to study
how responses take shape from one stage to another in the visual system.
A c k n o w l e d g me n t s
This work was supported by the Swiss National Science Foundation and by the
James S McDonnell Foundation 21st Century Research Award in Bridging Brain,
Mind & Behavior.
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1,694 | 254 | 44
Beer and Chiel
Neural Implementation of Motivated Behavior:
Feeding in an Artificial Insect
Randall D. Beerl,2 and Hillel J. Chiel 2
Departments of 1 Computer Engineering and Science, and 2Biology
and the Center for Automation and Intelligent Systems Research
Case Western Reserve University
Cleveland, OH 44106
ABSTRACT
Most complex behaviors appear to be governed by internal motivational states or drives that modify an animal's responses to its
environment. It is therefore of considerable interest to understand
the neural basis of these motivational states. Drawing upon work
on the neural basis of feeding in the marine mollusc Aplysia, we
have developed a heterogeneous artificial neural network for controlling the feeding behavior of a simulated insect. We demonstrate
that feeding in this artificial insect shares many characteristics with
the motivated behavior of natural animals.
1
INTRODUCTION
While an animal's external environment certainly plays an extremely important role
in shaping its actions, the behavior of even simpler animals is by no means solely
reactive. The response of an animal to food, for example, cannot be explained only
in terms of the physical stimuli involved. On two different occasions, the very same
animal may behave in completely different ways when presented with seemingly
identical pieces of food (e.g. hungrily consuming it in one case and ignoring or
even avoiding it in another). To account for these differences, behavioral scientists
hypothesize internal motivational states or drives which modulate an animal's response to its environment. These internal factors playa particularly important role
in complex behavior, but are present to some degree in nearly all animal behavior. Behaviors which exhibit an extensive dependence on motivational variables are
termed motivated behaviors.
Neural Implementation or Motivated Behavior: Feeding in an Artificial Insect
While a rigorous definition is difficult to state, behaviors spoken of as motivated generally exhibit some subset of the following six characteristics (Kupfermann, 1974):
(1) grouping and sequencing of component behaviors in time, (2) goal-directedness:
the sequence of component behaviors generated can often be understood only by reference to some internal goal, (3) spontaneity: the behavior can occur in the absence
of any recognizable eliciting stimuli, (4) changes in responsiveness: the effect of a
motivational state varies depending upon an animal's level of arousal, (5) persistence: the behavior can greatly outlast any initiating stimulus, and (6) associative
learning.
Motivational states are pervasive in mammalian behavior. However, they have
also proven to be essential for explaining the behavior of simpler animals as well.
Unfortunately, the explanatory utility of these internal factors is limited by the fact
that they are hypothetical constructs, inferred by the theorist to intervene between
stimulus and action in order to account for otherwise inexplicable responses. What
might be the neural basis of these motivational states?
In order to explore this question, we have drawn upon work on the neural basis
of feeding in the marine mollusc Aplysia to implement feeding in a simulated insect. Feeding is a prototypical motivated behavior in which attainment of the goal
object (food) is clearly crucial to an animal's survival. In this case, the relevant
motivational state is hunger. When an animal is hungry, it will exhibit a sequence
of appetitive behaviors which serve to identify and properly orient the animal to
food. Once food is available, consummatory behaviors are generated to ingest it.
On the other hand, a satiated animal may ignore or even avoid sensory stimuli
which suggest the presence of food (Kupfermann, 1974).
This effort is part of a larger project aimed at designing artificial nervous systems
for the flexible control of complete autonomous agents (Beer, 1989). In addition
to feeding, this artificial insect is currently capable of locomotion (Beer, Chiel, and
Sterling, 1989; Chiel and Beer, 1989), wandering, and edge-following, and possesses
a simple behavioral hierarchy as well. A central theme of this work has been the
utilization of biologically-inspired architectures in our neural network designs. To
support this capability, we make use of model neurons which capture some of the
intrinsic properties of nerve cells.
The simulated insect and the environment in which it exists is designed as follows. The insect has six legs, and is capable of statically stable locomotion and
turning. Its head contains a mouth which can open and close, and its mouth and
two antennae possess tactile and chemical sensors. The insect possesses an internal
energy supply which is depleted at a fixed rate. The simulated environment also
contains unmovable obstacles and circular food patches. The food patches emit an
odor whose intensity is proportional to the size of the patch. As this odor diffuses
through the environment, its intensity falls off as the inverse square of the distance
from the center of the patch. Whenever the insect's mouth closes over a patch of
food, a fixed amount of energy is transferred from the patch to the insect.
45
46
Beer and Chiel
Anlenna Chemical Sensor
Anlenna Chemical Sensor
left Turn
Righi Turn
Feeding Arousal
Energy Sensor
Figure 1: Appetitive Controller
2
APPETITIVE COMPONENT
The appetitive component of feeding is responsible for getting a hungry insect to a
food patch. To accomplish this task, it utilizes the locomotion, wandering, and edgefollowing capabilities of the insect. The interactions between the neural circuitry
underlying these behaviors and the feeding controller presented in this paper are
described elsewhere (Beer, 1989). Assuming that the insect is already close enough
to a food patch that the chemical sensors in its antennae can detect an odor signal,
there are two separate issues which must be addressed by this phase of the behavior.
First, the insect must use the information from the chemical sensors in its antennae
to turn itself toward the food patch as it walks. Second, this orientation should only
occur when the insect is actually in need of energy. Correspondingly, the appetitive
neural controller (Figure 1) consists of two distinct components.
The orientation component is comprised of the upper six neurons in Figure 1. The
odor signals detected by the chemical sensors in each antenna (ACS) are compared
(by LOS and ROS), and the difference between them is used to generate a turn
toward the stronger side by exciting the corresponding turn interneuron (LT or RT)
by an amount proportional to the size of the difference. These turn interneurons
connect to the motor neurons controlling the lateral extension of each front leg.
The second component is responsible for controlling whether or not the insect ac-
Neural Implementation of Motivated Behavior: Feeding in an Artificial Insect
tually orients to a nearby patch of food. This decision depends upon its internal
energy level, and is controlled by the bottom three neurons in Figure 1. Though the
odor gradient is continuously being sensed, the connections to the turn interneurons
are normally disabled, preventing access of this information to the motor apparatus which turns the insect. As the insect's energy level falls, however, so does the
activity of its energy sensor (ES). This decreasing activity gradually releases the
spontaneously active feeding arousal neuron (FA) from inhibition. When activity
in FA becomes sufficient to fire the search command neuron (SC), the connections
between the odor strength neurons and the turn neurons are enabled by gating
connections from SC, and the insect begins to orient to food.
3
CONSUMMATORY COMPONENT
Once the appetitive controller has successfully oriented the insect to food, the consummatory component of the behavior is triggered. This phase consists of rhythmic
biting movements which persist until sufficient food has been ingested. Like the appetitive phase, consummatory behavior should only be released when the insect is
in need of energy. In addition, an animal's interest in feeding (its feeding arousa~,
may be a function of more than just its energy requirements. Other factors, such as
the exposure of an animal to the taste, odor, or tactile sensations of food, can significantly increase its feeding arousal. This relationship between feeding and arousal,
in which the very act of feeding further enhances an animal's interest in feeding,
leads to a form of behavioral hysteresis. Once food is encountered, an animal may
feed well beyond the internal energy requirements which initiated the behavior. In
many animals, this hysteresis is thought to playa role in the patterning of feeding
behavior into discrete meals rather than continuous grazing (Susswein, Weiss, and
Kupfermann, 1978). At some point, of course, the ingested food must be capable
of overriding the arousing effects of consummatory behavior, or the animal would
never cease to feed.
The neural controller for the consummatory phase of feeding is shown in Figure
2. When chemical (MCS) and tactile (MTS) sensors in the mouth signal that
food is present (FP), and the insect is sufficiently aroused to feeding (FA), the
consummatory command neuron (CC) fires. The conjunction of tactile and chemical
signals is required in order to prevent attempts to ingest nonfood patches and, due
to the diffusion of odors, to prevent biting from beginning before the food patch
is actually reached. Once CC fires, it triggers the bite pacemaker neuron (BP)
to generate rhythmic bursts which cause a motor neuron (MO) to open and close
the mouth. Because the threshold of the consummatory command neuron (CC) is
somewhat lower than that of the search command neuron (SC), an insect which
is not sufficiently aroused to orient to food may nevertheless consume food that is
directly presented to its mouth.
The motor neuron controlling the mouth also makes an excitatory connection onto
the feeding arousal neuron (FA), which in turn makes an excitatory modulatory
synapse onto the connection between the consummatory command neuron (CC)
47
48
Beer and Chiel
Mouth Chemical Sensor
Mouth Tact~e Sensor
Energy Sensor
Mouth Open
Figure 2: Consummatory Controller
and the bite pacemaker (BP). The net effect of these excitatory connections is a
positive feedback loop: biting movements excite FA, which causes BP to cause more
frequent biting movements, which further excites FA until its activity saturates.
This neural positive feedback loop is inspired by work on the neural basis of feeding
arousal maintenance in Aplysia (Weiss, Chiel, Koch, and Kupfermann, 1986).
As the insect consumes food, its energy level begins to rise. This leads to increased
activity in ES which both directly inhibits FA, and also decreases the gain of the
positive feedback loop via an inhibitory modulatory synapse onto the connection
between MO and FA. At some point, these inhibitory effects will overcome the
positive feedback and activity in FA will drop low enough to terminate the feeding
behavior. This neural mechanism is based upon a similar one hypothesized to
underlie satiation in Aplysia (Weiss, Chiel, and Kupfermann, 1986).
4
RESULTS
With the neural controllers described above, we have found that feeding behavior
in the artificial insect exhibits four of the six characteristics of motivated behavior
which were described by Kupfermann (1974):
Neural Implementation or Motivated Behavior: Feeding in an Artificial Insect
Grouping and sequencing of behavior in time. When the artificial insect
is "hungry", it generates appetitive and consummatory behaviors with the proper
sequence, timing, and intensity in order to obtain food.
Goal-Directedness. Regardless of its environmental situation, a hungry insect
will generate movements which serve to obtain food. Therefore, the behavior of a
hungry insect can only be understood by reference to an internal goal. Due to the
internal effects of the energy sensor (ES) and feeding arousal (FA) neurons on the
controllers, the insect's external stimuli are insufficient to account for its behavior.
Changes in responsiveness due to a change in internal state. While a
hungry insect will attempt to orient to and consume any nearby food, a satiated
one will ignore it. In addition, once a hungry insect has consumed sufficient food, it
will simply walk over the food patch which initially attracted it. We will examine
the arousal and satiation of feeding in this artificial insect in more detail below.
Persistence. If a hungry insect is removed from food before it has fed to satiation,
its feeding arousal will persist, and it will continue to exhibit feeding movements.
One technique that has been applied to the study of feeding arousal in natural animals is the examination of the time interval between successive bites as an animal
feeds under various conditions. In Aplysia, for example, the interbite interval progressively decreases as an animal begins to feed (showing a build-up of arousal),
and increases as the animal satiates. In addition, the rate of rise and fall of arousal
depends upon the initial degree of satiation (Susswein, Weiss, and Kupfermann,
1978).
In order to examine the role of feeding arousal in the artificial insect, we performed
a similar set of experiments. Food was directly presented to insects with differing degrees of initial satiation, and the time interval between successive bites was
recorded for the entire resulting consummatory response. Above an energy level
of approximately 80% of capacity, insects could not be induced to bite. Below
this level, however, insects began to consume the food. As these insects fed, the
interbite interval decreased as their feeding arousal built up until some minimum
interval was achieved (Figure 3). The rate of build-up of arousal was slowest for
those insects with the highest initial degree of satiation. In fact, an insect whose
energy level was already 75% of capacity never achieved full arousal. As the feeding
insects neared satiation, their interbite interval increased as arousal waned. It is
interesting to note that, regardless of the initial degree of satiation, all insects in
which biting was triggered fed until their energy stores were approximately 99%
full. The appropriate number of bites to achieve this were generated in all cases.
What is the neural basis of these arousal and satiation phenomena? Clearly, the
answer lies in the interactions between the internal energy sensor and the positive
feedback loop mediated by the feeding arousal neuron, but the precise nature of
the interaction is not at all clear from the qualitative descriptions of the neural
controllers given earlier. In order to more carefully examine this interaction, we
produced a phase plot of the activity in these two neurons under the experimental
49
50
Beer and Chiel
u
CD
en
E
.......
ca
-
t=
:s...
-
500
400
.
..
300
CD
.5
CD
CD
.5
200
Q
25% Satiation
50% Satiation
60% Satiation
75'Y. Satiation
100
0
0
10
20
30
40
50
Bite Number
Figure 3: Build-Up of Arousal and Satiation
conditions described above (Figure 4).
An insect with a full complement of energy begins at the lower right-hand corner of
the diagram, with maximum activity in ES and no activity in FA. As the insect's
energy begins to fall, it moves to the left on the ES axis until the inhibition from
ES is insufficient to hold FA below threshold. At this point, activity in FA begins
to increase. Since the positive feedback loop is not yet active because no biting has
occurred, a linear decrease in energy results in a linear increase in FA activity. If
no food is consumed, the insect continues to move along this line toward the upper
left of the diagram until its energy is exhausted.
However, if biting is triggered by the presence of food at the mouth, the relationship
between FA and ES changes drastically. As the insect begins consuming food,
activity in FA initially increases as arousal builds up, and then later decreases as
the insect satiates. Each "bump" corresponds to the arousing effects on FA of one
bite via the positive feedback loop and to the small increase of energy from the food
consumed in that bite. Trajectories are shown for energy levels of 25%, 50%, 60%,
65%, and 75% of capacity. The shape of these trajectories depend upon the activity
level of FA and the gain of the positive feedback loop in which it is embedded,
both of which in turn depend upon the negative feedback from the energy sensor.
We must therefore conclude that, even in this simple artificial insect, there is no
single neural correlate to "hunger". Instead, this motivational state is the result of
the complex dynamics of interaction between the feeding arousal neuron and the
internal energy sensor.
References
Beer, R. D. (1989). Intelligence as Adaptive Behavior: An Experiment in Computational Neuroethology. Ph.D. Dissertation, Dept. of Computer Engineering and
Science, Case Western lleserve University. Also available as Technical Report TR
Neural Implementation of Motivated Behavior: Feeding in an Artificial Insect
>
>
.....
~
.-o
?
?
u.
ES Activity
Figure 4: Phase Plot of FA vs. ES Activity
89-118, Center for Automation and Intelligent Systems Research.
Beer, R. D., Chiel, H. J. and Sterling, L. S. (1989). Heterogeneous Neural Networks for Adaptive Behavior in Dynamic Environments. In D.S. Touretzky (Ed.),
Advances in Neural Information Processing Systems 1 (pp. 577-585). San Mateo,
CA: Morgan Kaufmann Publishers.
Chiel, H. J. and Beer, R. D. (1989). A lesion study of a heterogeneous neural
network for hexapod locomotion. Proceedings of the International Joint Conference
on Neural Networks (IJCNN 89), pp. 407-414.
Kupfermann, I. J. (1974). Feeding behavior in Aplysia: A simple system for the
study of motivation. Behavioral Biology 10:1-26.
Susswein, A. J., Weiss, K. R. and Kupfermann, 1. (1978). The effects of food arousal
on the latency of biting in Aplysia. J. Compo Physiol. 123:31-41.
Weiss, K. R., Chiel, II. J., Koch, U. and Kupfermann, 1. (1986). Activity of an
identified histaminergic neuron, and its possible role in arousal of feeding behavior
in semi-intact Aplysia. J. Neuroscience 6(8):2403-2415.
Weiss, K. R., Chiel, II. J. and Kupfermann, I. (1986). Sensory function and gating
of histaminergic neuron C2 in Aplysia. J. Neuroscience 6(8):2416-2426.
SI
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1,695 | 2,540 | Gaussian Process Latent Variable Models for
Visualisation of High Dimensional Data
Neil D. Lawrence
Department of Computer Science,
University of Sheffield,
Regent Court, 211 Portobello Street,
Sheffield, S1 4DP, U.K.
[email protected]
Abstract
In this paper we introduce a new underlying probabilistic model for principal component analysis (PCA). Our formulation interprets PCA as a
particular Gaussian process prior on a mapping from a latent space to
the observed data-space. We show that if the prior?s covariance function constrains the mappings to be linear the model is equivalent to PCA,
we then extend the model by considering less restrictive covariance functions which allow non-linear mappings. This more general Gaussian process latent variable model (GPLVM) is then evaluated as an approach to
the visualisation of high dimensional data for three different data-sets.
Additionally our non-linear algorithm can be further kernelised leading
to ?twin kernel PCA? in which a mapping between feature spaces occurs.
1 Introduction
Visualisation of high dimensional data can be achieved through projecting a data-set onto
a lower dimensional manifold. Linear projections have traditionally been preferred due
to the ease with which they can be computed. One approach to visualising a data-set in
two dimensions is to project the data along two of its principal components. If we were
forced to choose a priori which components to project along, we might sensibly choose
those associated with the largest eigenvalues. The probabilistic reformulation of principal
component analysis (PCA) also informs us that choosing the first two components is also
the choice that maximises the likelihood of the data [11].
1.1 Integrating Latent Variables, Optimising Parameters
Probabilistic PCA (PPCA) is formulated as a latent variable model: given a set centred of
-dimensional data
and denoting the latent variable associated with each datapoint we may write the likelihood for an individual data-point under the PPCA model
as
"!
where is Gaussian distributed with unit covariance,
, and
can then be found1 by as
. The solution for
suming that is i.i.d. and maximising the likelihood of the data-set,
where
T
is the
design matrix.
Probabilistic principal component analysis and other latent variable models, such as factor
analysis (FA) or independent component analysis (ICA), require a marginalisation of the
latent variables and optimisation of the parameters. In this paper we consider the dual
approach of marginalising and optimising each . This probabilistic model also turns
out to be equivalent to PCA.
1.2 Integrating Parameters, Optimising Latent Variables
where is the th
By first specifying a prior distribution,
row of the matrix , and integrating over we obtain a marginalised likelihood for ,
354
where ,
then
!
T <
9 :!;!
>
4
?
$#&% ('*+ )
,
'
T
D#E%
4
,
#F@CB
6
tr ,
#"
+.-0/21
=
and !
#A@CB
"
T
87
(1)
. The corresponding log-likelihood is
4
6
tr ,
#"
T
(2)
Now that the parameters are marginalised we may focus on optimisation of the likelihood
with respect to the ! . The > gradients of (2) with respect to ! may be found as,
G
G
!
H :,
;
T
,
4
!
,
!
which implies that at our solution
"
;
T
,
!
H!
some algebraic manipulation of this formula [11] leads to
PRQ
Q
!
JILK
MON
T
where ILK is an
matrix ( is the dimension of the latent space) whose columns
QSTQ
6 T
, M is a
diagonal matrix whose U th element is VXW
are eigenvectors
of
4
Y[Z8\
]
+
^ ]:_ a`
, where b W is the U th eigenvalue of ;
T
, and N
is an arbitrary
QcdQ
orthogonal matrix2 . Note that the eigenvalue problem we have developed can easily be
shown to be equivalent to that solved in PCA (see e.g. [10]), indeed the formulation of
PCA in this manner is a key step in the development of kernel PCA [9] where ; T is
replaced with a kernel. Our probabilistic PCA model shares an underlying structure with
[11] but differs in that where they optimise we marginalise and where they marginalise we
optimise. The marginalised likelihood we are optimising in (1) is recognised as the product
of
independent Gaussian processes where the (linear) covariance function is given by
T<
e!6!
. Therefore a natural extension is the non-linearisation of the mapping from
latent space to the data space through the introduction of a non-linear covariance function.
1
As can the solution for f but since the solution for g
is not dependent on f we will disregard
it.
2
For independent component analysis the correct rotation matrix h must also be found, here we
have placed no constraints on the orientation of the axes so this matrix cannot be recovered.
2 Gaussian Process Latent Variable Models
We saw in the previous section how PCA can be interpreted as a Gaussian process ?mapping3 ? from a latent space to a data space where the locale of the points in latent space is
determined by maximising the Gaussian process likelihood with respect to ! . We will refer to models of this class as Gaussian process latent variable models (GPLVM).
Principal
[
inner prodcomponent analysis is a GPLVM where the process prior is based on the
uct matrix of ! , in this section we develop an alternative GPLVM by considering a prior
which allows for non-linear processes, specifically we focus on the popular ?RBF kernel?
which takes the form
Y 4
#
-
/ 1
4
T
4
<
_
where is the element in the
th row and th column of , , is a scale parameter and
denotes the Kronecker delta. Gradients of (2) with respect to the latent points can be
found through combining
>
G
G
,
,
6
T
,
4
,
with
\ via the chain rule. These gradients may be used in combination with (2) in a
non-linear optimiser such as scaled conjugate gradients (SCG) [7] to obtain a latent variable
representation of the data. Furthermore gradients with respect to the parameters of the
kernel matrix may be computed and used to jointly optimise ! , , and . The solution
for ! will naturally not be unique; even for the linear case described above the solution is
subject to an arbitrary rotation, here we may expect multiple local minima.
2.1 Illustration of GPLVM via SCG
To illustrate a simple Gaussian process latent variable model we turn to the ?multi-phase oil
flow? data [2]. This is a twelve dimensional data-set containing data of three known classes
corresponding to the phase of flow in an oil pipeline: stratified, annular and homogeneous.
In this illustration, for computational reasons, the data is sub-sampled to 100 data-points.
Figure 1 shows visualisations of the data using both PCA and our GPLVM algorithm which
required 766 iterations of SCG. The ! positions for the GPLVM model were initialised using PCA (see http://www.dcs.shef.ac.uk/~neil/gplvm/ for the MATLAB
code used).
The gradient based optimisation of the RBF based GPLVM?s latent space shows results
which are clearly superior (in terms of greater separation between the different flow domains) to those achieved by the linear PCA model. Additionally the use of a Gaussian
process to perform our ?mapping? means that there is uncertainty in the positions of the
points in the data space. For our formulation the level of uncertainty is shared across all 4
dimensions and thus may be visualised in the latent space. In Figure 1 (and subsequently)
this is done through varying the intensity of the background pixels.
Unfortunately, a quick analysis of the complexity ofD
the
algorithm shows that each gradient
operation, rendering the algorithm
step requires an inverse of the kernel matrix, an
impractical for many data-sets of interest.
3
Strictly speaking the model does not represent a mapping as a Gaussian process ?maps? to a
distribution in data space rather than a point.
4
This apparent weakness in the model may be easily rectified to allow different levels of uncertainty for each output dimension, our more constrained model allows us to visualise this uncertainty
in the latent space and is therefore preferred for this work.
0.3
2.4
0.2
0.2
2.2
0.15
2
0.1
0.1
1.8
0.05
0
1.6
0
?0.1
1.4
?0.05
1.2
?0.2
?0.1
1
?0.15
?0.3
0.8
?0.2
?0.4
0.6
?0.25
0.4
?0.5
?0.8
?0.6
?0.4
?0.2
0
0.2
0.4
0.6
?0.2
?0.1
0
0.1
0.2
0.3
Figure 1: Visualisation of the Oil data with (a) PCA (a linear GPLVM) and (b) A GPLVM which
uses an RBF kernel. Crosses, circles and plus signs represent stratified, annular and homogeneous
flows respectively. The greyscales in plot (b) indicate the precision with which the manifold was
,
expressed in data-space for that latent point. The optimised parameters of the kernel were
and f
.
2.2 A Practical Algorithm for GPLVMs
There are three main components to our revised, computationally efficient, optimisation
process:
Sparsification. Kernel methods may be sped up through sparsification, i.e. representing
the data-set by a subset, , of ! points known as the active set. The remainder, the inactive
set, is denoted by . We make use of the informative vector machine [6] which selects
points sequentially according to the reduction in the posterior process?s entropy that they
induce.
Latent Variable Optimisation. A point from the inactive set, U , can be shown to project
into the data space as a Gaussian distribution
W W
W
W W
(3)
W where , denotes the kernel matrix developed from
whose mean is W T ,
4
the U th column of
the active set and W is a column vector consisting of the elements from
T
, that correspond to the active set. The variance is W
W Note
W W
W ,
that since W does not appear in the inverse, gradients with respect to W do not depend on
other data in . We can therefore independently optimise the likelihood of each W with
respect to each W . Thus the full set !
can be optimised with one pass through the data.
Kernel Optimisation. The likelihood of the active set is given by
3 4
#&%
'
+
"
,
which
can be optimised5 with respect to ,
!
#"
- /21
+
`
and
, 7
T
(4)
with gradient evaluations costing
Algorithm 1 summarises the order in which we implemented these steps. Note that whilst
we never optimise points in the active set, we repeatedly reselect the active set so it is
5
In practice we looked for MAP solutions for all our optimisations, specifying a unit covariance
Gaussian prior for the matrix and using
, 0f and
for , f and respectively.
Algorithm 1 An algorithm for modelling with a GPLVM.
Require: A size for the active set, ! . A number of iterations, .
Initialise ! through PCA.
for iterations do
Select a new active set using the IVM algorithm.
Optimise (4) with respect to the parameters of , using scaled conjugate gradients.
Select a new active set.
for Each point not in active set, U . do
Optimise (3) with respect to W using scaled conjugate gradients.
end for
end for
unlikely that many points
remain in their original location. For all the experiments that
iterations
. The experiments were
follow we used
and an active set of size !
"
6
run on a ?one-shot? basis
so we cannot make statements as to" the effects that significant
modification of these parameters would have. We present results on three data-sets: for the
oil flow data (Figure 2) from the previous section we now make use of all 1000 available
points and we include a comparison with the generative topographic mapping (GTM) [4].
1
2
0.8
1.8
1.5
0.6
1.6
1
0.4
0.5
0.2
1.4
0
0
1.2
?0.5
?0.2
?1
?0.4
1
?1.5
?0.6
0.8
?2
?0.8
?1
?1
?0.8
?0.6
?0.4
?0.2
0
0.2
0.4
0.6
0.8
?2.5
?3
1
?2
?1
0
1
2
3
Figure 2: The full oil flow data-set visualised with (a) GTM with 225 latent points laid out on a
grid and with 16 RBF nodes and (b) an RBF based GPLVM. The parameters of the latent
variable model were found to be
,f
and
. Notice how the GTM artificially
?discretises? the latent space around the locations of the 225 latent points.
We follow [5] in our 2-D visualisation of a sub-set of 3000 of the digits 0-4 (600 of each
digit) from a
greyscale version of the USPS digit data-set (Figure 3).
"
"
Finally we modelled
a face data-set [8] consisting of 1965 images from a video sequence
digitised at #
# . Since the images are originally from a video sequence we might expect
the underlying dimensionality of the data to be one ? the images are produced in a smooth
way over time which can be thought of as a piece of string embedded in a high (560)
dimensional pixel space. We therefore present ordered results from a 1-D visualisation in
Figure 4 .
All the code used for performing the experiments is available from http://www.dcs.
6
By one-shot we mean that, given the algorithm above, each experiment was only run once with
one setting of the random seed and the values of and given. If we were producing a visualisation
for only one dataset this would leave us open to the criticism that our one-shot result was ?lucky?.
However we present three data-sets in what follows and using a one-shot approach in problems with
multiple local minima removes the temptation of preferentially selecting ?prettier? results.
0.8
4
0.75
0.7
2
0.65
0.6
0
0.55
?2
0.5
0.45
?4
0.4
?4
?2
0
2
4
Figure 3: The digit images visualised in the 2-D latent space. We followed [5] in plotting images in
a random order but not plotting any image which would overlap an existing image. 538 of the 3000
digits are plotted. Note how little space is taken by the ?ones? (the thin line running from (-4, -1.5) to
(-1, 0)) in our visualisation, this may be contrasted with the visualisation of a similar data-set in [5].
We suggest this is because ?ones? are easier to model and therefore do not require a large region in
latent space.
shef.ac.uk/~neil/gplvm/ along with avi video files of the 1-D visualisation and
results from two further experiments on the same data (a 1-D GPLVM model of the digits
and a 2-D GPLVM model of the faces).
3 Discussion
Empirically the RBF based GPLVM model gives useful visualisations of a range of datasets. Strengths of the method include the ability to optimise the kernel parameters and to
generate fantasy data from any point in latent space. Through the use of a probabilistic
process we can obtain error bars on the position of the manifolds which can be visualised
by imposing a greyscale image upon the latent space.
When Kernels Collide: Twin Kernel PCA The eigenvalue problem which provides the
maxima of (2) with respect to ! for the linear kernel is exploited in kernel PCA. One could
consider a ?twin kernel? PCA where both e!6! T < and 6 T are replaced by kernel
functions. Twin kernel PCA could no longer be undertaken with an eigenvalue decomposition but Algorithm 1 would still be a suitable mechanism with which to determine the
values of ! and the parameters of ! ?s kernel.
Figure 4: Top: Fantasy faces from the 1-D model for the face data. These faces were created by
taking 64 uniformly spaced and ordered points from the latent space and visualising the mean of their
distribution in data space. The plots above show this sequence unfolding (starting at the top left and
moving right). Ideally the transition between the images should be smooth. Bottom: Examples from
the data-set which are closest to the corresponding fantasy images in latent space. Full sequences of
2000 fantasies and the entire dataset are available on the web as avi files.
>
Stochastic
neighbor
embedding. Consider that (2) could be written as
,
,
@CB
"!
where we have introduced a vector, , of length ,
>
6 T
,
, as ,
,
is constant7
and we have redefined
. The entropy of
in ! , we therefore may add it to to obtain
KL
,
@CB
!
,
,
(5)
which is recognised Kullback-Leibler (KL) divergence between the two distributions.
Stochastic neighbor embedding (SNE) [5] also minimises this KL divergence to visualise
data. However in SNE the vector is discrete.
Generative topographic mapping. The Generative topographic mapping [3] makes use
of a radial basis function network to perform the mapping from latent space to observed
space. Marginalisation of the latent space is achieved with an expectation-maximisation
to be of full rank, this is not true in general but can be forced
.
7
Computing the entropy requires
by adding ?jitter? to
, e.g.
(EM) algorithm. A radial basis function network is a special case of a generalised linear
model and can be interpreted as a Gaussian process. Under this interpretation the GTM
becomes GPLVM with a particular covariance function. The special feature of the GTM
is the manner in which the latent space is represented, as a set of uniformly spaced delta
functions. One could view the GPLVM as having a delta function associated with each
data-point: in the GPLVM the positions of the delta functions are optimised, in the GTM
each data point is associated with several different fixed delta functions.
4 Conclusions
We have presented a new class of models for probabilistic modelling and visualisation
of high dimensional data. We provided strong theoretical grounding for the approach by
proving that principal component analysis is a special case. On three real world data-sets
we showed that visualisations provided by the model cluster the data in a reasonable way.
Our model has an advantage over the various spectral clustering algorithms that have been
presented in recent years in that, in common with the GTM, it is truly generative with
an underlying probabilistic interpretation. However it does not suffer from the artificial
?discretetisation? suffered by the GTM. Our theoretical analysis also suggested a novel
non-linearisation of PCA involving two kernel functions.
Acknowledgements We thank Aaron Hertzmann for comments on the manuscript.
References
[1] S. Becker, S. Thrun, and K. Obermayer, editors. Advances in Neural Information Processing
Systems, volume 15, Cambridge, MA, 2003. MIT Press.
[2] C. M. Bishop and G. D. James. Analysis of multiphase flows using dual-energy gamma densitometry and neural networks. Nuclear Instruments and Methods in Physics Research, A327:580?
593, 1993.
[3] C. M. Bishop, M. Svens?n, and C. K. I. Williams. GTM: a principled alternative to the SelfOrganizing Map. In Advances in Neural Information Processing Systems, volume 9, pages
354?360. MIT Press, 1997.
[4] C. M. Bishop, M. Svens?n, and C. K. I. Williams. GTM: the Generative Topographic Mapping.
Neural Computation, 10(1):215?234, 1998.
[5] G. Hinton and S. Roweis. Stochastic neighbor embedding. In Becker et al. [1], pages 857?864.
[6] N. D. Lawrence, M. Seeger, and R. Herbrich. Fast sparse Gaussian process methods: The
informative vector machine. In Becker et al. [1], pages 625?632.
[7] I. T. Nabney.
Netlab:
Algorithms for Pattern
in Pattern Recognition. Springer,
Berlin,
2001.
http://www.ncrg.aston.ac.uk/netlab/.
Recognition.
Advances
Code available from
[8] S. Roweis, L. K. Saul, and G. Hinton. Global coordination of local linear models. In T. G.
Dietterich, S. Becker, and Z. Ghahramani, editors, Advances in Neural Information Processing
Systems, volume 14, pages 889?896, Cambridge, MA, 2002. MIT Press.
[9] B. Sch?lkopf, A. J. Smola, and K.-R. M?ller. Kernel principal component analysis. In Proceedings 1997 International Conference on Artificial Neural Networks, ICANN?97, page 583,
Lausanne, Switzerland, 1997.
[10] M. E. Tipping. Sparse kernel principal component analysis. In T. K. Leen, T. G. Dietterich,
and V. Tresp, editors, Advances in Neural Information Processing Systems, volume 13, pages
633?639, Cambridge, MA, 2001. MIT Press.
[11] M. E. Tipping and C. M. Bishop. Probabilistic principal component analysis. Journal of the
Royal Statistical Society, B, 6(3):611?622, 1999.
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1,696 | 2,541 | GPPS: A Gaussian Process Positioning System
for Cellular Networks
Anton Schwaighofer?, Marian Grigoras?, Volker Tresp, Clemens Hoffmann
Siemens Corporate Technology, Information and Communications
81730 Munich, Germany
http://www.igi.tugraz.at/aschwaig
Abstract
In this article, we present a novel approach to solving the localization
problem in cellular networks. The goal is to estimate a mobile user?s
position, based on measurements of the signal strengths received from
network base stations. Our solution works by building Gaussian process
models for the distribution of signal strengths, as obtained in a series
of calibration measurements. In the localization stage, the user?s position can be estimated by maximizing the likelihood of received signal
strengths with respect to the position. We investigate the accuracy of
the proposed approach on data obtained within a large indoor cellular
network.
1
Introduction
Cellular networks form the basis of modern wireless communication infrastructure. Examples include GSM and UMTS networks for mobile phones, wireless LAN (WLAN) for
computer networks, and DECT for cordless phones. Within these networks, location-based
services (services that are tailored specifically to the current position of the mobile user)
have great potential. Examples of such services are guiding the user through a building
or city, delivering the time-table of buses at the nearest bus stop, or simply answering the
user?s query ?Where am I??. All such services crucially depend on methods to accurately
estimate the position of the mobile user within the network (?localization?, ?positioning?).
In this article, we present a novel approach to obtain position estimates for the mobile user.
Most importantly, this method is based solely on infrastructure that is already present in
a typical cellular network, and thus leads to minimal extra cost. Furthermore, we focus
on indoor networks, where a number of specific problems needs to be addressed. Since
our approach relies heavily on Gaussian process models, we call it the ?Gaussian process
positioning system? (GPPS).
We proceed by introducing the localization problem in detail in Sec. 1.1, and by giving a
brief overview of previous approaches. Sec. 2 follows with a description of the Gaussian
process positioning system (GPPS). Sec. 3 shows how the required calibration stage of
the system can be performed in an optimal manner. Sec. 4 presents an evaluation of the
? Also with the Institute for Theoretical Computer Science, Graz University of Technology, Austria
GPPS in a DECT network environment. We show that the GPPS gives accurate location
estimates, in particular when only very few calibration measurements are available.
1.1
Problem Description
Our overall goal is to develop a localization system for indoor cellular networks, that is (in
order to minimize cost) based solely on existing standard networking hardware. Location
estimates can be based on different characteristics of the radio signal received at the mobile
station (i.e., the laptop in a WLAN network, or the phone in a DECT network). Yet,
in most hardware, the only available information about the radio signal is the received
signal strength. Information like phase or propagation time from the base station requires
additional hardware, and can thus not be used.
In general, estimating the user?s position based only on measurements of the signal strength
is known to be a very challenging task [7], in particular in indoor networks. Due to reflections, refraction, and scattering of the electromagnetic waves along structures of the
building, the received signal is only a distorted version of the transmitted signal. Noise
and co-channel interference further corrupt the signals [4]. Furthermore, when using standard hardware, we must expect a high level of measurement noise for the signal strength.
Changes in the environment can also have a strong impact on signal propagation. For example, in a WLAN environment [1], it has been noted that shielding by a single person can
attenuate the signal by up to ?3.5 dBm. Also, the localization system ought to be robust,
since base stations may fail, be switched off, or may be temporarily shielded for unknown
reasons. In these cases, a sensible localization system should not draw the conclusion that
the user is far from the respective base station.
Due to the complex signal propagation behaviour, almost all previous approaches to indoor
localization use an initial calibration stage. Calibration here means that signal strengths
received from the network base stations are measured at a number of points inside the
building. Systems differ in their ways of using this calibration data. In principle, two
basic approaches can be used here. In a ?forward modelling? approach, a model of signal
strength as a function of position is built first. The localization procedure then tries to
find the location which best agrees with the measured signal strengths. Alternatively, the
mapping from signal strengths to position can be modelled directly (?inverse modelling?).
The RADAR system [1], one of the first indoor localization systems, is an inverse modelling approach using a nearest neighbor technique. [7] build simple probabilistic models
from the calibration data (forward modelling), in conjunction with maximum likelihood
position estimation. Bayesian networks have been considered by [2], with states of node
corresponding to different locations (using coarse discretization). Discrete locations, yet
with a finer grid, are also considered in [5], in an approach inspired by robot navigation.
2
The Gaussian Process Positioning System
The difficulties of indoor localization, as mentioned in Sec. 1.1, call for a probabilistic
method for localization. The key idea of the Gaussian process positioning system (GPPS)
is to use Gaussian process models for the signal strength received from each base station,
and to obtain position estimates via maximum likelihood, i.e. by searching for the position
which best fits the measured signal strengths.
Consider a cellular network with a total of B base stations. Assume that, for each of base
stations, we have a probabilistic model that describes the distribution of received signal
strength. More formally, we denote by p j (s j | t) the likelihood of receiving a signal strength
s j from the j-th base station on position t.
With the models p j (s j | t), j = 1, . . . , B given, localization can be done in a straight-forward
way. The user reports a vector s (of length B) of signal strength measurements for all base
stations. It may occur that no signal is received from some base stations (indicated by
/ e.g., because the user is too far from this base station, or due to hardware failure.
s j = 0),
In the GPPS, the estimated position ?t is computed by maximizing the joint likelihood1 with
respect to the unknown position,
?t = arg max
t
?
j:s j 6=0/
p j (s j | t).
(1)
In the above equation, we only use the likelihood contributions of those base stations that
are actually received. Alternatively, one could use a very low signal strength as a default
value for each base station that is not received [7]. We found that this can give high errors
if a base station close to the user fails, since now the low default value indicates that one
should expect the user to be far from the base station. Thus, by using the above expression,
we also obtain a certain degree of robustness with respect to hardware failures or other
unexpected effects.
Yet, we still need to define and build suitable base station models p j (s j | t), j = 1, . . . , B.
In the GPPS, we use Gaussian process (GP) models for this task, where each base station
model is estimated from the calibration data. Gaussian processes are particularly useful
here for several reasons. Firstly, one obtains a full predictive distribution, as opposed to the
point estimate output by other regression approaches. Secondly, GPs are a nonparametric
method that can flexibly adapt to the complex signal propagation behaviour observed in
indoor cellular networks.
Mind that this approach opens a wide range of possibilities for further extensions. Due
to particular project requirements, we currently only use the maximum likelihood position
estimate in Eq. (1) (?one-shot localization? without error estimates). Instead of the implicitly assumed uninformative prior in Eq. (1), one could, for example, specify an informative
prior based on known previous positions of the user, in conjunction with a motion model.
Subsequently, the complete posterior distribution p(t | s) can be evaluated for localization.
In the following sections, we will describe the GP models in more detail, and also discuss
the choice of kernel function, which is of great importance in order to build an accurate
localization system.
2.1
Gaussian Process Models for Signal Strengths
In the GPPS, a Gaussian process (GP) approach is used for the models p j (s j | t) that describe the signal strength received from a single base station j. Details on GP models can
be found, for example, in [6]; we only give a brief summary here.
Recall from Sec. 1.1 that the proposed GPPS is based on a set calibration measurements,
where the signal strength is measured at a number of points spread over the area to be
covered. Consider now the calibration data for a single base station j. We denote this calibration data by D j = {(xi , yi )}Ni=1 , meaning that a signal strength of yi has been measured
on point xi , with a total of N calibration measurements.
For simplicity of computation, we use a GP model with Gaussian noise, i.e., the measured
signal strength yi is composed of a ?true? signal strength s(xi ) plus independent Gaussian (measurement) noise ei of variance ?2 , with yi = s(xi ) + ei . The Gaussian process
assumption for the true signal s implies that the true signal strengths for all calibration
1 Assuming
independence of the individual measurements. One could also use a solution inspired
from co-kriging, that takes into account the full dependence between signals received from different
base stations. We did not consider this solution for reasons of efficiency.
points (s(x1 ), . . . , s(xN )) are jointly Gaussian distributed, with zero mean and covariance
matrix K. K itself is given by the kernel (covariance) function k, with Kmn = k(xm , xn ),
m, n = 1, . . . , N.
Given the calibration data D j , the predictive distribution for the signal strength s j received
on some arbitrary point t turns out to be Gaussian. With v(t) = (k(t, x1 ), . . . , k(t, xN ))> ,
y = (y1 , . . . , yN )> and Q = K + ?2 I, mean and variance of the prediction are
E(s j | D j , t) = v(t)> Q?1 y
(2)
>
?1
var(s j | D j , t) = k(t, t) ? v(t) Q v(t)
(3)
Using these expressions for the predictive distribution (a univariate Gaussian) in Eq. (1)
becomes straight forward. Also, gradients of the likelihood with respect to the position t
can be derived easily [8]. Thus, the position estimate, Eq. (1), can be computed easily using
either some standard optimization routine, or by evaluating the likelihood grid-based in the
area of interest.
An important issue is also the choice of noise variance ?2 and the parameters ? of the kernel
function k (which we have not explicitly denoted above) . We set them by maximizing the
marginal likelihood of the calibration data with respect to the model parameters, which
turns out to be [6]
? = arg max ? log det Q ? y> Q?1 y .
(?? 2 , ?)
(4)
?2 ,?
? are set individually for each base station.
The model parameters (?? 2 , ?)
2.2
The Mat?ern Kernel Function
In our GPPS application, with a 2-dimensional input space for the GP models, the choice
of an appropriate kernel function is a more critical issue if compared to typical machine
learning applications with many input dimensions. For the commonly used squared exponential kernel, k(x, x0 ) = exp(?wkx ? x0 k2 ), it has been argued [9] that sample paths of
such GP models are ?infinitely smooth?, thus often leading to unreasonably low predictive
variance. In GPPS, we instead use the Mat?ern class of kernel functions [9], which allows a
continuous parameterization of the smoothness of the sample paths via its parameter ?. Its
functional form is
? ?
?
2 ?z
0
K? (2 ?z)
(5)
k(x, x ) = M? (z) =
?(?)
where ?(?) is the Gamma function and K? (r) is the modified Bessel function of the second
kind of degree ?. The parameter ? determines the smoothness (fractal dimension) of the
sample paths and can be estimated from the data using Eq. (4). We use an isotropic kernel
function with length scale w, thus z2 = wkx ? x0 k2 .
2.3
Learning GP Models with Mat?ern Kernel
For efficient solutions of Eq. (4), we require derivatives of the Mat?ern kernel function
Eq. (5) with respect to all its parameters ?, w. Numerical gradients, as used for example by
[9], require a large number of evaluations of the Bessel functions and thus lead to a huge
computational overhead. To compute the derivatives analytically, we use
??(?)
= ?(?)?(?) and
??
?K? (z)
1
= ? (K??1 (z) + K?+1 (z))
?z
2
(6)
where ?(?) is the Polygamma function of order 0. To the best of our knowledge, there is
no closed form expression for the derivative of the Bessel function K? (z) with respect to its
? (z)
degree ?. We approximate this by ?K??
= DK? (z) ? ??1 (K?+? (z) ? K? (z)). Using these
identities, we find for the gradients of the Mat?ern function, Eq. (5),
? ? ?
?
?
2 ? ?z
?M? (z) ?
= M? (z) ?
K??1 (2 ?z) + K?+1 (2 ?z)
?z
z
?(?)
?
?M? (z)
1
= M? (z)
+ log ?z ? ?(?)
(7)
??
2
? ?
?
?
?
2 ?z
z
+
? ? K??1 (2 ?z) + K?+1 (2 ?z) + DK? (2 ?z) .
?(?)
2 ?
Based on the above equations, derivatives of Eq. (4) with respect to the model parameters
?2 , ?, w can be computed using standard matrix algebra, see [6].
3
Optimal Calibration and Model Building
In order to make the GPPS, as presented in Sec. 2, a practical system, two further issues need to be solved. Firstly, it must be noted that taking calibration measurements is
a very time-consuming (thus, expensive) task. The number of calibration data must thus
be kept as low as possible, while retaining high localization accuracy. This question has
been addressed in the literature under the name optimal design. [3] showed that?in a
2-dimensional space?hexagonal sampling design yields optimal results in terms of integral mean square error when the covariance structure of the underlying Gaussian process
is unknown. We also adopt this optimal design for the GPPS system when evaluating it in
Sec. 4.
Secondly, we assumed a GP model with zero mean in Sec. 2.1, which clearly does not fit
the propagation law of radio signals. In the actual GPPS, the GP mean is a linear function
of the distance to the base station (when signal strength is given on a logarithmic scale).
The overall process of building the GPPS is summarized is follows. Starting point is the
calibration data, with a total of C measurements. On calibration point xi , i ? {1, . . . ,C}, we
receive a signal strength of ci j from base station j, j ? {1, . . . , B}, or ci j = 0/ if base station
j has not been received at xi (for example, due to signal obstruction). Signal strength is
measured in dB, all model fitting is thus done on a logarithmic scale.
The calibration data is then split into subsets D j containing those points where base station
/ corresponding to D j introduced
j has actually been received, i.e., D j = {(xi , ci j ) : ci j 6= 0},
in Sec. 2.1. For each base station, that is, for each data D j , we proceed as follows:
1. Often, the exact position of base station j is not known.2 In this case, we use a
simple estimate for the base station position, that is the average of the 3 calibration
points xi with maximum signal strength yi . This estimate is rather crude, yet we
found it to give sensible results in all of the configurations we have considered. In
particular with sparse calibration measurements, more sophisticated estimates for
the base station position are difficult to come up with.
2. Compute the distance of each calibration point to the base station (using either
the exact or the estimated position obtained in step 1). As the mean function of
the GP model, we fit a linear model3 to the received signal strength as a function
of distance to the base station. Subtract the value of the mean function from the
2 When
setting up the network, or after modifying the network by moving base stations, the base
station positions are often not recorded.
3 Alternatively, one could also use a procedure similar to universal kriging, and combine fitting of
the mean function with learning the parameters of the kernel function, see Eq. (4).
original measurements, and use the modified values in the subsequent GP model
fitting procedure.
3. Use Eq. (4) to find optimal parameters for the GP model, which are the noise
variance ?2 , the Mat?ern smoothness parameter ? and the input length scale w.
4
Evaluation in a DECT Network
We tested the accuracy of the GPPS in a large DECT cellular network. In a large assembly hall of 250 ? 180 meters, measurements of signal strengths received from DECT base
stations were made on 650 points spread over the hall. In this environment, moving robots,
metal constructions, corridors, office cubicles, etc., are all affecting the signal propagation.
We observed a very high fluctuation of received signals (up to ?10 dB when repeating measurements, while the total signal range is only ?30 to ?90 dB), both due to measurement
noise, and due to dynamical changes of the environment.
We compare the GPPS with a nearest neighbor based localization system (abbreviated by
NNLoc in the following), that is quite similar to the RADAR [1] approach.4 This system
finds the calibration measurements that best match the signal strength received at test stage.
The best matches are used in a weighted triangulation scheme to compute the location
estimate. This method requires careful fine tuning of parameters, and we have to omit
details for brevity here.
Dense Calibration Points In a first experiment, we investigate the achievable precision
of location estimates when using the full set of calibration measurements. We evaluate both
the GPPS and the nearest neighbor based method in a 5fold cross validation scheme. The
total set of measurements is split up into five equally sized parts, where four of these parts
were used as the calibration set. The resulting positioning system is tested on the fifth part
left out. This is repeated five times, so that each point is being used as the test point exactly
once. We found that, in this setting, the nearest neighbor based method NNLoc works very
fine, and provides an average localization error of 7 meters. The GPPS performs slightly
worse, with an average error of 7.5 meters. With the GPPS, localization is typically based
on around 15 base stations, that is, 15 likelihood terms contributing to Eq. (1).
Unfortunately, such a high number of calibration measurements is unlikely to be available
in practice. Taking calibration measurements is a very costly process, in particular if larger
areas need to be covered. Thus, one is very much interested in keeping the number of
calibration points as low as possible.
Experiments with Sparse Calibration Points In the second experimental setup, we aim
at building the positioning system with only a minimal number of calibration points. Again,
5fold cross validation was performed. After splitting the data into five parts, we select
subsets of C? = 100, 50, 25, 12 points, either at random or simulating the optimal design,
from the union of four of these parts. The localization system is built based on these
C? points and evaluated on the fifth part of the data. In order to simulate a near-optimal
design (see Sec. 3), we superimpose a hexagonal grid with C? points on the area under
consideration. Out of the given calibration measurements, we select those C? points that are
closest (in terms of Euclidean distance) to the grid points.
In Fig. 1 we plot the localization accuracy, averaged over the 5fold cross validation,
of the GPPS and the nearest neighbor based system built on only C? calibration points,
4 We
also investigated localization using Eq. (1) with a simplistic propagation model, where the
expected signal (on log scale) is a linear function of the distance to the base station. Yet, this approach
lead to very poor localization accuracy, and is thus not considered in more detail here.
Figure 1: Mean localization error of the GPPS and the NNLoc method, as a function of
the number of calibration points used. Vertical bars indicate ?1 standard deviation of the
mean localization error. The calibration points are either selected at random, or according
to an optimal design criterion
C? ? {100, 50, 25, 12} calibration measurement. It can be clearly seen that the GPPS system
(with optimal design) achieves a high precision for its location estimates, even when using
only a minimal number of calibration measurements. With only 12 calibration measurements, GPPS achieves an average error of around 17 meters, while the competing method
reaches only 29 meters at best. In this setting, the average distance in between calibration
measurements is around 75 meters. Both the NNLoc system and the GPPS system show
large improvements of performance when selecting the calibration points according to the
optimal design, instead of a purely random fashion. Also, note that the localization error
of the GPPS system degrades only slowly when the number of calibration measurements
is reduced. In contrast, the curves for the nearest neighbor based method show a sharper
increase of positioning error.
It is worth noticing that the choice of kernel functions has a strong impact on the localization accuracy of the GPPS. In Fig. 2(a), we also plot a comparison of the GPPS with
either the Mat?ern kernel, Eq. (5), or an RBF kernel of the form k(x, x0 ) = exp(?wkx ? x0 k).
GP models with RBF kernels tend to be over-optimistic [9] about the predictive variance,
Eq. (3), which in turn leads to overly tight position estimates. Thus, the accuracy of GPPS
with RBF kernel is clearly inferior to that of GPPS with Mat?ern kernel. It is also interesting to consider different methods for selecting the calibration points. Fig. 2(b) plots
the accuracy obtained with GPPS, when calibration points are either placed randomly, on a
hexagonal grid (the theoretically optimal procedure) or on a square grid. Somehow counterintuitively, a square grid for calibration gives a performance that is just as good or even
worse than a random grid. In contrast, localization with NNLoc performs about the same
with either hexagonal or square grid (this is not plotted in the figure).
5
Conclusions
In this article, we presented a novel approach to solving the localization problem in indoor
cellular network networks. Gaussian process (GP) models with the Mat?ern kernel function
were used as models for individual base stations, so that location estimates could be computed using maximum likelihood. We showed that this new Gaussian process positioning
system (GPPS) can provide sufficiently high accuracy when used within a DECT network.
(a) GPPS using either the Mat?ern or the
RBF kernel function
(b) GPPS with calibration measurements
placed either randomly, on a square grid, or
on a hexagonal grid (optimal design)
Figure 2: Average localization error of the GPPS method with different kernel function
(left) and different methods for placing calibration points (right)
A particular advantage of the GPPS system is that it can be based on only a small number of calibration measurements, and yet retain high accuracy. Furthermore, we showed
how calibration points can be optimally chosen in order to provide high accuracy position
estimates.
Acknowledgments Anton Schwaighofer gratefully acknowledges support through an
Ernst-von-Siemens scholarship.
References
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[4] Hashemi, H. The indoor radio propagation channel. Proceedings of the IEEE, 81(7):943?968,
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1,697 | 2,542 | Image Reconstruction by Linear Programming
Koji Tsuda?? and Gunnar R?atsch??
Max Planck Institute for Biological Cybernetics
Spemannstr. 38, 72076 T?ubingen, Germany
?
AIST CBRC, 2-43 Aomi, Koto-ku, Tokyo, 135-0064, Japan
?
Fraunhofer FIRST, Kekul?estr. 7, 12489 Berlin, Germany
?
{koji.tsuda,gunnar.raetsch}@tuebingen.mpg.de
Abstract
A common way of image denoising is to project a noisy image to the subspace of admissible images made for instance by PCA. However, a major
drawback of this method is that all pixels are updated by the projection,
even when only a few pixels are corrupted by noise or occlusion. We propose a new method to identify the noisy pixels by 1 -norm penalization
and update the identified pixels only. The identification and updating of
noisy pixels are formulated as one linear program which can be solved
efficiently. Especially, one can apply the ?-trick to directly specify the
fraction of pixels to be reconstructed. Moreover, we extend the linear
program to be able to exploit prior knowledge that occlusions often appear in contiguous blocks (e.g. sunglasses on faces). The basic idea is
to penalize boundary points and interior points of the occluded area differently. We are able to show the ?-property also for this extended LP
leading a method which is easy to use. Experimental results impressively
demonstrate the power of our approach.
1 Introduction
Image denoising is an important subfield of computer vision, which has extensively been
studied (e.g. [2, 6, 1, 9]). The aim of image denoising is to restore the image corrupted by
noise as close as possible to the original one. When one does not have any prior knowledge about the distribution of images, the image is often denoised by simple smoothing
(e.g. [2, 1]). When one has a set of template images, it is preferable to project the noisy image to the linear manifold made by PCA, which is schematically illustrated in Fig. 1 (left).
One can also construct a nonlinear manifold, for instance by kernel PCA, requiring additional computational costs [6]. The projection amounts to finding the closest point in the
manifold according to some distance. Instead of using the standard Euclidean distance (i.e.
the least squares projection), one can adopt a robust loss such as Huber?s loss as the distance, which often gives a better result (robust projection [9]). However, a major drawback
of these projection approaches is that all pixels are updated by the projection. However,
typically only a few pixels are corrupted by noise, thus non-noise pixels should best be left
untouched.
This paper proposes a new denoising approach by linear programming, where the 1 -norm
regularizer is adopted for automatic identification of noisy pixels ? only these are updated.
The identification and updating of noisy pixels are neatly formulated as one linear program.
The theoretical advantages of linear programming lie in duality and optimality conditions.
By considering both primal and dual problems at the same time, one can construct effective
and highly principled optimizers such as interior point methods. Also, the optimality con-
||?||1 < C
Least
Squares
Projection
Robust
projection
Off-manifold
Solution
On-Manifold
Solution
Figure 1: Difference between projection methods (left) and our LP method (right).
ditions enables us to predict important properties of the optimal solution before we actually
solve it. In particular, we can explicitly specify the fraction of noisy pixels by means of the
?-trick originally developed for SVMs [8] which was later applied to Boosting [7].
In some cases the noisy pixels are not scattered over the image (?impulse noise?), but
form a considerably large connected region (?block noise?), e.g. face images occluded by
sunglasses. By using the prior knowledge that the noisy pixels form blocks, we should be
able to improve the denoising performance. Several ad-hoc methods have been proposed
so far (e.g. [9]), but we obviously need a more systematic way. We will show that a very
simple modification of the linear program has the effect that we can control how blockshape like the identified and reconstructed region is. In the experimental section we will
show impressive results on face images from the MPI face data base corrupted by impulse
and block noises.
2 Image Denoising by Linear Programming
Let {tj }Jj=1 be the set of vectors in N , which have been derived for instance by principal
component analysis. The linear manifold of admissible images is described as
J
T = t|t=
?j tj , ?j ?
j=1
Now we would like to denoise a noisy image x ? N . Let us describe the denoised image
? . In order that the denoised image x
? is similar to admissible images, x
? should be close
as x
to the manifold:
J
?,
?j t j ? 1 ,
(1)
min d1 x
?
j=1
?,
where d1 is a distance between two images. Also, we have to constrain x to be close to x
otherwise the denoised image becomes completely independent from the original image:
x, x) ? 2 ,
d2 (?
(2)
where d2 is another distance. A number of denoising methods can be produced by choosing
different distances and changing how to minimize the two competing objectives (1) and (2).
In projection methods, 1 is simply set to zero and 2 is minimized with d2 being set to the
Euclidean distance or a robust loss.
A Linear Programming Formulation Our wish is that most pixels of x stay unchanged
? , in other words, the difference vector ? = x
? ? x should be sparse. For this purpose,
in x
d2 is chosen as the 1 -norm, as it is well known that the 1 -norm constraints produce sparse
solutions (e.g. [7]). Also for d 1 , the ? -norm is especially interesting as it leads to linear
programming. We design the optimization problem as follows:
J
?j t j
(3)
min
x + ? ?
j=1
?,?
?1 ? C,
?
(4)
N
where x? = maxi |xi |, ?1 = i=1 |?i | and C is a constant to determine the sparseness, i.e. the solution ? tends to become more sparse as C decreases. Geometrically, this
? within the
optimization problem is explained as Fig. 1 (right). The constraint (4) keeps x
which
is closest to
1 -sphere centered on x. The optimization finds a point in the sphere,
the linear manifold. As a side effect, we have another solution j ?j tj on the manifold.
We call the former the ?off-manifold solution? and the latter ?on-manifold solution?. Here,
we are mainly concerned with the off-manifold solution, because of the sparsity.
Let us actually formulate (3) as a linear programming problem. It is equivalent to
1 N
min
|?n | + ?
n=1
?,?,
N
J
xn + ?n ?
?j tjn ? , n = 1, . . . , N,
(5)
j=1
where ? is a regularization parameter. Still this problem is not linear programming because
of |?n | in the objective function. Next let us restate ? as follows:
?
? = ?+ ? ?? , ?+
n = 1, . . . , N.
n , ?n ? 0,
Then (5) is rewritten as the following linear programming problem:
1 N
?
(?+
(6)
min
n + ?n ) + ?
n=1
N
?? ,?,
J
?
+
?
?+
?j tjn ? , n = 1, . . . , N. (7)
n , ?n ? 0, xn + ?n ? ?n ?
j=1
?
Here we used the well known fact that either ? +
n or ?n is zero at the optimum.
The ?-Trick In the above optimization problem, the regularization constant ? should
be determined to control the fraction of updated pixels. Interestingly, ? has an intuitive
meaning as follows: Let N p denote the number of nonzero elements in ?. Furthermore
let Nc be the number of ?crucial pixels? which are not updated, but the corresponding
constraint constraints (7) are met as equalities. If one of these pixels is modified, then it
will likely lead to a different solution, while changing any of the other N ? N p ? Nc pixels
locally does not change the optimal solution.
Proposition 1. Suppose the optimal is greater than 0. Then the number of nonzero
elements Np in the optimal ? is
1. upper bounded by ?N , i.e. N p ? ?N and
2. lower bounded by ?N ? N c , i.e. Np ? ?N ? Nc .
The proof is a special case of the one of Proposition 2 and is omitted. The slack in the bound
only comes from N c . In practice we usually observed small values of N c . We suspect that
its value is related to J ? the number of basis vectors.
In terms of images, one can bound the anticipated fraction of noise pixels by ?. In contrast,
the constant C in (4) specifies the sum of noise magnitudes, which is in practice rather
difficult to figure out.
3 Dealing with Block Noises
Preliminaries When noises are clustered as blocks, this prior knowledge is considered
to lead to an increased denoising performance. So far we could only control the number
of modified pixels which corresponds to the area of reconstruction. In this section we also
consider the length of the boundary of the identified pixels. For instance, consider the three
occlusion patterns in Figure 2. The pixel is white, when it is identified as noisy/occluded
and black otherwise. In the first case (left) the occlusion forms a block, in the second case
the letters ?lp? and in the third case the pixels are randomly distributed. The covered area
is the same for all three cases.
S ? = 130, S + = 256
S ? = 280, S + = 552
S ? = 1987, S + = 2725
5
5
5
10
10
10
15
15
15
20
20
20
25
25
25
30
30
30
35
35
35
40
40
40
45
45
45
50
50
55
50
55
10
20
30
40
50
60
55
10
20
30
40
50
60
10
20
30
40
50
60
Figure 2: Three occlusion patterns with different degrees of having a block shape.
We will now define two measures of how much an occlusion pattern mismatches the block
shape. It is related to the length of the boundary. Note that optimal ?block? shapes have
shortest boundaries. (It depends on the metric what will be optimal.)
We distinguish between two types of penalties: first, the ones which occur when a reconstructed pixel is a neighbor of an untouched pixel (?boundary point?) and second, if a
reconstructed pixel is neighbor of another such pixel, but the corrections are in different
directions (?inversion point?). We have two definitions for our scores, which we will later
relate to the solution of our extended linear program. The differences between the two
scores S ? and S + are only in subtle details in how to count boundary points and inversion
points:
? Let Nb? be the number of pixels n which satisfy: (a) ? n = 0 and there exists
m ? G(n) such that ?m = 0 (outer boundary point) or (b) ? n = 0 and for all
m ? G(n) holds ?m = 0 (single pixel change). Let N i? be the number of pixels
n with ?n ?m < 0 for at least one m ? G(n) and ? n ?m ? 0 for all m ? G(n)
(single inversion point). The first score is computed as S ? := Nb? + 2Ni? .
? Let Nb+ be the number of pixels n which satisfy: (a) ? n = 0 and there exists
m ? G(n) such that ?m = 0 (outer boundary point) or (b) ? n = 0 and there
exists m ? G(n) with ?m = 0 (inner boundary point). Let N i+ be the number of
pixels n with ?n ?m < 0 for at least one m ? G(n) (inversion point). Then the
second score is computed as S + := Nb+ + 2Ni+ .
The main difference between the two scores is that S + counts the length of the inner and
outer boundary, while S ? only counts the outer boundary.
The Extended LP The question is how we can introduce these definitions into a linear
program, which somehow penalizes these scores. As we will show in the following proposition, it turns out that it is enough to penalize the differences between neighboring ??s. We
introduce a new set of variables (the ??s) which account for these differences and which
are linearly penalized. We control the contribution of the ??s with the one of the ??s by
introducing a new parameter ? ? (0, 1) ? if ? = 0, then the original LP is recovered:
min
??0,?,?0,?
? N
1 ? ? N
?n +
|?n | + ?
n=1
n=1
N
N
J
xn + ?n ?
?j tj,n ? for all n = 1, . . . , N
j=1
|?n ? ?m | ? ?n
(8)
for all m ? G(n)
We will show in the experimental part that these novel constraints lead to substantial improvements for block noises. The analysis of this linear program is considerably more
difficult than of the previous one. However, we will show that the ?-trick still works in a
generalized manner with some subtleties. We will show in the following Proposition that
LP (8) trades-off the area N p with the penalty scores S ? and S + :
Proposition 2. Let Nc the number of crucial pixels and N p the number of updated pixels
(as before). Assume the optimal is greater 0. Then holds:
1. The ?-weighted average between area of the occlusion and score S ? is not greater
than ?N , i.e.
(9)
(1 ? ?)Np + ?S ? ? ?N
1
2. If ? < 2+|G|
, then the ?-weighted average between area of the occlusion and
+
score S is not smaller than ?N minus 2N c , i.e.
(1 ? ?)Np + ?S + > ?N ? 2Nc ,
(10)
where |G| := maxn |G(n)|
Note that the slackness in (10) again only comes from the number of crucial
1
points Nc . The restriction ? < 2+|G|
only concerns the second part and and
not the functioning of the LP in practice. It can be made less restrictive, but
this goes beyond the scope of this paper. Due to space limitations we have to
omit the proof. It is found in a technical report, which can be downloaded from
http://www.kyb.tuebingen.mpg.de/publications/pdfs/pdf2420.pdf.
4 Denoising by QP and Robust Statistics
A characteristic of the LP method is that the ? -norm is used as d1 . But other choices are
of course possible. For example, when the squared loss is adopted as d 1 , the optimization
problem (3) is rewritten as
2
J
1 N
xn + ?n ?
?j tjn + ?|?n |.
(11)
min
n=1
j=1
?,? N
This is a quadratic program (QP), which can also be solved by standard algorithms. In
our experience, QP takes longer time to solve than LP and the denoising performance is
more or less the same. Furthermore the ?-trick does not hold for QP. Nevertheless, it is
interesting to take a close look at the QP method as it is more related to existing robust
statistical approaches [2, 9]. The QP can partially be solved analytically with respect to ?:
J
N
(12)
? xn ?
?j tjn ,
min
n=1
?
where ? is the Huber?s loss
j=1
t2
N
? N2? ? t ? N2?
|t| ?
otherwise.
Thus, the on-manifold solution of (11) corresponds to the robust projection by the Huber?s
loss. In other words, ? is considered as a set of slack variables in the robust projection. It
is worthwhile to notice another choice of slack variables proposed in [2]:
2
J
1 N
1
zn xn ?
?j tjn + ?
.
(13)
min
n=1
j=1
z,?
2?
2zn
0 ? zn ? 1, n = 1, . . . , N.
Here the slack variables are denoted as z, which is called the outlier process [2]. Notice ?
J
is a regularization constant. Let us define g n = xn ? j=1 ?j tjn . Then the inside problem
with respect to zn can be analytically solved, and we have the reduced problem as
J
N
(14)
h? xn ?
?j tjn
min
?(t) =
?
n=1
N ?2
4
j=1
2
t
where h? (t) is again the Huber?s loss function: h ? (t) = 2?
+ ?2 if |t| < ? and |t| if |t| ? ?.
The outlier process tells one which pixels are ignored, but it does not directly represent
the denoised image. From the viewpoint of denoising, our slack variables ? seem to make
more sense.
5 Experiments
We applied our new methods and the standard methods to the MPI face database [3, 4].
This dataset has 200 face images (100 males and 100 females) and each image is rescaled
to 44?64. The images are artificially corrupted by impulse and block noises. As impulse
noises, 20% of the pixels are chosen randomly and set to 0. For block noises, a rectangular
region (10% of the pixels) is set to zero to hide the eyes. We hide the same position for
all images, but the position of the rectangle is not known to our algorithm. The task is to
recover the original image based on the remaining 199 images (i.e. l.o.o. cross validation).
Our linear program is compared against the least squares projection and the robust projection using Huber?s loss (i.e. the on-manifold solution of QP). One could also apply the
non-convex robust losses for better robustness, e.g. Tukey?s biweight, Hampel, GemanMcClure, etc [2]. On the other hand, we could also use the non-convex regularizers which
are ?steeper? than the 1 -norm for greater sparsity [5]. However, we will not trade convexity with denoising performance here, because local minima often put practitioners into
trouble. As a reference, we also consider an idealistic denoising method, to which we give
the true position of noises. Here, the pixel values of noisy positions are estimated by the
least squares projection only with respect to the non-noise pixels. Then, the estimated pixel
values are plugged back into the original image. The linear manifold is made by PCA from
the remaining 199 images. The number of principal components is determined such that
the idealistic method performs the best. For impulse and block noise images, it turned out
to be 110 and 30, respectively.
The reconstruction errors of LP and QP for impulse noises are shown in Fig. 4. Here, the
reconstruction error is measured by the 2 -norm between the images. Also an example of
denoising is shown in Fig. 3. Both in LP and QP, the off-manifold solution outperforms
a: original image
c: least squares
proj. (702)
b:noisy image
d: Off?Manifold
?=0.4 (454)
Figure 3: A typical result of denoising impulse noise. (a) An original face image. (b) The image
corrupted by impulse noise. (c) Reconstruction by the least squares projection to the PCA basis. The
number in (?) shows the reconstruction error. (d) Reconstruction by the LP (off-m.) when ? = 0.4.
the on-manifold one, which confirms our intuition that it is effective to keep most pixels
unchanged. Compared with the least squares projection, the difference is so large that one
can easily see it in the reconstructed images (Fig. 3). Notably, the off-manifold solutions of
LP and QP (cf. the solid curves in Fig. 4, left and right) performed significantly better than
the on-manifold solution of QP, which corresponds to the robust projection using Huber?s
loss (cf. the dashed curve in Fig. 4 right).
800
800
LP
700
700
600
600
500
500
400
400
300
300
0.2
0.3
0.4
0.5
?
0.6
0.7
0.8
0.9
-2 -1.8 -1.6 -1.4 -1.2 -1
log10 ?
QP
-0.8 -0.6 -0.4 -0.2
Figure 4: Reconstruction errors of LP and QP methods for impulse noise. The solid and dashed lines
corresponds to the off-manifold and on-manifold solutions. The flat lines correspond to the least
squares projection and the unrealistic setting where the correct positions of noises are given.
LP (? = 0)
LP (? = 0.5)
1600
1500
on-manifold LP
1000
off-manifold LP
500
?
0.5
1400
PCA
2000
average reconstruction error
average reconstruction error
2500
PCA
1200
0.4
on-manifold
solution
1000
upper bound (10)
off-manifold
solution
0.3
800
0.2
lower bound (11)
PCA with known area
600
0.1
PCA with known area
0
0
0.2
0.4
0.6
parameter ?
0.8
1
400
0
0.2
0.4
0.6
parameter ?
0.8
1
Figure 5: Reconstruction errors of the LP method for block noises.
(Left) the reconstruction error of the ?plain? LP, where the block
constraints are not taken into account (? = 0). The right plot
shows the improvement for increased ? and fixed ? = 1/2.
0
0
0.2
?
0.4
0.6
Figure 6: Illustration of Prop. 2:
For ? = 0.15 we compute the
lower and upper bound of ?N for
different ??s.
The results for block noises are shown in Fig. 5, where we again averaged over the 200 faces
(using l.o.o. cross validation for the construction of the PCA basis). In the left figure, we
measure the reconstruction error for various ??s with fixed ? = 0, i.e. the block constraints
are not taken into account. As in the case with impulse noise, the error is smaller than
that of the least squares regression (PCA projection), and the minimum is attained around
? = 1/2. Moreover, we investigated how the error is further reduced by increasing ? from
0. As shown in the right figure, we obtain a significant improvement. Actually, there is
not much room for improvements, since even the idealistic case where the position of the
occlusion is know is not much better.
An example of reconstructed images are shown in Fig. 7. Here we have shown variables
? and ? as well. When ? = 0, nonzero ??s appear not only in occluded part but also
for instance along the face edge (Fig. 7:e). When ? = 1/2, nonzero ??s are more concentrated in the occluded part, because the block constraints suppress a isolated nonzero
values (Fig. 7:h). In Fig. 7:i, one can see high ??s in the edge pixels of occluded region,
which indicates that the block constraints are active for those pixels.
Finally we empirically verify Proposition 2. In Fig. 6 we plot the lower and upper bound of
? as given in Proposition 2 for different values of ?. Observe that the difference between
lower and upper bound is quite small.
6 Concluding Remarks
In summary, we have presented a new image denoising method based on linear programming. Our main idea is to introduce sparsity by detaching the solution slightly from the
manifold. The on-manifold solution of our method is related to existing robust statistical
approaches. Remarkably, our method can deal with block noises while retaining the convexity of the optimization problem (every linear program is convex). Existing approaches
(e.g. [9]) tend to rely on non-convex optimization to include the prior knowledge that the
noises form blocks. Perhaps surprisingly, our convex approach can solve this problem to a
great extent. We are looking forward to apply the linear programming to other computer
vision problems which involve combinatorial optimization, e.g. image segmentation. Also,
it is interesting to explore the limitations of convex optimization, since ? naturally ? convex
optimization cannot solve every problem. Nevertheless, according to our experience in this
work, we feel that the power of convex optimization is not fully exploited.
Acknowledgment The authors gratefully acknowledge A. Graf for preparing the face
image dataset. We would like to thank B. Sch?olkopf, J. Weston, T. Takahashi, T. Kurita,
S. Akaho and Chan-Kyoo Park for fruitful discussions.
a: original image
b: noisy image
f: Off Manifold
?=0.5 (654)
c: least squares
proj. (1512)
g: On Manifold
?=0.5 (708)
d: Off Manifold
?=0 (1106)
h: ? [?=0.5]
e: ? [?=0]
i: ? [?=0.5]
Figure 7: A typical result of denoising block noises (? = 0.5). The numbers in (?) in (c),(d),(f),(g)
show the reconstruction errors. The image (d) shows the denoising result when the block constraints
are not taken into account (? = 0, ? = 1/2). This result improves by imposing the block constraints
(? = 1/2, ? = 1/4) as shown in (f) and (g), which are the off and on-manifold solutions, respectively.
The images (e),(h) and (i) show the parameter values obtained as the result of linear programming
(see the text for details).
References
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1,698 | 2,543 | Machine Learning Applied to Perception:
Decision-Images for Gender Classification
Felix A. Wichmann and Arnulf B. A. Graf
Max Planck Institute for Biological Cybernetics
T?ubingen, Germany
[email protected]
Eero P. Simoncelli
Howard Hughes Medical Institute
Center for Neural Science
New York University, USA
?
Heinrich H. Bulthoff
and Bernhard Sch?olkopf
Max Planck Institute for Biological Cybernetics
T?ubingen, Germany
Abstract
We study gender discrimination of human faces using a combination
of psychophysical classification and discrimination experiments together
with methods from machine learning. We reduce the dimensionality of
a set of face images using principal component analysis, and then train a
set of linear classifiers on this reduced representation (linear support vector machines (SVMs), relevance vector machines (RVMs), Fisher linear
discriminant (FLD), and prototype (prot) classifiers) using human classification data. Because we combine a linear preprocessor with linear
classifiers, the entire system acts as a linear classifier, allowing us to visualise the decision-image corresponding to the normal vector of the separating hyperplanes (SH) of each classifier. We predict that the female-tomaleness transition along the normal vector for classifiers closely mimicking human classification (SVM and RVM [1]) should be faster than
the transition along any other direction. A psychophysical discrimination experiment using the decision images as stimuli is consistent with
this prediction.
1
Introduction
One of the central problems in vision science is to identify the features used by human
subjects to classify visual stimuli. We combine machine learning and psychophysical techniques to gain insight into the algorithms used by human subjects during visual classification of faces. Comparing gender classification performance of humans to that of machines
has attracted considerable attention in the past [2, 3, 4, 5]. The main novel aspect of our
study is to analyse the machine algorithms to make inferences about the features used by
human subjects, thus providing an alternative to psychophysical feature extraction techniques such as the ?bubbles? [6] or the noise classification image [7] techniques. In this
?machine-learning-psychophysics research? we first we train machine learning classifiers
on the responses (labels) of human subjects to re-create the human decision boundaries by
learning machines. Then we look for correlations between machine classifiers and sev-
eral characteristics of subjects? responses to the stimuli?proportion correct, reaction times
(RT) and confidence ratings. Ideally this allows us to find preprocessor-classifier pairings
that are closely aligned with the algorithm employed by the human brain for the task at
hand. Thereafter we analyse properties of the machine closest to the human?in our case
support vector machines (SVMs), and to slightly lesser degree, relevance vector machines
(RVMs)?and make predictions about human behaviour based on machine properties.
In the current study we extract a decision-image containing the information relevant for
~ is the image corresponding
classification by the machine classifiers. The decision-image W
to a vector w
~ orthogonal to the SH of the classifier. The decision-image has the same
dimensionality as the (input-) images?in our case 256 ? 256?whereas the normal vector
lives in the (reduced dimensionality) space after preprocessing?in our case in 200 ? 1
after Principal Component Analysis (PCA). Second, we use w
~ of the classifiers to generate
novel stimuli by adding (or subtracting) various ?amounts? (?w)
~ to a genderless face in
w
~
PCA space. The novel stimuli, images, I(?) are generated as I(?) = P CA?1 ? kwk
~ . We
predict that the female-to-maleness transition along the vectors normal to the SHs, w
~ SVM
and w
~ RVM , should be significantly faster than those along the normal vectors of machine
classifiers that do not correlate as well with human subjects. A psychophysical gender
discrimination experiment confirms our predictions: the female-to-maleness axis of the
SVM and, to a smaller extent, RVM, are more closely aligned with the human female-tomaleness axis than those of the prototype (Prot) and a Fisher linear discriminant (FLD)
classifier.
2
Preprocessing and Machine Learning Methods
We preprocessed the faces using PCA. PCA is a good preprocessor in the current context since we have previously shown that in PCA-space strong correlations exist between
man and machine [1]. Second, there is evidence that the PCA representation may be
biologically-plausible [8]. The face stimuli were taken from the gender-balanced Max
Planck Institute (MPI) face database1 composed of 200 greyscale 256 ? 256-pixel frontal
2
views of human faces, yielding a data matrix X ? R200?256 . For the gender discrimination task we adhere to the following convention for the class labels: y = ?1 for females
and y = +1 for males. We consider no dimensionality reduction and keep all 200 components of the PCA. This implies that the reconstruction of the data from the PCA analysis
? T ?X
? = EB where E ? R200?200 is the matis perfect and we can write: E = XB
rix of the encodings (each row is a PCA vector in the space of reduced dimensionality),
2
? the centered data matrix. The comB ? R200?256 is the orthogonal basis matrix and X
bination of the encoding matrix E with the true class labels y of the MPI database yields
the true dataset, whereas its combination with the class labels yest by the subjects yields
the subject dataset.
To model classification in human subjects we use methods from supervised machine learning. In particular, we consider linear classifiers where classification is done using a SH
defined by its normal vector w
~ and offset b. Furthermore the normal vector w
~ of our
classifiers can then be writtenPas a linear combination of the input patterns ~xi with suitable coefficients ?i as w
~ =
xi . We define the distance of a pattern to the SH as
i ?i ~
~ xi+b
?(~x) = hw|~
.
Note
that
in
our
experiments
the ~xi are the PCA coefficients of the imkwk
~
2
ages, that is ~xi ? R200 , whereas the images themselves are in R256 . For the subject dataset
we chose the mean values of w,
~ b and w
~ ? over all subjects.
1
The MPI face database is located at http://faces.kyb.tuebingen.mpg.de
2.1 Machine Classifiers
The Support Vector Machine (SVM, [9, 10]) is a state-of-the-art maximum margin algorithm based on statistical learning theory. SVMs have an intuitive geometrical interpretation: they classify by maximizing the margin separating both classes while minimizing
the classification error.
The Relevance Vector Machine (RVM, [11]) is a probabilistic Bayesian classifier. It optimises the expansion coefficients of a SV-style decision function using a hyperprior which
favours sparse solutions.
Common classifiers in neuroscience, cognitive science and psychology are variants of the
Prototype classifier (Prot, [12]). Their popularity is due to their simplicity: they classify
according to the nearest mean-of-class prototype; in the simplest form all dimensions are
weighted equally but variants exist that weight the dimensions inversely proportional the
class variance along the dimensions. As we cannot estimate class variance along all 200
dimensions from only 200 stimuli, we chose to implement the simplest Prot with equal
weight along all dimensions.
The Fisher linear discriminant classifier (FLD, [13]) finds a direction in the dataset which
allows best linear separation of the two classes. This direction is then used as the normal
vector of the separating hyperplane. In fact, FLD is arguably a more principled whitened
?1
variant of the Prot classifier: Its weight vector can be written as w
~ = SW
(~
?+ ?~
?? ), where
?1
SW is the within class covariance matrix of the two classes, and ?? are the class means.
Consequently, if we disregard the constant offset b, we can write the decision function as
?1/2
?1/2
?1
hw|~
~ xi = hSW
(~
?+ ? ?
~ ? )|~xi = hSW (~
?+ ? ?
~ ? )|SW ~xi, which is a prototype classifier
?1/2
using the prototypes ?
~ ? after whitening the space with SW .
2.2 Decision-Images and Generalised Portraits
? = EB and the linear classifier (SVM, RVM,
We combine the linear preprocessor (PCA) X
Prot, FLD) y(~x) = hw|~
~ xi + b to yield a linear classification system: ~y = w
~ T E T + ~b where
~b = b~1. We define the decision-image as the vector W
~ effectively used for classification as:
~ TX
? T + ~b. We then have w
~ TX
?T ? w
?T = W
~ TX
? T where B ?1
~y = W
~ T ET = W
~ T B ?T X
is the pseudo-inverse of B. For the last condition, we obtain a definition of the decision2
~ = B ?1 w
image W
~ ? R256 . In the case of PCA where B ?1 = B T , we simply have
~ = B T w.
W
~
~ for the four classifiers, SVM, RVM, Prot and FLD.
Figure 1 shows the decision-images W
The decision-images in the first row are those obtained if the classifiers are trained on the
true dataset; those in the second row if trained on the subject dataset, marked on the right
hand side of the figure by ?true data? and ?subj data?, respectively. Decision-images are
represented by a vector pointing to the positive class and can thus be expected to have male
attributes (the negative of it looks female). Both dark and light regions are more important
for classification than the grey regions. Inspection of the decision-images is instructive. For
the prototype learner, the eye and beard regions are most important. SVM, RVM and FLD
have somewhat more ?holistic? decision-images. Equally instructive is the comparison of
the optimal decision-images of the machine classifiers in row one (0 to 1% classification
error for SVM, RVM and FLD) and those trained on the subject labels in row two (the
average subject error is 16 % when classifying the faces; the machines attempt to re-create
the decision boundaries of the subjects and thus show similar mis-classification errors).
The decision-images for the subject dataset are slightly more ?face-like? and less holistic
than those obtained using the true labels; the eye and mouth regions are more strongly
emphasised. This trend is true across all classifiers. This suggest that human subjects base
their gender classification strongly on the eye and mouth regions of the face?clearly a
sub-optimal strategy as revealed by the more holistic true dataset SVM, RVM and FLD
decision-images.
A decision-image thus represents a way to extract the visual cues and features used by human subjects during visual classification without using a priori assumptions or knowledge
about the task at hand.
SVM
RVM
Prot
FLD
trained
on
?
W
true
data
?
W
subj
data
~ for each classifier for both the true and the subject dataset; all
Figure 1: Decision-images W
images are rescaled to [0, 1] and their means set to 128 for illustration purposes (different
scalers for different images).
~ ? . The generalised portraits W
~ ? can be
We can also define generalised portraits2 W
seen as ?summary? faces in each class reflecting the decision rule of the classifier. They
can be viewed as an extension of the concept of a prototype: they are the prototype
of the P
faces the classifier
bases its decision
~ can be written as:
P
P on. We note that w
w
~ =
?
~
x
=
?
~
x
?
|?
|~
x
.
This allows to define
i
i
i
i
i
i
i
i| sign(?i )=+1
i| sign(?i )=?1
~
the generalized portraits as W? which are computed by inverting the PCA transformation
on the patterns w
~? =
P
xi
i| sign(?i )=?1 ?i ~
P
.
i| sign(? )=?1 ?i
The vector w
~ ? is constrained to be in the convex
i
hull of the respective data in order to yield a ?viewable? portrait. The generalised portraits for the SVM, RVM and FLD together with the Prot, where the prototype is the same
as thePgeneralisedPportrait, are shown inP
figure 2. We also note that w
~ can be written as
w
~ = i ?i ~xi = i| sign(?i )=+1 ?i ~xi ? i| sign(?i )=?1 |?i |~xi .
~ + are males whereas
The generalised portraits can be associated with the correct class: W
~ ? are females. The SVM and the FLD use patterns close to the SH for classification
W
and hence their decision-images appear androgynous, whereas Prot and RVM tend to use
patterns distant from the SH resulting in more female and male generalised portraits. Comparison of the optimal, true, generalised portraits to those based on the subject labels shows
that classification has become more difficult: generalised portraits have moved closer to
each other in gender space, narrowing the distance between the classes and thereby diminishing the gender typicality of the generalised portraits for all classifiers.
3
Human Gender Discrimination along the Decision-Image Axes
The decision-images introduced in section 2.2 are based purely on machine learning, albeit
on labels provided by human subjects in the case of the subject dataset. Our previous paper
[1] reported that the subjects? responses to the faces?proportion correct, reaction times
2
This term was introduced by [14] with the idea in mind that when trained on a set of portraits of
members of a family, one would obtain a ?generalized? portrait which captures the essential features
of the family as a superposition of all family members.
SVM
RVM
Prot
FLD
trained
on
?
W
+
true
data
?
W
?
true
data
?
W+
subj
data
?
W?
subj
data
~ ? for each classifier for both the true and the subject
Figure 2: Generalised portraits W
dataset; all images are rescaled to [0, 1] and their means set to 128 for illustration purposes
(different scalers for different images). [Unfortunately the downsampling (low-pass filtering) of the faces necessary to fit them in the figure makes all the faces somewhat more
androgynous than they are viewed at full resolution.]
(RT) and confidence ratings?correlated very well with the distance of the stimuli to their
separating hyperplane (SH) for support and relevance vector machines (SVMs, RVMs) but
not for simple prototype (Prot) classifier. If these correlations really implied that SVM
and RVM capture some crucial aspects of human internal face representation the following
prediction must hold: already for small |?| ISVM (?) and IRVM (?) should look male/female
whereas |?| IProt (?) and IFLD (?) should only be perceptually male/female for larger |?|.
In other words: the female-to-maleness axis of SVM and RVM should be closely aligned
to those of our subjects whereas that is not expected to be the case for FLD and Prot.
3.1 Psychophysical Methods
Four observers?one of the authors (FAW) with extensive psychophysical training and
three na??ve subjects paid for their participation?took part in a standard, spatial (left versus
right) two-alternative forced-choice (2AFC) discrimination experiment. Subjects were
presented with two faces I(??) and I(?) and had to indicate which face looked more
male. Stimuli were presented against the mean luminance (50 cd/m2 ) of a carefully linearised Clinton Monoray CRT driven by a Cambridge Research Systems VSG 2/5 display
controller. Neither male nor female faces changed the mean luminance. Subjects viewed
the screen binocularly with their head stabilised by a headrest. The temporal envelope of
stimulus presentation was a modified Hanning window (a raised cosine function with rise
and fall times of 500 ms and a plateau time of 1000 ms). The probability of the female
face being presented on the left was 0.5 on each trial and observers indicated whether they
0.8
FLD
Prot
RVM
SVM
0.7
0.6
0.5
a. FAW
0.4
0.3
0.2
0.05
0.0 9
0.4
0.8
1.4
length of normalised decision image vector ? W / ||W||
@75% correct
@90% correct
c. FJ
1.4
1
2
1
@75% correct
FLD
RVM
Prot
FLD
RVM
@75% correct
@90% correct
FLD
FLD
RVM
0.6
Prot
0.6
Prot
1
RVM
1
Prot
1.4
FLD
e. KT
1.4
RVM
d. HM
Prot
Prot
@90% correct
1.8
Prot
1.8
FLD
RVM
FLD
Prot
0.6
RVM
threshold elevation re. SVM
@90% correct
1.8
2.5
3
RVM
threshold elevation re. SVM
@75% correct
b. FAW
FLD
proportion correct gender identification
1
0.9
threshold elevation re. SVM
2
@90% correct
@75% correct
f. pooled
1. 5
1
RVM
Prot
FLD
RVM
Prot
FLD
0. 5
Figure 3: a. Shows raw data and fitted psychometric functions for one observer (FAW).
b?e. For each of four observers the threshold elevation for the RVM, Prot and FLD
decision-image relative to that of the SVM; results are shown for both 75 and 90% correct together with 68%-CIs. f. Same as in b?e but pooled across observers.
thought the left or right face was female by touching the corresponding location on a Elo
TouchSystems touch-screen immediately in front of the display; no feedback was provided.
Trials were run in blocks of 256 in which eight repetitions of eight stimulus levels
(??1 . . . ? ?8 ) for each of the four classifiers were randomly intermixed. The na??ve subjects required approximately 2000 trials before their performance stabilised; thereafter they
did another five to six blocks of 256 trials. All results presented below are based on the
trials after training; all training trials were discarded.
3.2 Results and Discussion
Figure 3a shows the raw data and fitted psychometric functions for one of the observers.
Proportion correct gender identification on the y-axis is plotted against ? on the x-axis
on semi-logarithmic coordinates. Psychometric functions were fitted using the psignifit
toolbox for Matlab which implements the constrained maximum-likelihood method described in [15]. 68%-confidence intervals (CIs), indicated by horizontal lines at 75 and
90-% correct in figure 3a, were estimated by the BCa bootstrap method also implemented
in psignifit [16]. The raw data appear noisy because each data point is based on only eight
trials. However, none of fitted psychometric functions failed various Monte Carlo based
goodness-of-fit tests [15].
To summarise the data we extracted the ? required for two performance levels
(?thresholds?), 75 and 90% correct, together with their corresponding 68%-CIs. Figure 3b?
e shows the thresholds for all four observers normalised by ?SVM (the ?threshold elevation?
re. SVM). Thus values larger than 1.0 for RVM, Prot and FLD indicate that more of the
corresponding decision-images had to be added for the human observers to be able to discriminate females from males. In figure 3f we pool the data across observers as the main
trend, poorer performance for Prot and FLD compared to SVM and RVM, is apparent for
all four observers. The difference between SVM and RVM is small; going along the direction of both Prot and FLD, however, results in a much ?slower? transition from female-tomaleness.
The psychophysical data are very clear: all observers require a larger ? for Prot and FLD;
the length ratio ranges from 1.2 to nearly 3.0, and averages to around 1.7 across observers.
In the pooled data all the differences are statistically significant but even at the individual
subject level all differences are significant at the 90% performance level, and five of eight
are significant at the 75% performance level. It thus appears that SVM and RVM capture
more of the psychological face-space of our human observers than Prot and FLD. From
our results we cannot exclude the possibility that some other direction might have yielded
even steeper psychometric functions, i.e. faster female-to-maleness transitions, but we can
conclude that the decision-images of SVM and RVM are closer to the decision-images
used by human subjects than those of Prot and FLD. This is exactly as predicted by the
correlations between proportion correct, RTs and confidence ratings versus distance to the
hyperplane reported in [1]?high correlations for SVM and RVM, low correlations for Prot.
4
Summary and Conclusions
We studied classification and discrimination of human faces both psychophysically as well
as using methods from machine learning. The combination of linear preprocessor (PCA)
and classifier (SVM, RVM, Prot and FLD) allowed us to visualise the decision-images of
a classifier corresponding to the vector normal to the SH of the classifier. Decision-images
can be used to determine the regions of the stimuli most useful for classification simply
by analysing the distribution of light and dark regions in the decision-image. In addition
we defined the generalised portraits to be the prototypes of all faces used by the classifier
to obtain its classification. For the SVM this is the weighted average of all the support
vectors (SVs), for the RVM the weighted average of all the relevance vectors (RVs), and
for the Prot it is the prototype itself. The generalised portraits are, like the decision-images,
another useful visualisation of the categorisation algorithm of the machine classifier.
However, the central result of our paper is the corroboration of the machine-learningpsychophysics research methodology. In the machine-learning-psychophysics research we
substitute a very hard to analyse complex system (the human brain) by a reasonably complex system (learning machine) that is complex enough to capture essentials of our human
subjects? behaviour but is nonetheless amenable to close analysis. From the analysis of
the machines we then derive predictions for human subjects which we subsequently test
psychophysically.
Given the success in predicting the steepness of the female-to-male transition of the w
~ SVM
~
-axis we believe that the decision-image WSVM captures some of the essential characteristics of the human decision algorithm.
Acknowledgements The authors would like to thank Bruce Henning, Frank J?akel, Ulrike
von Luxburg and Christian Wallraven for helpful comments and suggestions. In addition
we thank Frank J?akel for supplying us with the code to run the touch-screen experiment.
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1,699 | 2,544 | A Large Deviation Bound
for the Area Under the ROC Curve
Shivani Agarwal? , Thore Graepel? , Ralf Herbrich? and Dan Roth?
?
?
Dept. of Computer Science
University of Illinois
Urbana, IL 61801, USA
Microsoft Research
7 JJ Thomson Avenue
Cambridge CB3 0FB, UK
{sagarwal,danr}@cs.uiuc.edu
{thoreg,rherb}@microsoft.com
Abstract
The area under the ROC curve (AUC) has been advocated as an evaluation criterion for the bipartite ranking problem. We study large deviation properties of the AUC; in particular, we derive a distribution-free
large deviation bound for the AUC which serves to bound the expected
accuracy of a ranking function in terms of its empirical AUC on an independent test sequence. A comparison of our result with a corresponding
large deviation result for the classification error rate suggests that the test
sample size required to obtain an -accurate estimate of the expected accuracy of a ranking function with ?-confidence is larger than that required
to obtain an -accurate estimate of the expected error rate of a classification function with the same confidence. A simple application of the
union bound allows the large deviation bound to be extended to learned
ranking functions chosen from finite function classes.
1
Introduction
In many learning problems, the goal is not simply to classify objects into one of a fixed
number of classes; instead, a ranking of objects is desired. This is the case, for example, in
information retrieval problems, where one is interested in retrieving documents from some
database that are ?relevant? to a given query or topic. In such problems, one wants to return
to the user a list of documents that contains relevant documents at the top and irrelevant
documents at the bottom; in other words, one wants a ranking of the documents such that
relevant documents are ranked higher than irrelevant documents.
The problem of ranking has been studied from a learning perspective under a variety of
settings [2, 8, 4, 7]. Here we consider the setting in which objects come from two categories, positive and negative; the learner is given examples of objects labeled as positive
or negative, and the goal is to learn a ranking in which positive objects are ranked higher
than negative ones. This captures, for example, the information retrieval problem described
above; in this case, the training examples consist of documents labeled as relevant (positive) or irrelevant (negative). This form of ranking problem corresponds to the ?bipartite
feedback? case of [7]; for this reason, we refer to it as the bipartite ranking problem.
Formally, the setting of the bipartite ranking problem is similar to that of the binary classification problem. In both problems, there is an instance space X and a set of two
class labels Y = {?1, +1}. One is given a finite sequence of labeled training examples
S = ((x1 , y1 ), . . . , (xM , yM )) ? (X ? Y)M , and the goal is to learn a function based on
this training sequence. However, the form of the function to be learned in the two problems
is different. In classification, one seeks a binary-valued function h : X ?Y that predicts
the class of a new instance in X . On the other hand, in ranking, one seeks a real-valued
function f : X ? R that induces a ranking over X ; an instance that is assigned a higher
value by f is ranked higher than one that is assigned a lower value by f .
The area under the ROC curve (AUC) has recently gained some attention as an evaluation
criterion for the bipartite ranking [3]. Given a ranking function f : X ?R and a finite
data sequence T = ((x1 , y1 ), . . . , (xN , yN )) ? (X ? Y)N containing m positive and n
? ; T ), can be expressed as
negative examples, the AUC of f with respect to T , denoted A(f
the following Wilcoxon-Mann-Whitney statistic [3]:
X
X
1
1
? ;T) =
(1)
I{f (xi )>f (xj )} + I{f (xi )=f (xj )} ,
A(f
mn
2
{i:yi =+1} {j:yj =?1}
where I{?} denotes the indicator variable whose value is one if its argument is true and
zero otherwise. The AUC of f with respect to T is thus simply the fraction of positivenegative pairs in T that are ranked correctly by f , assuming that ties are broken uniformly
at random.1
The AUC is an empirical quantity that evaluates a ranking function with respect to a particular data sequence. What does the empirical AUC tell us about the expected performance
of a ranking function on future examples? This is the question we consider. The question
has two parts, both of which are important for machine learning practice. First, what can be
said about the expected performance of a ranking function based on its empirical AUC on
an independent test sequence? Second, what can be said about the expected performance of
a learned ranking function based on its empirical AUC on the training sequence from which
it is learned? We address the first question in this paper; the second question is addressed
in [1].
We start by defining the expected ranking accuracy of a ranking function (analogous to
the expected error rate of a classification function) in Section 2. Section 3 contains our
large deviation result, which serves to bound the expected accuracy of a ranking function
in terms of its empirical AUC on an independent test sequence. Our conceptual approach
in deriving the large deviation result for the AUC is similar to that of [9], in which large
deviation properties of the average precision were considered. Section 4 compares our
bound to a corresponding large deviation bound for the classification error rate. A simple
application of the union bound allows the large deviation bound to be extended to learned
ranking functions chosen from finite function classes; this is described in Section 5.
2
Expected Ranking Accuracy
We begin by introducing some notation. As in classification, we shall assume that all
examples are drawn randomly and independently according to some (unknown) underlying distribution D over X ? Y. The notation D+1 and D?1 will be used to denote the
class-conditional distributions DX|Y =+1 and DX|Y =?1 , respectively. We shall find it convenient to decompose a data sequence T = ((x1 , y1 ), . . . , (xN , yN )) ? (X ? Y)N into
two components, TX = (x1 , . . . , xN ) ? X N and TY = (y1 , . . . , yN ) ? Y N . Several
of our results will involve the conditional distribution DTX |TY =y for some label sequence
y = (y1 , . . . , yN ) ? Y N ; this distribution is simply Dy1 ? . . . ? DyN .2 As a final note of
1
In [3], a slightly simpler form of the Wilcoxon-Mann-Whitney statistic is used, which does not
account for ties.
2
Note that, since the AUC of a ranking function f with respect to a data sequence T ? (X ? Y)N
is independent of the ordering of examples in the sequence, our results involving the conditional
distribution DTX |TY =y for some label sequence y = (y1 , . . . , yN ) ? Y N depend only on the number
m of positive labels in y and the number n of negative labels in y. We state our results in terms of
m
n
the distribution DTX |TY =y ? Dy1 ? . . . ? DyN only because this is more general than D+1
? D?1
.
convention, we use T ? (X ? Y)N to denote a general data sequence (e.g., an independent
test sequence), and S ? (X ? Y)M to denote a training sequence.
Definition 1 (Expected ranking accuracy). Let f : X ?R be a ranking function on X .
Define the expected ranking accuracy (or simply ranking accuracy) of f , denoted by A(f ),
as follows:
n
o
1
A(f ) = EX?D+1 ,X 0 ?D?1 I{f (X)>f (X 0 )} + I{f (X)=f (X 0 )} .
2
The ranking accuracy A(f ) defined above is simply the probability that an instance drawn
randomly according to D+1 will be ranked higher by f than an instance drawn randomly
according to D?1 , assuming that ties are broken uniformly at random. The following simple lemma shows that the empirical AUC of a ranking function f is an unbiased estimator
of the expected ranking accuracy of f :
Lemma 1. Let f : X ?R be a ranking function on X , and let y = (y1 , . . . , yN ) ? Y N be
any finite label sequence. Then
n
o
? ;T)
ET |T =y A(f
= A(f ) .
X
Y
Proof. Let m be the number of positive labels in y, and n the number of negative labels in
y. Then from the definition of the AUC (Eq. (1)) and linearity of expectation, we have
o
n
? ;T)
ETX |TY =y A(f
o
n
X
X
1
1
=
EXi ?D+1 ,Xj ?D?1 I{f (Xi )>f (Xj )} + I{f (Xi )=f (Xj )}
mn
2
{i:yi =+1} {j:yj =?1}
=
1
mn
X
X
A(f )
{i:yi =+1} {j:yj =?1}
= A(f ) .
3
t
u
Large Deviation Bound
We are interested in bounding the probability that the empirical AUC of a ranking function
f with respect to a (random) test sequence T will have a large deviation from its expected
ranking accuracy. In other words, we are interested in bounding probabilities of the form
n
o
?
P A(f
; T ) ? A(f ) ?
for given > 0. Our main tool in deriving such a large deviation bound will be the following powerful concentration inequality of McDiarmid [10], which bounds the deviation of
any function of a sample for which a single change in the sample has limited effect:
Theorem 1 (McDiarmid, 1989). Let X1 , . . . , XN be independent random variables with
Xk taking values in a set Ak for each k. Let ? : (A1 ? ? ? ? ? AN ) ?R be such that
sup
xi ?Ai ,x0k ?Ak
|?(x1 , . . . , xN ) ? ?(x1 , . . . , xk?1 , x0k , xk+1 , . . . , xN )|
? ck .
Then for any > 0,
P {|?(X1 , . . . , XN ) ? E{?(X1 , . . . , XN )}| ? }
? 2e?2
2
/
PN
k=1
c2k
.
Note that when X1 , . . . , XN are independent bounded random variables with Xk ? [ak , bk ]
PN
with probability one and ?(X1 , . . . , XN ) = k=1 Xk , McDiarmid?s inequality (with ck =
bk ? ak ) reduces to Hoeffding?s inequality. Next we define the following quantity which
appears in several of our results:
Definition 2 (Positive skew). Let y = (y1 , . . . , yN ) ? Y N be a finite label sequence of
length N ? N. Define the positive skew of y, denoted by ?(y), as follows:
?(y)
=
1
N
X
1.
{i:yi =+1}
The following can be viewed as the main result of this paper. We note that our results are
all distribution-free, in the sense that they hold for any distribution D over X ? Y.
Theorem 2. Let f : X ?R be a fixed ranking function on X and let y = (y1 , . . . , yN ) ?
Y N be any label sequence of length N ? N. Then for any > 0,
o
n
2
?
? 2e?2?(y)(1??(y))N .
; T ) ? A(f ) ?
PTX |TY =y A(f
Proof. Let m be the number of positive labels in y, and n the number of negative labels in
y. We can view TX = (X1 , . . . , XN ) ? X N as a random vector; given the label sequence
y, the random variables X1 , . . . , XN are independent, with each Xk taking values in X .
Now, define ? : X N ?R as follows:
? (x1 , . . . , xN )
= A? (f ; ((x1 , y1 ), . . . , (xN , yN ))) .
Then, for each k such that yk = +1, we have the following for all xi , x0k ? X :
?(x1 , . . . , xN ) ? ?(x1 , . . . , xk?1 , x0k , xk+1 . . . , xN )
1 X
1
=
I{f (xk )>f (xj )} + I{f (xk )=f (xj )} ?
mn
2
{j:yj =?1}
!
1
I{f (x0k )>f (xj )} + I{f (x0k )=f (xj )}
2
?
=
1
n
mn
1
.
m
Similarly, for each k such that yk = ?1, one can show for all xi , x0k ? X :
?(x1 , . . . , xN ) ? ?(x1 , . . . , xk?1 , x0k , xk+1 . . . , xN )
?
1
.
n
Thus, taking ck = 1/m for k such that yk = +1 and ck = 1/n for k such that yk = ?1,
and applying McDiarmid?s theorem, we get for any > 0,
n
n
o
o
2
1 2
1 2
?
? ; T ) ?
PTX |TY =y A(f
; T ) ? ETX |TY =y A(f
? 2e?2 /(m( m ) +n( n ) ) . (2)
Now, from Lemma 1,
n
o
? ;T)
ETX |TY =y A(f
= A(f ) .
Also, we have
1 2
)
m( m
1
=
+ n( n1 )2
1
m
1
+
1
n
=
mn
= ?(y)(1 ? ?(y))N .
m+n
Substituting the above in Eq. (2) gives the desired result.
t
u
We note that the result of Theorem 2 can be strengthened so that the conditioning is only
on the numbers m and n of positive and negative labels, and not on the specific label vector
y.3 From Theorem 2, we can derive a confidence interval interpretation of the bound that
gives, for any 0 < ? ? 1, a confidence interval based on the empirical AUC of a ranking
function (on a random test sequence) which is likely to contain the true ranking accuracy
with probability at least 1 ? ?. More specifically, we have:
Corollary 1. Let f : X ?R be a fixed ranking function on X and let y = (y1 , . . . , yN ) ?
Y N be any label sequence of length N ? N. Then for any 0 < ? ? 1,
s
(
)
ln 2?
?
? ?.
PTX |TY =y A(f ; T ) ? A(f ) ?
2?(y)(1 ? ?(y))N
2
Proof. This follows directly from Theorem 2 by setting 2e?2?(y)(1??(y))N = ? and solving for .
t
u
Theorem 2 also allows us to obtain an expression for a test sample size that is sufficient to
obtain, for 0 < , ? ? 1, an -accurate estimate of the ranking accuracy with ?-confidence:
Corollary 2. Let f : X ?R be a fixed ranking function on X and let 0 < , ? ? 1. Let
y = (y1 , . . . , yN ) ? Y N be any label sequence of length N ? N. If
ln 2?
N ?
,
2?(y)(1 ? ?(y))2
o
n
then
?
; T ) ? A(f ) ?
? ?.
PTX |TY =y A(f
2
Proof. This follows directly from Theorem 2 by setting 2e?2?(y)(1??(y))N ? ? and solving for N .
t
u
Figure 1 illustrates the dependence of the above expression for the sufficient test sample
size on the the accuracy parameter and positive skew ?(y) for different values of ?.
The confidence interval of Corollary 1 can in fact be generalized to remove the conditioning
on the label vector completely:
Theorem 3. Let f : X ?R be a fixed ranking function on X and let N ? N. Then for any
0 < ? ? 1,
s
(
)
ln 2?
?
PT ?DN A(f ; T ) ? A(f ) ?
? ?.
2?(TY )(1 ? ?(TY ))N
Proof. For T ? (X ? Y)N and 0 < ? ? 1, define the proposition
s
(
)
ln 2?
?
?(T, ?) ?
.
A(f ; T ) ? A(f ) ?
2?(TY )(1 ? ?(TY ))N
Then for any 0 < ? ? 1, we have
PT {?(T, ?)} = ET I?(T,?)
n
o
= ETY ETX |TY =y I?(T,?)
n
o
= ETY PTX |TY =y {?(T, ?)}
? ETY {?}
= ?.
3
(by Corollary 1)
Our thanks to an anonymous reviewer for pointing this out.
t
u
Figure 1: The test sample size N (based on Corollary 2) sufficient to obtain an -accurate estimate
of the ranking accuracy with ?-confidence, for various values of the positive skew ? ? ?(y) for some
label sequence y, for (left) ? = 0.01 and (right) ? = 0.001.
Note that the above ?trick? works only once we have gone to a confidence interval; an
attempt to generalize the bound of Theorem 2 in a similar way gives an expression in which
the final expectation is not easy to evaluate. Interestingly, the above proof does not even
require a factorized distribution DTY since it is built on a result for any fixed label sequence
y. We note that the above technique could also be applied to generalize the results of [9] in
a similar manner.
4
Comparison with Large Deviation Bound for Error Rate
Our use of McDiarmid?s inequality in deriving the large deviation bound for the AUC of
a ranking function is analogous to the use of Hoeffding?s inequality in deriving a large
deviation bound for the error rate of a classification function. (e.g., see [6, Chapter 8]). The
need for the more general inequality of McDiarmid in our derivations arises from the fact
that the empirical AUC, unlike the empirical error rate, cannot be expressed as a sum of
independent random variables.
Given a classification function h : X ?Y, let L(h) denote the expected error rate of h:
L(h) = EXY ?D I{h(X)6=Y } .
Similarly, given a classification function h : X ?Y and a finite data sequence T =
? T ) denote the empirical error rate of h
((x1 , y1 ), . . . , (xN , yN )) ? (X ? Y)N , let L(h;
with respect to T :
N
X
? T) = 1
L(h;
I{h(xi )6=yi } .
N i=1
Then the large deviation bound obtained via Hoeffding?s inequality for the classification
error rate states that for a fixed classification function h : X ?Y and for any N ? N, > 0,
n
o
2
?
PT ?DN L(h;
T ) ? L(h) ?
? 2e?2N .
(3)
Comparing Eq. (3) to the bound of Theorem 2, we see that the AUC bound differs from the
error rate bound by a factor of ?(y)(1 ? ?(y)) in the exponent. This difference translates
into a 1/(?(y)(1 ? ?(y))) factor difference in the resulting sample size bounds: given
0 < , ? ? 1, the test sample size sufficient to obtain an -accurate estimate of the expected
accuracy of a ranking function with ?-confidence is 1/(?(y)(1??(y))) times larger than the
corresponding test sample size sufficient to obtain an -accurate estimate of the expected
error rate of a classification function with the same confidence. For ?(y) = 1/2, this means
a sample size larger by a factor of 4; as the positive skew ?(y) departs from 1/2, the factor
grows larger (see Figure 2).
Figure 2: The test sample size bound for the AUC, for positive skew ? ? ?(y) for some label
sequence y, is larger than the corresponding test sample size bound for the classification error rate by
a factor of 1/(?(1 ? ?)).
5
Bound for Learned Ranking Functions Chosen from Finite Classes
The large deviation result of Theorem 2 bounds the expected accuracy of a ranking function
in terms of its empirical AUC on an independent test sequence. A simple application of the
union bound allows the result to be extended to bound the expected accuracy of a learned
ranking function in terms of its empirical AUC on the training sequence from which it is
learned, in the case when the learned ranking function is chosen from a finite function class.
More specifically, we have:
Theorem 4. Let F be a finite class of real-valued functions on X and let fS ? F denote
the ranking function chosen by a learning algorithm based on the training sequence S. Let
y = (y1 , . . . , yM ) ? Y M be any label sequence of length M ? N. Then for any > 0,
o
n
2
?
;
S)
?
A(f
)
? 2|F|e?2?(y)(1??(y))M .
PSX |SY =y A(f
S
S ?
Proof. For any > 0, we have
o
n
?
PSX |SY =y A(f
S ; S) ? A(fS ) ?
?
? PSX |SY =y max A(f ; S) ? A(f ) ?
f ?F
n
o
X
?
; S) ? A(f ) ?
?
PSX |SY =y A(f
(by the union bound)
f ?F
2
? 2|F|e?2?(y)(1??(y))M
(by Theorem 2) .
t
u
As before, we can derive from Theorem 4 expressions for confidence intervals and sufficient training sample size. We give these here without proof:
Corollary 3. Under the assumptions of Theorem 4, for any 0 < ? ? 1,
s
(
)
ln |F| + ln 2?
?
PSX |SY =y A(fS ; S) ? A(fS ) ?
? ?.
2?(y)(1 ? ?(y))M
Corollary 4. Under the assumptions of Theorem 4, for any 0 < , ? ? 1, if
1
2
ln |F| + ln
,
M ?
2?(y)(1 ? ?(y))2
?
then
o
n
?
? ?.
PSX |SY =y A(f
S ; S) ? A(fS ) ?
Theorem 5. Let F be a finite class of real-valued functions on X and let fS ? F denote
the ranking function chosen by a learning algorithm based on the training sequence S. Let
M ? N. Then for any 0 < ? ? 1,
s
(
)
ln |F| + ln 2?
?
PS?DM A(fS ; S) ? A(fS ) ?
? ?.
2?(SY )(1 ? ?(SY ))M
6
Conclusion
We have derived a distribution-free large deviation bound for the area under the ROC curve
(AUC), a quantity used as an evaluation criterion for the bipartite ranking problem. Our result parallels the classical large deviation result for the classification error rate obtained via
Hoeffding?s inequality. Since the AUC cannot be expressed as a sum of independent random variables, a more powerful inequality of McDiarmid was required. A comparison with
the corresponding large deviation result for the error rate suggests that, in the distributionfree setting, the test sample size required to obtain an -accurate estimate of the expected
accuracy of a ranking function with ?-confidence is larger than the test sample size required
to obtain a similar estimate of the expected error rate of a classification function. A simple
application of the union bound allows the large deviation bound to be extended to learned
ranking functions chosen from finite function classes.
A possible route for deriving an alternative large deviation bound for the AUC could be
via the theory of U-statistics; the AUC can be expressed as a two-sample U-statistic, and
therefore it may be possible to apply specialized results from U-statistic theory (see, for
example, [5]) to the AUC.
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