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-tE5T4oBgHgl3EQfRw5F/content/tmp_files/2301.05523v1.pdf.txt

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+arXiv:2301.05523v1  [cond-mat.str-el]  13 Jan 2023
+Bi layer properties in the Bi–FeNi GMR-type structures
+probed by spectroscopic ellipsometry
+Natalia Kovaleva,1, ∗ Dagmar Chvostova,2 Ladislav Fekete,2 and Alexandr Dejneka2, †
+1Lebedev Physical Institute, Russian Academy of Sciences
+Leninsky prospect 53, Moscow 119991, Russia
+2Institute of Physics, Academy of Sciences of the Czech Republic
+Na Slovance 2, Prague 18221, Czech Republic
+(Dated: January 16, 2023)
+Abstract
+Bismuth (Bi) having a large atomic number is characterized by the strong spin-orbit coupling
+(SOC) and is a parent compound of many 3D topological insulators (TIs). The ultrathin Bi films
+are supposed to be 2D TIs possessing the nontrivial topology, which opens the possibility of devel-
+oping new efficient technologies in the field of spintronics. Here we aimed at studying the dielectric
+function properties of ultrathin Bi/FeNi periodic structures using spectroscopic ellipsometry. The
+[Bi(d)–FeNi(1.8 nm)]N GMR-type structures were grown by rf sputtering deposition on Sitall-glass
+(TiO2) substrates. The ellipsometric angles Ψ(ω) and ∆(ω) were measured for the grown series
+(d=0.6, 1.4, 2.0, 2.5 nm, N = 16) of the multilayered film samples at room temperature for four an-
+gles of incidence of 60◦, 65◦, 70◦, and 75◦ in a wide photon energy range of 0.5-6.5 eV. The measured
+ellipsometric angles, Ψ(ω) and ∆(ω), were simulated in the framework of the corresponding mul-
+tilayer model. The complex (pseudo)dielectric function spectra of the Bi layer were extracted.
+The GMR effects relevant for the studied Bi–FeNi MLF systems are estimated from the optical
+conductivity zero-limit (optical GMR effect). The obtained results demonstrate that the Bi layer
+possesses the surface metallic conductivity induced by the SOC effects, which is strongly enhanced
+on vanishing the semimetallic-like phase contribution on decreasing the layer thickness, indicating
+its nontrivial 2D topology properties.
+∗Electronic address: [email protected]
+†Electronic address: [email protected]
+1
+
+I.
+INTRODUCTION
+The relativistic effect of spin–orbit (SOC) coupling is involved in the so-called Rashba
+effect [1]. This phenomenon arises from the apparent loss of crystalline inversion symmetry
+near the surface or heterojunction leading to the lifting of the spin degeneracy and generating
+spin-polarized surface metallic states. In this respect, 3D (2D) topological insulators (TIs)
+also exhibit spin-polarized surface metallic states due to SOC. However, contrary to the
+Rashba effect, the surface metallic bands of a TI are determined by its bulk characteristics.
+The TIs host metallic surface states in a bulk energy gap, which are topologically protected.
+The surface (or interface) states of TIs can be topologically trivial or nontrivial. In the latter
+case, for example, electrons cannot be backscattered by impurities. Bismuth (Bi) having a
+large atomic number is characterized by the strong SOC and is a parent compound of many
+3D TIs, such as Bi1−xSbx or Bi2Se3, even although 3D bulk Bi itself is topologically trivial.
+The specific feature of the electronic band structure of bulk Bi having R¯3m rhombohedral
+symmetry [2–4] is its inverted band gaps at both the Γ and M points of the Brillouin zone
+due to the strong SOC. The uniqueness of Bi films associated with the surface metallic states
+[5, 6] and the semiconductor-to-metal transition [7, 8] are well documented in the literature.
+Theoretical analyses predict a 1-bilayer (BL) Bi(111) film to be a 2D TI [9, 10].
+If
+there is no or weak inter-BL coupling, a stack of the odd-even 1-BL films will exhibit
+nontrivial to trivial oscillations of topology (where the topological number ν [11] is equal
+to 1 or 0, respectively). However, for the nontrivial topology in a stack of the 1-BL films,
+the intermediate inter-BL coupling strength, which is, for example, higher than the van der
+Waals strengths, is a mandatory condition. The direct (Γ point) and indirect band gap values
+were calculated by Liu et al. as a function of the Bi film thickness [12]. It was established
+that below 4BLs the film is a semiconductor with the direct gap open at the Γ point and
+the positive indirect band gap leading to nontrivial topology peculiar for an intrinsic 2D TI.
+Above 4BLs the indirect band gap becomes negative resulting in a semiconductor- semimetal
+transition due to overlapping of two bands at the Fermi level around the Γ and M points.
+This suggests that the Bi films from 5 to 8 BLs represent a 2D TI situated between two
+trivial metallic surfaces [12].
+A comprehensive study of the associated SOC effects in ultrathin Bi layers opens the pos-
+sibility of developing new efficient technologies in the field of spintronics. For this purpose,
+2
+
+here we aimed at studying the dielectric function properties of ultrathin periodic structures
+Bi/Ni79Fe21, prepared by rf sputter deposition, which is one of the most common technologies
+used to grow coatings and multilayered films (MLFs) exhibiting a giant magnetoresistance
+(GMR) effect for various existing and modern nanotechnological applications. Earlier, we
+have demonstrated that electronic band structure and surface electronic properties of ul-
+trathin Bi layers in real GMR-type (Bi–FeNi)N MLF structures incorporating nanoisland
+FeNi layers can be successfully studied by spectroscopic ellipsometry (SE) [13]. Here, by ap-
+plying the elaborated SE approach, we investigate (Bi–FeNi) MLFs, where the thickness of
+the FeNi layer was 1.8 nm, corresponding to the FeNi layer structural percolation threshold
+[14, 15], and the Bi spacer layer was 0.6, 1.4, 2.0, and 2.5 nm thick, incorporating about 2,
+4, 6, and 8 Bi(012)-type planes, respectively. We found that the Bi spacer layers have the
+metallic surface conductivity, which demonstrates strongly enhanced metallicity properties
+on vanishing the Bi semimetallic-like phase contribution on decreasing the layer thickness,
+which can be constructive in finding new nontrivial 2D topology properties of the (Bi–FeNi)
+GMR-type structures for their different nanotechnological applications.
+II.
+MATERIALS AND METHODS
+The (Bi–FeNi)N MLFs were prepared in a sputter deposition system by cathode sput-
+tering from 99.95% pure Bi and Fe21Ni79 targets in an alternative way. The base pressure
+in a sputter deposition chamber was 2×10−6 Torr. The multilayers were deposited at ap-
+proximately 80 ◦C in an argon atmosphere of 6× 10−4 Torr on to insulating glassy Sitall
+(TiO2) substrates. We utilized the substrates having typical dimensions 15 × 5 × 0.6 mm3.
+The nominal thicknesses of the FeNi and Bi layers were controlled by the layer deposition
+times in accordance with the material deposition rates. A series consisting of four MLF
+samples was prepared. In the series of the grown (Bi–FeNi)N samples, the nominal thickness
+of the FeNi layer was 1.8 nm and the Bi layer thickness was of 0.6, 1.4, 2.0, and 2.5 nm, the
+number N of the periodically repeated Bi/FeNi layers was 16. The thickness of the FeNi
+layer was chosen to be 1.8 nm matching the structural percolation threshold [14, 15]. The
+Bi layer thicknesses were chosen in such a way that the conditions for ferromagnetic (FM)
+or antiFM coupling in the GMR-type structures would be optimized. To prevent degra-
+dation, the deposited (Bi–FeNi)16/Sitall samples were covered with the 2.1 nm-thick Al2O3
+3
+
+FIG. 1: AFM images (a–d) 5 × 5 µm2 and (e–h) 1 × 1 µm2 of the Al2O3/(Bi–FeNi)16/Sitall MLF
+samples, where the nominal Al2O3 and FeNi layer thicknesses are 2.1 and 1.8 nm and the nominal Bi
+layer thicknesses are 0.6, 1.4, 2.0, and 2.5 nm, respectively. The estimated surface RMS roughness
+values are in (a-d) 3.6, 3.0, 3.1, and 5.2 nm and in (e-h) 3.2, 2.6, 2.7, and 5.3 nm, respectively.
+(i,j) The typical height profiles for the MLF samples with the nominal Bi layer thicknesses of 0.6
+and 2.5 nm, respectively.
+layer. The related [Bi–FeNi(0.8,1.2 nm)]N samples prepared by rf sputtering deposition onto
+the Sitall substrates under similar conditions were investigated by X-ray diffraction (XRD)
+as well as by X-ray reflectivity (XRR) experimental techniques in our previous study (see
+Supplementary online information for the article [13]). The XRR spectra proved a good
+periodicity and consistency with the corresponding nominal thicknesses of the FeNi and Bi
+slices in the Bi/FeNi MLF structures, as well as relatively low interface roughness between
+the constituent layers. The XRD characterization suggests (012)-type Bi plane orientation,
+where the interlayer distance is 3.28 ˚A. It follows from this that in the studied MLF struc-
+tures the Bi layers with a thickness corresponding to 0.6, 1.4, 2.0, and 2.5 nm incorporate
+4
+
+D!2SUce ()
+0
+500
+300
+400
+eoo
+100
+008
+a0o
+0001
+SO
+10
+JO
+(uw
+SO
+30D!2Sce (u)
+0
+500
+300
+400
+ooa
+100
+008
+a0o
+0001
+2
+Heiapr (uw)
+0
+2HGidpf 2GU20l
+mn 0.00s
+-34'2 UW
+S4'0 UWHGidpf 2Gu20l
+my o.r
+mn 8.84-
+401 UwHGidpf 26GU20l
+mn 0.00s
+mn 8.07-
+mn 8.eHGidpf 26GU20l
+my o.7
+mn c.er.HGidpf 26GU20l
+mn 0.00s
+n r.ar-
+JI'SUwHeiauf 26u20l
+my o.1
+mn &.rr-
+mn c.srHGidpf 26u20l
+mn 0.00s
+mn r.rr-
+mn f.crHeidpf 26u20k
+my o.r
+-531UWtwo, four, six, and eight Bi(012)-type planes, respectively.
+In the present study, the surface morphology of the Bi–FeNi(1.8 nm) MLF samples, pre-
+pared by rf sputtering deposition on the Sitall (TiO2) substrates, was studied at room
+temperature using an ambient AFM (Bruker, Dimension Icon) in the Peak Force Tapping
+mode with ScanAsyst Air tips (Bruker, k=0.4 N/m, nominal tip radius 2 nm). The SE mea-
+surements for the investigated Al2O3/(Bi–FeNi)16/Sitall samples were performed at room
+temperature in a wide photon energy range of 0.5 – 6.5 eV using a J.A. Woollam VUV-VASE
+ellipsometer (see the scheme illustrating the SE study of the (Bi–FeNi)N MLFs in Fig. 1(a)
+of Ref. [13]). The measured ellipsometry spectra are represented by real values of the angles
+Ψ(ω) and ∆(ω), which are defined through the complex Fresnel reflection coefficients for
+light-polarized parallel rp and perpendicular rs to the plane of incidence, tan Ψ ei∆ = rp
+rs .
+The ellipsometric angles, Ψ(ω) and ∆(ω), measured for the Bi–FeNi MLF samples were sim-
+ulated using the multilayer model simulation available in the J.A. Woollam VASE software
+[16]. From the multilayer model simulations, the (pseudo)dielectric function spectra of the
+ultrathin 0.6, 1.4, 2.0, and 2.5 nm Bi layers and 1.8 nm FeNi layer inside the Bi–FeNi MLF
+structures were extracted. The corresponding calculated optical conductivity spectra were
+analyzed.
+III.
+RESULTS
+A.
+Atomic force microscopy study
+The retrieved 5×5 µm2 and 1×1 µm2 AFM images of the Al2O3(2.1 nm)/[Bi(0.6, 1.4, 2.0,
+2.5 nm)–FeNi(1.8 nm)]N/Sitall multilayered films (where the given layer thicknesses corre-
+spond to their nominal values), presented in Figure 1a–h show discernable contrast because
+of the available surface hight deviations. The surface roughness of the Sitall glass (TiO2)
+substrates was investigated by us by AFM in our earlier publication [17]. The height profile
+of the Sitall substrates (see Fig. 2a of Ref. [17]) demonstrated the height deviation within
+the range 1-3 nm peculiar to the relatively large 0.3-1 µm lateral scale, which characterizes
+the Sitall substrate surface roughness. From the AFM measurements on the areas 5×5 µm2
+and 1×1 µm2 the root-mean square (RMS) surface roughness values were evaluated, which
+are presented in the caption to Figure 1.
+The corresponding RMS roughness values are
+5
+
+notably higher for the Al2O3(2.1 nm)/[Bi(2.5 nm)–FeNi(1.8 nm)]16/Sitall MLF sample. The
+smaller-scale (1 × 1 µm2) images clearly recognize a fine grainy-like structure of the surface
+morphology, which seems to be characteristic for all studied film samples (see Figure 1e–h).
+The typical grain size, being of about 50 nm, is notably larger for the FeNi(1.8 nm) – Bi MLF
+sample incorporating the 2.5 nm-thick Bi layers, and, following the estimated RMS rough-
+ness values, the average grain size decreases to about 20 nm with decreasing the Bi layer
+thickness to 1.4 nm. As one can see from the typical height profiles presented in Figure 1i,j,
+with decreasing the Bi layer thickness from 2.5 to about 0.6 nm, the surface morphology
+becomes highly irregular due to the formation of conglomerates of nanoislands separated by
+rather flat (relatively small roughness) areas of about 20 nm.
+B.
+Spectroscopic ellipsometry study of the ultrathin Bi–FeNi multilayer film samples
+The ellipsometric angles Ψ(ω) and ∆(ω) were measured for the prepared Al2O3/(Bi–
+FeNi)16/Sitall MLF samples at the angles of incidence of 60◦, 65◦, 70◦, and 75◦. Figure 2
+demonstrates the ellipsometric angles Ψ(ω) and ∆(ω) recorded at 65◦ and 70◦. To model
+the contributions from free charge carriers and interband optical transitions, the complex
+dielectric function ˜ε(ω) = ε1(ω) + iε2(ω) of the Bi and FeNi layers was interpreted in terms
+of the Drude and Lorentz parts, respectively,
+˜ε(E ≡ ℏω) = ǫ∞ −
+AD
+E2 + iEγD
++
+�
+j
+AjγjEj
+E2
+j − E2 − iEγj
+,
+(1)
+where ε∞ is the high-frequency dielectric constant, which takes into account the contribution
+from the higher-energy interband transitions. The fitted Drude parameters were AD and free
+charge carrier’s scattering rate γD. The fitted parameters of Lorentz bands were Ej, γj, and
+Aj of the band maximum energy, the full width at half maximum, and the ε2 band height,
+respectively. The obtained ellipsometric angles Ψ(ω) and ∆(ω) measured at different angles
+of incidence of 60◦, 65◦, 70◦, and 75◦ were fitted for each sample simultaneously using the
+J.A. Woollam VASE software [16] in the framework of the designed multilayer model. The
+multilayer model for the studied Al2O3/(Bi–FeNi)/Sitall multiayers was constructed as it is
+schematically presented in Figure 3, exactly so as the layers were deposited. In addition, we
+attempted to take into account the roughness properties of the surface by using the conven-
+tional approach of effective medium approximation (EMA) based on the (50% Al2O3–50%
+6
+
+FIG. 2: (a-d) Ellipsometric angles, Ψ(ω) and ∆(ω) (symbols), measured at the angles of incidence
+of 65◦ and 70◦ for the Al2O3/[Bi(d)–NiFe(1.8 nm)]16/Sitall multilayered films where the Bi spacer
+layer thicknesses d = 0.6, 1.4, 2.0, and 2.5 nm, respectively. The solid red, blue, green, and black
+curves show the corresponding simulation results for the angle 65◦ by the dielectric function model
+using Equation 1.
+7
+
+vacuum) Bruggeman model. The dispersion model for the Bi layers included three or four
+Lorentz terms as well as the Drude part. The dispersion model for the 1.8 nm permalloy
+layers incorporated in the studied MLF structures included the Drude term responsible for
+the free charge carrier contribution and one Lorentz oscillator to account for the most pro-
+nounced interband optical transition. In addition, the dielectric function spectra of the bare
+Sitall substrate derived from our earlier SE studies [18, 19] were introduced to the elabo-
+rated multilayer model. The dielectric response of the Al2O3 capping layer was represented
+by the tabular complex dielectric function spectra [20]. The thicknesses of the Bi and FeNi
+layers, as well as of the surface layers, were fitted. The unknown parameters were allowed to
+vary until the minimum of the mean squared error (MSE) is reached. The best simulation
+result for the studied [Bi(0.6, 1.4, 2.0, 2.5 nm)–FeNi(1.8 nm)]16 MLF samples corresponded
+to the lowest obtained MSE values of 0.3843, 0.297, 0.2934, and 0.4508, respectively. The
+good quality of the fit allowed us to estimate the actual Bi and FeNi layer thicknesses in the
+MLFs under study. The quality of the fit is demonstrated by Figure 2, where we plotted the
+measured ellipsometric angles along with the simulation results. The Drude and Lorentz
+parameters resulting from the simulation of the Al2O3/[Bi(d)–FeNi(1.8 nm)]16/Sitall MLF
+samples are given in Tables I and II, and the resulting ε1(ω) and ε2(ω) parts of the Bi and
+FeNi (pseudo)dielectric function spectra are presented in Figure 4.
+From Figure 4a,b one can see that the complex (pseudo)dielectric functions of the 0.6, 1.4,
+2.0, and 2.5 nm thick Bi spacers inside the investigated Bi–FeNi MLFs demonstrate metal-
+lic character. Moreover, the ε1(ω) function progressively decreases while the Bi thickness
+decreases from 2.5–2.0 to 1.4 nm and the ε2(ω) increases at low photon energies, respec-
+tively. According to our simulation results, we expect that the best metallicity properties
+are demonstrated by the Bi layer in the [Bi(1.4 nm)–NiFe(1.8 nm)]16 structure. At the same
+time, the complex (pseudo)dielectric functions of the thinnest 0.6 nm thick Bi layer look
+somewhat different. Here, in addition to the low-energy metallic Drude response identified
+by the characteristic behavior of the ε1(ω) and ε2(ω), the Lorentz band around 4–5 eV makes
+an essential contribution to the dielectric function response (the corresponding Drude (AD
+and γD) and Lorentz (Aj, Ej, and γj) parameters are listed in Table I). Next, being similar,
+the dielectric functions of the 1.8 nm thick permalloy layers in the [FeNi–Bi(1.4, 2.0, 2.5 nm)]
+MLFs are dominated by the ε2(ω) resonance and ε1(ω) antiresonance features, indicat-
+ing the predominant contribution from the Lorentz oscillator peaking at around 3 eV (see
+8
+
+(a)
+(b)
+(  )
+(d)
+c
+FIG. 3:
+The multilayer model applied for the simulation of the Al2O3/[Bi(0.6, 1.4, 2.0, and
+2.5 nm)–FeNi(1.8 nm)]16/Sitall samples. The Bi and FeNi thicknesses estimated from the model
+simulations in (a) 0.684±0.037 nm and 2.082±0.116 nm, (b) 1.408±0.574 nm and 1.780±0.65 nm,
+(c) 1.764±0.194 nm and 1.825±0.358 nm, and (d) 2.387±0.128 nm and 1.782±0.171 nm. Note good
+agreement between the thicknesses of the FeNi and Bi layers estimated from the model simula-
+tions and their respective nominal thickness values. The roughness and Al2O3 thicknesses esti-
+mated from the model simulations in (a) 0.00±3.85 nm and 1.283±2.37 nm, (b) 0.000±4.97 nm and
+4.967±2.17 nm, (c) 0.848±5.86 nm and 4.738±2.92 nm, and (d) 0.000±2.95 nm and 5.389±1.23 nm.
+Figure 4c,d).
+An upturn evident in the ε2(ω) at low photon energies indicates an addi-
+tional Drude contribution, which is relatively less pronounced. Following our simulation
+results, we expect the advanced metallicity properties of the FeNi layer in the [Bi(0.6 nm)–
+NiFe(1.8 nm)]16 structure (see the corresponding Drude (AD and γD) and Lorentz (Aj, Ej,
+and γj) parameters listed in Table II).
+Figure 5a–d presents the evolution of the Bi intralayer optical conductivity, σ1(ω) =
+ε2(ω)ω(cm−1)/60, upon decreasing the Bi spacer layer thickness in the [FeNi(1.8 nm) –
+Bi(2.5, 2.0, 1.4, 0.6 nm)]16 structures, and Figure 5e–h shows the associated optical conduc-
+tivity spectra of the 1.8 nm FeNi permalloy layer. Here, the contributions from the Drude
+and Lorentz oscillators following the multilayer model simulations using Equation 1 are evi-
+9
+
+woge
+0
+sti2(19VSl
+mna.Oid) 19vsl
+mn8.TingtS80.Sarmna.Oid
+19vsl↑80.03
+91503
+Clε8S.T2lonap0'000wog6
+0
+sti219Vsl
+mn8, Tingt
+S1081.1arS
+19Vsl
+p!↓'4uw80↓.13
+91503
+Cl2lonap
+40'000wogel
+0
+lsti219Vsl
+mn8, Tingt
+328.1armno.Sid1043
+91503
+Cl881.2lonap
+488.0woge
+0
+ti219Vsl
+mn8. Tingt
+4S81.1arIS入Gl
+mnc,.Sid188.S3
+91S03
+Cle88.2lonap
+40'000FIG. 4: The complex (pseudo)dielectric function spectra, ε2(ω) and ε1(ω), of the (a,b) Bi layers and
+(c,d) FeNi layers in the [Bi(d)–FeNi(1.8 nm)]16 structures shown for the Bi layer nominal thickness
+values d = 0.6, 1.4, 2.0, and 2.5 nm by solid red, blue, green, and black curves, respectively.
+TABLE I: Drude-Lorentz parameters for the Bi spacer layer in the [Bi(0.6, 1.4, 2.0, 2.5 nm)–
+NiFe(1.8 nm)]16 multilayered films obtained from the model simulations of the dielectric functions
+by using Equation 1. The values of Ej, γj, and γD are given in eV, and optical conductivity limit
+σ1(ω→0) in Ω−1·cm−1.
+Parameters 0.6 nm
+1.4 nm
+2.0 nm
+2.5 nm
+Drude
+AD
+46.(9)±4
+66.(7)±4
+24.(5)±4
+25.(1)±2
+γD
+1.2(5)±0.09 1.51(0)±0.06 2.7(2)±0.4
+3.1(3)±0.2
+σ1(ω→0)
+6300±540
+8970±540
+3290±540
+3370±270
+Lorentz
+E1
+–
+0.45(8)±0.05 0.35(9)±0.01 0.38(6)±0.004
+oscillator
+A1
+–
+15.(0)±6
+96.(0)±10
+70.(8)±2
+γ1
+–
+0.52(6)±0.09 0.79(1)±0.02 0.67(6)
+Lorentz
+E2
+4.67
+5.31(5)±0.03 5.08(7)±0.04 4.77(5)±0.04
+oscillator
+A2
+10.2(7)±0.6 2.53(2)±0.05 1.2(5)±0.1
+0.67(6)±0.08
+γ2
+4.2(1)±0.07 3.99(3)±0.07 3.4(7)±0.2
+2.5(5)±0.2
+Lorentz
+E3
+11.1
+7.8
+7.7
+7.7
+oscillator
+A3
+7.2
+4.1
+4.1
+4.1
+γ3
+8.9
+2.8
+2.8
+2.8
+10
+
+TABLE
+II:
+Drude-Lorentz
+parameters
+for
+the
+1.8 nm
+thick
+NiFe
+layer
+in
+the
+[Bi(0.6, 1.4, 2.0, 2.5 nm)–NiFe]16 multilayered films obtained from the simulations of the model
+dielectric function described by Equation 1. The values of E1, γ1, and γD are given in eV, and
+optical conductivity limit σ1(ω→0) in Ω−1·cm−1.
+Parameters 0.6 nm
+1.4 nm
+2.0 nm
+2.5 nm
+Drude
+AD
+33.(8)±2
+15.(0)±1
+21.(7)±2
+13.(1)±2
+γD
+0.876(5)±0.04 2.8(2)±0.3 3.4(2)±0.4 3.1(3)±0.2
+σ1(ω→0)
+4540±270
+2020±130
+2920±270
+1760±270
+Lorentz
+E1
+1.87
+3.32
+3.32
+3.32
+oscillator
+A1
+14.76
+14.28
+15.23
+14.74
+γ1
+3.62
+5.88
+5.65
+5.95
+dently demonstrated. The optical conductivity spectra of the Bi and FeNi layers follow the
+main trends identified in their complex dielectric function spectra presented in Figure 4.
+IV.
+DISCUSSION
+Initially, we would like to discuss GMR effects relevant for the studied MLF sys-
+tems.
+Our simulations of the dielectric functions for the 1.8 nm-thick NiFe layer inside
+the [Bi(0.6,1.4,2.0,2.5 nm)–NiFe(1.8 nm)] MLFs show the presence of the Drude term com-
+plemented with the pronounced Lorentz band located at around 2–3 eV (see Table II). From
+the corresponding optical conductivity spectra presented in Figure 5e–h one can notice that
+the associated Drude dc limit, σ1ω→0, displays an oscillating character (in agreement with the
+results deduced for the corresponding Drude parameter AD, see Table II and Figure 6). We
+can expect that the Bi spacer thicknesses for which the FeNi layers are preferentially antiFM
+coupled in the studied MLFs are around 1.4 and 2.5 nm implying that the [Bi(1.4,2.5 nm)–
+NiFe(1.8 nm)]16 film structures will exhibit a drop in the resistance (being negative magne-
+toresistance) when exposed to an external magnetic field. It is well known from the literature
+that the first antiFM maximum exhibits negative magnetoresistance of about 20%, while
+the second antiFM maximum decreases to about 10%, and the presence of the third antiFM
+maximum cannot confidently be retrieved (see, for example, Ref. [21] and references therein).
+11
+
+FIG. 5: The intralayer optical conductivity, σ1(ω) = ε2(ω)ω[cm−1]/60, for the (a-d) Bi layers and
+(e-h) FeNi layers in the [Bi(d)–FeNi(1.8 nm)]16 structures shown for the Bi layer nominal thickness
+values d = 2.5, 2.0, 1.4, and 0.6 nm by solid curves (a,e) black, (b,f) green, (c,g) blue, and (d,h)
+red, respectively. The contributions from the Drude term and the Lorentz oscillator in (a-d) are
+displayed by the yellow and cyan shaded area. In (e-h) the Drude term for the FeNi layers is
+displayed by the magenta shaded area. Shown by the dotted curves are the summary of the Drude
+and Lorentz contributions.
+12
+
+Using a simple model of a two-current series resistor [22], the magnetoresistance ∆R
+R can be
+estimated as
+∆R
+R = 100%
+(α − β)2
+4
+�
+α +
+dBi
+dF eNi
+� �
+β +
+dBi
+dF eNi
+�,
+(2)
+where dBi and dF eNi are the thicknesses of Bi and FeNi layers, and α =
+↓ρF eNi
+ρBi
+and β =
+↑ρF eNi
+ρBi
+are the ratios of the resistivity in the FeNi layer to that in the Bi layer in the spin down
+and spin up current channel, respectively. Exploiting values for ρ = σ−1
+1ω→0 estimated for
+the 1.4 nm Bi and 1.8 nm FeNi layers from the current model simulations (see Table I and
+II), namely, ρBi=
+1
+8970Ω·cm, ↓ρF eNi=
+1
+2020Ω·cm and ↑ρF eNi=
+1
+4540Ω·cm (the latter estimate is
+given by the FM coupling for the 0.6 nm Bi spacer), we obtain α=4.4 and β=2.0. Then,
+using Equation (2) we have ∆R
+R =10%. This means that the 1.4 nm Bi spacer corresponds
+to the second antiFM maximum. Following the same approach for the 2.5 nm Bi spacer,
+where ρBi=
+1
+3370Ω·cm, ↓ρF eNi=
+1
+1760Ω·cm and ↑ρF eNi=
+1
+2920Ω·cm (corresponding to the FM cou-
+pling for the 2.0 nm Bi spacer), we obtain α=1.9 and β=1.2. Using Equation (2), we have
+∆R
+R =1.4%, which may correspond to the very weakly pronounced third antiFM maximum.
+From the analysis presented above, we may expect that the first antiFM maximum corre-
+sponding to the magnetoresistance of about 20% occurs for the Bi spacer thickness of about
+0.9 nm, which is in agreement with the results presented in Ref. [21].
+Further,
+in
+the
+XRD
+patterns
+of
+the
+investigated
+Al2O3/[Bi(1.4,2.0,2.5 nm)–
+NiFe(1.8 nm)]16/Sitall film samples, the peak of the R¯3m crystalline Bi phase is identi-
+fied at 2θ ≈ 26.2◦ suggesting (012) orientation of the Bi layers, which is characterized by
+the interlayer distance of 3.28 ˚A. Using STM and reflection high-energy electron diffraction
+(RHEED) techniques, it was shown that initial growth of Bi(012)-type films occurs in the
+form of islands with the height increment of about 6.6 ˚A, indicating even-number layer sta-
+bility leading to the laterally flat morphology of the Bi(012)-type islands [23]. Consequently,
+we can expect that the 0.6, 1.4, 2.0, and 2.5 nm Bi spacer layers in the investigated MLFs
+incorporate about 2, 4, 6, and 8 (012)-type Bi planes, respectively.
+The model simulations for the [Bi(2.5, 2.0 nm)–FeNi(1.8 nm)]16 film samples reveal that
+the low-energy dielectric function of the Bi intralayers has competing contributions from the
+Drude term and from the intense Lorentz band around 0.36–0.39 eV with a ε2 maximum
+height of 70–100 (see Table I). The Drude and Lorentz contributions are more clearly pro-
+nounced in the corresponding optical conductivity spectra (see Figure 5a,b). The obtained
+13
+
+Drude and Lorentz parameters are in excellent agreement with those deduced in our pre-
+vious study [13] for the Bi spacer layer incorporated in the [Bi(2.5, 2.0 nm)–NiFe(1.2 nm)]16
+structures under study. The pronounced Lorentz band found at low photon energies for
+Bi single crystals (rhombohedral symmetry, space group R¯3m) [24, 25] and bulk Bi layers
+[26, 27] is characteristic of the semimetallic-like electronic band structure due to the con-
+tributions from the interband transitions near the Γ point, Γ+
+6 – Γ−
+6 and Γ+
+45 – Γ−
+6 [2], and
+near the T point, T−
+6 – T−
+45 [4]. The estimated values (see Table I) of the Drude dc limit
+σ1ω→0 (2750–3830 Ω−1·cm−1) as well as the free charge carrier’s γD (2.3–3.3 eV) are consis-
+tent with those peculiar for the metallic surface states related to the Rashba SOC in Bi(111)
+films, σ1ω→0 = 2300 Ω−1·cm−1 and γD = 2.0 eV) [6]. Meanwhile, the model simulation for
+the [Bi(1.4 nm)–NiFe(1.8 nm)]16 structure indicates that for the 1.4 nm Bi layer the Drude
+dc limit significantly increases to 8970±540 Ω−1·cm−1, while the γD essentially decreases to
+1.50±0.06 eV. In this case, the Lorentz band is nearly suppressed. The associated found
+Drude parameters for the ultrathin Bi layer inside the [Bi(0.6 nm)–NiFe(1.8 nm)]16 structure
+are slightly different, namely, σ1ω→0 = 6300±540 Ω−1·cm−1 and γD = 1.2±0.1 eV, and the
+Lorentz band is not present clearly (see Figure 5c,d and Table I).
+Thus, we have discovered that, on the one hand, the optical conductivity spectra spectra
+of the 2.0 and 2.5 nm thick Bi spacer layers in the (Bi–FeNi) MLFs incorporating 8 and 6
+Bi(012)-type monolayers, respectively, have contributions from the pronounced low-energy
+Lorentz oscillator and from the free charge carrier Drude term (for details, see Figure 5a,b
+and Table I). Here, the presence of the low-energy Lorentz band points on the Bi semimetallic
+phase contribution, and the parameters obtained for the Drude conductivity indicate that
+its origin can be associated with the surface metallic states [6].
+Therefore, the 2.0 and
+2.5 nm Bi layers can be associated with the semimetallic Bi phase sandwiched between two
+metallic layers on the top and bottom surfaces. On the other hand, the contribution from
+the intrinsic Lorentz band is strongly suppressed for the 1.4 and 0.6 nm layers, where the
+Drude conductivity displays notably improved metallicity properties, as one can see from
+the optical conductivity spectra shown in Figure 5c,d (for details, see Table I).
+From the above discussion of the obtained results, we can conclude that the Bi layer
+consisting of 4 Bi(012)-type monolayers represents a kind of crossover regarding the contri-
+butions from the semimetallic Bi phase and/or surface metallic-like states. Here we noticed
+some similarity with the theory results presented for the ultrathin Bi(111) layers by Liu
+14
+
+et al. [12]. There, it was established that below 4 Bi(111) BLs the film is a semiconduc-
+tor with the direct gap open at the Γ point and the positive indirect band gap, leading
+to nontrivial Z2 topology (ν=1) peculiar for an intrinsic 2D TI. Hovewer, above 4 Bi(111)
+BLs, the indirect band gap becomes negative resulting in a semiconductor-semimetal tran-
+sition due to overlapping of two bands at the Fermi level around the Γ and M points. It
+is argued by Liu et al. [12] that the Bi layers consisting of 5 to 8 Bi(111) BLs represent
+a 2D TI suited between two “trivial” metallic surfaces [12]. This means that for the sur-
+face considered as an individual 2D system its Z2 number is trivial (ν=0). The surface
+bands have no contribution to the nontrivial Z2 topology and, therefore, these trivial metal-
+lic surfaces are not robust and can easily be removed by surface defects or impurities. It
+was found by us [13] that the Bi layers in the [Bi(2.0, 2.5 nm)–NiFe(0.8 nm)] multilayers,
+incorporating the nanoisland permalloy layer, exhibit bulk-like semimetallic properties of
+the electronic band structure, although the surface (Drude) metallic conductivity is absent
+there (see Fig. 4(d) of Ref. [13]). Indeed, strong magnetic and spatial disorder induced by
+magnetic FeNi nanoislands, as well as long-range many-body interactions between magnetic
+moments of permalloy nanoislands [17], may lead to specific localization of free charge car-
+riers [28]. However, the surface conductivity (or interface) states for the 1.4 nm layer in
+the Bi–FeNi(1.8 nm) multilayers may be topologically nontrivial and, in this case, the elec-
+trons cannot be backscattered by impurities. Here, the Drude dc limit is 8970±540 Ω·cm−1
+and the scattering rate γD=1.5±0.06 eV. We found that the 0.6 nm thick Bi layer exhibits
+somewhat different Drude dc limit (6300±540 Ω·cm−1) and γD (1.2±0.1 eV), see Table I and
+Figure 6, which can be attributed to the discontinuous nanoisland structure of this layer.
+Finally, we would like to note that it will be challenging to investigate dc transport
+and superconductivity properties of the ultrathin Bi films possessing 2D TI surface states
+following the approach presented in Ref. [29], where the subkelvin superconductivity without
+any external stimuli was discovered in 3D TI Cd3As2 films [30, 31].
+V.
+CONCLUSIONS
+In summary, using wide-band (0.5-6.5 eV) spectroscopic ellipsometry we studied the
+optical properies of the [Bi(0.6, 1.4, 2.0, 2.5 nm)–NiFe(1.8˙nm)]16 MLFs prepared by rf
+sputtering. The XRD analysis suggested that the 0.6, 1.4, 2.0, and 2.5 nm Bi layers in the
+15
+
+FIG. 6: (a,b) Parameters of the Drude term (AD and γD) for the Bi (filled symbols) and FeNi
+(empty symbols) layers in the [Bi(0.6, 1.4, 2.0, 2.5 nm)–FeNi(1.8 nm)] MLF structures.
+studied MLFs correspond to about two, four, six, and eight Bi(012)-type monolayers,
+respectively.
+From the multilayer model simulations of the measured ellipsometric data,
+we extracted the Bi and FeNi layer dielectric functions. The dielectric function for the 2.0
+and 2.5 nm Bi spacer layers are represented by the Drude resonance due to the surface
+states and the low-energy Lorentz band peaking at around 0.3-0.4 eV. The pronounced
+Lorentz band is characteristic of the semimetallic bulk-like Bi electronic zone structure
+due to the contributions from the interband transitions near the Γ point. We discovered
+that the 2.0 and 2.5 nm Bi spacer layers can be associated with the semimetallic Bi phase
+sandwiched between two trivial (where the topology number ν=0) metallic layers on the
+top and bottom surfaces. The contribution from the low-photon-energy Lorentz band is
+strongly suppressed for the 1.4 and 0.6 nm Bi layers, where the Drude conductivity displays
+notably improved metallicity properties. This indicates that the Bi layer consisting of 4
+Bi(012)-type monolayers represents a kind of crossover regarding the contributions from
+the semimetallic Bi phase and/or surface metallic-like states.
+Therefore, the properties
+of Bi layers below 4 monolayers may be associated with nontrivial topology (where the
+topology number ν=1) peculiar for an intrinsic 2D TI. We expect that the Bi layers having
+16
+
+thickness of 0.9 nm will exhibit maximal GMR effect of about 20% in the (Bi-FeNi) MLFs,
+where the Drude dc limit is about 8970±540 Ω·cm−1. These states may be protected from
+backscattering, which makes them promising in spintronic devices and quantum computing.
+Acknowledgement
+We thank F.A. Pudonin for providing us with the Bi/FeNi multilayer film samples
+and O. Pacherova for their XRD analysis.
+We thank A. Muratov for participation in
+the spectroscopic ellipsometry measurements. This work was supported by the European
+Structural and Investment Funds and the Czech Ministry of Education, Youth, and Sports
+(Project No. SOLID21, Cz.02.1.01/0.0/0.0/16−019/0000760).
+Declaration of competing interest
+The authors declare no conflict of interest.
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+TinyHD: Efficient Video Saliency Prediction with Heterogeneous Decoders using
+Hierarchical Maps Distillation
+Feiyan Hu1, Simone Palazzo2, Federica Proietto Salanitri2, Giovanni Bellitto2, Morteza Moradi2,
+Concetto Spampinato2, Kevin McGuinness1
+1 Insight SFI Research Centre for Data Analytics, Dublin City University, Dublin, Ireland
+{feiyan.hu, kevin.mcguinness}@dcu.ie
+2 PeRCeiVe Lab, University of Catania, Catania, Italy
+{simone.palazzo, concetto.spampinato}@unict.it
+Abstract
+Video saliency prediction has recently attracted atten-
+tion of the research community, as it is an upstream task
+for several practical applications.
+However, current so-
+lutions are particularly computationally demanding, espe-
+cially due to the wide usage of spatio-temporal 3D convolu-
+tions. We observe that, while different model architectures
+achieve similar performance on benchmarks, visual varia-
+tions between predicted saliency maps are still significant.
+Inspired by this intuition, we propose a lightweight model
+that employs multiple simple heterogeneous decoders and
+adopts several practical approaches to improve accuracy
+while keeping computational costs low, such as hierarchi-
+cal multi-map knowledge distillation, multi-output saliency
+prediction, unlabeled auxiliary datasets and channel re-
+duction with teacher assistant supervision. Our approach
+achieves saliency prediction accuracy on par or better than
+state-of-the-art methods on DFH1K, UCF-Sports and Hol-
+lywood2 benchmarks, while enhancing significantly the ef-
+ficiency of the model.
+1. Introduction
+Video saliency prediction aims at estimating patterns of
+human attention during free-viewing of dynamic scenes,
+to emulate the capabilities of the human visual system of
+quickly analyzing and interpreting the surrounding environ-
+ment. Due to its several practical applications [7, 5, 28, 33,
+9, 40, 12, 27], it is an active area of research in computer
+vision. However, the solution to this problem is not triv-
+ial, for several reasons. First, attention mechanisms in the
+human visual system are not fully known, so it is not clear
+how to emulate them. Also, it requires complex modeling
+of both visual features and their motion and interaction: an
+object with striking visual patterns may be shadowed by a
+TASED
+100% 84%
+72%
+ViNet
+84% 100% 75%
+HD2S
+72%
+TASED
+75%
+ViNet
+100%
+HD2S
+CC Metric
+TASED
+100% 73%
+57%
+ViNet
+73% 100% 58%
+HD2S
+57%
+TASED
+58%
+ViNet
+100%
+HD2S
+SIM Metric
+Figure 1: Measuring prediction similarity among video
+saliency prediction models TASED, ViNet and HD2S on
+DHF1K validation set.
+bland element of the scene that starts moving in a pecu-
+liar way. Finally, modeling the temporal dimension may
+become computationally expensive, especially with current
+deep learning methods based on spatio-temporal 3D convo-
+lutions, thus limiting the applicability to low-power devices.
+Many solutions have been proposed, based on different
+assumptions on how to capture video saliency. It is inter-
+esting to note that, in spite of the remarkably different re-
+search directions followed by the variety of works in the
+literature, top results over video saliency prediction bench-
+marks are very close [1, 3, 36], suggesting that predictions
+of different models are similar. We assessed the validity
+of this conclusion by comparing three of the best perform-
+ing methods on the DHF1K dataset [36] — TASED [25],
+HD2S [1] and ViNet [16] — not in terms of their scores
+on summary metrics, but in terms of the relative similarity
+of the predicted saliency maps. To illustrate our findings,
+Fig. 1 shows pairwise similarities between predicted maps
+over two common metrics, Linear Correlation Coefficient
+(CC) and Similarity (SIM). Although the three approaches
+achieve similar scores on both metrics on DHF1K (between
+0.470 and 0.511 for CC, and between 0.361 and 0.406 for
+SIM), the same metrics computed between each other are
+relatively low, compared to what one would expect given
+their similarity to the saliency ground truth. A visual in-
+spection of the saliency maps generated by methods under
+arXiv:2301.04619v1  [cs.CV]  11 Jan 2023
+
+comparison confirms this behavior: Fig. 2 shows that it is
+common to find cases where each approach produces re-
+markably different saliency maps.
+Notably, all three methods — TASED, HD2S and ViNet
+— are encoder-decoder networks and share the same en-
+coder, S3D [38], while employing different decoding strate-
+gies (a U-Net–like approach for TASED and ViNet, a hi-
+erarchical map aggregation for HD2S). This suggests that
+a key factor underlying differences between current video
+saliency prediction approaches lies in the way encoded fea-
+tures are processed in the decoding path, leading to models
+that learn specific (and often exclusive) representations. We
+strengthen this hypothesis by experimenting with another
+decoding strategy, exemplified by DLA [39], which com-
+bines hierarchical decoding with complex feature interac-
+tions: the results, also included in Fig. 2, show yet another
+saliency prediction pattern, while using the same encoder
+network, S3D. These results lead us to hypothesize that dif-
+ferent model architectures introduce different inductive bi-
+ases, which are more suitable to recognize certain patterns
+more than others, thus requiring to increase model capacity
+in order to generalize well to multiple saliency dynamics.
+Indeed, the size of weights of models in our analysis range
+between 82 MB and 116 MB, and the size of the top ten
+models in the DHF1K leaderboard1 is on average 238 MB.
+Given these premises, instead of increasing the complex-
+ity of a single decoding strategy, it may be more efficient
+to employ multiple simpler architectures with fewer param-
+eters, relying on each architecture’s capability to attend to
+different salient regions and combining their results. Hence,
+we propose TinyHD, a lightweight, efficient and heteroge-
+neous multi-decoder architecture for video saliency predic-
+tion. The proposed method is inspired by encoder-decoder
+architectures, but introduces the adoption of heterogeneous
+decoding strategies in order to reduce the complexity of
+each decoder, increasing efficiency (the weights of the re-
+sulting model take only 16 MB) and improving the accu-
+racy of predictions, as we show in our experiments. Fur-
+thermore, along the direction of reducing computational
+costs while retaining high accuracy, we also introduce a
+novel knowledge distillation approach, based on exploiting
+a teacher with multiple hierarchical predictions: this allows
+the model to freely learn its own features, since no explicit
+conditioning on representations is enforced, while at the
+same time receiving a supervision signal that encodes in-
+formation at different layers of abstraction.
+Experiments confirm that our model can generate high-
+quality predictions with low computational costs and model
+size (only 16 MB). We assess the impact of our heteroge-
+neous multi-decoder strategy by carrying out extensive ab-
+lation studies and comparing alternative architectures. We
+also demonstrate the effectiveness of our knowledge dis-
+1https://mmcheng.net/videosal/
+tillation strategy, compared to the employment of a non-
+hierarchical teacher. To summarize our contributions:
+• We propose a decoding strategy for video saliency pre-
+diction which combines heterogeneous decoders to ex-
+ploit the specific pattern analysis capabilities, while re-
+ducing the overall model complexity. To our knowl-
+edge, we are first to propose multiple saliency maps
+output using 3D CNN to improve model efficiency.
+• We employ a knowledge distillation approach based
+on a hierarchical teacher, providing saliency maps es-
+timated from different abstraction layers.
+• Extensive experiments show that our model achieves
+state-of-the-art performance on the DHF1K bench-
+mark, at lower computational costs of current meth-
+ods. Ablation studies support the motivations for our
+decoding and knowledge distillation strategies.
+2. Related Work
+The main contributions of the proposed approach con-
+sist of a novel heterogeneous multi-decoder scheme, which
+combines lightweight versions of common decoding strate-
+gies, and a multi-objective knowledge distillation approach.
+In this section, we briefly present the state-of-the-art on
+these topics.
+Decoding
+strategies
+for
+video
+saliency
+prediction.
+Among recent methods from the state-of-the-art for video
+saliency prediction, leveraging encoder-decoder networks
+can be considered a mainstream approach; however, sev-
+eral architectural variations have been proposed for feature
+sharing between encoder and decoder and for output recon-
+struction. As shown in the taxonomy presented in Fig. 3, a
+simpler class of approaches employs independent encoder
+and decoder, with no feature sharing between the two paths.
+Among these, approaches based on recurrent layers typ-
+ically model temporal dynamics at the bottleneck of the
+architecture [18, 37, 22]. Non-recurrent architectures, in-
+stead, model time by means of 3D convolutions [42, 38, 4],
+Other approaches employ architectures similar to U-Net by
+introducing skip connections that encourage feature shar-
+ing between encoder and decoder.
+TASED [25] aggre-
+gates spatio-temporal features through the use of auxiliary
+pooling for reducing the temporal dimension. ViNet [16]
+integrates S3D features from multiple hierarchical levels
+by employing trilinear interpolation and 3D convolutions.
+UNISAL [6] proposes a multi-objective unified framework
+for both 2D and 3D saliency with domain-specific modules
+and a lightweight recurrent architecture to handle temporal
+dynamics; While single-decoder approaches are common,
+multi-decoder output integration has recently attracted in-
+terest. DVA [35] and HD2S [1] fuse maps predicted by
+independent decoders operating at different abstraction lev-
+els. RecSal [30] predicts multiple saliency maps in a multi-
+objective training framework. Recent works introduce more
+
+(a) Frame
+(b) GT
+(c) TASED
+(d) ViNet
+(e) HD2S
+(f) DLA
+Figure 2: Examples of video saliency maps from state-of-the-art methods. Although they achieve very similar performance
+on popular metrics, remarkable differences can be seem in the learned saliency patterns.
+Figure 3: A taxonomy of decoding strategies commonly
+employed in video saliency prediction.
+Subfigures(top-
+left, top-right, bottom-left, bottom-right): Independent en-
+coder and decoder, with no feature sharing between the two
+paths [4, 18, 22, 37, 42]; U-Net–like architecture, with fea-
+tures sharing between encoder and decoder [6, 16, 19, 25];
+Deep Layer Aggregation [39]; Hierarchical intermediate
+map aggregation [1, 30, 35].
+complex feature interactions among decoding paths, where
+high-resolution features are affected by deeper high-level
+features, as in DLA [39] and TSFP-Net [3]. All of the ap-
+proaches presented employ either a single-decoder archi-
+tecture or a homogeneous multi-decoder one, where differ-
+ences between decoders lie in the number of layers rather
+than in their structure. In our work, we propose an architec-
+ture which combines heterogeneous decoder structures, in
+order to better exploit their distinctive saliency prediction
+properties and thus increase computational efficiency.
+Knowledge distillation for visual saliency prediction.
+Knowledge distillation [13, 11] is commonly employed
+to train an efficient student model from a more complex
+teacher model, with higher accuracy than when training
+the student directly from dataset labels.
+Several knowl-
+edge distillation approaches have been recently proposed
+for video saliency prediction.
+SKD-DVA [20] proposes
+spatio-temporal knowledge distillation with two teachers
+and two students, with each pair focusing on either spatial
+or temporal transfer. SV2T-SS [41] distills corresponding
+features of teacher and student (implemented as encoder-
+decoder networks), based on first- and second-order feature
+statistics transfer. UVA-DVA [10] employs separate spatial
+and temporal teachers, whose knowledge is transferred to
+a single student model, which then fuses the resulting fea-
+tures in the final saliency prediction, achieving reasonable
+accuracy at impressive speed. Leveraging knowledge dis-
+tillation for video salient object detection is the main theme
+of the work in [34]. The knowledge distillation setting pro-
+posed in our work differs from existing techniques in two
+main aspects: 1) we define a multi-objective distillation tar-
+get on saliency maps directly; 2) we employ a hierarchical
+model as a teacher in order to further capture differences in
+saliency patterns extracted at multiple scales.
+3. Methodology
+3.1. Overview
+The overall architecture of the proposed saliency predic-
+tion network with knowledge distillation is shown in Fig. 4.
+Following the taxonomy introduced in Sect. 2, a shared en-
+coder extracts multi-level features that are then processed
+by three parallel decoding architectures: decoder 1 (D1)
+implements hierarchical intermediate maps aggregation (in-
+spired by HD2S); decoder 2 (D2) employs a U-Net–like ap-
+proach; decoder 3 (D3) is based on deep layer aggregation
+concepts (as in DLA [39]).
+The hierarchical aggregation decoder (i.e., decoder 1 in
+Fig. 4) produces four intermediate saliency maps from fea-
+tures extracted at different encoder layers; then, the set of
+
+DVSS三三三三predictions from all decoders are fused into the final pre-
+diction. At training time, we compute a supervised loss
+by comparing the final prediction to the ground-truth map,
+and a knowledge distillation loss on the final prediction and
+the intermediate maps extracted by D1 (all losses are based
+on Kullback-Leibler divergence between saliency maps; see
+Sect. 3.3). In order to have a correspondence between inter-
+mediate maps produced by D1 and teacher maps, we em-
+ploy HD2S as a teacher, since it naturally and semantically
+matches the decoder’s hierarchical structure.
+3.2. Encoder structure
+Depthwise separable convolutions are widely used for
+efficient network design, as in MobileNetV2 [31], com-
+monly pre-trained on ImageNet and used as a backbone for
+lightweight models. In order to adapt it as a 3D video fea-
+ture extractor, we follow the kernel inflation approach in-
+troduced in [2] and already employed for static [14] and
+dynamic [6] saliency prediction. A 2D convolutional ker-
+nel of size Cin × Cout × H × W can be inflated into a 3D
+kernel of size Cin × Cout × T × H × W by replicating
+its weights along the temporal dimension T. This simple
+trick provides a convenient initialization that responds to
+common spatial patterns and can be gradually adapted to
+temporal dynamics during training, eliminating the burden
+of learning basic spatial structures from scratch. Given the
+inflated MobileNetV2 encoder, we follow the approach in
+FastSal [14] to extract four blocks of concatenated feature
+from the whole set of layers.
+3.2.1
+Decoder structure and multiple prediction
+The set of heterogeneous decoders, employed in our model,
+includes D1 (hierarchical map aggregation), D2(U-Net–
+like) and D3 (deep layer aggregation). Our realizations of
+each of these approaches are designed to process the four
+input streams of features extracted by the encoder.
+D1
+produces four intermediate saliency maps, while D2 and
+D3 produce a map each. The fusion layer that computes
+the final output map is implemented as a 1×1 convolu-
+tion of the predicted maps.
+Architectural details are re-
+ported in the supplementary materials. As an additional ef-
+ficiency consideration, we note that the high computation
+cost of many state-of-the-art approaches due to multiple-
+input/single-output (MISO) prediction, where a sequence
+of frames is used to predict a single saliency map, usu-
+ally referring to the last frame. This provides a full con-
+text of previous frames to the model, but also means that,
+in order to predict N saliency maps (without interpolation),
+N forward passes are also required, with a proportional in-
+crease of computational power. In order to further improve
+efficiency, we implement a multiple-input/multiple-output
+(MIMO) schema for output generation, by designing de-
+coders that predict a number of saliency maps equal to the
+number of frames provided to the encoder. MIMO decoders
+can intrinsically make use of the similarity between con-
+secutive saliency maps, and employ this information to re-
+duce the computational power required to generate the same
+number of saliency maps by MISO decoders. Of course, the
+downside is that each frame has a different amount of sur-
+rounding context; however, in our experiments this has little
+impact on our model’s accuracy.
+3.3. Knowledge distillation
+Given the presence of multiple decoders in our model,
+one of which also produces intermediate saliency maps,
+choosing a distillation approach to supervise the student’s
+training is not trivial. As illustrated in Fig. 4, we carry out
+knowledge distillation by extracting intermediate and final
+outputs from a hierarchical teacher to supervise intermedi-
+ate of one student decoder and final student outputs. Our
+design of the distillation process is guided by several ob-
+servations. First and foremost, it is necessary to provide a
+training signal at the very output of the model, in order to
+train the final fusion layer. Second, carrying out distilla-
+tion at the representation level, by enforcing similarity be-
+tween teacher and student features, defeats the purpose of
+having multiple decoders that are meant to recognize their
+own distinctive saliency patterns and should therefore be
+free to independently learn their own features. Also, us-
+ing saliency maps directly ensures that the output and target
+have the same size, so that the use of adaptation layers to
+match feature size of student’s and teacher’s can be avoided.
+We therefore choose to use intermediate saliency maps from
+a hierarchical teacher, HD2S [1], since this makes it possi-
+ble in a natural way to affect the model at different depths
+of the encoder, without providing as strong a training signal
+as internal features.
+We formalize our knowledge distillation procedure as
+follows. Let V be the space of video sequences and S be the
+space of saliency maps (whether for the entire sequence or
+for a single frame); let M be a family of models such that
+each element in M is a function M : V → Sn+1, which
+provides n intermediate and one output saliency maps. We
+thus define a teacher T ∈ M and student S ∈ M. For
+simplicity, the notations Si and Ti will indicate the i-th map
+generated by, respectively, the student and teacher; indexes
+from 1 to n will denote intermediate maps, while index n+1
+will refer to the final output. Saliency map distance is mea-
+sured by Kullback-Leibler (KL) divergence:
+LKL (x, y) =
+�
+i
+yi log yi
+xi
+,
+(1)
+with i iterating over spatial locations of the saliency maps.
+At each training iteration, we sample a video sequence
+v ∈ V and its ground-truth saliency s ∈ S.
+The em-
+
+CNN
+Block 1
+CNN
+Block 2
+CNN
+Block 3
+CNN
+Block 4
+Feature Aggregation
+Decoder 1
+Decoder 2
+Decoder 3
+6 Intermediate 
+Saliency 
+predictions
+CNN
+Block 4
+Prediction
+Teacher
+Prediction
+Ground 
+Truth
+Teacher 
+Network
+Student 
+Network
+CNN
+Block 1
+CNN
+Block 2
+CNN
+Block 3
+Input
+Input
+Blocks of Convolutional layers used for 
+encoders and decoders. Student network 
+structures are detailed in supplementary 
+material and Teacher network is the same as 
+HD2S.
+Features from all layers of mobileNet V2 are 
+reorganized and aggregated to features from 
+4 abstraction level.
+4 intermediate maps from student network 
+generated from decoder 1 (similar to HD2S )
+1 intermediate map from student network 
+generated from decoder 2 (U-Net like)
+1 intermediate map from student network 
+generated from decoder 3 (DLA like)
+Final predictions generated by fusing all 6 
+intermediate maps
+4 intermediate maps from teacher 
+network(HD2S )
+Final predictions generated by teacher 
+network (pseudo labels)
+Ground truth saliency maps
+KL divergence 
+losses are 
+computed and 
+back propagated 
+through student 
+network
+Figure 4: Overview of the proposed multi-decoder architecture with hierarchical knowledge distillation.
+ployed loss function aims at minimizing the KL divergence
+between student and teacher maps (both intermediate and
+final) and between (final) student and ground-truth maps:
+L =
+n+1
+�
+i=1
+LKL
+�
+Si (v) , Ti (v)
+�
++ LKL
+�
+Sn+1 (v) , s
+�
+. (2)
+3.3.1
+Training with auxiliary dataset
+The usage of unlabeled auxiliary datasets in a knowledge
+distillation setting has been shown to help boost perfor-
+mance [21, 32, 15]. Following this approach, we introduce
+a new video distribution W, and extend the loss function
+with a term that measures the distance between student’s
+predicted saliency maps and “pseudo-labels” (which are, in
+fact, also maps) provided by the teacher. As a result, given
+a pair of input videos v ∈ V and w ∈ W, the new loss
+function becomes:
+L =
+n+1
+�
+i=1
+LKL
+�
+Si (v) , Ti (v)
+�
++
+n+1
+�
+i=1
+LKL
+�
+Si (w) , Ti (w)
+�
++LKL
+�
+Sn+1 (v) , s
+�
+.
+(3)
+3.3.2
+Channel reduction with teacher assistant
+Previous works have shown that, with a suitable network de-
+sign, it is possible to decrease the number of channels in the
+encoder’s layers, in order to reduce the computational cost,
+without an excessive loss in accuracy [8]. Our channel re-
+duction strategy applies multiple knowledge distillation it-
+erations: at each of them, a new student is initialized by av-
+eraging the weights of each pair of consecutive kernels into
+a new kernel. Although kernel ordering is essentially ran-
+dom, this approach has been shown to provide a meaningful
+initialization to the new student. Additionally, we also ex-
+plore the “teacher assistant” [26] distillation strategy: rather
+than using the original teacher to perform knowledge dis-
+tillation on reduced-channel students, we employ the full-
+capacity student (i.e., before any channel reduction) as a
+new teacher. As a result, by combining the channel reduc-
+tion and teacher assistant, we encourage the model to distill
+more information while reducing computational cost.
+4. Experiments
+4.1. Datasets and Metrics
+We conduct experiments on DHF1K [36],
+UCF-
+Sports [29, 24] and Hollywood2 [23, 24] datasets, com-
+monly employed to evaluate video saliency prediction.
+DHF1K contains 1000 videos split into 600/100/300 for
+training, validation, and test (unreleased).
+Eye fixations
+are collected from 17 participants in free-viewing experi-
+ments. UCF-Sports is a task-driven dataset that includes
+150 videos (103 for training, 47 for test) covering 9 sport
+activities. Participants were asked to identify the activity in
+each video sequence. Hollywood2 includes 1707 videos ex-
+tracted from 69 movies and categorized between 12 action
+classes. At data collection, 3 observers are free-viewing, 12
+observers are asked recognize the action, and 4 observers
+are asked to recognize the scene. 823 videos are used for
+training and 884 for test.
+We also employ the Kinetic-
+400 [17] action recognition benchmark as auxiliary dataset,
+used by the teacher to generate additional training inputs
+with pseudo-labels. For evaluation purposes, we report re-
+sults in terms of the standard metrics for video saliency
+prediction [35]: AUC-Judd (AUC-J), AUC-Borji (AUC-B),
+Linear Correlation Coefficient (CC), Normalized Scanpath
+Saliency (NSS), and Similarity Metric (SIM).
+
+Table 1: Comparison with SoA on the DHF1K and Hollywood2 test set in both the MISO and MIMO settings.
+(a) Prediction accuracy and computational cost on the DHF1K test set. GMACs are
+estimated for 16 frames, hence the ×16 multiplication for MISO models. Models
+marked with a ∗ are image saliency models.
+Models
+AUC-J SIM sAUC
+CC
+NSS
+GMACs
+#params
+Multi-input/single-output (MISO) prediction
+SalGAN∗
+0.866
+0.262 0.709 0.370 2.043
+45.73×16
+31.92M
+FastSal∗
+0.887
+0.293 0.712 0.426 2.330
+2.64×16
+2.47M
+3DSal
+0.850
+0.321 0.623 0.356 1.996 136.45×16
+46.15M
+TASED
+0.895
+0.361 0.712 0.470 2.667
+91.75×16
+21.26M
+ViNet
+0.908
+0.381 0.729 0.511 2.872 115.28×16
+31.1M
+HD2S
+0.908
+0.406 0.700 0.503 2.812
+11.08×16
+29.8M
+TinyHD-S
+0.909
+0.396 0.714 0.505 2.921
+5.57×16
+3.94M
+Multi-input/multi-output (MIMO) prediction
+SalEMA
+0.890
+0.466 0.667 0.449 2.574
+640.16×1
+31.79M
+STRA-Net
+0.895
+0.355 0.663 0.458 2.558
+266.01×3
+168.02M
+UNISAL
+0.901
+0.390 0.691 0.490 2.776
+19.42×1
+3.71M
+TinyHD-M
+0.905
+0.387 0.707 0.493 2.819
+7.95×1
+3.92M
+(b) Prediction accuracy on Hollywood2
+Models
+AUC-J SIM
+CC
+NSS
+Multi-input/single-output prediction
+ACLNet
+0.913
+0.757 0.623 3.086
+SalSAC
+0.931
+0.529 0.670 3.356
+TASED
+0.918
+0.507 0.646 3.302
+ViNet
+0.930
+0.550 0.693 3.730
+HD2S
+0.936
+0.551 0.670 3.352
+TinyHD-S
+0.935
+0.561 0.690 3.815
+Multi-input/multi-output prediction
+SalEMA
+0.919
+0.487 0.613 3.186
+STRA-Net
+0.923
+0.536 0.662 3.478
+UNISAL
+0.934
+0.542 0.673 3.901
+TinyHD-M
+0.934
+0.553 0.686 3.744
+Frame
+GT
+Output
+D1 (1)
+D1 (2)
+D1 (3)
+D1 (4)
+D2
+D3
+Figure 5: Examples of video saliency maps predicted by the proposed model, as well as intermediate maps by multiple
+decoders. Values between parentheses indicate one of the intermediate saliency maps by decoder D1.
+4.2. Training procedure
+Models are trained for 200 epochs using mini-batch
+stochastic gradient descent, with a mini-batch size of 12.
+The initial learning rate is 0.01, and it is reduced by a factor
+of 0.1 at epochs 100, 150, and 180. Input sequence length
+is 16 frames, spatially resized to 192×256. We carry out
+data augmentation by means of random horizontal flips; in
+our experiments, spatial resize and cropping do not lead to
+significant benefits. When the teacher assistant strategy is
+employed for channel reduction, we perform two additional
+knowledge distillation, each time training a new student net-
+work whose encoder contains, respectively, half and a quar-
+ter of the original number of channel at each encoder layer.
+4.3. Performance comparison with state-of-the-art
+models
+In these experiments, we report results of our model
+in both the MISO and MIMO configurations (respectively,
+TinyHD-S and TinyHD-M), trained with the auxiliary un-
+labeled dataset but without channel reduction that using the
+teacher assistant strategy (which introduces trade-offs be-
+tween accuracy and computational costs that will be dis-
+cussed later).
+We also report the number of multiply-
+accumulate operations (MAC) carried out by each method2
+to generate a 16-frame saliency sequence.
+Results on
+2Values are computed from official implementations when available
+and from our own implementations otherwise.
+
+18-PAR4
+JASONDUFNER
+SNDT
+484
+LIVEDHF1K are shown in Table 1a.
+In the MISO configu-
+ration, our model is on par with state-of-the-art methods
+(and even better on NSS), but only employs a fraction of
+the their computational cost. In the MIMO configuration,
+our method sets a new state of the art, outperforming (on
+four metrics out of five) also UNISAL, which has a sim-
+ilar number of parameters but is about twice as demand-
+ing in terms of GMACs. Fig. 5 presents a few examples
+of saliency predictions by our model3. For each example,
+we also show the intermediate maps provided by each de-
+coder. Qualitatively, our model predicts reasonable saliency
+regions, sometimes identifying additional elements not in-
+cluded in the ground truth (e.g., the third example). In-
+termediate maps also exhibit a certain variability, although
+similar patterns can be found in pairs (e.g., maps 1-2 and
+maps 3-4 from D1, and maps from D2 and D3). In general,
+the highest-level map from D1 (the fourth) mostly affects
+the output prediction: this is expected, since the correspond-
+ing architecture matches the teacher’s. However, the fusion
+layer includes all information from intermediate maps, as
+shown in the last example, where two salient areas identi-
+fied by the highest-level map from D1 are discarded.
+Table 1b and 2 report results on Hollywood2 and UCF-
+Sports. While the model performs very well on the for-
+mer, especially in the more efficient MIMO setting, ViNet
+and UNISAL achieve higher accuracy on UCF-Sports. This
+may be due to the lower performance of the HD2S teacher
+on that specific dataset, and to the arguable suitability of
+UCF-Sports as a video saliency prediction benchmark: the
+vast majority of its videos has fewer than 100 frames, and
+user fixations are driven by action classification, rather than
+free-viewing saliency [1].
+4.4. Ablation studies
+In order to experimentally substantiate our architectural
+and methodological choices, we carry out a set of abla-
+tion studies on each component of the model. the results
+of these experiments are reported on the DHF1K validation
+set, since testing set is not publicly available. First, we as-
+sess the effect of our heterogeneous multi-decoder strategy,
+evaluating the model’s performance under several decoder
+configurations. We carry out this experiment in the MISO
+configuration, which achieves higher accuracy, as shown in
+Table 1a. In order to demonstrate the importance of com-
+bining different decoder architectures, Table 3 reports re-
+sults when using homogeneous decoders in our architec-
+ture. Table 3 show that the heterogeneous approach gen-
+erally performs better than configurations with a single de-
+coder type, most remarkably in the NSS metric. For the
+sake of completeness, we also show configurations where
+a smaller number of homogeneous decoders are employed;
+3More examples are provided in the supplementary materials, as well
+as a visual comparison with state-of-the-art models.
+Table 2: Performance comparison on UCF-Sports in both
+the MISO and MIMO settings.
+Models
+AUC-J SIM
+CC
+NSS
+Multi-input/single-output prediction
+ACLNet
+0.897 0.406 0.510 2.567
+3DSal
+0.881 0.478 0.590 2.802
+TASED
+0.899 0.469 0.582 2.920
+ViNet
+0.924 0.522 0.673 3.620
+HD2S
+0.904 0.507 0.604 3.114
+TinyHD-S
+0.918 0.510 0.624 3.280
+Multi-input/multi-output prediction
+SalEMA
+0.906 0.431 0.544 2.638
+STRA-Net
+0.910 0.479 0.593 3.018
+UNISAL
+0.918 0.523 0.644 3.381
+TinyHD-M 0.911 0.499 0.609 3.234
+Table 3: Performance of our architecture with homoge-
+neous decoders, on the DHF1K validation set.
+Number
+of parameters of models with homogeneous decoders are:
+D1×1 (2.55M), D2×1 (3.57M). D3×1 (2.53M); D1×2
+(2.75M), D2×2 (4.78M). D3×2 (2.70M); D1×3 (2.95M),
+D2×3 (6.00M). D3×3 (2.88M); TinyHD-S (3.94M).
+Decoder
+AUC-J AUC-B
+CC
+NSS
+SIM GMACs
+D1×1
+0.8993 0.8210 0.4881 2.8163 0.3939 3.55×16
+D2×1
+0.9040 0.8235 0.4837 2.7976 0.3820 2.88×16
+D3×1
+0.9034 0.8248 0.4836 2.7851 0.3794 2.45×16
+D1×2
+0.8998 0.8195 0.4882 2.8256 0.3928 5.45×16
+D2×2
+0.9046 0.8251 0.4855 2.8117 0.3806 4.11×16
+D3×2
+0.9046 0.8239 0.4864 2.8095 0.3819 3.24×16
+D1×3
+0.9013 0.8253 0.4922 2.8420 0.3924 7.35×16
+D2×3
+0.9049 0.8266 0.4847 2.8042 0.3774 5.33×16
+D3×3
+0.9047 0.8242 0.4845 2.7967 0.3799 4.03×16
+TinyHD-S 0.9075 0.8244 0.4945 2.8735 0.3887 5.57×16
+these setups are, of course, more computationally efficient,
+but exhibit lower performance on average in the accuracy
+metrics.
+In the second part of our ablation study, we evaluate of
+the impact of our knowledge distillation strategy. Table 4 re-
+ports the results obtained by the proposed model, in MISO
+configuration, when trained on ground-truth maps only,
+and when gradually adding knowledge distillation terms on
+DHF1K and on Kinetics-400, using HD2S as teacher. The
+full loss setting achieves better performance on average —
+as previously. This is most evident in the NSS metric.
+
+Table 4: Impact of loss terms on our model in the MISO
+configuration, starting from training on ground-truth (GT)
+maps only, and gradually adding knowledge distillation
+terms on DHF1K (target dataset or TD) and on Kinetics-
+400 (auxiliary dataset or AD), using HD2S as a teacher.
+Loss term
+AUC-J AUC-B
+CC
+NSS
+SIM
+GT maps
+0.9033 0.8286 0.4864 2.7680 0.3765
++ K.D. on TD
+0.9058 0.8237 0.4875 2.8182 0.3846
++ K.D. on AD 0.9075 0.8244 0.4945 2.8735 0.3887
+4.5. Channel reduction with teacher assistant
+Finally, we investigate further reducing computational
+costs by means of our channel reduction strategy: mul-
+tiple distillation steps are carried out, with each student
+progressively halving its number of encoding and decod-
+ing features, as described in Sect. 3.3.2. We also evalu-
+ate the performance of this approach when training on the
+original teacher (HD2S) and when using the “teacher as-
+sistant” technique, with the full-capacity student used as a
+teacher. Table 5 reports results, on both MISO and MIMO
+settings, after one and two reduction steps steps, respec-
+tively resulting in models with half (marked as × 1
+2) and a
+quarter (marked as × 1
+4) of the original number of convolu-
+tional features (marked as ×1). Rows with “+TA” denote
+the use of the full-capacity student as teacher for knowl-
+edge distillation, rather than HD2S. As expected, channel
+reduction introduces a trade-off between retaining the ac-
+curacy of the original model and reducing computational
+costs. As multiply-accumulate operations and model pa-
+rameters are significantly reduced, accuracy also decreases,
+most evidently in the NSS and, to a smaller extent, in the
+SIM metrics. It is noteworthy that configurations employing
+a teacher assistant outperform the counterpart using HD2S.
+5. Conclusions
+In this work, starting from the observation that differ-
+ent encoder-decoder architectures recognize specific video
+saliency patterns, we propose a heterogeneous multi-
+decoder architecture that leverages simpler versions of
+state-of-the-art decoding strategies to achieve high predic-
+tion accuracy at a fraction of the computational cost. We
+train our model in a multi-target knowledge distillation set-
+ting, where a hierarchical decoder is used as a teacher to
+supervise a matching internal decoder in our model and the
+output prediction; additionally, we employ semi-supervised
+learning on an unlabeled auxiliary dataset to further im-
+prove model generalization. Our model sets new state-of-
+the-art performance when employed in a multi-input/multi-
+output setting, while being significantly more efficient in
+terms of floating-point operations and number of parame-
+Table 5: Performance of the proposed model when employ-
+ing channel reduction and teacher assistant distillation.
+(a) Number of parameters of models with reduced channels and
+GMACs reported on generating 16 output saliency maps.
+GMACs
+#params
+Models
+×1
+× 1
+2
+× 1
+4
+×1
+× 1
+2
+× 1
+4
+TinyHD-S 89.12 59.52 37.44 3.94M 1.37M 513.1k
+TinyHD-M 7.95
+6.92
+4.06 3.92M 1.37M 515.3k
+(b) Performance of channel reduction reported on DHF1K valida-
+tion set in both the MISO and MIMO settings.
+Models
+AUC-J AUC-B
+CC
+NSS
+SIM
+Multi-input/single-output prediction
+TinyHD-S×1
+0.9075 0.8244 0.4945 2.8735 0.3887
+TinyHD-S× 1
+2
+0.9038 0.8331 0.4754 2.7194 0.3641
++TA
+0.9052 0.8330 0.4805 2.7317 0.3684
+TinyHD-S× 1
+4
+0.9005 0.8285 0.4560 2.5830 0.3514
++TA
+0.9018 0.8318 0.4667 2.6329 0.3569
+Multi-input/multi-output prediction
+TinyHD-M×1 0.9050 0.8239 0.4880 2.8178 0.3844
+TinyHD-M× 1
+2 0.9016 0.8272 0.4687 2.6718 0.3612
++TA
+0.9021 0.8307 0.4718 2.6726 0.3630
+TinyHD-M× 1
+4 0.8980 0.8294 0.4487 2.5257 0.3438
++TA
+0.8999 0.8333 0.4564 2.5581 0.3478
+ters. We further push the limits of our model by applying
+a channel reduction procedure through multiple distillation
+steps and using the full-capacity student as a teacher, ac-
+cording to the “teacher assistant” paradigm. In the resulting
+model, the number of floating-point operations is approx-
+imately halved compared to the full-capacity version, and
+the number of parameters becomes as small as about 500k,
+taking about 2.4 MB storage space without compression.
+Acknowledgments
+This publication has been financially supported by:
+Science Foundation Ireland (SFI) under grant number
+SFI/12/RC/2289 P2;
+Regione Sicilia,
+Italy,
+RehaStart
+project (grant identifier:
+PO FESR 2014/2020, Azione
+1.1.5, N. 08ME6201000222, CUP G79J18000610007);
+University of Catania, Piano della Ricerca di Ateneo,
+2020/2022, Linea 2D; MIUR, Italy, Azione 1.2 “Mobilit`a
+dei Ricercatori” (grant identifier: Asse I, PON R&I 2014-
+2020, id. AIM 1889410, CUP: E64I18002520007).
+
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+The non-linear perturbation of a black hole by
+gravitational waves. III. Newman-Penrose constants
+J Frauendiener1, A Goodenbour2 and C Stevens2
+1Department of Mathematics and Statistics, University of Otago, Dunedin 9016, New
+Zealand
+2Department of Mathematics and Statistics, University of Canterbury, Christchurch
+8041, New Zealand
+E-mail: [email protected],
+[email protected], [email protected]
+Abstract.
+In this paper we continue our study of the non-linear response of a
+Schwarzschild black hole to an ingoing gravitational wave by computing the Newman-
+Penrose (NP) constants. The NP constants are five complex, supertranslation-invariant
+quantities defined on null infinity I + and although put forward in the 60’s, they
+have never been computed in a non-stationary setting. We accomplish this through
+a numerical implementation of Friedrich’s generalized conformal field equations whose
+semi-global evolution yields direct access to I +. Generalizations of the NP constants’
+integral expressions are made to allow their computation in a more general gauge
+that better suits the output of a numerical evolution. Canonical methods of fixing
+inherent degrees of freedom in their definitions are discussed. The NP constants are
+then computed for a variety of different ingoing wave profiles in axisymmetry, and then
+with no symmetry assumptions in 3+1 for which all five are non-zero.
+Submitted to: Class. Quantum Grav.
+arXiv:2301.05268v1  [gr-qc]  12 Jan 2023
+
+The non-linear perturbation of a black hole by gravitational waves
+2
+1. Introduction
+Gravitational waves are a robust prediction of general relativity. The existence of wave
+solutions to the field equations has been known since the early days of the theory,
+but there was doubt that wave-like behaviour occurred generically outside of overly-
+symmetric exact solutions [25]. The way we characterise radiation today emerged out of
+the work of Bondi [4], Sachs [26], Newman and Penrose [18, 19, 21]. Penrose’s procedure
+of conformal compactification [22] succinctly encodes the asymptotic fall-off conditions
+hard-coded by Bondi and Sachs. The conformal boundary, I , emerges as the natural
+place to define gravitational radiation.
+This
+picture
+of
+gravitational
+radiation
+owes
+a
+great
+debt
+to
+Maxwell’s
+electromagnetism.
+The isolation of radiative degrees of freedom by Bondi and
+Sachs echoes an earlier analysis of electromagnetic radiation via the Liénard-Wiechert
+potential and Penrose’s conformal compactification is premised on the conformal-
+invariance of zero-rest-mass fields, a fact which was shown for a Maxwell field much
+earlier [1, 6].
+This paper is focused on the explicit calculation of another such import from
+Maxwell electromagnetism, the Kirchhoff integral formula. Generalised to fields of spin-
+s in Minkowski space, it is called the generalised Kirchhoff-d’Adhémar formula and
+relates the value of the field at a point to an integral over an arbitrary smooth cut of its
+(past) light cone. Formally applied to the spin-2 Weyl spinor on the conformal boundary
+I + of an asymptotically flat spacetime, the Kirchhoff-d’Adhémar formula yields a set
+of five complex supertranslation invariant quantities on I + which are ostensibly the
+components of the Weyl spinor at timelike infinity. These are the NP constants [20],
+whose physical interpretation has proved elusive for over half a century.
+Nevertheless, much has been said about these constants in the intervening years.
+In their original paper, Newman and Penrose make the argument that the existence
+of the constants has non-trivial physical significance [20, 24]. It has since been shown
+that the vanishing of the NP constants distinguishes between fundamentally different
+late-time behaviour of self-gravitating waves [16]. They play an important role in the
+early radiative properties of isolated systems close to spatial infinity [17]. Another line
+of analysis has found that the NP constants appear as subleading BMS charges [15].
+However, as far as we are aware, the NP constants have never been explicitly computed
+in a general space-time.
+Explicit numerical computation of quantities at the conformal boundary without
+the use of limiting procedures can be done by employing conformal compactification in
+a numerical scheme. Friedrich’s conformal field equations regularly extend the Einstein
+equations to include the conformal boundary [10–12]. Recently, an initial boundary
+value problem (IBVP) framework for the generalised conformal field equations (GCFE)
+was presented [3]. This framework puts I + within the computational domain, allowing
+for the non-linear perturbation of black hole space-times.
+Because the computational domain includes at least a portion of the future null
+
+The non-linear perturbation of a black hole by gravitational waves
+3
+boundary, quantities defined there can be computed with local differential geometrical
+methods. Most asymptotic quantities, including the NP constants, are defined on a
+2-dimensional cut of I +, therefore one can see how a quantity evolves along the set of
+successive cuts. However, in the literature, these quantities are often defined in terms of
+a very specific set of coordinates, frame, and conformal factor. These choices are usually
+incompatible with the requirements of the numerical scheme. Therefore, to compute a
+quantity at null infinity with this scheme, it must be written in a conformally invariant
+way.
+The aim of this paper is to use the numerical framework provided by the IBVP
+formulation of the GCFE to compute for the first time, the NP constants explicitly on
+I +. The case considered is the non-linear perturbation of a Schwarzschild space-time
+by gravitational waves. The reader is referred to [3] for details of the numerical scheme
+and checks of correctness such as constraint convergence tests.
+The layout of the paper is as follows: Section 2 summarizes the IBVP framework
+for the GCFE. Section 3 presents the NP constants and proves their supertranslation-
+invariance in the general form required for their computation. Section 4 presents the
+details of aligning the frame of the GCFE with the frame in which the NP constants are
+defined and discusses the details of their calculation. Section 5 presents numerical checks
+of correctness and results for a range of initial wave profiles and Section 6 concludes
+with a brief discussion. We follow conventions of Penrose and Rindler [24] throughout.
+2. Overview of the GCFE and its numerical IBVP implementation
+We implement Friedrich’s generalized conformal field equations analogously to previous
+papers in this series [8] and here just give a brief overview.
+The conformal field equations are a regular extension of the Einstein equations
+defined on a physical space-time to another conformally related Lorentzian manifold,
+related by a conformal factor Θ, where the points at ’infinity’ of the physical space-time
+are given by Θ = 0. Imposing the conformal Gauß gauge on the GCFE [12] yields a
+system of evolution equations of which most are ordinary differential equations except
+those governing the components of the gravitational tensor which form a symmetric
+hyperbolic system. These evolution equations are complemented by a set of constraint
+equations which are preserved by the evolution. The associated IBVP is completed by
+constraint preserving boundary conditions [3] which are used to generate fully non-linear
+gravitational dynamics.
+The Schwarzschild space-time of mass m written in isotropic coordinates is again
+used as the initial space-time. The specific choice of conformal Gauß gauge given by
+Friedrich [13] is used by which regular coordinates, frame and conformal factor up to
+and beyond null infinity can be defined.
+Our numerical implementation is capable of general 3 + 1 dimensional simulations
+and we use this capability to generate a complete set of non-trivial NP constants going
+beyond the axisymmetric case.
+
+The non-linear perturbation of a black hole by gravitational waves
+4
+For all simulations presented here (excluding convergence tests) we use coordinates
+{t, r, θ, φ}‡.
+The spatial coordinates are discretized into equidistant points in the
+intervals r ∈ [m/2, m/2 + 2m], θ ∈ [0, π) and φ ∈ [0, 2π] with 401, 33 and 64
+points respectively. The temporal discretization is also equidistant in this study with
+timestep given by dt = dr/2 giving a Courant-Friedrichs-Lewy number of 0.5. The MPI-
+parallelized Python package COFFEE [7] contains all the necessary numerical methods
+to evolve this initial boundary value problem. The standard explicit Runge-Kutta fourth
+order method is used to march in time, Strand’s fourth order summation-by-parts finite
+difference operator (third order on the boundary) [27] is used to approximate radial
+derivatives and the simultaneous-approximation-method [5] is used to stably impose
+maximally dissipative boundary conditions.
+Finally, we use spin-weighted spherical
+harmonics to allow for fast and accurate angular derivatives through a pseudo-spectral
+implementation of Penrose’s ð-calculus [2].
+Regridding is also performed, whereby
+regions outside of future null infinity are chopped away from the computational domain
+to maintain a stable evolution. This is also performed inside the black hole to avoid the
+singularity.
+3. Newman-Penrose constants
+The Kirchhoff-d’Adhémar construction in Minkowski space expresses a solution to the
+zero-rest-mass field equations at a point P as an integral of an arbitrary smooth cut
+of the (past) light cone of P [23]. It is a conformally invariant construction and so
+we may apply it specifically to relate a zero-rest-mass field φAB...L at the future (past)
+timelike infinity of a conformally rescaled spacetime to a cut of its light cone, future
+(past) null infinity. Because the construction is invariant with respect to the cut of the
+light cone on which it is evaluated, it gives a set of supertranslation invariant constants
+corresponding to the 2s+1 components of the zero-rest-mass field at timelike infinity. In
+a spacetime with matter, timelike infinity becomes a singular point of the conformally
+rescaled manifold, but the integrand of the Kirchoff-d’Adhémar construction, being
+evaluated on null infinity, remains regular and so applied to the spin-2 Weyl spinor, we
+are left with a set of five complex supertranslation invariant quantities defined on any
+asymptotically flat spacetime. These are absolutely conserved in the sense that they
+remain constant even with non-vanishing news.
+The physical set-up of the Kirchoff-d’Adhémar construction is as follows. Consider
+a point P with a (past or future) light cone I and an arbitrary smooth 3-dimensional
+null hypersurface N intersecting I . The intersection has spherical topology and is
+labeled C .
+The Kirchhoff-d’Adhémar construction will take the form of an integral
+evaluated on C whose value is independent of the intersecting null hypersurface N and
+thus of the specific intersection C . In fact, any smooth cut of the light cone I can be
+‡ The axisymmetric numerical implementation is analogous to the proceeding outline but without the
+φ-direction and with optimized spin-weighted spherical harmonic transformations and corresponding
+ð-calculus calculations.
+
+The non-linear perturbation of a black hole by gravitational waves
+5
+said to have come about by the intersection of I with some null hypersurface N .
+The method of proof consists in showing that the Kirchhoff-d’Adhémar integral
+evaluated on two arbitrary cuts of I denoted C and C ′ gives the same result by treating
+C − C ′ as the oriented boundary of a 3-dimensional section of the light cone I . The
+generalised Stokes’ theorem can be used to relate cut invariance to the vanishing of a
+related integral on the region between the cuts. This is explained in detail in [20, 24]
+At each point of an intersection C there are two distinguished null directions,
+one along the generators of the light cone I and the other along the intersecting null
+hypersurface so it is advantageous to use the GHP formalism [14]. We can choose a
+spin-frame such that oA points along the intersecting hypersurface, and ιA along I ,
+normalised so that oAιA = 1.
+Formally, the Kirchhoff-d’Adhémar formula reads
+φ[U]
+��
+P=
+�
+C
+Uþcφ d2C
+(3.1)
+where φ := φAB...LoAoB . . . oL, U is a weighted scalar satisfying
+(i) ¯ðcU = 0,
+and
+(ii) þ′
+cU = 0.
+(3.2)
+and the derivative operators with a subscript ’c’ are conformally weighted operators of
+the cGHP formalism as introduced in [9].
+In Minkowski spacetime, in a gauge where each cut C is represented as a unit
+2-sphere, U will be a component of the spinor ιAιB . . . ιL, i.e., one of the 2s + 1 spin-
+weighted spherical harmonics −sYsm with spin-weight −s.
+In this gauge, these are
+constant, thus trivially propagating along the light cone.
+In a curved spacetime, U is a generalisation of these spin-weighted spherical
+harmonics to a topological but not necessarily metric sphere.
+In this general case,
+there are still 2s + 1 independent solutions of (3.2(i)) for U.
+4. Calculating the NP constants
+In the remainder of this work we focus on the gravitational NP constants, i.e., with
+φABCD = ψABCD, the conformally rescaled Weyl spinor. Many system variables of the
+GCFE are components of spinors with respect to a certain spin-frame. In general, this
+spin-frame, and its associated null tetrad,does not agree with the frame adapted to I
+that was used in the definition of the NP constants but we can use null rotations to
+transform between the GCFE null frame and the frame adapted to I herein referred
+to as a Bondi frame§.
+§ This is not strictly correct, since we are referring here only to a single cut, whereas the standard
+usage of the term Bondi frame refers to an entire system of cuts parametrised by the retarded Bondi
+time.
+
+The non-linear perturbation of a black hole by gravitational waves
+6
+A null rotation mixes one component of a spin-frame into another. For example, a
+null rotation of oA around ιA is given by
+oA → oA + Y ιA,
+ιA → ιA
+(4.1)
+thus keeping ιA fixed, where Y is a function of the spacetime coordinates.
+If we
+denote the Bondi spin-frame by OA and IA, the GCFE frame by oA and ιA, and the
+corresponding Bondi and GCFE null-tetrad vectors by capital and lowercase letters
+respectively, then we may transform between frames by two null successive rotations
+(first fixing oA and then the new ιA) which have the combined form
+OA = oA + Y (ιA + XoA),
+IA = ιA + XoA.
+(4.2)
+The null rotation functions X and Y are determined by the conditions
+∇aΘ = −ANa,
+M a∇at = 0,
+(4.3)
+where the scaling A is fixed given the conformal factor Θ and the above expression of the
+adapted spin-frame. These conditions impose that N a points along the null generators of
+I and that the complex vector M a lies within the t = const. cuts of I . Appropriately
+fixing the freedom in how the frame propagates along the timelike conformal geodesics
+of the conformal Gauß gauge allows one to satisfy the second condition automatically,
+yielding X = 0. The first condition gives us the value of the null rotation function Y
+on I +.
+The transformation between the GCFE frame and the Bondi frame is then known
+and so we may write components with respect to the Bondi frame in terms of components
+with respect to the GCFE frame which are known numerically. As a simple example,
+the third component of the gravitational spinor is written in the Bondi frame as
+ψABCDOAIBICID = ψABCD(oA + Y ιA)ιBιCιD = ψ3 + Y ψ4.
+Both the GCFE and the NP constants are defined with respect to a spin and boost
+covariant formalism and so a properly weighted expression with respect to one frame
+results in a properly weighted expression with respect to another.
+The same process is used to compute the area-form in terms of numerically available
+quantities.
+4.1. Fixing the behaviour of the frame off I +
+With the above two null rotations, the frame on I + is fixed, but we also have some
+freedom to choose how our frame changes as we move away from I +. The presentation
+of I + in the proof of supertranslation invariance makes use of this and takes κ = 0
+since the intersecting null hypersurface is foliated by a null geodetic congruence [24].
+The Bondi frame so far is only fixed on I + and in order to achieve κ = 0 we need
+to enforce that DoA ∝ oA which means that we need to determine the null rotation
+
+The non-linear perturbation of a black hole by gravitational waves
+7
+function Y away from I +. Suppose, we have fixed Y on I + with the above procedure,
+then we can get the required result with a third null rotation that becomes the identity
+on I +
+oA → ˆoA = oA + ZιA,
+ιA → ˆιA = ιA,
+where Z = O(Θ),
+(4.4)
+recalling the conformal factor Θ. Under this transformation,
+ˆκ = κ − DZ,
+(4.5)
+where D := La∇a and choosing Z so that κ = DZ on I + we obtain ˆκ = 0 there.
+Although the transformation becomes the identity on I +, we must worry about
+derivatives of the frame. In the Kirchhoff-d’Adhémar integral, we have a derivative of
+the form Dφ where φ = φAB..LoAoB..oL (2s indices). Under this null rotation,
+ˆD ˆφ
+���
+I + = D ˆφ
+(4.6)
+= D(φA..L(oA + ZιA)..(oL + ZιL))
+(4.7)
+= Dφ + 2sκφ1
+(4.8)
+since the derivative of any term containing powers of Z higher than one will vanish
+on I +.
+4.2. Computing U
+In the NP constant integrand (3.1), the active component would appear to be the term
+þcφ since this brings information of the arrival of the field φ at I +, while the quantity
+U appears to be somewhat inert, being used to project out certain pieces of information
+from the integrand. Before the jump was made to curved spacetime, the Kirchhoff-
+d’Adhémar integral could represent the value of the field φ at timelike infinity. In this
+case, the quantity U was replaced by components of ιAιB...ιL which are spin-weighted
+spherical harmonics in an appropriate frame. When curved spacetime is introduced,
+the U takes the role of these components and so different choices of U which satisfy the
+underlying equations (3.2), represent what would have been components of φ at timelike
+infinity. The job of U is to lend its spin, boost, and conformal weight to the expression,
+and so provide alignment between cuts of I + allowing for comparison from cut to cut.
+To compute U we must solve the “constraint equation” ¯ðcU = 0 on a cut∥ and evolve
+it along the null generators of I + with the evolution equation þ′
+cU = 0. Following
+methods from our earlier paper [8], we can expand these operators written in the Bondi
+frame in terms of the numerically implemented, standard operators ˜ð, ˜ð′, to obtain
+A˜ðU + B˜ð′U + CU = 0.
+(4.9)
+∥ This term is justified since, by considering the commutator [þ′
+c, ¯ðc] one can show that any U satisfying
+the evolution equation þ′
+cU = 0 will satisfy ¯ðcU = 0 on every cut if it satisfies this equation on a single
+cut. In this sense, the constraint is propagated by the evolution.
+
+The non-linear perturbation of a black hole by gravitational waves
+8
+Expanding the known coefficients A, B, and C, and the unknown function U in terms of
+spin-weighted spherical harmonics, and using the well-known relationship for products
+of spin-weighted spherical harmonics in terms of Clebsch-Gordon coefficients (see [23])
+results in a system of homogeneous linear equations for the spectral coefficients of the
+function U. There are five linearly independent solutions which span the solution space
+to the constraint equation for U.
+The evolution equation can similarly be written in terms of numerically available
+quantities as,
+A∂tU + B˜ðU + C˜ð′U + DU = 0,
+(4.10)
+and may be evolved along I + by the method of lines given an initial solution to the
+constraint equation. An adaptive fourth-order Runge-Kutta method is used since the
+numerical output is not linearly spaced in t.
+4.3. Fixing a basis in the solution space U
+It is clear that solution U of (3.2(ii)) leads to another solution αU provided that α is
+a complex constant on the cut C . More generally, any complex linear combination of
+solutions will also be a solution. Thus, changing the basis of the solution space U of
+(3.2(ii)) will also change the individual values of the five NP constants. Therefore, these
+do not, by themselves, carry independent physical information. Only the combination
+of the values of the integrals together with the knowledge of the basis of U carries the
+full information.
+This means that in order to compare NP constants across different spacetimes we
+need to make sure that we specify “the same basis” for the solution space for each
+spacetime. There are several ways to do this: two complicated ones which are also
+physically relevant, and a third easier one which not as physically meaningful but much
+more pragmatic.
+The first idea that comes to mind is to first conformally rescale the metric on the cut
+to make it into a unit-sphere, and, in a second step, to change the coordinate system by
+a Möbius transformation so that it becomes a standard polar coordinate system on the
+unit-sphere. In this situation, the solutions of (3.2(ii)) are the standard spin-weighted
+spherical harmonics Ym := −2Y2m with −2 ≤ m ≤ 2.
+While the first step is rather straightforward, the second step leads to a Poisson-type
+equation on the sphere with a δ-like source term which is difficult (but not impossible)
+to treat numerically. In addition, after the coordinate transformation all quantities must
+be transformed which may introduce several numerical errors.
+The second way to introduce a basis in U is to make use of the fact that the standard
+spin 2 spherical harmonics Ym form an irreducible representation of the group SU(2).
+They each are an eigenvector of an infinitesimal generator with different eigenvalue,
+and they are obtained one from the other by the action of two ladder operators (very
+much akin to the angular momentum algebra of quantum mechanics). Fixing one of
+them as being annihilated by one of the ladder operators, one can generate the others
+
+The non-linear perturbation of a black hole by gravitational waves
+9
+by successive application of the other ladder operator. This fixes the complete basis in
+terms of the first vector and leaves the freedom of scaling with one complex number.
+This can be almost fixed by normalising the vectors with respect to an appropriate
+Hermitian product, leaving the remaining freedom of a single phase.
+In principle, this program could be carried out but it is very cumbersome. First,
+one needs to find the infinitesimal generators of the group action.
+This leads to a
+series of elliptic equations to be solved on the sphere. Next, one needs to use these
+generators to determining the function that is killed by one of the ladder operators,
+which is again an elliptic equation on the sphere, and then generate the other functions
+by successive application of the other ladder operator. As an alternative, one could solve
+the eigenvalue problem for the third operator. Obviously, this procedure is numerically
+quite involved and prone to inaccuracies due to successive numerical differentiation.
+For this reason, we use a third method to fix a “universal” basis of U , and we
+use the “universal structure” that is available to us, namely our numerical setup which
+is the same for every spacetime that we compute. Recall that our method is based
+on concentric round spheres and that every function we compute can be expanded as
+a linear combination of spin-weighted spherical harmonics defined with respect to the
+numerical round spheres. Therefore, we proceed as follows: first, on an initial cut, we
+compute five linearly independent solutions (uk)k=−2:2 of (3.2(ii)). These have the form
+uk =
+2
+�
+m=−2
+cm
+kYm + Zl>2,
+k = −2 : 2
+where Zl>2 stands for terms with higher values of l.
+Then a straightforward linear
+combination of these solutions leads to the universal basis Uk which is defined by
+Uk = Yk + Zl>2
+where Zl>2 again stands for higher l terms. We can interpret this basis as being the
+deformation of the standard basis provided by the Ym due to the impact of the incoming
+gravitational wave. If there was no gravitational wave, then the cut would be spherically
+symmetric and the yk would agree with the standard basis. The basis, thus defined, is
+then propagated along I + using the evolution equation (3.2(i)). In this process, the
+form of the yk will change. This process of fixing a “universal basis” of U leaves no
+further freedom (except, of course, for the free phase inherent in the definition of the
+spin-weighted spherical harmonics).
+4.4. Integrating the Newman-Penrose integrand
+At this stage, all elements of the Newman-Penrose integral (3.1) have expressions in
+terms of known quantities. Integration can be performed against the basis Uk as defined
+in 4.3 by simply computing the s = l = m = 0 spectral coefficient of the complete
+integrand and dividing by 2π. The theory shows these five complex numbers, obtained
+on a cut, come out the same independently of which cut was chosen for their evaluation.
+In the next section we present numerical results that showcase these properties.
+
+The non-linear perturbation of a black hole by gravitational waves
+10
+5. Numerical Results
+Using the above procedure, the NP constants were computed with data on I + from a
+numerically evolved spacetime modelling the non-linear perturbation of a Schwarzschild
+black hole by an incoming gravitational wave pulse.
+Because the NP constants are
+evaluated on each cut of I + defined as the intersection with a t = const. hypersurface,
+we obtain five complex numbers for every t.
+The initial mass of the black hole is m = 0.5 for all simulations considered.
+5.1. Ingoing wave
+The ingoing pulse is defined as the choice of free data q0 of the lightlike, ingoing
+characteristic variable on the outer boundary. This is chosen to be a linear combination
+of the spin-weighted spherical harmonics 2Y2m for m = 0, 1, 2 and with amplitudes a, b
+and c respectively. The choices of these amplitudes will vary in the upcoming sections.
+This gives the wave profile on the outer boundary
+q0(t, θ, φ) =
+�
+�
+�
+[4a
+�
+2π
+15 2Y20 − 2b
+�
+5
+π 2Y21 + 2c�π
+5 2Y22] sin8(8πt)
+t ≤ 1
+8
+0
+t > 1
+8
+.
+5.2. Checks of correctness
+We have demonstrated in several papers [3, 8] that the solutions computed by the GCFE
+system converge at the correct order for the 2 + 1 axisymmetric case. Here we show
+that this is also true in the general case of 3 + 1 dimensions. We also show that the NP
+constants converge to constant values on I +.
+(a) θ = π/2 and φ = π.
+(b) r = 0.978 and φ = π.
+(c) r = 0.978 and θ = π/2.
+Figure 1:
+The imaginary part of a constraint equation from the Bianchi identity
+when
+a
+single
+spatial
+coordinate
+is
+fixed
+at
+t
+=
+1
+for
+r, θ, φ
+resolutions
+of {101, 9, 16}, {201, 17, 32} and {401, 33, 64} (denoted by Res1, Res2 and Res3
+respectively). The dashed vertical line represents I + in the radial plot. The curves
+from top to bottom correspond to increasing resolution.
+We use an ingoing wave profile proportional to 2Y22 with a = b = 0, c = i to allow
+excitation in the φ-direction. Fig. 1 demonstrates convergence in all spatial directions
+at t = 1.
+
+0
+Resl
+Res3
+Res2
+log10 lErrorl
+.5
+.10
+-15
+0.4
+0.6
+0.8
+1.0
+r0
+Res1
+Res3
+Res2
+log1o lErrorl
+-5
+-10
+-15
+0
+1
+2
+30
+Res1
+Res3
+Res2
+log1o lErrorl
+-5
+-10
+-15
+0
+2
+4
+6
+0The non-linear perturbation of a black hole by gravitational waves
+11
+Focusing now on the NP constants, we demonstrate how they approach constant
+values along I +. Fig. 2 shows a convergence test of the discrepancy from constancy
+of the single non-vanishing NP constant in axisymmetry, choosing amplitudes a = 1,
+b = c = 0, along I + as the spatial resolution is increased. The resolution in r (number
+of intervals) is denoted rres and corresponds to an equivalently scaled resolution θres along
+the θ-direction. The coarsest values are rres = 100 and θres = 8, and the resolutions are
+doubled in successive simulations.
+0.8
+1.0
+1.2
+1.4
+1.6
+Conformal time t
+−8
+−6
+−4
+−2
+0
+log10 Error
+rres = 100
+rres = 200
+rres = 400
+Figure 2: Convergence of the log10 difference between the magnitude of the NP constant
+at time t and at the initial cut with increasing spatial resolution.
+5.3. Variable ingoing amplitude
+An superficial glance at (3.1) would suggest that due to the linearity of the integrand
+and the zero-rest-mass field equations with respect to φ, scaling the amplitude of the
+ingoing wave would just scale the NP constants linearly. However, this turns out not
+to be the case because φ satisfies the Bianchi equation into which φ enters non-linearly
+through the connection coefficients of the covariant derivative [23, §5.7]. Hence, it is
+interesting to numerically probe the scaling of the NP constants as the ingoing wave
+profile is scaled.
+We performed four simulations in axisymmetry with the amplitudes b = c = 0 and
+a taking the values 1,2,5, and 10. These are evolved up to t = 1.77 at which point the
+system variables start to diverge due to the close ‘conformal’ proximity to i+ at t ≈ 1.79.
+Table 1 shows the corresponding single NP constant for each amplitude as well as the
+relative error from a linear fit through the origin. Fitting the NP constants to the ansatz
+
+The non-linear perturbation of a black hole by gravitational waves
+12
+αaβ yields α ≈ 0.53865 and β ≈ 0.99803. Fig. 3 shows the log10 deviation of the NP
+constant value from the value on the initial cut for each amplitude as a function of t.
+The deviation from a linear fit is orders of magnitude greater than the error. This is due
+to the amplitude of the initial wave profile entering into the field equations non-linearly
+resulting in a non-linear relationship between initial amplitude and Newman-Penrose
+constant.
+Amplitude
+1
+2
+5
+10
+NPC
+0.53882
+1.07638
+2.68405
+5.36222
+Rel. Err.
+0
+0.00116
+0.00373
+0.00482
+Table 1: The one non-vanishing NP constant for different ingoing wave amplitudes and
+the deviation from a linear fit through the origin. This deviation is orders of magnitude
+larger than the error for each. This is a result of the amplitude entering non-linearly
+into the field equations.
+0.8
+1.0
+1.2
+1.4
+1.6
+Conformal time t
+−10
+−9
+−8
+−7
+−6
+−5
+−4
+log10 error
+a = 1
+a = 2
+a = 5
+a = 10
+Figure 3: The log10 difference of the NP constant from the value on an initial cut as
+a function of conformal time t for a variable amplitude of the initial wave profile as
+a measure of deviation from constancy due to error. Cumulative error grows as we
+integrate along I +.
+5.4. Deviation from axisymmetry
+In a general asymptotically flat spacetime there are five complex NP constants
+corresponding to the five independent solutions to the equations (3.2). In axisymmetry,
+
+The non-linear perturbation of a black hole by gravitational waves
+13
+these collapse to only one independent solution, when the frame and coordinates also
+respect the symmetry, because then only the m = 0 modes of a spherical harmonic
+expansion remain.
+We can investigate the collapse of five NP constants into one by using the initial
+wave profile given by (5.1) and using a = i, b = c = ϵ i, where ϵ parametrises a deviation
+from axisymmetry.
+Six simulations were run for this wave profile for ϵ = 0, 1, 2, 3, 4, 5. Fig. 4 shows
+the magnitudes of the corresponding NP constants. Although we do have access to the
+full ten real degrees of freedom (five complex degrees of freedom) for each simulation,
+the trends can be seen in the behaviour of the magnitudes. Fig. 5 shows the same
+quantities but separated by the value of m of the corresponding U so that individual
+trends can be seen.
+We can see that for ϵ = 0, there is only one non-trivial constant
+0
+1
+2
+3
+4
+5
+Amplitude ϵ
+0.00
+0.25
+0.50
+0.75
+1.00
+1.25
+1.50
+NPC
+m = −2
+m = −1
+m = 0
+m = 1
+m = 2
+Figure 4: The magnitudes of the five complex NP constants as a function of a parameter
+ϵ which breaks axisymmetry in the initial wave profile. For ϵ > 0 all constants are non-
+zero although most are small.
+corresponding to the axisymmetric m = 0 mode, but as non-axisymmetric modes are
+introduced for ϵ > 0, all five constants take on non-trivial values and grow with ϵ.
+6. Discussion
+In this paper, we continue our numerical investigation into the non-linear perturbation
+of a Schwarzschild black hole using an initial boundary value problem for the general
+conformal field equations. This novel numerical scheme allows us to include I + within
+the computational domain and so compute quantities there directly. Thereby, we have
+computed the NP constants for the first time in a physically realistic spacetime.
+The gauge quantities of the system were fixed by numerical needs which implies
+that we were unable to directly use the very specific set of coordinates, frame, and
+
+The non-linear perturbation of a black hole by gravitational waves
+14
+0
+2
+4
+0.0
+0.5
+1.0
+1.5
+m = −2
+0
+2
+4
+0.0
+0.5
+1.0
+1.5
+m = −1
+0
+2
+4
+0.540
+0.541
+m = 0
+0
+2
+4
+Amplitude ϵ
+0.00
+0.02
+0.04
+NPC
+m = 1
+0
+2
+4
+0.000
+0.005
+0.010
+m = 2
+Figure 5: The magnitudes of the five complex NP constants as a function of a parameter
+ϵ which breaks axisymmetry in the initial wave profile split by the value of m of the
+corresponding U. Note that m is a label on spherical harmonics, not a mass.
+conformal factor typically used when defining quantities at I +. To compute physical
+quantities such as the NP constants with data from the numerical simulation, we need
+an explicitly conformally invariant expression for the quantity. However, this concept
+of conformal invariance is a rather special one, and it might be appropriate to highlight
+it again here.
+Physical quantities make reference to the physical metric ˜gab of the
+spacetime. In our context, the physical metric is represented as ˜gab = Ω−2gab, i.e., in
+terms of another metric in the same conformal class and the conformal factor relating
+the two. By conformal invariance we do not mean invariance under ˜gab �→ θ2˜gab, but
+rather the invariance under (gab, Ω) �→ (θ2gab, θΩ), which corresponds to the free choice
+of the splitting of ˜gab into a conformal and a scale part.
+For example, the recent analysis of the Bondi-Sachs energy-momentum in this
+framework involved generalising the procedure of constructing a basis of translations
+with respect to which components of the Bondi-Sachs 4-vector may be taken.
+The
+standard procedure of choosing the first four spherical harmonics is certainly not
+conformally invariant in this sense.
+This led to an invariant characterisation of the
+Lorentzian metric on the space of BMS translations [9]. Of course, the existence of this
+metric still leaves the freedom of Lorentz transformations for the choice of the basis.
+We run into the same problem when defining a basis for the quantity U which,
+when integrated against the Newman-Penrose integrand, gives the linearly independent
+NP constants. Again, it is the solution space which is defined in a conformally invariant
+way. But in this case there is no obvious inner product that one could use to select a
+basis. Even if there was one, the basis would still be defined only up to the appropriate
+
+REFERENCES
+15
+(pseudo-) orthogonal transformations. We circumvent the non-uniqueness of the basis
+in this case by refering to the universal structure that is imposed on the problem by
+the numerical setup as explained in Sec. 4.3. This seems to be the best way to ensure
+comparability across the different space-times that we investigate.
+Acknowledgments
+Supported by the Marsden Fund Council from Government funding, managed by Royal
+Society Te Ap¯arangi.
+The authors would like to thank L. Escobar for sharing the general form of his
+SWSH code.
+We wish to acknowledge the use of New Zealand eScience Infrastructure (NeSI) high
+performance computing facilities, consulting support and/or training services as part of
+this research. New Zealand’s national facilities are provided by NeSI and funded jointly
+by NeSI’s collaborator institutions and through the Ministry of Business, Innovation &
+Employment’s Research Infrastructure programme. URL https://www.nesi.org.nz.
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+

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+Approximate Quantum Compiling for Quantum Simulation: A
+Tensor Network based approach
+Niall F. Robertson1, Albert Akhriev1, Jiri Vala2,3 and Sergiy Zhuk1
+1 IBM Quantum, IBM Research Europe - Dublin, IBM Technology Campus, Dublin 15, Ireland
+2 Maynooth University, Maynooth, Ireland
+3 Tyndall National Institute, Cork, Ireland
+Abstract
+The simulation of quantum spin chains is a promising candidate for the demonstration of quantum
+advantage. One of the main obstacles to achieving this is the noise that arises from implementing
+the deep circuits that appear in standard quantum time evolution algorithms. Compiling these deep
+circuits into shallower ones is thus a key issue that we address in this work. We use a Tensor Network
+based approach to Approximate Quantum Compiling to produce short depth quantum circuits that
+simulate the time evolution of the Heisenberg spin chain on up to 100 qubits. Furthermore, we run
+these short depth circuits on a ibmq-mumbai - a 27 qubit device - and show that the accuracy of
+the measured observables is significantly improved after applying our Tensor Network compilation
+scheme.
+1
+Introduction
+The simulation of quantum many-body systems is a task of immense scientific interest. The study of
+quantum dynamics, in particular, allows for the study of thermalisation, many-body localisation, Hub-
+bard model physics and the applicability of field theory to out-of-equilibrium phenomena. In all of these
+fields there are many open scientific questions whose answers are likely to demand accurate simulation
+of quantum dynamics. However, the classical computational requirements of a brute-force approach to
+quantum dynamical simulations scales exponentially in the size of the system. Approximate techniques
+such as Tensor Networks are thus often called upon. Tensor Networks represent one of the best set of
+tools available to simulate time evolution and can also be applied to other problems such as ground state
+calculations [1] and machine learning [2, 3].
+Matrix Product States (MPS) are a particular type of Tensor Network that are particularly suited to
+describe quantum systems in one dimension. They form a key component of modern implementations of
+the well known Density Matrix Renormalisation Group (DMRG) algorithm used to find the ground state
+of local Hamiltonians. The DMRG algorithm was designed many years before [4] it was realised that it
+could be understood as a variational optimisation algorithm where a Matrix Product State is used as an
+Ansatz for the ground state [5]. This insight shed light on the reasons behind the spectacular success of
+DMRG; the ground states of local Hamiltonians are only weakly entangled and so too are Matrix Product
+States. More precisely, the bipartite entanglement entropy S of the ground state of a local Hamiltonian
+satisfies an area law, meaning that the entanglement entropy is proportional to the area of the boundary
+of the two subsystems in the bipartition. In 1D, this means that the entanglement entropy is independent
+of the system size [6]. This is in contrast to typical states in Hilbert space whose entanglement structures
+satisfy a volume law. Matrix Product States are also known to satisfy an area law [5] and thus have the
+same entanglement structure as the ground state by design.
+Since the weak entanglement of ground states of local Hamiltonians allow for their efficient storage as
+Matrix Product States, it is natural to ask if this is also possible for states that are generated by time
+evolution as these states are no longer necessarily weakly entangled. It turns out that for many physical
+systems of interest, entanglement entropy increases linearly until it saturates, at which point an MPS
+1
+arXiv:2301.08609v1  [quant-ph]  20 Jan 2023
+
+will no longer be an efficient representation of the state. However, if the initial state is weakly entangled
+then the MPS representation can be used to store the state at early times. A paradigmatic example of
+this scenario is a quantum quench, whereby a quantum system is initially prepared in the ground state of
+some local Hamiltonian, the parameters of the Hamiltonian are subsequently changed very rapidly and
+the system then evolves according to Schr¨odinger’s equation. The TEBD algorithm (Time Evolving Block
+Decimation) can be used to simulate time evolution after a quantum quench; the state is stored as an MPS
+and this MPS is updated as a function of time. Despite the success of DMRG, TEBD and other Tensor
+Network algorithms, these approaches are not without limitations. The memory requirements to store
+an MPS is characterised by the bond dimension, given by the dimension of the largest matrix used in the
+description of the state. For constant approximation error ϵ this bond dimension increases exponentially
+with the entanglement entropy and thus with time. Therefore, for a fixed maximum bond dimension,
+the error ϵ increases exponentially with time. This limits the applicability of Tensor Network algorithms
+to short time simulations. A quantum algorithm however, does not in principle suffer from this issue -
+the key difference between a quantum and a classical device being the ability to store highly entangled
+states. A quantum computer therefore has the potential to simulate quantum many-body systems for
+long times. The accurate simulation of the time evolution of 1D quantum systems is thus a promising
+route for the demonstration of quantum advantage in the short term. One such quantum algorithm is
+Trotterisation, where a discrete time step dt is used and the time evolution operator is approximated as
+a quantum circuit with an error that scales polynomially in dt. The depth of the quantum circuit used
+in such an approach increases with decreasing dt, leading to a trade-off between the noise arising from
+using deep circuits and the decreasing accuracy of the approximation when dt is increased. A number
+of variational quantum algorithms for the simulation of time evolution have therefore been developed
+that aim to use shallower circuits [7, 8, 9, 10]. Each of these approaches suffer from a number of issues
+such as convergence, runtime and limited device connectivity. As a result, it has been argued that such
+variational approaches are not practical for use on near term quantum hardware [11].
+One approach that aims to overcome the issue of deep circuits is Approximate Quantum Compiling
+[12, 13, 14], where one defines a parametric circuit of fixed depth and uses techniques from optimisation
+to minimise the distance between the parametric circuit and the target circuit of interest - where distance
+is defined by some carefully chosen metric. In principle, this approach can lead to short depth circuits
+that implement the target circuit of interest within some error tolerance. In practice, a classical imple-
+mentation of such an approach [14] is limited to act on a small number of qubits due to the exponential
+scaling of the Hilbert space with the number of qubits.
+Here we develop a new approach to quantum simulation that combines Matrix Product States, Approx-
+imate Quantum Compiling and Trotterisation to produce short depth quantum circuits that implement
+the time evolution operator of the Heisenberg spin chain. This approach is scalable thanks to the im-
+mense power of Matrix Product States. Figure 1 shows a schematic of our approach: first we apply
+Trotterisation classically for the maximum length of time for which we can still store the state as an
+MPS. We then apply a Matrix Product State implementation of Approximate Quantum Compiling to
+squeeze the circuit (purple box in the figure) to find a much shallower circuit that still reproduces the
+same state as Trotterisation, up to some small error in the fidelity. We then use the squeezed circuit as
+the input for the Trotter circuit which can now generate a quantum state beyond what can be stored
+classically.
+2
+Setup
+2.1
+The model
+We will consider the XXX spin-chain - a paradigmatic model for quantum magnetism - defined by the
+Hamiltonian:
+HXXX = −
+L−1
+�
+i=0
+hi,i+1 = −
+L−1
+�
+i=0
+�
+Sx
+i Sx
+i+1 + Sy
+i Sy
+i+1 + Sz
+i Sz
+i+1
+�
+,
+(1)
+where Sx, Sy and Sz are written in terms of Pauli matrices as Sx = σx
+2 , Sy = σy
+2 and Sz = σz
+2 . The
+Hamiltonian in (1) is a prototypical example of an integrable 1D model and its dynamical behaviour has
+been studied extensively [15], including on a quantum computer [16]. The time evolution of a quantum
+2
+
+Compress with AQC
+...
+...
+...
+...
+...
+...
+...
+...
+Figure 1: Schematic of our approach: Trotterisation is applied classically (purple box) and then a Matrix
+Product State implementation of Approximate Quantum Compiling is applied to compress the first part
+of the circuit. Standard Trotterisation is then applied on a quantum device afterwards to simulate longer
+times, i.e. times which are beyond what is possible classically.
+Rz(θ)
+Rz( π
+2 )
+Rz(− π
+2 )
+Ry(φ)
+Ry(λ)
+Figure 2: Implementation of two site operator ei(ασx⊗σx+βσy⊗σy+γσz⊗σz) as a quantum circuit. We have
+the correspondences θ = π
+2 − 2γ, φ = 2α − π
+2 and λ = π
+2 − 2β. The Hamiltonian in (1) corresponds to
+the case α = β = γ = dt
+state |ψ(t)⟩ is governed by the Schr¨odinger equation:
+|ψ(t)⟩ = e−iHXXXt |ψ(0)⟩
+(2)
+where |ψ(0)⟩ is the wavefunction at time t = 0. In this work, we will consider the N´eel state, written as:
+|↑↓↑↓ ... ↑↓⟩ where ↑ and ↓ represent up and down spins respectively. The N´eel state for n spins is simply
+implemented on n qubits as |1010...10⟩.
+The time evolution operator U(t) ≡ e−iHt can be executed as a quantum circuit in a resource efficient
+way; we first write the Hamiltonian in (1) as HXXX = H1 + H2 where H1 = − �
+i odd
+hi,i+1 and H2 =
+− �
+i even
+hi,i+1. Note that all operators in a given sum commute with all other operators in their respective
+sums. We then define the Suzuki-Trotter time evolution operator Utrot(dt) in the following way:
+U(1)
+trot(dt) =
+L/2−1
+�
+j=0
+U2j,2j+1(dt)
+L/2−1
+�
+j=1
+U2j−1,2j(dt) = e−iHXXZdt + O(dt2)
+(3)
+where Ujk(dt) = e−ihjkdt. The exact time evolution operator U(t) is thus approximated by m repeated
+applications of Utrot(dt =
+t
+m), i.e. U(t) ≈ Um
+trot(dt =
+t
+m). As discussed in [16], each Ujk(dt) appearing in
+(3) can be implemented by the quantum circuit with just three CNOTs as in Figure 2. We can reduce
+the error in the Trotter formula in equation (3) by using higher order expressions [17]. It turns out that
+the second order Trotter formula can be implemented on a quantum circuit with only one extra layer in
+the circuit [16]. We have:
+U(2)
+trot(dt) =
+L/2−1
+�
+j=0
+U2j,2j+1
+�dt
+2
+� L/2−1
+�
+j=1
+U2j−1,2j (dt)
+L/2−1
+�
+j=0
+U2j,2j+1
+�dt
+2
+�
+= e−iHXXZdt + O(dt2)
+(4)
+which can be implemented on a quantum device by the circuit in Figure 4.
+3
+
+U(dt)
+U(dt)
+U(dt)
+U(dt)
+U(dt)
+U(dt)
+U(dt)
+U(dt)
+U(dt)
+U(dt)
+U(dt)
+U(dt)
+U(dt)
+U(dt)
+U(dt)
+Figure 3: First order Trotter circuit acting on six qubits.
+U( dt
+2 )
+U(dt)
+U(dt)
+U( dt
+2 )
+U(dt)
+U(dt)
+U(dt)
+U( dt
+2 )
+U(dt)
+U(dt)
+U( dt
+2 )
+U(dt)
+U(dt)
+U(dt)
+U( dt
+2 )
+U(dt)
+U(dt)
+U( dt
+2 )
+Figure 4: Second order Trotter circuit acting on six qubits.
+4
+
+A(1)
+A(2)
+A(3)
+A(4)
+A(5)
+A(6)
+Figure 5: Graphical representation of an MPS. There are two matrices A(i) for each qubit at position i.
+A(1)
+A(2)
+A(3)
+A(4)
+A(5)
+A(6)
+B(1)
+B(2)
+B(3)
+B(4)
+B(5)
+B(6)
+Figure 6: The inner product ⟨ψ1|ψ2⟩ of two Matrix Product States - see equations (11) and (6).
+2.2
+Matrix Product States
+An arbitrary quantum state on n qubits can be written in terms of complex variables cj1,...,jn, the number
+of which scales as 2n:
+|ψ⟩ =
+�
+{j1,...,jn}
+cj1,...,jn |j1, ..., jn⟩
+(5)
+where the sum is over all configurations of the binary variables j1, ..., jn. The bipartite entanglement
+entropy of an arbitrary quantum state picked at random from Hilbert space satisfies a volume law which,
+as was discussed in the introduction, is distinct from area law entanglement in which case the entanglement
+entropy of two regions after the bipartition of the system is proportional to the area of the boundary of
+the system. A small subset of states in Hilbert space satisfies an area law. The coefficients cj1,...,jn of
+such states have a certain structure that we can take advantage of to study classically. Any state |ψ⟩ can
+be written in the following way:
+cj1,...,jn = A(1)
+j1 · A(2)
+j2 ... · A(n)
+jn
+(6)
+where the Aj are χj × χj+1 dimensional matrices. Quantum states of the form (6) are known as Matrix
+Product States (MPS). The maximum value of χj is referred to as the bond dimension of the MPS. We
+can represent an MPS graphically as in Figure 5. We associate one matrix A(i) to each qubit. Note
+that for each qubit i we have two matrices. We thus have a total of 2n matrices to keep track of. The
+bond dimension χj can be seen as a measure of the entanglement between the two subsystems when a
+bipartition is made at qubit j. Therefore, states in Hilbert space that satisfy an area law - and therefore
+have a low bond dimension in their MPS representation - can be efficiently stored as Matrix Product
+States. States that satisfy a volume law will have a bond dimension that is exponential in the number of
+qubits. We will consider in this work the non-trivial dynamics governed by equation (2). As discussed in
+the introduction, the bipartite entanglement entropy of a ground state of a one-dimensional Hamiltonian
+that has a gap between its ground state and its excited state is independent of the size of the subsystems.
+The ground state of such a system - and hence the initial state in our setup - can be efficiently stored
+as an MPS. One can then use an algorithm such as TEBD (Time Evolving Block Decimation) [18] to
+update the MPS as a function of time to study the dynamics of the system. However, the entanglement
+entropy of the state increases linearly with time, hence the bond dimension χ that is required to keep
+the error constant diverges exponentially with time. To simulate for longer times, a quantum computer
+would be needed. In section 2.3, we will discuss how Matrix Product States can be leveraged to reduce
+the resource requirements for this simulation problem when implemented on a quantum device.
+2.3
+Matrix Product States applied to Approximate Quantum Compiling
+j
+k
+Ry(θ1)
+Rz(θ2)
+Ry(θ3)
+Rx(θ4)
+Figure 7: CNOT block forms the basic building block of our circuit ansatz.
+5
+
+Rz(θ1)
+Ry(θ2)
+Rz(θ3)
+Rz(θ4)
+Ry(θ5)
+Rz(θ6)
+Rz(θ7)
+Ry(θ8)
+Rz(θ9)
+Rz(θ10)
+Ry(θ11)
+Rz(θ12)
+Figure 8: Parameterised circuit inspired by the structure of the first order Trotter circuit in Figure 3.
+Rz(θ1)
+Ry(θ2)
+Rz(θ3)
+Rz(θ4)
+Ry(θ5)
+Rz(θ6)
+Rz(θ7)
+Ry(θ8)
+Rz(θ9)
+Rz(θ10)
+Ry(θ11)
+Rz(θ12)
+Figure 9: Parameterised circuit inspired by the structure of the second order Trotter circuit in Figure 4.
+Approximate quantum compiling (AQC) involves the design of a parametric quantum circuit with
+fixed depth - the parameters are then adjusted to bring it as close as possible to the target, where “close”
+is defined via some carefully chosen metric, see below. As discussed in [12], one can use so-called CNOT
+blocks to construct a natural circuit Ansatz. A CNOT block is a CNOT gate followed by single qubit
+rotations (see Figure 7). A block with a CNOT gate acting on a “control” qubit j and “target” qubit
+k is written as CUjk(θ1, θ2, θ3, θ4). For a given hardware connectivity, one can then write down a fully
+parameterised circuit as:
+Vct(θ) =CUct(L) (θ3n+4L−3, . . . , θ3n+4L) · · · CUct(1) (θ3n+1, . . . , θ3n+4)
+[Rz (θ1) Ry (θ2) Rz (θ3)] ⊗ · · · ⊗ [Rz (θ3n−2) Ry (θ3n−1) Rz (θ3n)]
+(7)
+The position of the CNOT blocks in the parameterised circuit can be customised to suit the particular
+target circuit that one is interested in. Here we are interested in finding a circuit that implements the
+unitary time evolution operator as in equation (2). We thus consider a structure inspired by the first and
+second-order Trotter circuits in Figures 3 and 4 respectively. Recall that each block U(dt) in Figures 3
+and 4 represents the 2-qubit sub-circuit with three CNOTs in Figure 2; it is therefore natural to consider
+a circuit Ansatz with sub-circuits each with three CNOT blocks as in Figures 8 and 9, such that the
+circuit Ansatz mimics the structure of the first and second order Trotter circuits. In the notation of [14],
+the parameterised circuits in Figures 8 and 9 correspond to n = 4 qubits, l = 2 layers and b = 3 CNOT
+blocks in each layer. In both Figure 8 and Figure 9 there are three rotation gates acting on each qubit at
+the beginning of the circuit. In the examples considered in this work we will take the initial state to be |0⟩
+- the initial rotation gate Rz(θ) is redundant for these cases but is necessary for more general initial states.
+One can define the distance between the target and parameterised circuit via a number of different
+metrics. Here we use a cost function based on the Hilbert-Schmidt test:
+Cstate
+hs
+= 1 − | ⟨0| V †(θ) |ψ0⟩ |2
+(8)
+The goal of AQC is to tune the parameters θ to minimise the cost function under consideration. Note
+that here we are considering the application of AQC to state preparation as opposed to full circuit com-
+pilation. More precisely, this means that our cost function is designed such that it is minimised when the
+action of V (θ) on the initial state |0⟩ produces a state that is as close as possible to a target state |ψ0⟩
+(up to some global phase). This is in contrast to the situation where one starts with some target circuit
+U and the cost function is designed to bring the full matrix V (θ) as close as possible to U.
+6
+
+As pointed out in [19], the gradient of the cost function in (8) vanishes exponentially. This observation
+lead to the distinction between global and local cost functions; local cost functions have only polynomially
+vanishing gradients in some cases of interest - see [19, 20, 14] for details. As was shown in [14], the Hilbert-
+Schmidt test - which is a global cost function - can be turned into a local one by adding several “bit-flip”
+terms which increases the magnitude of the gradient:
+Cstate
+lhs
+= 1−| ⟨0| V †(θ) |ψ0⟩ |2 −
+�n − 1
+n
+�
+n
+�
+j=1
+| ⟨0| XjV †(θ) |ψ0⟩ |2
+−
+�n − 2
+n
+� �
+j<k
+| ⟨0| XjXkV †(θ) |ψ0⟩ |2 − ... − 1
+n
+�
+j<k<l<...
+| ⟨0| XjXkXl...V †(θ) |ψ0⟩ |2
+(9)
+Convergence of the cost function can be significantly improved by adding these terms, however the
+computational cost of calculating the gradient becomes prohibitive. It was demonstrated in [14] that this
+can be overcome by truncating the expression in (9) to get:
+C(1)
+L (α) = 1 − | ⟨0| V †(θ) |ψ0⟩ |2 − α
+n
+�
+j=1
+| ⟨0| XjV †(θ) |ψ0⟩ |2
+(10)
+where α is a parameter that can be tuned throughout the optimisation procedure - a scheme to implement
+this tuning effectively was demonstrated in [14]. In (10), we have only kept 1 “bit-flip” term, i.e. we have
+dropped all terms with more than one NOT operators Xi. As discussed in [14], one can obtain higher
+order expressions C(k)
+L
+with more “bit-flip” terms included - doing so induces a larger gradient in the cost
+function but increases the computational burden.
+Note that each term in (9) or (10) is an overlap of quantum states, and since the overlap of two MPS
+can be calculated very efficiently the architecture of Matrix Product States can be leveraged to calculate
+the cost function and solve the approximate quantum compilation problem for large numbers of qubits.
+Consider for example two quantum states |ψ1⟩ and |ψ2⟩:
+|ψ1⟩ =
+�
+{j1,...,jn}
+c(1)
+j1,...,jn |j1, ..., jn⟩
+|ψ2⟩ =
+�
+{j1,...,jn}
+c(2)
+j1,...,jn |j1, ..., jn⟩
+(11)
+As discussed in section 2.2, for weakly entangled states the coefficients c(1)
+j1,...,jn and c(2)
+j1,...,jn are not all
+independent and can be represented efficiently as Matrix Product States - see Figure 5:
+c(1)
+j1,...,jn = A(1)
+j1 · A(2)
+j2 ... · A(n)
+jn
+c(2)
+j1,...,jn = B(1)
+j1 · B(2)
+j2 ... · B(n)
+jn
+(12)
+We want to calculate the quantity:
+f(|ψ1⟩ , |ψ2⟩) = || ⟨ψ1|ψ2⟩ ||2
+(13)
+The overlap of two MPS, and hence the fidelity f in (13) can be calculated efficiently by “contracting”
+the Tensor Network shown in Figure 6.
+3
+Results
+We now present the results of our simulations of the Schr¨odinger equation in equation (2) using the
+second order Trotter formula in equation (4). First let’s define and clarify some notation:
+• |a1⟩: The state generated by the optimised parametric circuit in Figure 9.
+• Number of layers l in Ansatz: in Figures 8 and 9 there are l = 2 layers.
+• |t1⟩: The state generated by the Trotter circuit in Figure 4.
+7
+
+Figure 10: 50 qubits: fidelities of the parametric circuit and the Trotter circuit with the ”ground truth”
+obtained by Tensor Network simulations. The two circuits are of identical length but the parametric
+circuit achieves a significantly higher fidelity at late times.
+Figure 11: 50 qubits: the maximum number of layers in the parametric circuit is 12 while it is 18 for the
+Trotter circuit. Despite this, the parametric circuit achieves a higher fidelity.
+• Number of Trotter steps: the analogue of the number of layers in the Ansatz circuits. In Figure 3
+and 4 there are 3 Trotter steps.
+• |t1 gt⟩: the “ground truth” generated by classical Tensor Network simulations of deep Trotter
+circuits, i.e. extremely small time steps. We take dt = 0.04 to generate the ground truth state
+while |t1⟩ is generated with a time step of dt = 0.4.
+All circuits considered here take the form of the second-order Trotter structure. More precisely, in each
+graph we use the labels “Trotter” and “Ansatz” and these circuits have the structure in Figures 4 and 9
+respectively. In Figures 10, 11 and 12 we plot fidelity vs evolution time for the 50 qubit XXX Hamiltonian
+and we compare the result of the Trotter circuit vs the AQC circuit. In Figure 10 we can see that the
+fidelity of the Trotter circuit decays rapidly while the fidelity of the AQC circuit remains above 0.99.
+Note that the Trotter circuit and the AQC circuit are of equal depth. In Figures 11 and 12 we compare
+short depth circuits generated by AQC with deep Trotter circuits. We observe that the AQC circuits
+can achieve comparable or better fidelities with much shorter depth. In particular, in Figure 12 the two
+fidelities are almost identical but the AQC circuit is half the depth of the Trotter circuit. We plot the
+8
+
+Fidelity: lal> vs Itl_gt> and Itl> vs Itl gt>, num.qubits: 50
+1.00
+0.99
+0.98
+0.97
+0.96
+fidelity
+0.95
+0.94
+0.93
+0.92
+Ansatz
+0.91
+Trotter
+0.90
+1.2
+2.4 
+3.6
+4.8
+6.0
+7.2
+evolution time
+13
+６
+9
+12
+15
+18
+number of layers in ansatz
+13
+６
+９
+12
+15
+18
+number of Trotter stepsFidelity: lal> vs Itl_gt> and Itl> vs Itl gt>, num.qubits: 50
+1.00
+0.99
+0.98
+0.97
+0.96
+fidel
+0.95
+0.94
+0.93
+0.92
+0.91
+Ansatz
+Trotter
+0.90
+1.2
+2.4 
+3.6
+4.8
+6.0
+7.2
+evolution time
+2
+12
+4
+6
+8
+10
+number of layers in ansatz
+13
+６
+9
+12
+15
+18
+number of Trotter stepsFigure 12: 50 qubits: the maximum number of layers in the parametric circuit is 9 while it is 18 for the
+Trotter circuit. Both circuits achieve very similar fidelities despite the parametric circuit being half the
+depth of the Trotter circuit.
+Figure 13: 100 qubits: The maximum number of layers in the parametric circuit and the Trotter circuit
+is 18. It can be seen that the fidelity of the Trotter circuit decays rapidly but that of the parametric
+circuit remains high.
+same data for 100 qubits in Figures 13 and 14.
+Now we would like to consider how these results affect the implementation on a real quantum device.
+We consider a 20 qubit spin-chain on the 27 qubit device ibmq-mumbai. First we plot the fidelity results
+for 20 qubits in Figure 15. We ran the resulting parametric circuit on ibmq-mumbai and in Figure 16
+we plot the expectation values ⟨ψ(t)|Sz
+0|ψ(t)⟩ as obtained from the quantum device using the parametric
+AQC circuit, the Trotter circuit and from a classical Tensor Network simulation. We observe that this
+observable is more accurate when obtained with the AQC circuit due to its reduced depth. Note that
+the difference between the results from the simulation and the results from the quantum device would be
+greatly reduced after applying error mitigation [21, 22]. We have not attempted to apply error mitigation
+to either circuit as this would be outside the scope of this work.
+We expect that, since our Tensor
+Network compilation scheme greatly reduces the noise of the circuit, any error mitigation scheme would
+be enhanced by our approach.
+9
+
+Fidelity: lal> vs |tl_gt> and Itl> vs Itl_gt>, num.qubits: 50
+1.00
+Ansatz
+0.99
+Trotter
+0.98
+0.97
+0.96
+fideli
+0.95
+0.94
+0.93
+0.92
+0.91
+0.90
+1.2
+2.4 
+3.6
+4.8
+6.0
+7.2
+evolution time
+2
+4
+8
+９
+6
+number of layers in ansatz
+13
+６
+9
+12
+15
+18
+number of Trotter stepsFidelity: lai) and Iti> vs. ground-truth state, Ngqubits: 1o0
+1.00
+0.99
+0.98
+0.97
+0.96
+0.95
+p!
+0.94
+0.93
+0.92
+0.91
+Ansatz
+Trotter
+0.90
+1.2
+2.4 
+3.6
+4.8
+6.0
+7.2
+evolution time
+13
+６
+9
+12
+15
+18
+number of layers in ansatz
+13
+6
+９
+12
+15
+18
+number of Trotter stepsFigure 14: 100 qubits: The maximum number of layers in the parametric circuit is 12 while it is 18 for
+the Trotter circuit.
+Figure 15: The maximum depth of the parametric circuit is half that of the Trotter circuit - there are 9
+and 18 layers respectively. These 20 qubit circuits were implemented on ibmq-mumbai - see Figure 16.
+4
+Discussion
+In this paper we applied Tensor Network methods to Quantum Compiling and demonstrated their efficacy
+on the 27 qubit device ibmq-mumbai. Our method is similar in spirit to [23] where Matrix Product States
+were used to prepare the initial state for VQE to find the ground state of some Hamiltonian - here we
+use Matrix Product States to prepare a short depth quantum circuit that simulates the time evolution of
+a 1D Hamiltonian. We chose the XXX Hamiltonian in equation (1) because it has been well studied, but
+we would be particularly interested to apply the compilation methods developed here to non-integrable
+systems by e.g. adding a random field to the Hamiltonian in (1) and studying phenomena of scientific
+interest such as many-body localisation.
+We have shown results of our simulations on up to 100 qubits. In principle we can significantly increase
+the number of qubits and the length of time to which we apply our MPS compilation scheme; the limiting
+factor at present seems to be the particular implementation that we apply to SVD and to calculate the
+gradient. We believe that both of these can be improved significantly, in particular by using an efficient
+parallel implementation - this is the subject of ongoing work. In our current framework we use the Qiskit
+10
+
+Fidelity: lai) and Iti> vs. ground-truth state, Ngqubits: 1o0
+1.00
+Ansatz
+0.99
+Trotter
+0.98
+0.97
+0.96
+0.95
+p!
+0.94
+0.93
+0.92
+0.91
+0.90 
+1.2
+2.4 
+3.6
+4.8
+6.0
+7.2
+evolution time
+2
+12
+4
+6
+8
+10
+number of layers in ansatz
+13
+６
+9
+12
+15
+18
+number of Trotter stepsFidelity: lai> and Iti> vs. ground-truth state, Nqubits: 20
+1.000
+0.995
+0.990
+0.985
+0.980
+0.975
+fide
+0.970
+0.965
+0.960
+Ansatz
+0.955
+Trotter
+0.950
+1.2
+2.4 
+3.6
+4.8
+6.0
+7.2
+evolution time
+2
+4
+９
+6
+7
+8
+number of layers in ansatz
+13
+６
+９
+12
+15
+18
+number of Trotter stepsFigure 16: The expectation value of Sz
+0 vs time for a chain of 20 qubits as measured on the 27 qubit
+quantum device ibmq-mumbai.
+The circuit produced from our MPS implementation of AQC uses is
+shallower than the Trotter circuit, and thus produces an expectation value that is much closer to the true
+value plotted in the blue curve, obtained by classical Tensor Network simulations.
+MPS package which is designed for generic situations in which long range connectivity may be required
+and thus does not take advantage of the short range structure of the circuits in Figures 8 and 9.
+5
+Acknowledgements
+This work was funded by the Disruptive Technologies Innovation Fund (DTIF), by Enterprise Ireland,
+under project number DTIF2019-090 (project QCoIR) and also supported by IBM Quantum.
+11
+
+0.0 -
++
+-0.1
++
+0.2
+0.3
+-0.4
+Simulation
+-0.5
+ibm mumbai: Trotter circuit
++
+ibm mumbai: AQC circuit
+-0.6.
+0.D
+0.5
+15
+2D
+25
+3.D
+3.5
+4.D
+tReferences
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+Matrix product states represent ground states faithfully.
+Physical review b, 73(9):094423, 2006.
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+13
+

1tFAT4oBgHgl3EQfkB3r/content/tmp_files/load_file.txt

@@ -0,0 +1,460 @@
+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFAT4oBgHgl3EQfkB3r/content/2301.08609v1.pdf,len=459
+page_content='Approximate Quantum Compiling for Quantum Simulation: A  Tensor Network based approach  Niall F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFAT4oBgHgl3EQfkB3r/content/2301.08609v1.pdf'}
+page_content=' Robertson1, Albert Akhriev1, Jiri Vala2,3 and Sergiy Zhuk1  1 IBM Quantum, IBM Research Europe - Dublin, IBM Technology Campus, Dublin 15, Ireland  2 Maynooth University, Maynooth, Ireland  3 Tyndall National Institute, Cork, Ireland  Abstract  The simulation of quantum spin chains is a promising candidate for the demonstration of quantum  advantage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFAT4oBgHgl3EQfkB3r/content/2301.08609v1.pdf'}
+page_content=' One of the main obstacles to achieving this is the noise that arises from implementing  the deep circuits that appear in standard quantum time evolution algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFAT4oBgHgl3EQfkB3r/content/2301.08609v1.pdf'}
+page_content=' Compiling these deep  circuits into shallower ones is thus a key issue that we address in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFAT4oBgHgl3EQfkB3r/content/2301.08609v1.pdf'}
+page_content=' We use a Tensor Network  based approach to Approximate Quantum Compiling to produce short depth quantum circuits that  simulate the time evolution of the Heisenberg spin chain on up to 100 qubits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFAT4oBgHgl3EQfkB3r/content/2301.08609v1.pdf'}
+page_content=' Furthermore, we run  these short depth circuits on a ibmq-mumbai - a 27 qubit device - and show that the accuracy of  the measured observables is significantly improved after applying our Tensor Network compilation  scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFAT4oBgHgl3EQfkB3r/content/2301.08609v1.pdf'}
+page_content='  1  Introduction  The simulation of quantum many-body systems is a task of immense scientific interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFAT4oBgHgl3EQfkB3r/content/2301.08609v1.pdf'}
+page_content=' The study of  quantum dynamics, in particular, allows for the study of thermalisation, many-body localisation, Hub-  bard model physics and the applicability of field theory to out-of-equilibrium phenomena.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFAT4oBgHgl3EQfkB3r/content/2301.08609v1.pdf'}
+page_content=' In all of these  fields there are many open scientific questions whose answers are likely to demand accurate simulation  of quantum dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFAT4oBgHgl3EQfkB3r/content/2301.08609v1.pdf'}
+page_content=' However, the classical computational requirements of a brute-force approach to  quantum dynamical simulations scales exponentially in the size of the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFAT4oBgHgl3EQfkB3r/content/2301.08609v1.pdf'}
+page_content=' Approximate techniques  such as Tensor Networks are thus often called upon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFAT4oBgHgl3EQfkB3r/content/2301.08609v1.pdf'}
+page_content=' Tensor Networks represent one of the best set of  tools available to simulate time evolution and can also be applied to other problems such as ground state  calculations [1] and machine learning [2, 3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFAT4oBgHgl3EQfkB3r/content/2301.08609v1.pdf'}
+page_content='  Matrix Product States (MPS) are a particular type of Tensor Network that are particularly suited to  describe quantum systems in one dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFAT4oBgHgl3EQfkB3r/content/2301.08609v1.pdf'}
+page_content=' They form a key component of modern implementations of  the well known Density Matrix Renormalisation Group (DMRG) algorithm used to find the ground state  of local Hamiltonians.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFAT4oBgHgl3EQfkB3r/content/2301.08609v1.pdf'}
+page_content=' The DMRG algorithm was designed many years before [4] it was realised that it  could be understood as a variational optimisation algorithm where a Matrix Product State is used as an  Ansatz for the ground state [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFAT4oBgHgl3EQfkB3r/content/2301.08609v1.pdf'}
+page_content=' This insight shed light on the reasons behind the spectacular success of  DMRG;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFAT4oBgHgl3EQfkB3r/content/2301.08609v1.pdf'}
+page_content=' the ground states of local Hamiltonians are only weakly entangled and so too are Matrix Product  States.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFAT4oBgHgl3EQfkB3r/content/2301.08609v1.pdf'}
+page_content=' More precisely, the bipartite entanglement entropy S of the ground state of a local Hamiltonian  satisfies an area law, meaning that the entanglement entropy is proportional to the area of the boundary  of the two subsystems in the bipartition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFAT4oBgHgl3EQfkB3r/content/2301.08609v1.pdf'}
+page_content=' In 1D, this means that the entanglement entropy is independent  of the system size [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFAT4oBgHgl3EQfkB3r/content/2301.08609v1.pdf'}
+page_content=' This is in contrast to typical states in Hilbert space whose entanglement structures  satisfy a volume law.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFAT4oBgHgl3EQfkB3r/content/2301.08609v1.pdf'}
+page_content=' Matrix Product States are also known to satisfy an area law [5] and thus have the  same entanglement structure as the ground state by design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFAT4oBgHgl3EQfkB3r/content/2301.08609v1.pdf'}
+page_content='  Since the weak entanglement of ground states of local Hamiltonians allow for their efficient storage as  Matrix Product States, it is natural to ask if this is also possible for states that are generated by time  evolution as these states are no longer necessarily weakly entangled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFAT4oBgHgl3EQfkB3r/content/2301.08609v1.pdf'}
+page_content=' It turns out that for many physical  systems of interest, entanglement entropy increases linearly until it saturates, at which point an MPS  1  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFAT4oBgHgl3EQfkB3r/content/2301.08609v1.pdf'}
+page_content='08609v1  [quant-ph]  20 Jan 2023  will no longer be an efficient representation of the state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFAT4oBgHgl3EQfkB3r/content/2301.08609v1.pdf'}
+page_content=' However, if the initial state is weakly entangled  then the MPS representation can be used to store the state at early times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFAT4oBgHgl3EQfkB3r/content/2301.08609v1.pdf'}
+page_content=' A paradigmatic example of  this scenario is a quantum quench, whereby a quantum system is initially prepared in the ground state of  some local Hamiltonian, the parameters of the Hamiltonian are subsequently changed very rapidly and  the system then evolves according to Schr¨odinger’s equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFAT4oBgHgl3EQfkB3r/content/2301.08609v1.pdf'}
+page_content=' The TEBD algorithm (Time Evolving Block  Decimation) can be used to simulate time evolution after a quantum quench;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFAT4oBgHgl3EQfkB3r/content/2301.08609v1.pdf'}
+page_content=' the state is stored as an MPS  and this MPS is updated as a function of time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFAT4oBgHgl3EQfkB3r/content/2301.08609v1.pdf'}
+page_content=' Despite the success of DMRG, TEBD and other Tensor  Network algorithms, these approaches are not without limitations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFAT4oBgHgl3EQfkB3r/content/2301.08609v1.pdf'}
+page_content=' The memory requirements to store  an MPS is characterised by the bond dimension, given by the dimension of the largest matrix used in the  description of the state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFAT4oBgHgl3EQfkB3r/content/2301.08609v1.pdf'}
+page_content=' For constant approximation error ϵ this bond dimension increases exponentially  with the entanglement entropy and thus with time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFAT4oBgHgl3EQfkB3r/content/2301.08609v1.pdf'}
+page_content=' Therefore, for a fixed maximum bond dimension,  the error ϵ increases exponentially with time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFAT4oBgHgl3EQfkB3r/content/2301.08609v1.pdf'}
+page_content=' This limits the applicability of Tensor Network algorithms  to short time simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFAT4oBgHgl3EQfkB3r/content/2301.08609v1.pdf'}
+page_content=' A quantum algorithm however, does not in principle suffer from this issue -  the key difference between a quantum and a classical device being the ability to store highly entangled  states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFAT4oBgHgl3EQfkB3r/content/2301.08609v1.pdf'}
+page_content=' A quantum computer therefore has the potential to simulate quantum many-body systems for  long times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFAT4oBgHgl3EQfkB3r/content/2301.08609v1.pdf'}
+page_content=' The accurate simulation of the time evolution of 1D quantum systems is thus a promising  route for the demonstration of quantum advantage in the short term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFAT4oBgHgl3EQfkB3r/content/2301.08609v1.pdf'}
+page_content=' One such quantum algorithm is  Trotterisation, where a discrete time step dt is used and the time evolution operator is approximated as  a quantum circuit with an error that scales polynomially in dt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFAT4oBgHgl3EQfkB3r/content/2301.08609v1.pdf'}
+page_content=' The depth of the quantum circuit used  in such an approach increases with decreasing dt, leading to a trade-off between the noise arising from  using deep circuits and the decreasing accuracy of the approximation when dt is increased.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFAT4oBgHgl3EQfkB3r/content/2301.08609v1.pdf'}
+page_content=' A number  of variational quantum algorithms for the simulation of time evolution have therefore been developed  that aim to use shallower circuits [7, 8, 9, 10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFAT4oBgHgl3EQfkB3r/content/2301.08609v1.pdf'}
+page_content=' Each of these approaches suffer from a number of issues  such as convergence, runtime and limited device connectivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFAT4oBgHgl3EQfkB3r/content/2301.08609v1.pdf'}
+page_content=' As a result, it has been argued that such  variational approaches are not practical for use on near term quantum hardware [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFAT4oBgHgl3EQfkB3r/content/2301.08609v1.pdf'}
+page_content='  One approach that aims to overcome the issue of deep circuits is Approximate Quantum Compiling  [12, 13, 14], where one defines a parametric circuit of fixed depth and uses techniques from optimisation  to minimise the distance between the parametric circuit and the target circuit of interest - where distance  is defined by some carefully chosen metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFAT4oBgHgl3EQfkB3r/content/2301.08609v1.pdf'}
+page_content=' In principle, this approach can lead to short depth circuits  that implement the target circuit of interest within some error tolerance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFAT4oBgHgl3EQfkB3r/content/2301.08609v1.pdf'}
+page_content=' In practice, a classical imple-  mentation of such an approach [14] is limited to act on a small number of qubits due to the exponential  scaling of the Hilbert space with the number of qubits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFAT4oBgHgl3EQfkB3r/content/2301.08609v1.pdf'}
+page_content='  Here we develop a new approach to quantum simulation that combines Matrix Product States, Approx-  imate Quantum Compiling and Trotterisation to produce short depth quantum circuits that implement  the time evolution operator of the Heisenberg spin chain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFAT4oBgHgl3EQfkB3r/content/2301.08609v1.pdf'}
+page_content=' This approach is scalable thanks to the im-  mense power of Matrix Product States.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFAT4oBgHgl3EQfkB3r/content/2301.08609v1.pdf'}
+page_content=' Figure 1 shows a schematic of our approach: first we apply  Trotterisation classically for the maximum length of time for which we can still store the state as an  MPS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFAT4oBgHgl3EQfkB3r/content/2301.08609v1.pdf'}
+page_content=' We then apply a Matrix Product State implementation of Approximate Quantum Compiling to  squeeze the circuit (purple box in the figure) to find a much shallower circuit that still reproduces the  same state as Trotterisation, up to some small error in the fidelity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFAT4oBgHgl3EQfkB3r/content/2301.08609v1.pdf'}
+page_content=' We then use the squeezed circuit as  the input for the Trotter circuit which can now generate a quantum state beyond what can be stored  classically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFAT4oBgHgl3EQfkB3r/content/2301.08609v1.pdf'}
+page_content='  2  Setup  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFAT4oBgHgl3EQfkB3r/content/2301.08609v1.pdf'}
+page_content='1  The model  We will consider the XXX spin-chain - a paradigmatic model for quantum magnetism - defined by the  Hamiltonian:  HXXX = −  L−1  �  i=0  hi,i+1 = −  L−1  �  i=0  �  Sx  i Sx  i+1 + Sy  i Sy  i+1 + Sz  i Sz  i+1  �  ,  (1)  where Sx, Sy and Sz are written in terms of Pauli matrices as Sx = σx  2 , Sy = σy  2 and Sz = σz  2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFAT4oBgHgl3EQfkB3r/content/2301.08609v1.pdf'}
+page_content=' The  Hamiltonian in (1) is a prototypical example of an integrable 1D model and its dynamical behaviour has  been studied extensively [15], including on a quantum computer [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFAT4oBgHgl3EQfkB3r/content/2301.08609v1.pdf'}
+page_content=' The time evolution of a quantum  2  Compress with AQC  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFAT4oBgHgl3EQfkB3r/content/2301.08609v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFAT4oBgHgl3EQfkB3r/content/2301.08609v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFAT4oBgHgl3EQfkB3r/content/2301.08609v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFAT4oBgHgl3EQfkB3r/content/2301.08609v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFAT4oBgHgl3EQfkB3r/content/2301.08609v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFAT4oBgHgl3EQfkB3r/content/2301.08609v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFAT4oBgHgl3EQfkB3r/content/2301.08609v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFAT4oBgHgl3EQfkB3r/content/2301.08609v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFAT4oBgHgl3EQfkB3r/content/2301.08609v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFAT4oBgHgl3EQfkB3r/content/2301.08609v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFAT4oBgHgl3EQfkB3r/content/2301.08609v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFAT4oBgHgl3EQfkB3r/content/2301.08609v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFAT4oBgHgl3EQfkB3r/content/2301.08609v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFAT4oBgHgl3EQfkB3r/content/2301.08609v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFAT4oBgHgl3EQfkB3r/content/2301.08609v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFAT4oBgHgl3EQfkB3r/content/2301.08609v1.pdf'}
+page_content='  Figure 1: Schematic of our approach: Trotterisation is applied classically (purple box) and then a Matrix  Product State implementation of Approximate Quantum Compiling is applied to compress the first part  of the circuit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFAT4oBgHgl3EQfkB3r/content/2301.08609v1.pdf'}
+page_content=' Standard Trotterisation is then applied on a quantum device afterwards to simulate longer  times, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFAT4oBgHgl3EQfkB3r/content/2301.08609v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFAT4oBgHgl3EQfkB3r/content/2301.08609v1.pdf'}
+page_content=' times which are beyond what is possible classically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFAT4oBgHgl3EQfkB3r/content/2301.08609v1.pdf'}
+page_content='  Rz(θ)  Rz( π  2 )  Rz(− π  2 )  Ry(φ)  Ry(λ)  Figure 2: Implementation of two site operator ei(ασx⊗σx+βσy⊗σy+γσz⊗σz) as a quantum circuit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFAT4oBgHgl3EQfkB3r/content/2301.08609v1.pdf'}
+page_content=' We have  the correspondences θ = π  2 − 2γ, φ = 2α − π  2 and λ = π  2 − 2β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFAT4oBgHgl3EQfkB3r/content/2301.08609v1.pdf'}
+page_content=' The Hamiltonian in (1) corresponds to  the case α = β = γ = dt  state |ψ(t)⟩ is governed by the Schr¨odinger equation:  |ψ(t)⟩ = e−iHXXXt |ψ(0)⟩  (2)  where |ψ(0)⟩ is the wavefunction at time t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFAT4oBgHgl3EQfkB3r/content/2301.08609v1.pdf'}
+page_content=' In this work, we will consider the N´eel state, written as:  |↑↓↑↓ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFAT4oBgHgl3EQfkB3r/content/2301.08609v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFAT4oBgHgl3EQfkB3r/content/2301.08609v1.pdf'}
+page_content=' ↑↓⟩ where ↑ and ↓ represent up and down spins respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFAT4oBgHgl3EQfkB3r/content/2301.08609v1.pdf'}
+page_content=' The N´eel state for n spins is simply  implemented on n qubits as |1010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFAT4oBgHgl3EQfkB3r/content/2301.08609v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFAT4oBgHgl3EQfkB3r/content/2301.08609v1.pdf'}
+page_content='10⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFAT4oBgHgl3EQfkB3r/content/2301.08609v1.pdf'}
+page_content='  The time evolution operator U(t) ≡ e−iHt can be executed as a quantum circuit in a resource efficient  way;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFAT4oBgHgl3EQfkB3r/content/2301.08609v1.pdf'}
+page_content=' we first write the Hamiltonian in (1) as HXXX = H1 + H2 where H1 = − �  i odd  hi,i+1 and H2 =  − �  i even  hi,i+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFAT4oBgHgl3EQfkB3r/content/2301.08609v1.pdf'}
+page_content=' Note that all operators in a given sum commute with all other operators in their respective  sums.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFAT4oBgHgl3EQfkB3r/content/2301.08609v1.pdf'}
+page_content=' We then define the Suzuki-Trotter time evolution operator Utrot(dt) in the following way:  U(1)  trot(dt) =  L/2−1  �  j=0  U2j,2j+1(dt)  L/2−1  �  j=1  U2j−1,2j(dt) = e−iHXXZdt + O(dt2)  (3)  where Ujk(dt) = e−ihjkdt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFAT4oBgHgl3EQfkB3r/content/2301.08609v1.pdf'}
+page_content=' The exact time evolution operator U(t) is thus approximated by m repeated  applications of Utrot(dt =  t  m), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFAT4oBgHgl3EQfkB3r/content/2301.08609v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFAT4oBgHgl3EQfkB3r/content/2301.08609v1.pdf'}
+page_content=' U(t) ≈ Um  trot(dt =  t  m).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFAT4oBgHgl3EQfkB3r/content/2301.08609v1.pdf'}
+page_content=' As discussed in [16], each Ujk(dt) appearing in  (3) can be implemented by the quantum circuit with just three CNOTs as in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFAT4oBgHgl3EQfkB3r/content/2301.08609v1.pdf'}
+page_content=' We can reduce  the error in the Trotter formula in equation (3) by using higher order expressions [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFAT4oBgHgl3EQfkB3r/content/2301.08609v1.pdf'}
+page_content=' It turns out that  the second order Trotter formula can be implemented on a quantum circuit with only one extra layer in  the circuit [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFAT4oBgHgl3EQfkB3r/content/2301.08609v1.pdf'}
+page_content=' We have:  U(2)  trot(dt) =  L/2−1  �  j=0  U2j,2j+1  �dt  2  � L/2−1  �  j=1  U2j−1,2j (dt)  L/2−1  �  j=0  U2j,2j+1  �dt  2  �  = e−iHXXZdt + O(dt2)  (4)  which can be implemented on a quantum device by the circuit in Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFAT4oBgHgl3EQfkB3r/content/2301.08609v1.pdf'}
+page_content='  3  U(dt)  U(dt)  U(dt)  U(dt)  U(dt)  U(dt)  U(dt)  U(dt)  U(dt)  U(dt)  U(dt)  U(dt)  U(dt)  U(dt)  U(dt)  Figure 3: First order Trotter circuit acting on six qubits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFAT4oBgHgl3EQfkB3r/content/2301.08609v1.pdf'}
+page_content='  U( dt  2 )  U(dt)  U(dt)  U( dt  2 )  U(dt)  U(dt)  U(dt)  U( dt  2 )  U(dt)  U(dt)  U( dt  2 )  U(dt)  U(dt)  U(dt)  U( dt  2 )  U(dt)  U(dt)  U( dt  2 )  Figure 4: Second order Trotter circuit acting on six qubits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFAT4oBgHgl3EQfkB3r/content/2301.08609v1.pdf'}
+page_content='  4  A(1)  A(2)  A(3)  A(4)  A(5)  A(6)  Figure 5: Graphical representation of an MPS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFAT4oBgHgl3EQfkB3r/content/2301.08609v1.pdf'}
+page_content=' There are two matrices A(i) for each qubit at position i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFAT4oBgHgl3EQfkB3r/content/2301.08609v1.pdf'}
+page_content='  A(1)  A(2)  A(3)  A(4)  A(5)  A(6)  B(1)  B(2)  B(3)  B(4)  B(5)  B(6)  Figure 6: The inner product ⟨ψ1|ψ2⟩ of two Matrix Product States - see equations (11) and (6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFAT4oBgHgl3EQfkB3r/content/2301.08609v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFAT4oBgHgl3EQfkB3r/content/2301.08609v1.pdf'}
+page_content='2  Matrix Product States  An arbitrary quantum state on n qubits can be written in terms of complex variables cj1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFAT4oBgHgl3EQfkB3r/content/2301.08609v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFAT4oBgHgl3EQfkB3r/content/2301.08609v1.pdf'}
+page_content=',jn, the number  of which scales as 2n:  |ψ⟩ =  �  {j1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFAT4oBgHgl3EQfkB3r/content/2301.08609v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFAT4oBgHgl3EQfkB3r/content/2301.08609v1.pdf'}
+page_content=',jn}  cj1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFAT4oBgHgl3EQfkB3r/content/2301.08609v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFAT4oBgHgl3EQfkB3r/content/2301.08609v1.pdf'}
+page_content=',jn |j1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFAT4oBgHgl3EQfkB3r/content/2301.08609v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFAT4oBgHgl3EQfkB3r/content/2301.08609v1.pdf'}
+page_content=', jn⟩  (5)  where the sum is over all configurations of the binary variables j1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFAT4oBgHgl3EQfkB3r/content/2301.08609v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFAT4oBgHgl3EQfkB3r/content/2301.08609v1.pdf'}
+page_content=', jn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFAT4oBgHgl3EQfkB3r/content/2301.08609v1.pdf'}
+page_content=' The bipartite entanglement  entropy of an arbitrary quantum state picked at random from Hilbert space satisfies a volume law which,  as was discussed in the introduction, is distinct from area law entanglement in which case the entanglement  entropy of two regions after the bipartition of the system is proportional to the area of the boundary of  the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFAT4oBgHgl3EQfkB3r/content/2301.08609v1.pdf'}
+page_content=' A small subset of states in Hilbert space satisfies an area law.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFAT4oBgHgl3EQfkB3r/content/2301.08609v1.pdf'}
+page_content=' The coefficients cj1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFAT4oBgHgl3EQfkB3r/content/2301.08609v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFAT4oBgHgl3EQfkB3r/content/2301.08609v1.pdf'}
+page_content=',jn of  such states have a certain structure that we can take advantage of to study classically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFAT4oBgHgl3EQfkB3r/content/2301.08609v1.pdf'}
+page_content=' Any state |ψ⟩ can  be written in the following way:  cj1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFAT4oBgHgl3EQfkB3r/content/2301.08609v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFAT4oBgHgl3EQfkB3r/content/2301.08609v1.pdf'}
+page_content=',jn = A(1)  j1 · A(2)  j2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFAT4oBgHgl3EQfkB3r/content/2301.08609v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFAT4oBgHgl3EQfkB3r/content/2301.08609v1.pdf'}
+page_content=' · A(n)  jn  (6)  where the Aj are χj × χj+1 dimensional matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFAT4oBgHgl3EQfkB3r/content/2301.08609v1.pdf'}
+page_content=' Quantum states of the form (6) are known as Matrix  Product States (MPS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFAT4oBgHgl3EQfkB3r/content/2301.08609v1.pdf'}
+page_content=' The maximum value of χj is referred to as the bond dimension of the MPS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFAT4oBgHgl3EQfkB3r/content/2301.08609v1.pdf'}
+page_content=' We  can represent an MPS graphically as in Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFAT4oBgHgl3EQfkB3r/content/2301.08609v1.pdf'}
+page_content=' We associate one matrix A(i) to each qubit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFAT4oBgHgl3EQfkB3r/content/2301.08609v1.pdf'}
+page_content=' Note  that for each qubit i we have two matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFAT4oBgHgl3EQfkB3r/content/2301.08609v1.pdf'}
+page_content=' We thus have a total of 2n matrices to keep track of.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFAT4oBgHgl3EQfkB3r/content/2301.08609v1.pdf'}
+page_content=' The  bond dimension χj can be seen as a measure of the entanglement between the two subsystems when a  bipartition is made at qubit j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFAT4oBgHgl3EQfkB3r/content/2301.08609v1.pdf'}
+page_content=' Therefore, states in Hilbert space that satisfy an area law - and therefore  have a low bond dimension in their MPS representation - can be efficiently stored as Matrix Product  States.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFAT4oBgHgl3EQfkB3r/content/2301.08609v1.pdf'}
+page_content=' States that satisfy a volume law will have a bond dimension that is exponential in the number of  qubits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFAT4oBgHgl3EQfkB3r/content/2301.08609v1.pdf'}
+page_content=' We will consider in this work the non-trivial dynamics governed by equation (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFAT4oBgHgl3EQfkB3r/content/2301.08609v1.pdf'}
+page_content=' As discussed in  the introduction, the bipartite entanglement entropy of a ground state of a one-dimensional Hamiltonian  that has a gap between its ground state and its excited state is independent of the size of the subsystems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFAT4oBgHgl3EQfkB3r/content/2301.08609v1.pdf'}
+page_content='  The ground state of such a system - and hence the initial state in our setup - can be efficiently stored  as an MPS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFAT4oBgHgl3EQfkB3r/content/2301.08609v1.pdf'}
+page_content=' One can then use an algorithm such as TEBD (Time Evolving Block Decimation) [18] to  update the MPS as a function of time to study the dynamics of the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFAT4oBgHgl3EQfkB3r/content/2301.08609v1.pdf'}
+page_content=' However, the entanglement  entropy of the state increases linearly with time, hence the bond dimension χ that is required to keep  the error constant diverges exponentially with time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFAT4oBgHgl3EQfkB3r/content/2301.08609v1.pdf'}
+page_content=' To simulate for longer times, a quantum computer  would be needed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFAT4oBgHgl3EQfkB3r/content/2301.08609v1.pdf'}
+page_content=' In section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFAT4oBgHgl3EQfkB3r/content/2301.08609v1.pdf'}
+page_content='3, we will discuss how Matrix Product States can be leveraged to reduce  the resource requirements for this simulation problem when implemented on a quantum device.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFAT4oBgHgl3EQfkB3r/content/2301.08609v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFAT4oBgHgl3EQfkB3r/content/2301.08609v1.pdf'}
+page_content='3  Matrix Product States applied to Approximate Quantum Compiling  j  k  Ry(θ1)  Rz(θ2)  Ry(θ3)  Rx(θ4)  Figure 7: CNOT block forms the basic building block of our circuit ansatz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFAT4oBgHgl3EQfkB3r/content/2301.08609v1.pdf'}
+page_content='  5  Rz(θ1)  Ry(θ2)  Rz(θ3)  Rz(θ4)  Ry(θ5)  Rz(θ6)  Rz(θ7)  Ry(θ8)  Rz(θ9)  Rz(θ10)  Ry(θ11)  Rz(θ12)  Figure 8: Parameterised circuit inspired by the structure of the first order Trotter circuit in Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFAT4oBgHgl3EQfkB3r/content/2301.08609v1.pdf'}
+page_content='  Rz(θ1)  Ry(θ2)  Rz(θ3)  Rz(θ4)  Ry(θ5)  Rz(θ6)  Rz(θ7)  Ry(θ8)  Rz(θ9)  Rz(θ10)  Ry(θ11)  Rz(θ12)  Figure 9: Parameterised circuit inspired by the structure of the second order Trotter circuit in Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFAT4oBgHgl3EQfkB3r/content/2301.08609v1.pdf'}
+page_content='  Approximate quantum compiling (AQC) involves the design of a parametric quantum circuit with  fixed depth - the parameters are then adjusted to bring it as close as possible to the target, where “close”  is defined via some carefully chosen metric, see below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFAT4oBgHgl3EQfkB3r/content/2301.08609v1.pdf'}
+page_content=' As discussed in [12], one can use so-called CNOT  blocks to construct a natural circuit Ansatz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFAT4oBgHgl3EQfkB3r/content/2301.08609v1.pdf'}
+page_content=' A CNOT block is a CNOT gate followed by single qubit  rotations (see Figure 7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFAT4oBgHgl3EQfkB3r/content/2301.08609v1.pdf'}
+page_content=' A block with a CNOT gate acting on a “control” qubit j and “target” qubit  k is written as CUjk(θ1, θ2, θ3, θ4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFAT4oBgHgl3EQfkB3r/content/2301.08609v1.pdf'}
+page_content=' For a given hardware connectivity, one can then write down a fully  parameterised circuit as:  Vct(θ) =CUct(L) (θ3n+4L−3, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFAT4oBgHgl3EQfkB3r/content/2301.08609v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFAT4oBgHgl3EQfkB3r/content/2301.08609v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFAT4oBgHgl3EQfkB3r/content/2301.08609v1.pdf'}
+page_content=' , θ3n+4L) · · · CUct(1) (θ3n+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFAT4oBgHgl3EQfkB3r/content/2301.08609v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFAT4oBgHgl3EQfkB3r/content/2301.08609v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFAT4oBgHgl3EQfkB3r/content/2301.08609v1.pdf'}
+page_content=' , θ3n+4)  [Rz (θ1) Ry (θ2) Rz (θ3)] ⊗ · · · ⊗ [Rz (θ3n−2) Ry (θ3n−1) Rz (θ3n)]  (7)  The position of the CNOT blocks in the parameterised circuit can be customised to suit the particular  target circuit that one is interested in.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFAT4oBgHgl3EQfkB3r/content/2301.08609v1.pdf'}
+page_content=' Here we are interested in finding a circuit that implements the  unitary time evolution operator as in equation (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFAT4oBgHgl3EQfkB3r/content/2301.08609v1.pdf'}
+page_content=' We thus consider a structure inspired by the first and  second-order Trotter circuits in Figures 3 and 4 respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFAT4oBgHgl3EQfkB3r/content/2301.08609v1.pdf'}
+page_content=' Recall that each block U(dt) in Figures 3  and 4 represents the 2-qubit sub-circuit with three CNOTs in Figure 2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFAT4oBgHgl3EQfkB3r/content/2301.08609v1.pdf'}
+page_content=' it is therefore natural to consider  a circuit Ansatz with sub-circuits each with three CNOT blocks as in Figures 8 and 9, such that the  circuit Ansatz mimics the structure of the first and second order Trotter circuits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFAT4oBgHgl3EQfkB3r/content/2301.08609v1.pdf'}
+page_content=' In the notation of [14],  the parameterised circuits in Figures 8 and 9 correspond to n = 4 qubits, l = 2 layers and b = 3 CNOT  blocks in each layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFAT4oBgHgl3EQfkB3r/content/2301.08609v1.pdf'}
+page_content=' In both Figure 8 and Figure 9 there are three rotation gates acting on each qubit at  the beginning of the circuit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFAT4oBgHgl3EQfkB3r/content/2301.08609v1.pdf'}
+page_content=' In the examples considered in this work we will take the initial state to be |0⟩  the initial rotation gate Rz(θ) is redundant for these cases but is necessary for more general initial states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFAT4oBgHgl3EQfkB3r/content/2301.08609v1.pdf'}
+page_content='  One can define the distance between the target and parameterised circuit via a number of different  metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFAT4oBgHgl3EQfkB3r/content/2301.08609v1.pdf'}
+page_content=' Here we use a cost function based on the Hilbert-Schmidt test:  Cstate  hs  = 1 − | ⟨0| V †(θ) |ψ0⟩ |2  (8)  The goal of AQC is to tune the parameters θ to minimise the cost function under consideration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFAT4oBgHgl3EQfkB3r/content/2301.08609v1.pdf'}
+page_content=' Note  that here we are considering the application of AQC to state preparation as opposed to full circuit com-  pilation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFAT4oBgHgl3EQfkB3r/content/2301.08609v1.pdf'}
+page_content=' More precisely, this means that our cost function is designed such that it is minimised when the  action of V (θ) on the initial state |0⟩ produces a state that is as close as possible to a target state |ψ0⟩  (up to some global phase).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFAT4oBgHgl3EQfkB3r/content/2301.08609v1.pdf'}
+page_content=' This is in contrast to the situation where one starts with some target circuit  U and the cost function is designed to bring the full matrix V (θ) as close as possible to U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFAT4oBgHgl3EQfkB3r/content/2301.08609v1.pdf'}
+page_content='  6  As pointed out in [19], the gradient of the cost function in (8) vanishes exponentially.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFAT4oBgHgl3EQfkB3r/content/2301.08609v1.pdf'}
+page_content=' This observation  lead to the distinction between global and local cost functions;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFAT4oBgHgl3EQfkB3r/content/2301.08609v1.pdf'}
+page_content=' local cost functions have only polynomially  vanishing gradients in some cases of interest - see [19, 20, 14] for details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFAT4oBgHgl3EQfkB3r/content/2301.08609v1.pdf'}
+page_content=' As was shown in [14], the Hilbert-  Schmidt test - which is a global cost function - can be turned into a local one by adding several “bit-flip��  terms which increases the magnitude of the gradient:  Cstate  lhs  = 1−| ⟨0| V †(θ) |ψ0⟩ |2 −  �n − 1  n  �  n  �  j=1  | ⟨0| XjV †(θ) |ψ0⟩ |2  −  �n − 2  n  � �  j<k  | ⟨0| XjXkV †(θ) |ψ0⟩ |2 − .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFAT4oBgHgl3EQfkB3r/content/2301.08609v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFAT4oBgHgl3EQfkB3r/content/2301.08609v1.pdf'}
+page_content=' − 1  n  �  j<k<l<.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFAT4oBgHgl3EQfkB3r/content/2301.08609v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFAT4oBgHgl3EQfkB3r/content/2301.08609v1.pdf'}
+page_content='  | ⟨0| XjXkXl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFAT4oBgHgl3EQfkB3r/content/2301.08609v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFAT4oBgHgl3EQfkB3r/content/2301.08609v1.pdf'}
+page_content='V †(θ) |ψ0⟩ |2  (9)  Convergence of the cost function can be significantly improved by adding these terms, however the  computational cost of calculating the gradient becomes prohibitive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFAT4oBgHgl3EQfkB3r/content/2301.08609v1.pdf'}
+page_content=' It was demonstrated in [14] that this  can be overcome by truncating the expression in (9) to get:  C(1)  L (α) = 1 − | ⟨0| V †(θ) |ψ0⟩ |2 − α  n  �  j=1  | ⟨0| XjV †(θ) |ψ0⟩ |2  (10)  where α is a parameter that can be tuned throughout the optimisation procedure - a scheme to implement  this tuning effectively was demonstrated in [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFAT4oBgHgl3EQfkB3r/content/2301.08609v1.pdf'}
+page_content=' In (10), we have only kept 1 “bit-flip” term, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFAT4oBgHgl3EQfkB3r/content/2301.08609v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFAT4oBgHgl3EQfkB3r/content/2301.08609v1.pdf'}
+page_content=' we have  dropped all terms with more than one NOT operators Xi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFAT4oBgHgl3EQfkB3r/content/2301.08609v1.pdf'}
+page_content=' As discussed in [14], one can obtain higher  order expressions C(k)  L  with more “bit-flip” terms included - doing so induces a larger gradient in the cost  function but increases the computational burden.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFAT4oBgHgl3EQfkB3r/content/2301.08609v1.pdf'}
+page_content='  Note that each term in (9) or (10) is an overlap of quantum states, and since the overlap of two MPS  can be calculated very efficiently the architecture of Matrix Product States can be leveraged to calculate  the cost function and solve the approximate quantum compilation problem for large numbers of qubits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFAT4oBgHgl3EQfkB3r/content/2301.08609v1.pdf'}
+page_content='  Consider for example two quantum states |ψ1⟩ and |ψ2⟩:  |ψ1⟩ =  �  {j1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFAT4oBgHgl3EQfkB3r/content/2301.08609v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFAT4oBgHgl3EQfkB3r/content/2301.08609v1.pdf'}
+page_content=',jn}  c(1)  j1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFAT4oBgHgl3EQfkB3r/content/2301.08609v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFAT4oBgHgl3EQfkB3r/content/2301.08609v1.pdf'}
+page_content=',jn |j1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFAT4oBgHgl3EQfkB3r/content/2301.08609v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFAT4oBgHgl3EQfkB3r/content/2301.08609v1.pdf'}
+page_content=', jn⟩  |ψ2⟩ =  �  {j1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFAT4oBgHgl3EQfkB3r/content/2301.08609v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFAT4oBgHgl3EQfkB3r/content/2301.08609v1.pdf'}
+page_content=',jn}  c(2)  j1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFAT4oBgHgl3EQfkB3r/content/2301.08609v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFAT4oBgHgl3EQfkB3r/content/2301.08609v1.pdf'}
+page_content=',jn |j1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFAT4oBgHgl3EQfkB3r/content/2301.08609v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFAT4oBgHgl3EQfkB3r/content/2301.08609v1.pdf'}
+page_content=', jn⟩  (11)  As discussed in section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFAT4oBgHgl3EQfkB3r/content/2301.08609v1.pdf'}
+page_content='2, for weakly entangled states the coefficients c(1)  j1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFAT4oBgHgl3EQfkB3r/content/2301.08609v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFAT4oBgHgl3EQfkB3r/content/2301.08609v1.pdf'}
+page_content=',jn and c(2)  j1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFAT4oBgHgl3EQfkB3r/content/2301.08609v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFAT4oBgHgl3EQfkB3r/content/2301.08609v1.pdf'}
+page_content=',jn are not all  independent and can be represented efficiently as Matrix Product States - see Figure 5:  c(1)  j1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFAT4oBgHgl3EQfkB3r/content/2301.08609v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFAT4oBgHgl3EQfkB3r/content/2301.08609v1.pdf'}
+page_content=',jn = A(1)  j1 · A(2)  j2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFAT4oBgHgl3EQfkB3r/content/2301.08609v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFAT4oBgHgl3EQfkB3r/content/2301.08609v1.pdf'}
+page_content=' · A(n)  jn  c(2)  j1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFAT4oBgHgl3EQfkB3r/content/2301.08609v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFAT4oBgHgl3EQfkB3r/content/2301.08609v1.pdf'}
+page_content=',jn = B(1)  j1 · B(2)  j2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFAT4oBgHgl3EQfkB3r/content/2301.08609v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFAT4oBgHgl3EQfkB3r/content/2301.08609v1.pdf'}
+page_content=' · B(n)  jn  (12)  We want to calculate the quantity:  f(|ψ1⟩ , |ψ2⟩) = || ⟨ψ1|ψ2⟩ ||2  (13)  The overlap of two MPS, and hence the fidelity f in (13) can be calculated efficiently by “contracting”  the Tensor Network shown in Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFAT4oBgHgl3EQfkB3r/content/2301.08609v1.pdf'}
+page_content='  3  Results  We now present the results of our simulations of the Schr¨odinger equation in equation (2) using the  second order Trotter formula in equation (4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFAT4oBgHgl3EQfkB3r/content/2301.08609v1.pdf'}
+page_content=' First let’s define and clarify some notation:  |a1⟩: The state generated by the optimised parametric circuit in Figure 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFAT4oBgHgl3EQfkB3r/content/2301.08609v1.pdf'}
+page_content='  Number of layers l in Ansatz: in Figures 8 and 9 there are l = 2 layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFAT4oBgHgl3EQfkB3r/content/2301.08609v1.pdf'}
+page_content='  |t1⟩: The state generated by the Trotter circuit in Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFAT4oBgHgl3EQfkB3r/content/2301.08609v1.pdf'}
+page_content='  7  Figure 10: 50 qubits: fidelities of the parametric circuit and the Trotter circuit with the ”ground truth”  obtained by Tensor Network simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFAT4oBgHgl3EQfkB3r/content/2301.08609v1.pdf'}
+page_content=' The two circuits are of identical length but the parametric  circuit achieves a significantly higher fidelity at late times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFAT4oBgHgl3EQfkB3r/content/2301.08609v1.pdf'}
+page_content='  Figure 11: 50 qubits: the maximum number of layers in the parametric circuit is 12 while it is 18 for the  Trotter circuit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFAT4oBgHgl3EQfkB3r/content/2301.08609v1.pdf'}
+page_content=' Despite this, the parametric circuit achieves a higher fidelity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFAT4oBgHgl3EQfkB3r/content/2301.08609v1.pdf'}
+page_content='  Number of Trotter steps: the analogue of the number of layers in the Ansatz circuits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFAT4oBgHgl3EQfkB3r/content/2301.08609v1.pdf'}
+page_content=' In Figure 3  and 4 there are 3 Trotter steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFAT4oBgHgl3EQfkB3r/content/2301.08609v1.pdf'}
+page_content='  |t1 gt⟩: the “ground truth” generated by classical Tensor Network simulations of deep Trotter  circuits, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFAT4oBgHgl3EQfkB3r/content/2301.08609v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFAT4oBgHgl3EQfkB3r/content/2301.08609v1.pdf'}
+page_content=' extremely small time steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFAT4oBgHgl3EQfkB3r/content/2301.08609v1.pdf'}
+page_content=' We take dt = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFAT4oBgHgl3EQfkB3r/content/2301.08609v1.pdf'}
+page_content='04 to generate the ground truth state  while |t1⟩ is generated with a time step of dt = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFAT4oBgHgl3EQfkB3r/content/2301.08609v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFAT4oBgHgl3EQfkB3r/content/2301.08609v1.pdf'}
+page_content='  All circuits considered here take the form of the second-order Trotter structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFAT4oBgHgl3EQfkB3r/content/2301.08609v1.pdf'}
+page_content=' More precisely, in each  graph we use the labels “Trotter” and “Ansatz” and these circuits have the structure in Figures 4 and 9  respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFAT4oBgHgl3EQfkB3r/content/2301.08609v1.pdf'}
+page_content=' In Figures 10, 11 and 12 we plot fidelity vs evolution time for the 50 qubit XXX Hamiltonian  and we compare the result of the Trotter circuit vs the AQC circuit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFAT4oBgHgl3EQfkB3r/content/2301.08609v1.pdf'}
+page_content=' In Figure 10 we can see that the  fidelity of the Trotter circuit decays rapidly while the fidelity of the AQC circuit remains above 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFAT4oBgHgl3EQfkB3r/content/2301.08609v1.pdf'}
+page_content='99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFAT4oBgHgl3EQfkB3r/content/2301.08609v1.pdf'}
+page_content='  Note that the Trotter circuit and the AQC circuit are of equal depth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFAT4oBgHgl3EQfkB3r/content/2301.08609v1.pdf'}
+page_content=' In Figures 11 and 12 we compare  short depth circuits generated by AQC with deep Trotter circuits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFAT4oBgHgl3EQfkB3r/content/2301.08609v1.pdf'}
+page_content=' We observe that the AQC circuits  can achieve comparable or better fidelities with much shorter depth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFAT4oBgHgl3EQfkB3r/content/2301.08609v1.pdf'}
+page_content=' In particular, in Figure 12 the two  fidelities are almost identical but the AQC circuit is half the depth of the Trotter circuit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFAT4oBgHgl3EQfkB3r/content/2301.08609v1.pdf'}
+page_content=' We plot the  8  Fidelity: lal> vs Itl_gt> and Itl> vs Itl gt>, num.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFAT4oBgHgl3EQfkB3r/content/2301.08609v1.pdf'}
+page_content='qubits: 50  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFAT4oBgHgl3EQfkB3r/content/2301.08609v1.pdf'}
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+page_content='96  fidelity  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFAT4oBgHgl3EQfkB3r/content/2301.08609v1.pdf'}
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+page_content='92  Ansatz  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFAT4oBgHgl3EQfkB3r/content/2301.08609v1.pdf'}
+page_content='91  Trotter  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFAT4oBgHgl3EQfkB3r/content/2301.08609v1.pdf'}
+page_content='90  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFAT4oBgHgl3EQfkB3r/content/2301.08609v1.pdf'}
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+page_content='6  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFAT4oBgHgl3EQfkB3r/content/2301.08609v1.pdf'}
+page_content='8  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFAT4oBgHgl3EQfkB3r/content/2301.08609v1.pdf'}
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+page_content='2  evolution time  13  ６  9  12  15  18  number of layers in ansatz  13  ６  ９  12  15  18  number of Trotter stepsFidelity: lal> vs Itl_gt> and Itl> vs Itl gt>, num.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFAT4oBgHgl3EQfkB3r/content/2301.08609v1.pdf'}
+page_content='qubits: 50  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFAT4oBgHgl3EQfkB3r/content/2301.08609v1.pdf'}
+page_content='00  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFAT4oBgHgl3EQfkB3r/content/2301.08609v1.pdf'}
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+page_content='96  fidel  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFAT4oBgHgl3EQfkB3r/content/2301.08609v1.pdf'}
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+page_content='91  Ansatz  Trotter  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFAT4oBgHgl3EQfkB3r/content/2301.08609v1.pdf'}
+page_content='90  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFAT4oBgHgl3EQfkB3r/content/2301.08609v1.pdf'}
+page_content='2  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFAT4oBgHgl3EQfkB3r/content/2301.08609v1.pdf'}
+page_content='4  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFAT4oBgHgl3EQfkB3r/content/2301.08609v1.pdf'}
+page_content='6  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFAT4oBgHgl3EQfkB3r/content/2301.08609v1.pdf'}
+page_content='8  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFAT4oBgHgl3EQfkB3r/content/2301.08609v1.pdf'}
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+page_content='2  evolution time  2  12  4  6  8  10  number of layers in ansatz  13  ６  9  12  15  18  number of Trotter stepsFigure 12: 50 qubits: the maximum number of layers in the parametric circuit is 9 while it is 18 for the  Trotter circuit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFAT4oBgHgl3EQfkB3r/content/2301.08609v1.pdf'}
+page_content=' Both circuits achieve very similar fidelities despite the parametric circuit being half the  depth of the Trotter circuit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFAT4oBgHgl3EQfkB3r/content/2301.08609v1.pdf'}
+page_content='  Figure 13: 100 qubits: The maximum number of layers in the parametric circuit and the Trotter circuit  is 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFAT4oBgHgl3EQfkB3r/content/2301.08609v1.pdf'}
+page_content=' It can be seen that the fidelity of the Trotter circuit decays rapidly but that of the parametric  circuit remains high.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFAT4oBgHgl3EQfkB3r/content/2301.08609v1.pdf'}
+page_content='  same data for 100 qubits in Figures 13 and 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFAT4oBgHgl3EQfkB3r/content/2301.08609v1.pdf'}
+page_content='  Now we would like to consider how these results affect the implementation on a real quantum device.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFAT4oBgHgl3EQfkB3r/content/2301.08609v1.pdf'}
+page_content='  We consider a 20 qubit spin-chain on the 27 qubit device ibmq-mumbai.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFAT4oBgHgl3EQfkB3r/content/2301.08609v1.pdf'}
+page_content=' First we plot the fidelity results  for 20 qubits in Figure 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFAT4oBgHgl3EQfkB3r/content/2301.08609v1.pdf'}
+page_content=' We ran the resulting parametric circuit on ibmq-mumbai and in Figure 16  we plot the expectation values ⟨ψ(t)|Sz  0|ψ(t)⟩ as obtained from the quantum device using the parametric  AQC circuit, the Trotter circuit and from a classical Tensor Network simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFAT4oBgHgl3EQfkB3r/content/2301.08609v1.pdf'}
+page_content=' We observe that this  observable is more accurate when obtained with the AQC circuit due to its reduced depth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFAT4oBgHgl3EQfkB3r/content/2301.08609v1.pdf'}
+page_content=' Note that  the difference between the results from the simulation and the results from the quantum device would be  greatly reduced after applying error mitigation [21, 22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFAT4oBgHgl3EQfkB3r/content/2301.08609v1.pdf'}
+page_content=' We have not attempted to apply error mitigation  to either circuit as this would be outside the scope of this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFAT4oBgHgl3EQfkB3r/content/2301.08609v1.pdf'}
+page_content='  We expect that, since our Tensor  Network compilation scheme greatly reduces the noise of the circuit, any error mitigation scheme would  be enhanced by our approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFAT4oBgHgl3EQfkB3r/content/2301.08609v1.pdf'}
+page_content='  9  Fidelity: lal> vs |tl_gt> and Itl> vs Itl_gt>, num.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFAT4oBgHgl3EQfkB3r/content/2301.08609v1.pdf'}
+page_content='qubits: 50  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFAT4oBgHgl3EQfkB3r/content/2301.08609v1.pdf'}
+page_content='00  Ansatz  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFAT4oBgHgl3EQfkB3r/content/2301.08609v1.pdf'}
+page_content='99  Trotter  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFAT4oBgHgl3EQfkB3r/content/2301.08609v1.pdf'}
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+page_content='96  fideli  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFAT4oBgHgl3EQfkB3r/content/2301.08609v1.pdf'}
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+page_content='2  evolution time  2  4  8  ９  6  number of layers in ansatz  13  ６  9  12  15  18  number of Trotter stepsFidelity: lai) and Iti> vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFAT4oBgHgl3EQfkB3r/content/2301.08609v1.pdf'}
+page_content=' ground-truth state, Ngqubits: 1o0  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFAT4oBgHgl3EQfkB3r/content/2301.08609v1.pdf'}
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+page_content='91  Ansatz  Trotter  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFAT4oBgHgl3EQfkB3r/content/2301.08609v1.pdf'}
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+page_content='2  evolution time  13  ６  9  12  15  18  number of layers in ansatz  13  6  ９  12  15  18  number of Trotter stepsFigure 14: 100 qubits: The maximum number of layers in the parametric circuit is 12 while it is 18 for  the Trotter circuit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFAT4oBgHgl3EQfkB3r/content/2301.08609v1.pdf'}
+page_content='  Figure 15: The maximum depth of the parametric circuit is half that of the Trotter circuit - there are 9  and 18 layers respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFAT4oBgHgl3EQfkB3r/content/2301.08609v1.pdf'}
+page_content=' These 20 qubit circuits were implemented on ibmq-mumbai - see Figure 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFAT4oBgHgl3EQfkB3r/content/2301.08609v1.pdf'}
+page_content='  4  Discussion  In this paper we applied Tensor Network methods to Quantum Compiling and demonstrated their efficacy  on the 27 qubit device ibmq-mumbai.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFAT4oBgHgl3EQfkB3r/content/2301.08609v1.pdf'}
+page_content=' Our method is similar in spirit to [23] where Matrix Product States  were used to prepare the initial state for VQE to find the ground state of some Hamiltonian - here we  use Matrix Product States to prepare a short depth quantum circuit that simulates the time evolution of  a 1D Hamiltonian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFAT4oBgHgl3EQfkB3r/content/2301.08609v1.pdf'}
+page_content=' We chose the XXX Hamiltonian in equation (1) because it has been well studied, but  we would be particularly interested to apply the compilation methods developed here to non-integrable  systems by e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFAT4oBgHgl3EQfkB3r/content/2301.08609v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFAT4oBgHgl3EQfkB3r/content/2301.08609v1.pdf'}
+page_content=' adding a random field to the Hamiltonian in (1) and studying phenomena of scientific  interest such as many-body localisation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFAT4oBgHgl3EQfkB3r/content/2301.08609v1.pdf'}
+page_content='  We have shown results of our simulations on up to 100 qubits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFAT4oBgHgl3EQfkB3r/content/2301.08609v1.pdf'}
+page_content=' In principle we can significantly increase  the number of qubits and the length of time to which we apply our MPS compilation scheme;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFAT4oBgHgl3EQfkB3r/content/2301.08609v1.pdf'}
+page_content=' the limiting  factor at present seems to be the particular implementation that we apply to SVD and to calculate the  gradient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFAT4oBgHgl3EQfkB3r/content/2301.08609v1.pdf'}
+page_content=' We believe that both of these can be improved significantly, in particular by using an efficient  parallel implementation - this is the subject of ongoing work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFAT4oBgHgl3EQfkB3r/content/2301.08609v1.pdf'}
+page_content=' In our current framework we use the Qiskit  10  Fidelity: lai) and Iti> vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFAT4oBgHgl3EQfkB3r/content/2301.08609v1.pdf'}
+page_content=' ground-truth state, Ngqubits: 1o0  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFAT4oBgHgl3EQfkB3r/content/2301.08609v1.pdf'}
+page_content='00  Ansatz  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFAT4oBgHgl3EQfkB3r/content/2301.08609v1.pdf'}
+page_content='99  Trotter  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFAT4oBgHgl3EQfkB3r/content/2301.08609v1.pdf'}
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+page_content='95  p!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFAT4oBgHgl3EQfkB3r/content/2301.08609v1.pdf'}
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+page_content='2  evolution time  2  12  4  6  8  10  number of layers in ansatz  13  ６  9  12  15  18  number of Trotter stepsFidelity: lai> and Iti> vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFAT4oBgHgl3EQfkB3r/content/2301.08609v1.pdf'}
+page_content=' ground-truth state, Nqubits: 20  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFAT4oBgHgl3EQfkB3r/content/2301.08609v1.pdf'}
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+page_content='955  Trotter  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFAT4oBgHgl3EQfkB3r/content/2301.08609v1.pdf'}
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+page_content='4  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFAT4oBgHgl3EQfkB3r/content/2301.08609v1.pdf'}
+page_content='6  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFAT4oBgHgl3EQfkB3r/content/2301.08609v1.pdf'}
+page_content='8  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFAT4oBgHgl3EQfkB3r/content/2301.08609v1.pdf'}
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+page_content='2  evolution time  2  4  ９  6  7  8  number of layers in ansatz  13  ６  ９  12  15  18  number of Trotter stepsFigure 16: The expectation value of Sz  0 vs time for a chain of 20 qubits as measured on the 27 qubit  quantum device ibmq-mumbai.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFAT4oBgHgl3EQfkB3r/content/2301.08609v1.pdf'}
+page_content='  The circuit produced from our MPS implementation of AQC uses is  shallower than the Trotter circuit, and thus produces an expectation value that is much closer to the true  value plotted in the blue curve, obtained by classical Tensor Network simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFAT4oBgHgl3EQfkB3r/content/2301.08609v1.pdf'}
+page_content='  MPS package which is designed for generic situations in which long range connectivity may be required  and thus does not take advantage of the short range structure of the circuits in Figures 8 and 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFAT4oBgHgl3EQfkB3r/content/2301.08609v1.pdf'}
+page_content='  5  Acknowledgements  This work was funded by the Disruptive Technologies Innovation Fund (DTIF), by Enterprise Ireland,  under project number DTIF2019-090 (project QCoIR) and also supported by IBM Quantum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFAT4oBgHgl3EQfkB3r/content/2301.08609v1.pdf'}
+page_content='  11  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFAT4oBgHgl3EQfkB3r/content/2301.08609v1.pdf'}
+page_content='0 -  +  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFAT4oBgHgl3EQfkB3r/content/2301.08609v1.pdf'}
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+page_content='4  Simulation  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFAT4oBgHgl3EQfkB3r/content/2301.08609v1.pdf'}
+page_content='5  ibm mumbai: Trotter circuit  +  ibm mumbai: AQC circuit  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFAT4oBgHgl3EQfkB3r/content/2301.08609v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFAT4oBgHgl3EQfkB3r/content/2301.08609v1.pdf'}
+page_content='  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFAT4oBgHgl3EQfkB3r/content/2301.08609v1.pdf'}
+page_content='D  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFAT4oBgHgl3EQfkB3r/content/2301.08609v1.pdf'}
+page_content='5  15  2D  25  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFAT4oBgHgl3EQfkB3r/content/2301.08609v1.pdf'}
+page_content='D  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFAT4oBgHgl3EQfkB3r/content/2301.08609v1.pdf'}
+page_content='5  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFAT4oBgHgl3EQfkB3r/content/2301.08609v1.pdf'}
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+page_content='  [12] Liam Madden and Andrea Simonetto.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFAT4oBgHgl3EQfkB3r/content/2301.08609v1.pdf'}
+page_content=' Best approximate quantum compiling problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFAT4oBgHgl3EQfkB3r/content/2301.08609v1.pdf'}
+page_content=' ACM Trans-  actions on Quantum Computing, 3(2):1–29, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFAT4oBgHgl3EQfkB3r/content/2301.08609v1.pdf'}
+page_content='  [13] Liam Madden, Albert Akhriev, and Andrea Simonetto.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFAT4oBgHgl3EQfkB3r/content/2301.08609v1.pdf'}
+page_content=' Sketching the best approximate quantum  compiling problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFAT4oBgHgl3EQfkB3r/content/2301.08609v1.pdf'}
+page_content=' arXiv preprint arXiv:2205.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFAT4oBgHgl3EQfkB3r/content/2301.08609v1.pdf'}
+page_content='04025, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFAT4oBgHgl3EQfkB3r/content/2301.08609v1.pdf'}
+page_content='  [14] Niall F Robertson, Albert Akhriev, Jiri Vala, and Sergiy Zhuk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFAT4oBgHgl3EQfkB3r/content/2301.08609v1.pdf'}
+page_content=' Escaping barren plateaus in approx-  imate quantum compiling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFAT4oBgHgl3EQfkB3r/content/2301.08609v1.pdf'}
+page_content=' arXiv preprint arXiv:2210.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFAT4oBgHgl3EQfkB3r/content/2301.08609v1.pdf'}
+page_content='09191, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFAT4oBgHgl3EQfkB3r/content/2301.08609v1.pdf'}
+page_content='  [15] Lorenzo Piroli, Bal´azs Pozsgay, and Eric Vernier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFAT4oBgHgl3EQfkB3r/content/2301.08609v1.pdf'}
+page_content=' From the quantum transfer matrix to the quench  action: the loschmidt echo in xxz heisenberg spin chains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFAT4oBgHgl3EQfkB3r/content/2301.08609v1.pdf'}
+page_content=' Journal of Statistical Mechanics: Theory  and Experiment, 2017(2):023106, 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFAT4oBgHgl3EQfkB3r/content/2301.08609v1.pdf'}
+page_content='  [16] Adam Smith, MS Kim, Frank Pollmann, and Johannes Knolle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFAT4oBgHgl3EQfkB3r/content/2301.08609v1.pdf'}
+page_content=' Simulating quantum many-body  dynamics on a current digital quantum computer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFAT4oBgHgl3EQfkB3r/content/2301.08609v1.pdf'}
+page_content=' npj Quantum Information, 5(1):1–13, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFAT4oBgHgl3EQfkB3r/content/2301.08609v1.pdf'}
+page_content='  [17] David Layden.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFAT4oBgHgl3EQfkB3r/content/2301.08609v1.pdf'}
+page_content=' First-order trotter error from a second-order perspective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFAT4oBgHgl3EQfkB3r/content/2301.08609v1.pdf'}
+page_content=' Physical Review Letters,  128(21):210501, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFAT4oBgHgl3EQfkB3r/content/2301.08609v1.pdf'}
+page_content='  [18] Johannes Hauschild and Frank Pollmann.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFAT4oBgHgl3EQfkB3r/content/2301.08609v1.pdf'}
+page_content='  Efficient numerical simulations with tensor networks:  Tensor network python (tenpy).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFAT4oBgHgl3EQfkB3r/content/2301.08609v1.pdf'}
+page_content=' SciPost Physics Lecture Notes, page 005, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFAT4oBgHgl3EQfkB3r/content/2301.08609v1.pdf'}
+page_content='  [19] Sumeet Khatri, Ryan LaRose, Alexander Poremba, Lukasz Cincio, Andrew T Sornborger, and  Patrick J Coles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFAT4oBgHgl3EQfkB3r/content/2301.08609v1.pdf'}
+page_content=' Quantum-assisted quantum compiling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFAT4oBgHgl3EQfkB3r/content/2301.08609v1.pdf'}
+page_content=' Quantum, 3:140, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFAT4oBgHgl3EQfkB3r/content/2301.08609v1.pdf'}
+page_content='  [20] Marco Cerezo, Akira Sone, Tyler Volkoff, Lukasz Cincio, and Patrick J Coles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFAT4oBgHgl3EQfkB3r/content/2301.08609v1.pdf'}
+page_content=' Cost function depen-  dent barren plateaus in shallow parametrized quantum circuits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFAT4oBgHgl3EQfkB3r/content/2301.08609v1.pdf'}
+page_content=' Nature communications, 12(1):1–12,  2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFAT4oBgHgl3EQfkB3r/content/2301.08609v1.pdf'}
+page_content='  12  [21] Kristan Temme, Sergey Bravyi, and Jay M Gambetta.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFAT4oBgHgl3EQfkB3r/content/2301.08609v1.pdf'}
+page_content=' Error mitigation for short-depth quantum  circuits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFAT4oBgHgl3EQfkB3r/content/2301.08609v1.pdf'}
+page_content=' Physical review letters, 119(18):180509, 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFAT4oBgHgl3EQfkB3r/content/2301.08609v1.pdf'}
+page_content='  [22] Youngseok Kim, Christopher J Wood, Theodore J Yoder, Seth T Merkel, Jay M Gambetta, Kris-  tan Temme, and Abhinav Kandala.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFAT4oBgHgl3EQfkB3r/content/2301.08609v1.pdf'}
+page_content=' Scalable error mitigation for noisy quantum circuits produces  competitive expectation values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFAT4oBgHgl3EQfkB3r/content/2301.08609v1.pdf'}
+page_content=' arXiv preprint arXiv:2108.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFAT4oBgHgl3EQfkB3r/content/2301.08609v1.pdf'}
+page_content='09197, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFAT4oBgHgl3EQfkB3r/content/2301.08609v1.pdf'}
+page_content='  [23] Manuel S Rudolph, Jacob Miller, Jing Chen, Atithi Acharya, and Alejandro Perdomo-Ortiz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFAT4oBgHgl3EQfkB3r/content/2301.08609v1.pdf'}
+page_content=' Syn-  ergy between quantum circuits and tensor networks: Short-cutting the race to practical quantum  advantage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFAT4oBgHgl3EQfkB3r/content/2301.08609v1.pdf'}
+page_content=' arXiv preprint arXiv:2208.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFAT4oBgHgl3EQfkB3r/content/2301.08609v1.pdf'}
+page_content='13673, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFAT4oBgHgl3EQfkB3r/content/2301.08609v1.pdf'}
+page_content='  13' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFAT4oBgHgl3EQfkB3r/content/2301.08609v1.pdf'}

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+Pointwise Bi-Slant Submanifolds in Locally
+Conformal Kähler Manifolds Immersed as
+Warped Products∗
+Umar Mohd Khan & Viqar Azam Khan
+Department of Mathematics
+Aligarh Muslim University
+Aligarh-202002, India
+Email: [email protected], [email protected]
+Abstract
+We study immersions of pointwise bi-slant submanifolds of locally conformal
+Kähler manifolds as warped products. In particular, we establish characterisation
+theorem for a pointwise bi-slant submanifold of a locally conformal Kähler manifold
+to be immersed as a warped product and show that a necessary condition is that
+the Lee vector field B is orthogonal to the second factor and the warping function
+λ satisfies grad(ln λ) =
+1
+2BT , where BT denotes the tangential part of the Lee
+vector field. We also extend Chen’s inequality for the squared length of the second
+fundamental form to our case and study the corresponding equality case.
+1
+Introduction
+Vaisman introduced locally conformal Kähler (lcK) manifolds as a generalisation of Kähler
+manifolds [33, 34, 35, 36, 21, 37, 38]. An lcK manifold is a Hermitian manifold that can
+be written as the union of Kähler manifolds such that the lcK metric is locally conformal
+to these Kähler metrics. LcK manifolds are characterised by the existence of a globally
+defined closed 1-form ω, called the Lee form, such that the fundamental 2-form of the
+lcK metric satisfies dΩ = Ω ∧ ω. The Lee form and its associated Lee Vector field play
+an important part in the geometry of lcK manifolds.
+∗Keywords and phrases: warped product submanifolds; locally conformal Kähler manifolds (lcK);
+pointwise bi-slant submanifolds
+2020 AMS Subject Classification: 53C15, 53C40, 53C42, 53B25
+1
+arXiv:2301.05280v1  [math.GM]  11 Jan 2023
+
+From an extrinsic geometric standpoint, holomorphic and totally real submanifolds
+are important objects of study in the setting of almost Hermitian manifolds. Bejancu
+[5, 6] defined CR submanifolds as a generalisation of holomorphic and totally real sub-
+manifolds which were further studied by Chen [11, 12]. Later, Chen [13, 14] extended the
+class of holomorphic and totally real submanifolds by introducing the notion of slant sub-
+manifolds. The concept was further generalised to pointwise slant submanifolds [20] by
+the same author. The study of CR submanifolds and slant submanifolds was later gener-
+alised by several authors to semi-slant submanifolds, hemi-slant submanifolds(also called
+pseudo-slant submanifolds) and bi-slant submanifolds, in various ambient manifolds.
+Semi-slant submanifolds in almost Hermitian manifolds were studied by Papaghiuc
+[28].
+Cabrerizo et al.
+[9, 10] studied semi-slant submanifolds in Sasakian manifolds.
+Slant and semi-slant submanifolds in almost product Riemannian manifolds were studied
+in [2, 24, 29]. Hemi-slant submanifolds were also studied in nearly Kenmotsu manifolds
+[4], LCS-manifolds [3] and locally product Riemannian manifolds [31].
+Bishop and O’Neill [7] while studying examples of manifolds with negative sectional
+curvature, defined warped product manifolds by homothetically warping the product
+metric on a product manifold. Warped products are a natural generalisation of Rieman-
+nian products and they have found extensive applications in relativity. Most notably
+the Schwarzschild metric describing the gravitational field outside a spherical mass under
+certain assumptions and the Robertsen Walker metric (FLRW metric) are examples of
+warped product metrics. A natural example of warped product manifolds are surfaces
+of revolution. Hiepko [22] gave a characterisation for a Riemannian manifold to be the
+warped product of its submanifolds, generalising the deRham decomposition theorem for
+product manifolds. Later on Nölker [27] and Chen [15, 16, 19] initiated the study of
+extrinsic geometry of warped product manifolds.
+Chen [17, 18] initiated the study of CR submanifolds immersed as warped products in
+Kähler manifolds. He proved that given any holomorphic MT and totally real submanifold
+M⊥ of a Kähler manifold, every warped product of the form MT ×λ M⊥ in a Kähler
+manifold satisfies the inequality
+||h||2 ≥ 2n2||grad(ln λ)||2
+(1.1)
+where λ is the warping function, n2 is the dimension of M⊥, ||h||2 is the squared norm
+of the second fundamental form and grad(ln λ) is the gradient of ln λ. Bonanzinga and
+Matsumoto [8, 26, 25] continued the study in the setting of lcK manifolds. Nargis Ja-
+mal et al. [23] studied Generic warped products in lcK manifolds. Further studies of
+semi-slant and hemi-slant submanifolds of lcK manifolds were carried out in [1, 30, 32].
+Generic submanifolds, CR-submanifolds and pointwise semi-slant submanifolds immersed
+as warped products in lcK manifolds were studied by [23, 1].
+We continue the study by considering pointwise bi-slant submanifolds in an lcK man-
+ifold. In particular we give characterisation theorems and establish Chen’s inequality
+2
+
+for the squared norm of the second fundamental form of pointwise bi-slant submanifolds
+immersed as warped products in an lcK manifold.
+2
+Preliminaries
+Definition 2.1. A Hermitian Manifold (�
+M 2n, J, g) is said to be a locally conformal Kähler
+(lcK) manifold if there exists an open cover {Ui}i∈I of �
+M 2n and a family {fi}i∈I of C∞
+functions fi : Ui → R such that for each i ∈ I, the metric
+gi = e−fig|Ui
+(2.1)
+on Ui is a Kähler metric.
+Given an lcK manifold (�
+M 2n, J, g), let U, V denote smooth sections of T �
+M 2n, then
+the local 1-forms dfi glue up to a globally defined closed 1-form ω, called the Lee form,
+and it satisfies the following equation
+dΩ = Ω ∧ ω
+(2.2)
+where Ω(U, V ) = g(JU, V ) is the fundamental 2-form associated to (J, g).
+Denote by B the vector field equivalent to ω with respect to g, i.e. ω(U) = g(B, U).
+B is called the Lee vector field.
+Let ∇ denote the Levi-Civita connection of (�
+M 2n, g) and �∇i denote the Levi-Civita
+connection of the local metrics gi for all i ∈ I. Then �∇i glue up to a globally defined
+torsion-free linear connection �∇ on �
+M 2n given by
+�∇UV = ∇UV − 1
+2 {ω(U)V + ω(V )U − g(U, V )B}
+(2.3)
+where U, V ∈ T �
+M 2n and satisfying
+�∇g = ω ⊗ g
+(2.4)
+�∇ is called the Weyl connection of the lcK manifold (�
+M 2n, J, g). As gi are Kähler metrics,
+the almost complex structure J is parallel with respect to the Weyl connection, i.e.
+�∇J = 0. This gives
+∇UJV = J∇UV + 1
+2 {Θ(V )U − ω(V )JU − g(U, V )A + Ω(U, V )B}
+(2.5)
+Now as ω is a closed form on �
+M 2n, we have
+(∇Uω)V = (∇V ω)U
+(2.6)
+3
+
+Let M m be a Riemannian manifold isometrically immersed in an lcK manifold (�
+M 2n, J, g).
+Let U, V, W denote smooth sections of TM m and ξ, η denote smooth sections of T ⊥M m.
+The Gauss and Weingarten formulae with respect to the Riemannian connection of
+�
+M 2n are given as
+∇UV = ∇UV + h(U, V )
+(2.7)
+∇Uξ = −AξU + ∇⊥
+Uξ
+(2.8)
+where h is the second fundamental form, A is the shape operator and ∇, ∇⊥ are respec-
+tively the induced connections in the tangent bundle and the normal bundle of M m with
+respect to ∇.
+The Gauss and Weingarten formulae with respect to the Weyl connection of �
+M 2n are
+given as
+�∇UV = ˆ∇UV + �h(U, V )
+(2.9)
+�∇Uξ = −�AξU + �∇⊥
+Uξ
+(2.10)
+where �h is the second fundamental form, �A is the shape operator and ˆ∇, �∇⊥ are respec-
+tively the induced connections in the tangent bundle and the normal bundle of M m with
+respect to �∇.
+Let H denote the trace of h, then H is called the mean curvature vector of M m in
+(�
+M 2n, J, g) and is a smooth section of T ⊥M m. We say M m is a totally umbilic submanifold
+of (�
+M 2n, J, g), if h(U, V ) = g(U, V )H. We say M m is a totally geodesic submanifold of
+(�
+M 2n, J, g), if h(U, V ) = 0.
+Let BT, BN denote the tangential and normal components of the Lee vector field B.
+From (2.3), we have the following relations
+ˆ∇UV = ∇UV − 1
+2
+�
+ω(U)V + ω(V )U − g(U, V )BT�
+(2.11)
+�h(U, V ) = h(U, V ) + 1
+2g(U, V )BN
+(2.12)
+�AξU = AξU + 1
+2ω(ξ)U
+(2.13)
+�∇⊥
+Uξ = ∇⊥
+Uξ − 1
+2ω(U)ξ
+(2.14)
+Now define
+JU = PU + FU
+Jξ = tξ + fξ
+(2.15)
+where PU, tξ and FU, fξ are respectively the tangential and normal parts. Then, we have
+P 2 + tF = −I
+f 2 + Ft = −I
+FP + fF = 0
+tf + Pt = 0
+(2.16)
+4
+
+Define the covariant differentiation of P, F, t and f with respect to the Levi-Civita
+connection of �
+M 2n as
+(∇UP)V = ∇UPV − P∇UV
+(∇UF)V = ∇⊥
+UFV − F∇UV
+(∇Ut)ξ = ∇utξ − t(∇⊥
+Uξ)
+(∇Uf)ξ = ∇⊥
+Ufξ − f(∇⊥
+Uξ)
+(2.17)
+Then as �∇J = 0, using (2.11), (2.12), (2.13), (2.14) we have
+(∇UP)V = AFV U + th(U, V ) + 1
+2
+�
+Θ(V )U − ω(V )PU + g(PU, V )BT − g(U, V )AT�
+(∇UF)V = fh(U, V ) − h(U, PV ) + 1
+2
+�
+g(PU, V )BN − g(U, V )AN − ω(V )FU
+�
+(∇Ut)ξ = AfξU − PAξU + 1
+2
+�
+g(FU, ξ)BT − ω(ξ)PU + Θ(ξ)U
+�
+(∇Uf)ξ = −h(U, tξ) − FAξU + 1
+2
+�
+g(FU, ξ)BN − ω(ξ)FU
+�
+(2.18)
+Bishop and O’Neill [7] defined warped product as
+Definition 2.2. Let (M n1
+1 , g1) and (M n2
+2 , g2) be Riemmanian manifolds and let
+π1 : M1 × M2 → M1 and π2 : M1 × M2 → M2
+be the canonical projections.
+Let λ : M1 → (0, ∞) be a smooth function.
+Then the
+warped product manifold (M, g) = M1 × λM2 is defined as the manifold M1×M2 equipped
+with the Riemannian metric
+g = π⋆
+1g1 + λ2π⋆
+2g2
+(2.19)
+Warped product manifolds are a generalization of the usual product of two Riemannian
+manifolds. In fact we have the following characterisation theorem
+Theorem 2.1 ([22]). Let (M m, g) be a connected Riemannian manifold equipped with
+orthogonal, complementary, involutive distributions D1 and D2. Further let the leaves of
+D1 be totally geodesic and the leaves of D2 be extrinsic spheres in M m, where by extrinsic
+spheres we mean totally umbilic submanifolds such that the mean curvature vector is par-
+allel in the normal bundle. Then (M m, g) is locally a warped product (M, g) = M1 × λM2,
+where M1 and M2 respectively denote the leaves of D1 and D2 and λ : M1 → (0, ∞) is a
+smooth function such that grad(ln λ) is the mean curvature vector of M2 in M.
+Further, if (M m, g) is simply connected and complete, then (M m, g) is globally a
+warped product.
+For (M n1
+1 , g1), (M n2
+2 , g2) and (M, g) denote respectively the Levi-Civita connections by
+∇1, ∇2 and ∇. Given any smooth function λ : M1 → R, let grad(λ) denote the lift of
+the gradient vector field of λ to (M, g).
+5
+
+Theorem 2.2 ([22]). Given a warped product manifold (M, g) = M1 × λM2 of Riem-
+manian manifolds (M n1
+1 , g1) and (M n2
+2 , g2), we have for all X, Y ∈ L(M1) and Z, W ∈
+L(M2),
+∇XY = ∇1
+XY
+(2.20)
+∇XZ = ∇ZX = X(ln λ)Z
+(2.21)
+∇ZW = ∇2
+ZW − g(Z, W)grad(ln λ)
+(2.22)
+It follows from Lemma 2.2 that H = −grad(ln λ) is the mean curvature vector of M2 in
+M.
+3
+Pointwise Bi-Slant Submanifolds of lcK manifolds
+Let M m be a Riemannian manifold isometrically immersed in an lcK manifold (�
+M 2n, J, g).
+M m is said to be a pointwise bi-slant submanifold if it admits two orthogonal comple-
+mentary distributions Dθ1 and Dθ2, such that Dθ1 and Dθ2 are pointwise slant with slant
+angles θ1, θ2 ∈
+�
+0, π
+2
+�
+and θ1 ̸= θ2, i.e. P 2X = − cos2 θ1X, for every smooth vector field
+X ∈ Dθ1 and P 2Z = − cos2 θ2Z, for every smooth vector field Z ∈ Dθ2.
+The tangent bundle and the normal bundle of a pointwise bi-slant submanifold admits
+an orthogonal decomposition as
+TM m = Dθ1 ⊕ Dθ2
+T ⊥M m = FDθ1 ⊕ FDθ2 + µ
+(3.1)
+where µ is the orthogonal complementary distribution of FDθ1 ⊕ FDθ2 in T ⊥M m and is
+an invariant subbundle of T ⊥M m with respect to J. It is easy to observe that for i = 1, 2,
+PDθi = Dθi
+t(FDθi) = Dθi
+t(µ) = {0}
+f(FDθi) = FDθi
+f(µ) = µ
+(3.2)
+Let M m be a pointwise bi-slant manifold isometrically immersed in an lcK manifold
+(�
+M 2n, J, g) such that the distributions Dθ1, Dθ2 are both involutive. Let M 2n1
+θ1
+and M 2n2
+θ2
+respectively denote the leaves of Dθ1 and Dθ2, where 2n1 = dimR Dθ1 and 2n2 = dimR Dθ2.
+We say M m is a
+• mixed totally geodesic pointwise bi-slant submanifold if h(Dθ1, Dθ2) = {0}.
+• pointwise bi-slant product submanifold if M m can be expressed locally as Mθ1 × Mθ2.
+• pointwise bi-slant warped product submanifold if M m can be expressed locally as
+Mθ1 × λMθ2 for some smooth function λ : Mθ1 → (0, ∞).
+Let X, Y be smooth vector fields in Dθ1 and Z, W be smooth vector fields in Dθ2. Then
+we have,
+6
+
+Theorem 3.1. Let M m be a pointwise bi-slant submanifold of an lcK manifold �
+M 2n.
+• the slant distribution Dθ1 is involutive if and only if
+g(AFPXZ − AFXPZ, Y ) − g(AFPY Z − AFY PZ, X)
+= g(∇⊥
+XFY, FZ) − g(∇⊥
+Y FX, FZ)
+(3.3)
+• the leaves of the slant distribution Dθ1 are totally geodesic in M m if and only if
+ω(Dθ2) = {0} and g(AFPXZ − AFXPZ, Y ) + g(∇⊥
+Y FX, FZ) = 0
+(3.4)
+• the leaves of the slant distribution Dθ1 are totally umbilic in M m if and only if
+g(AFPXZ − AFXPZ, Y ) + g(∇⊥
+Y FX, FZ)
+= sin2 θ1
+�1
+2ω(Z) + g(H, Z)
+�
+g(X, Y )
+(3.5)
+for some smooth vector field H ∈ Dθ2.
+Proof. From (2.5) and (2.16), we have
+g(∇XY, Z) = g(∇XPY + ∇XFY − 1
+2g(X, Y )JB − 1
+2g(PX, Y )B, JZ)
+= −g(J∇XPY, Z) − g(AFY X, PZ) + g(∇⊥
+XFY, FZ)
+− 1
+2g(X, Y )g(B, Z) − 1
+2g(PX, Y )g(B, JZ)
+= −g(∇XJPY, Z) + 1
+2g(PX, PY )g(B, Z) − g(AFY X, PZ)
+− 1
+2g(X, Y )g(B, Z) + g(∇⊥
+XFY, FZ)
+= cos2 θ1g(∇XY, Z) + g(AFPY X, Z) + 1
+2 sin2 θ1g(X, Y )g(B, Z)
+− g(AFY X, PZ) + g(∇⊥
+XFY, FZ)
+i.e. we have
+sin2 θ1
+�
+g(∇XY, Z) + 1
+2g(X, Y )g(B, Z)
+�
+= g(AFPY Z − AFY PZ, X) + g(∇⊥
+XFY, FZ)
+Hence, the result follows.
+Similarly, we have
+7
+
+Theorem 3.2. Let M m be a pointwise bi-slant submanifold of an lcK manifold �
+M 2n.
+Then
+• the slant distribution Dθ2 is involutive if and only if
+g(AFPZX − AFZPX, W) − g(AFPWX − AFWPX, Z)
+= g(∇⊥
+ZFW, FX) − g(∇⊥
+WFZ, FX)
+(3.6)
+• the leaves of the slant distribution Dθ2 are totally geodesic in M m if and only if
+ω(Dθ1) = {0} and g(AFPZX − AFZPX, W) + g(∇⊥
+WFZ, FX) = 0
+(3.7)
+• the leaves of the slant distribution Dθ2 are totally umbilic in M m if and only if
+g(AFPZX − AFZPX, W) + g(∇⊥
+WFZ, FX)
+= sin2 θ2
+�1
+2ω(X) + g(H, X)
+�
+g(Z, W)
+(3.8)
+for some smooth vector field H ∈ Dθ1.
+Notations: Let Dθ1 and Dθ2 be the slant distributions on a pointwise bi-slant subman-
+ifold M m of an lcK manifold �
+M 2n such that both distributions are involutive and let
+Mθ1 and Mθ2 respectively denote the leaves of the distributions Dθ1 and Dθ2 respectively.
+Then Dθ1(p, q) = T(p,q)(Mθ1 × {q}) and Dθ2(p, q) = T(p,q)({p} × Mθ2). Let L(Mθ1) and
+L(Mθ2) respectively denote the set of lifts of vector fields from Mθ1 and Mθ2 to M. Then
+X ∈ L(Mθ1) if and only if X|{p}×Mθ2 is constant for every p ∈ Mθ1. Similarly, Z ∈ L(Mθ2)
+if and only if Z|Mθ1×{q} is constant for every q ∈ Mθ2. Also, if πθ1 : Mθ1 × Mθ2 → Mθ1
+and πθ2 : Mθ1 × Mθ2 → Mθ2 are the canonical projections, we have dπθ1(L(Mθ1)) = TMθ1
+and dπθ2(L(Mθ2)) = TMθ2. It is clear that a general vector field in Dθ1 (respectively Dθ2)
+need not be in L(Mθ1) (respectively L(Mθ2)).
+From here on we use X, Y to denote smooth vector fields in L(Mθ1) and Z, W to
+denote smooth vector fields in L(Mθ2)
+4
+Some Lemmas
+We give the following lemmas which will be used to prove our main results.
+Lemma 4.1. Given a pointwise bi-slant warped product submanifold M = Mθ1 × λMθ2
+in an lcK manifold (�
+M 2n, J, g), we have for all X, Y ∈ L(Mθ1) and Z, W ∈ L(Mθ2),
+g(h(X, Z), FY ) = g(h(Y, Z), FX)
+(4.1)
+8
+
+g(h(X, Z), FW) = g(h(X, W), FZ)
+(4.2)
+g(h(X, Y ), FZ) = g(h(X, Z), FY ) − 1
+2g(X, Y )g(B, FZ)
+(4.3)
+g(h(Z, W), FX) = g(h(X, Z), FW) − 1
+2g(Z, W)g(B, FX)
+(4.4)
+X(ln λ) = 1
+2g(B, X)
+(4.5)
+g(B, Z) = 0
+(4.6)
+Proof. For all X, Y ∈ L(Mθ1) and Z, W ∈ L(Mθ2), we have using (2.5) and (2.21),
+g(h(X, Z), FW) = g(∇XZ, JW − PW)
+= −g(J∇XZ, W) − g(∇XZ, PW)
+= −g(∇XJZ, W) − X(ln λ)g(Z, PW)
+= −g(∇XPZ, W) − g(∇XFZ, W) − X(ln λ)g(Z, PW)
+= −X(ln λ)g(PZ, W) + g(AFZX, W) − X(ln λ)g(Z, PW)
+= g(h(X, W), FZ)
+which gives (4.2). Repeating the above calculation, we have
+g(h(X, Z), FW) = g(∇ZX, JW − PW)
+= −g(J∇ZX, W) − g(∇ZX, PW)
+= −g(∇ZJX, W) − 1
+2g(JB, X)g(Z, W) − 1
+2g(B, X)g(JZ, W)
+− X(ln λ)g(Z, PW)
+= −g(∇ZPX, W) − g(∇ZFX, W) − 1
+2g(JB, X)g(Z, W)
+− 1
+2g(B, X)g(JZ, W) − X(ln λ)g(Z, PW)
+= −PX(ln λ)g(Z, W) + g(AFXZ, W) + 1
+2g(B, JX)g(Z, W)
+− 1
+2g(B, X)g(PZ, W) − X(ln λ)g(Z, PW)
+Using (4.2) and comparing symmetric and skew symmetric terms in Z and W we have,
+X(ln λ) = 1
+2g(B, X)
+which gives (4.5) and
+g(h(X, Z), FW) = g(h(Z, W), FX) − PX(ln λ)g(Z, W) + 1
+2g(B, PX + FX)g(Z, W)
+9
+
+which on substituting from (4.5) gives (4.4). Similarly,
+g(h(X, Z), FY ) = g(∇ZX, JY − PY )
+= −g(J∇ZX, Y ) − g(∇ZX, PY )
+= −g(∇ZJX, Y )
+= −g(∇ZPX, Y ) − g(∇ZFX, Y )
+= g(AFXZ, Y )
+= g(h(Y, Z), FX)
+which gives (4.1). Repeating the above calculation, we have
+g(h(X, Z), FY ) = g(∇XZ, JY − PY )
+= −g(J∇XZ, Y ) − g(∇XZ, PY )
+= −g(∇XJZ, Y ) − 1
+2g(JB, Z)g(X, Y ) − 1
+2g(B, Z)g(JX, Y )
+= −g(∇XPZ, Y ) − g(∇XFZ, Y ) − 1
+2g(JB, Z)g(X, Y )
+− 1
+2g(B, Z)g(JX, Y )
+= g(AFZX, Y ) + 1
+2g(B, JZ)g(X, Y ) − 1
+2g(B, Z)g(PX, Y )
+Using (4.1) and comparing symmetric and skew symmetric terms in X and Y we have,
+1
+2g(B, Z) = 0
+which gives (4.6) and
+g(h(X, Z), FY ) = g(h(X, Y ), FZ) + 1
+2g(B, PZ + FZ)g(X, Y )
+which on substituting from (4.6) gives (4.3).
+From (4.5) and (4.6) we have
+Corollary 4.2. Given a pointwise bi-slant warped product submanifold M = Mθ1 × λMθ2
+in an lcK manifold (�
+M 2n, J, g), we have the Lee vector field B is orthogonal to the second
+factor and the warping function λ satisfies grad(ln λ) =
+1
+2BT, where BT denotes the
+tangential part of the Lee vector field along M.
+10
+
+Remark 4.1. Given a pointwise bi-slant warped product submanifold Mθ1 × λMθ2 of an
+l.c.K manifold �
+M 2n, let {Xi, β1Xi}p
+i=1 and {Zj, β2PZj}q
+j=1 respectively be local orthonor-
+mal frames of TMθ1 and TMθ2. Then a local orthonormal frame of �
+M 2n is
+�
+�
+Xi = Xi, �
+PXi = β1PXi
+�
+∪
+�
+�
+Zj = Zj
+λ , �
+PZj = β2PZj
+λ
+�
+∪
+�
+�ξk, �
+Jξk
+�
+∪
+�
+�
+FXi = α1FXi, �
+FPXi = α1β1FPXi
+�
+∪
+�
+�
+FZj = α2FZj
+λ
+, �
+FPZj = α2β2FPZj
+λ
+�
+where αi = csc θi, βi = sec θi for i = 1, 2 and
+�
+�
+Xi �
+PXi : 1 ≤ i ≤ n1
+�
+is an orthonormal basis of Dθ1
+�
+�
+Zj, �
+PZj : 1 ≤ j ≤ n2
+�
+is an orthonormal basis of Dθ2
+�
+�
+FXi, �
+FPXi : 1 ≤ j ≤ n1
+�
+is an orthonormal basis of FDθ1
+�
+�
+FZj, �
+FPZj : 1 ≤ j ≤ n2
+�
+is an orthonormal basis of FDθ2
+�
+�ξk, �
+Jξk : 1 ≤ s ≤ n − 2n1 − 2n2
+2
+�
+is an orthonormal basis of µ
+However, while Zj, βPZj ∈ L(Mθ2) we have �
+Zj, �
+PZj /∈ L(Mθ2) in general, as λ is a
+function on Mθ1. Also, note that
+J
+�
+�
+Zj
+�
+= J
+�Zj
+λ
+�
+= PZj
+λ
++ FZj
+λ
+= cos θ2 �
+PZj + sin θ2 �
+FZj
+J
+�
+�
+PZj
+�
+= J
+�
+sec θ2
+PZj
+λ
+�
+= sec θ2P 2Zj
+λ
++ sec θ2FPZj
+λ
+= − cos θ2�
+Zj + sin θ2 �
+FPZj
+5
+Main Results
+We first give a characterisation for pointwise bi-slant warped product submanifolds of
+lcK manifolds.
+Theorem 5.1. Let M m be a pointwise bi-slant submanifold of an lcK manifold �
+M 2n.
+Then the following are equivalent
+1. M m is a pointwise bi-slant warped product submanifold Mθ1 × λMθ2 of �
+M 2n
+2. For every X, Y ∈ L(Mθ1) and Z, W ∈ L(Mθ2) we have
+ω(Dθ2) = {0} and g (AFPXZ − AFXPZ, Y ) + g(∇⊥
+Y FX, FZ) = 0
+11
+
+g(AFPZX − AFZPX, W) + g(∇⊥
+WFZ, FX)
+(5.1)
+= sin2 θ2
+�1
+2ω(X) − X(ln λ)
+�
+g(Z, W)
+for some smooth function λ : Mθ1 → (0, ∞).
+3. For every X ∈ L(Mθ1) and Z ∈ L(Mθ2) we have
+ω(Dθ2) = {0} and ∇XZ = ∇ZX = 1
+2ω(X)Z
+(5.2)
+Also, in this case we have the mean curvature vector H of Mθ2 in M m is
+H = −grad(ln λ) = −1
+2BT
+(5.3)
+where BT is the tangential component of B along M.
+Proof. (1)⇔(2) This follows from Theorem 3.1, Theorem 3.2 and grad(ln λ) ∈ L(Mθ1)
+which implies
+g(∇Z(grad(ln λ)), X) = ZX(ln λ) − g(grad(ln λ), ∇ZX)
+= [Z, X](ln λ) − ∇ZX(ln λ)
+= −∇XZ(ln λ)
+= g(Z, ∇X(grad(ln λ)))
+= 0
+as Z(ln λ) = 0 and Mθ1 is totally geodesic in M. Also, (5.3) follows from Lemma 4.1
+(4.5).
+(1)⇔(3) Let M = Mθ1 × λMθ2 be a pointwise bi-slant warped product submanifold. Then
+(5.2) and (5.3) follow from (2.21) and Lemma 4.1 (4.5).
+Conversely, let M m be a pointwise bi-slant submanifold of an lcK manifold �
+M 2n such
+that (5.2) holds. Then for all X, Y ∈ L(Mθ1) and Z, W ∈ L(Mθ2) we have
+g([X, Y ], Z) = g(∇XY − ∇Y X, Z)
+= −g(∇XZ, Y ) + g(∇Y Z, X)
+= 0
+which implies Dθ1 is involutive.
+g(∇XY, Z) = −g(∇XZ, Y ) = 0
+12
+
+which implies leaves of Dθ1 are totally geodesic in M.
+g([Z, W], X) = g(∇ZW − ∇WZ, X)
+= −g(∇ZX, W) + g(∇WX, Z)
+= −1
+2ω(X)g(Z, W) + 1
+2ω(X)g(W, Z) = 0
+which implies Dθ2 is involutive.
+g(∇ZW, X) = −g(∇ZX, W)
+= −1
+2ω(X)g(Z, W)
+= −1
+2g(Z, W)g(BT, X)
+which implies leaves of Dθ2 are totally umbilical in M with mean curvature vector − 1
+2BT.
+g
+�
+∇ZBT, X
+�
+= 1
+2ω
+�
+BT�
+g(Z, X) = 0
+which implies BT is parallel in the normal bundle of Mθ2 in M.
+Hence by Theorem 2.1 we have M = Mθ1 × λMθ2 is a pointwise bi-slant warped prod-
+uct submanifold.
+We conclude our study of pointwise bi-slant warped product submanifolds of lcK
+manifolds by giving an inequality for the squared norm of the second fundamental form.
+Theorem 5.2. Let M = Mθ1 × λMθ2 be a pointwise bi-slant warped product submanifold
+in an lcK manifold (�
+M 2n, J, g). Then the norm of the second fundamental form satisfies
+the inequality
+||h||2 ≥ n1
+2 sin2 θ2∥B|FDθ2∥2 + n2
+2 sin2 θ1∥B|FDθ1∥2 + sin θ2g (HDθ1|FDθ2, B|FDθ2)
++ sin θ1g (HDθ2|FDθ1, B|FDθ1)
+(5.4)
+where 2n1 = dimR Dθ1, 2n2 = dimR Dθ2 and HDθ1 and HDθ2 are respectively the compo-
+nents of the mean curvature vector H of M in �
+M 2n along Dθ1 and Dθ2.
+If equality holds then we have
+• Image(h) ⊆ (FDθ1 ⊕ FDθ2), and
+• M is minimal in �
+M 2n, if and only if, M is mixed-totally geodesic in �
+M 2n.
+13
+
+Proof.
+||h||2 =
+��h(Dθ1, Dθ1)
+��
+FDθ1
+��2 +
+��h(Dθ1, Dθ2)
+��
+FDθ1
+��2 +
+��h(Dθ2, Dθ2)
+��
+FDθ1
+��2
++
+��h(Dθ1, Dθ1)
+��
+FDθ2
+��2 +
+��h(Dθ1, Dθ2)
+��
+FDθ2
+��2 +
+��h(Dθ2, Dθ2)
+��
+FDθ2
+��2
++
+���h(Dθ1, Dθ1)
+��
+µ
+���
+2
++
+���h(Dθ1, Dθ2)
+��
+µ
+���
+2
++
+���h(Dθ2, Dθ2)
+��
+µ
+���
+2
+From (4.3) and Remark 4.1 we have
+g
+�
+h
+�
+�
+Xi, �
+Zp
+�
+, �
+FXj
+�
+=csc θ1
+λ
+�
+g (h (Xi, Xj) , FZp) + 1
+2δijg (B, FZp)
+�
+=⇒ sin θ1g
+�
+h
+�
+�
+Xi, �
+Zp
+�
+, �
+FXj
+�
+= sin θ2g
+�
+h
+�
+�
+Xi, �
+Xj
+�
+, �
+FZp
+�
++ 1
+2 sin θ2δijg
+�
+B, �
+FZp
+�
+Similarly,
+sin θ1g
+�
+h
+�
+�
+Xi, �
+PZp
+�
+, �
+FXj
+�
+= sin θ2g
+�
+h
+�
+�
+Xi, �
+Xj
+�
+, �
+FPZp
+�
++ 1
+2 sin θ2δijg
+�
+B, �
+FPZp
+�
+sin θ1g
+�
+h
+�
+�
+PXi, �
+Zp
+�
+, �
+FPXj
+�
+= sin θ2g
+�
+h
+�
+�
+PXi, �
+PXj
+�
+, �
+FZp
+�
++ 1
+2 sin θ2δijg
+�
+B, �
+FZp
+�
+sin θ1g
+�
+h
+�
+�
+PXi, �
+PZp
+�
+, �
+FPXj
+�
+= sin θ2g
+�
+h
+�
+�
+PXi, �
+PXj
+�
+, �
+FPZp
+�
++ 1
+2 sin θ2δijg
+�
+B, �
+FPZp
+�
+sin θ1g
+�
+h
+�
+�
+PXi, �
+Zp
+�
+, �
+FXj
+�
+= sin θ2g
+�
+h
+�
+�
+PXi, �
+Xj
+�
+, �
+FZp
+�
+sin θ1g
+�
+h
+�
+�
+PXi, �
+PZp
+�
+, �
+FXj
+�
+= sin θ2g
+�
+h
+�
+�
+PXi, �
+Xj
+�
+, �
+FPZp
+�
+sin θ1g
+�
+h
+�
+�
+Xi, �
+Zp
+�
+, �
+FPXj
+�
+= sin θ2g
+�
+h
+�
+�
+Xi, �
+PXj
+�
+, �
+FZp
+�
+sin θ1g
+�
+h
+�
+�
+Xi, �
+PZp
+�
+, �
+FPXj
+�
+= sin θ2g
+�
+h
+�
+�
+Xi, �
+PXj
+�
+, �
+FPZp
+�
+which implies
+��h(Dθ1, Dθ2)
+��
+FDθ1
+��2 = cos2 θ1
+��h(Dθ1, Dθ2)
+��
+FDθ1
+��2 + sin2 θ1
+��h(Dθ1, Dθ2)
+��
+FDθ1
+��2
+= cos2 θ1
+��h(Dθ1, Dθ2)
+��
+FDθ1
+��2 + sin2 θ2
+��h(Dθ1, Dθ1)
+��
+FDθ2
+��2
++ 1
+2 sin2 θ2
+�
+i,p
+�
+g
+�
+B, �
+FZp
+�2
++ g
+�
+B, �
+FPZp
+�2�
++ sin θ2
+�
+i,p
+�
+g
+�
+h
+�
+�
+Xi, �
+Xi
+�
+, �
+FZp
+�
+g
+�
+B, �
+FZp
+�
++g
+�
+h
+�
+�
+Xi, �
+Xi
+�
+, �
+FPZp
+�
+g
+�
+B, �
+FPZp
+�
+14
+
++g
+�
+h
+�
+�
+PXi, �
+PXi
+�
+, �
+FZp
+�
+g
+�
+B, �
+FZp
+�
++g
+�
+h
+�
+�
+PXi, �
+PXi
+�
+, �
+FPZp
+�
+g
+�
+B, �
+FPZp
+��
+i.e.
+sin2 θ1
+��h(Dθ1, Dθ2)
+��
+FDθ1
+��2 = sin2 θ2
+��h(Dθ1, Dθ1)
+��
+FDθ2
+��2 + 2n1
+4 sin2 θ2
+����B
+��
+FDθ2
+����
+2
++ sin θ2g
+��
+i
+�
+h
+�
+�
+Xi, �
+Xi
+�
++ h
+�
+�
+PXi, �
+PXi
+�������
+FDθ2
+, B|FDθ2
+�
+≥ n1
+2 sin2 θ2∥B|FDθ2∥2 + sin θ2g (HDθ1|FDθ2, B|FDθ2)
+As done above we have
+sin2 θ2
+��h(Dθ1, Dθ2)
+��
+FDθ2
+��2 = sin2 θ1
+��h(Dθ2, Dθ2)
+��
+FDθ1
+��2 + 2n2
+4 sin2 θ1
+����B
+��
+FDθ1
+����
+2
++ sin θ1g
+�
+��
+p
+�
+h
+�
+�
+Zp, �
+Zp
+�
++ h
+�
+�
+PZp, �
+PZp
+�������
+FDθ1
+, B|FDθ1
+�
+�
+≥ n2
+2 sin2 θ1∥B|FDθ1∥2 + sin θ1g (HDθ2|FDθ1, B|FDθ1)
+Combining we have (5.4).
+If equality holds in (5.4), then the only non-zero components of ||h|| are
+��h(Dθ1, Dθ1)|FDθ2
+��2
+,
+��h(Dθ1, Dθ2)|FDθ1
+��2 ,
+��h(Dθ2, Dθ2)|FDθ1
+��2 and
+��h(Dθ1, Dθ2)|FDθ2
+��2 .
+Also, from the calculations above we have,
+��h(Dθ1, Dθ1)|FDθ2
+��2 = 0 if and only if
+��h(Dθ1, Dθ2)|FDθ1
+��2 = 0
+and
+��h(Dθ2, Dθ2)|FDθ1
+��2 = 0 if and only if
+��h(Dθ1, Dθ2)|FDθ2
+��2 = 0
+Hence, the result follows.
+6
+Example
+Consider Cn = E2n, J, g0) where E2n is the Euclidean space of dimension 2n with coor-
+dinates (x1, . . . , xn, y1, . . . , yn) equipped with the standard Euclidean metric g0 and the
+canonical almost complex structure
+J(x1, . . . , xn, y1, . . . , yn) = (−y1, . . . , −yn, x1, . . . , xn)
+(6.1)
+15
+
+Then Cn = (E2n, J, g0) is a flat Kähler manifold.
+We use the following result about pointwise slant immersions.
+Theorem 6.1 ([20] (Proposition 2.2)). Given a Kähler manifold (�
+M 2n, J, g0), let Mθ be a
+pointwise slant submanifold. Then for any smooth function f : �
+M → (0, ∞), we have Mθ
+is again a pointwise slant submanifold of the globally conformal Kähler (gcK) manifold
+(�
+M 2n, J, e−fg0) with the same slant angle.
+Example 6.1. Let C4 = (E8, J, g0) be as defined above. Consider an open subset of
+E4 with u1u2 ̸= 1, u3u4 ̸= 1, (u1 − u2) ∈
+�
+0, π
+4
+�
+and (u3 − u4) ∈
+� π
+4, π
+2
+�
+. Define the
+4-dimensional submanifold M of C4 given by
+x1 = u1 cos u2
+,
+y1 = u1 sin u2
+x2 = u2 cos u1
+,
+y2 = u2 sin u1
+x3 = u3 cos u4
+,
+y3 = u3 sin u4
+x4 = u4 cos u3
+,
+y4 = u4 sin u3
+(6.2)
+An orthonormal frame of the tangent bundle TM of M is
+X1 =
+1
+�
+1 + u2
+2
+�
+cos u2
+∂
+∂x1
+− u2 sin u1
+∂
+∂x2
++ sin u2
+∂
+∂y1
++ u2 cos u1
+∂
+∂y2
+�
+X2 =
+1
+�
+1 + u2
+1
+�
+−u1 sin u2
+∂
+∂x1
++ cos u1
+∂
+∂x2
++ u1 cos u2
+∂
+∂y1
++ sin u1
+∂
+∂y2
+�
+X3 =
+1
+�
+1 + u2
+4
+�
+cos u4
+∂
+∂x3
+− u4 sin u3
+∂
+∂x4
++ sin u4
+∂
+∂y3
++ u4 cos u3
+∂
+∂y4
+�
+X4 =
+1
+�
+1 + u2
+3
+�
+−u3 sin u4
+∂
+∂x3
++ cos u3
+∂
+∂x4
++ u3 cos u4
+∂
+∂y3
++ sin u3
+∂
+∂y4
+�
+Then, M is a proper pointwise bi-slant submanifold with slant distributions given by
+Dθ1 = Span{X1, X2} and Dθ2 = Span{X3, X4}. Also, the slant angles are given by
+cos2 θ1 = (u1u2 − 1)2 cos2(u1 − u2)
+(1 + u2
+1)(1 + u2
+2)
+, and , cos2 θ2 = (u3u4 − 1)2 cos2(u3 − u4)
+(1 + u2
+3)(1 + u2
+4)
+It is straightforward to check that Dθ1 and Dθ2 are both involutive and totally geodesic
+in M. Let Mθ1 and Mθ2 be the leaves of Dθ1 and Dθ2 respectively. Then we have M is
+the Riemannian product M = Mθ1 × Mθ2 and the metric gM induced on M from C4 is
+given by
+gM = g1 + g2
+(6.3)
+where
+g1 = (1 + u2
+2)du2
+1 + (1 + u2
+1)du2
+2 , and , g2 = (1 + u2
+4)du2
+3 + (1 + u2
+3)du2
+4
+(6.4)
+16
+
+Now, for any non-constant positive smooth function f = f(x1, x2, y1, y2) on C4, depending
+only on coordinates x1, x2, y1, y2, consider the Riemannian metric ˜g = e−fg0, conformal
+to the standard metric g0. Then, �
+M = (E8, J, ˜g) is a globally conformal Kähler manifold
+and the metric on M induced from �
+M is the warped product metric
+˜gM = ˜g1 + e−fg2
+(6.5)
+where
+˜g1 = e−fg1
+(6.6)
+is conformal to g1 by the choice of f.
+Hence from Theorem 6.1 we have (M, ˜gM) is a proper pointwise bi-slant warped prod-
+uct submanifold of �
+M = (E8, J, ˜g).
+Also, as f = f(x1, x2, y1, y2) is a non-constant positive smooth function on C4, depend-
+ing only on coordinates x1, x2, y1, y2, from (6.2) we have that restricted to the submanifold
+M, the Lee form ω of �
+M is given by
+ω = df = ∂f
+∂u1
+du1 + ∂f
+∂u2
+du2
+(6.7)
+Hence, it follows that the Lee-vector field B is orthogonal to Dθ2 and the warping function
+λ = −e− f
+2 |M satisfies grad(ln λ) = 1
+2grad(f|M) = 1
+2BT.
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+20
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+PA-GM: POSITION-AWARE LEARNING OF EMBEDDING NETWORKS FOR DEEP GRAPH
+MATCHING
+Dongdong Chen1, Yuxing Dai2, Lichi Zhang1, Zhihong Zhang2, Edwin R. Hancock3
+1 Shanghai Jiao Tong University
+2 Xiamen University
+3 University of York
+ABSTRACT
+Graph matching can be formalized as a combinatorial opti-
+mization problem, where there are corresponding relation-
+ships between pairs of nodes that can be represented as edges.
+This problem becomes challenging when there are potential
+ambiguities present due to nodes and edges with high simi-
+larity, and there is a need to find accurate results for similar
+content matching. In this paper, we introduce a novel end-to-
+end neural network that can map the linear assignment prob-
+lem into a high-dimensional space augmented with node-level
+relative position information, which is crucial for improving
+the method’s performance for similar content matching. Our
+model constructs the anchor set for the relative position of
+nodes and then aggregates the feature information of the tar-
+get node and each anchor node based on a measure of rela-
+tive position. It then learns the node feature representation by
+integrating the topological structure and the relative position
+information, thus realizing the linear assignment between the
+two graphs. To verify the effectiveness and generalizability of
+our method, we conduct graph matching experiments, includ-
+ing cross-category matching, on different real-world datasets.
+Comparisons with different baselines demonstrate the supe-
+riority of our method. Our source code is available under
+https://github.com/anonymous.
+Index Terms— Graph Matching, Graph Embedding,
+Deep Neural Network
+1. INTRODUCTION
+Graph matching aims to establish the relationship between
+two or more graphs based on information derived from their
+nodes and edges [1].
+Due to its ability to better express
+and encode data relationships, graph matching has gained
+considerable traction in computer vision and related fields.
+Applications of graph matching techniques include image
+registration in medical image analysis [2], link analysis in
+social networks [3] and image extrapolation in computer
+vision[4]. Current methods for solving the graph matching
+problem can be divided into two main classes: a) learning-
+free or b) learning-based [5, 6]. In the first case, the critical
+Fig. 1. A failed example of mismatching the position of the
+animal dog’s ears in the two images.
+element is the mathematical method used for obtaining an
+approximate solution to an intrinsically NP-hard problem.
+On the other hand, learning-based methods aim to improve
+the solver with a number of methods including deep neural
+networks.
+However, these methods can only compute a similarity
+score for the whole graph, or rely on inefficient global match-
+ing procedures[7]. For example, [8] only considers the em-
+bedding of local information for nodes in the graph, leading
+to a tendency to inconsistently match similar nodes from dif-
+ferent regions of the graph, and thus resulted to have ambigu-
+ities. As is illustrated in Fig. 1, it shows that the node em-
+bedding representation, which relies only on local structural
+information and semantic node information, lacks sufficient
+discrimination to effectively resolve these ambiguities. As a
+result, it is difficult to distinguish the left and right ears of
+the dog in the example. From this, relative positional infor-
+mation is the key to the matching of diagrams, especially in
+cases where the graphs have similar semantic and structural
+content.
+In this paper, we introduce the idea of position-awareness
+to solve the above-mentioned problem in graph matching. We
+first construct the graph as input to the model with node fea-
+tures extracted from an image, and then extract a collection
+of anchor points as reference coordinates for each node in the
+graph. We further propose a corresponding position-aware
+node embedding algorithm to capture the relative positional
+arXiv:2301.01932v1  [cs.CV]  5 Jan 2023
+
+Xinformation of the nodes in the graph. With the node-wise
+graph embedding to hand, we identify a node permutation for
+node-to-node correspondence.
+The main contributions of this paper are as follows:
+• We propose a novel Position-Aware Learning of Em-
+bedding Network for Deep Graph Matching (PA-GM).
+To the best of our knowledge, there are no methods that
+consider the learning of an embedding augmented with
+relative positional information in graph matching tasks.
+This restriction hampers their applications in terms of
+matching accuracy.
+• We extract the relative positional information for the
+nodes in constructing the node embedding instead of
+the traditional neighborhood aggregation approach. We
+fully consider the positional information relevant to the
+matching of keypoints, which is proved to be effective
+in our experiments.
+• The proposed framework is both scalable and flexible,
+and benefits from the use of a relative position coef-
+ficient together with an alignment loss. Experiments
+show that the model achieves state-of-the-art results on
+real-world datasets.
+2. METHODS
+In this paper, we intend to resolve the graph matching prob-
+lem based on the supervised matching of graphs. Specifi-
+cally, we aim to learn an end-to-end model which can extract
+graph information and their matches through given pair-wise
+ground-truth correspondences for a set of graphs, and which
+can be further generalized to unseen graph pairs. The overall
+pipeline for graph matching using the position-aware embed-
+ding network is presented in Fig. 2.
+2.1. Problem Definition
+In this paper, we denote a graph by the triple G = (V, A, X)
+which consists of a finite set of nodes V , an adjacency ma-
+trix A, and a set of node attributes X extracted from im-
+ages using CNN-based models[9, 10]. We construct a vec-
+tor v ∈ {0, 1}nm×1 to indicate the match of vertices in the
+source graph Gs = (Vs, As, Xs) and the target graph Gt =
+(Vt, At, Xt).
+The vector has elements vi,j = 1 if vertex
+i ∈ Vs is matched to vertex j ∈ Vt, and vi,j = 0 if other-
+wise. It is worth noting that all the vertex matches are subject
+to one-to-one mapping constraints �
+j∈Vt vi,j = 1 ∀i ∈ Vs
+and �
+i∈Vs vi,j ≤ 1 ∀j ∈ Vt. Furthermore, we construct a
+square symmetric positive matrix M ∈ Rnm×nm as the affin-
+ity matrix, to encode the edge-to-edge affinity between two
+graphs in the off-diagonal elements. In this way, the two-
+graph matching between Gs and Gt can be formulated as an
+edge-preserving, quadratic assignment programming (QAP)
+problem [11]:
+argmax
+v
+v⊤Mv
+s.t.
+�
+j∈Vt
+vi,j = 1 ∀i ∈ Vs,
+�
+i∈Vs
+vi,j ≤ 1 ∀j ∈ Vt,
+(1)
+where v ∈ {0, 1}nm×1 and M ∈ Rnm×nm. The goal of
+graph matching is to establish a correspondence between two
+attributed graphs, which minimizes the sum of local and ge-
+ometric costs of assignment between the vertices of the two
+graphs.
+2.2. Position-Aware Node Embedding
+Pixels at neighboring positions in the image convey similar
+semantic information, therefore we need to distinguish them
+to resolve matching ambiguities. Here, we refer to the nodes
+used as reference positions named as anchors and propose an
+effective strategy to construct an anchor set P consisting of
+all nodes in the graph, denoted by P = V , serving as a sta-
+ble reference for all nodes. To combine information from the
+nodes and the anchors, we design the information aggregation
+mechanism as is shown in Fig.3.
+Considering that the amount of effective information
+transmission to the nodes by the anchors with different rela-
+tive distances is different, we first compute a relative position
+coefficient qv,u between a pair of nodes v and u as:
+qv,u =
+e−dv,u
+�
+i∈V e−dv,i ,
+(2)
+where dv,u is the shortest path distance between the two
+nodes. In practice, to effectively reduce the time complexity
+of calculating the shortest path and also to reduce interfer-
+ence from those anchor points that are distant from the node
+under consideration, we demand that the maximum shortest
+path distance does not exceed the ceiling value r. Otherwise,
+the value of the coefficient is set to infinity. We continue to
+compute the information aggregation function I(v, u)(l) for
+the l-th layer between node v and u as:
+I(v, u)(l) = qv,uCONCAT(h(l−1)
+v
+, h(l−1)
+u
+),
+(3)
+where h(l−1)
+v
+and h(l−1)
+u
+are the feature representation and po-
+sition representation in the (l − 1)-th layer, and are combined
+through the message aggregation function. We further aggre-
+gate the information from all node-anchor pairs to obtaining a
+new representation in a high dimensional space using the non-
+linear variation. This hidden representation can be computed
+using the update
+h(l)
+v
+= σ(AGG(I(v, u)(l)| ∀u ∈ V )W (l)),
+(4)
+
+Fig. 2. Overview of the end-to-end position embedding networks for deep graph matching. The blue source graph Gs and the
+green target graph Gt are extracted with the node-wise graph feature representation in high-level through the two frameworks
+of position-aware node embedding.
+Fig. 3. The position-aware embedding first aggregates the
+message of each target node hv and each anchor point hui in
+the anchor set via the aggregation function I(v, u) based on
+the relative position coefficients q(v, u). The representation
+matrix is then further aggregated using the learnable function
+AGG and is finally transformed in a non-linear way to obtain
+the feature output hv for the target node in the current layer.
+where AGG is typically a permutation-invariant function
+(e.g., sum), and W (l) is a learnable weight vector for the l-th
+layer.
+2.3. Graph Feature Matching
+Utilizing the proposed position-aware node embedding, the
+proposed scheme encodes each node with position informa-
+tion into a high-level embedding space. In this way, we can
+simplify the second-order affinity matrix of paired graphs to
+a be a linear one with position-aware learning of the embed-
+ding. With the final hidden representation of the source graph
+Hs ∈ Rn∗F ′ and the target graph Ht ∈ Rm∗F ′ to hand, we
+can obtain a soft correspondence between these graphs with
+a node-wise affinity matrix through the inner product of the
+embeddings of the two graphs being matched. Specifically, to
+satisfy the condition that the original graph maps injectively
+onto the target graph, we apply the Sinkhorn normalization
+[12] to obtain rectangular doubly-stochastic correspondence
+matrices.
+S = sinkhorn(HsHT
+t ).
+(5)
+We further employ the cross-entropy loss function as the
+permutation loss between the predicted permutation matrix
+and the ground truth:
+L = −
+�
+i∈Vs,j∈Vt
+�
+Sgt
+i,j log Si,j +
+�
+1 − Sgt
+i,j
+�
+log (1 − Si,j)
+�
+,
+(6)
+where Sgt is the ground truth permutation matrix. Conse-
+quently, the cross-entropy loss based on linear allocations can
+be learned end-to-end no matter the number of nodes and
+edges in the graph.
+3. EXPERIMENTS
+3.1. Experimental Setting
+Dataset. We use the Willow Object Class [13], consisting
+of 9963 images in total and is used as a benchmark by many
+methods to measure the accuracy of image classification
+and recognition algorithms.
+And IMC-PT-SparseGM [14]
+datasets, containing 16 object categories and 25061 images,
+which gather from 16 tourist attractions around the world.
+https://www.di.ens.fr/willow/research/graphlearning/.
+Implementation Details. The experiments are conducted
+using two GeForce GTX 1080 Ti GPUs. We employ a batch
+size of 8 in training and evaluate the results of our method af-
+ter 2000 epochs for each iteration. We employ the Adam [15]
+
+hs: hidden representation of source graph
+ht: hidden representation of target graph
+Position-Aware
+ Node embedding
+Iteration
+3
+Ground truth
+?
+(6)
+Update
+?
+①
+Invariant
+{1,2,3, ., 8]
+Gs
+Anchor Matrix
+Anchor-set
+Embedding layer
+Hs
+HT
+ZAND
+Accuracy
+Position-Aware
+ Node embedding
+Graph Feature Matching
+?
+Iteration
+4-3
+①2
+4-3
+6
+?
+①
+Update
+Invariant
+{1,2,3, ...,8]
+Gt
+:
+Anchor-set
+Anchor Matrix
+Embedding layerq(v,
+hu?
+AGG
+q(v, u3)
+W
+hv
+v,u1~p
+,up)
+(V,
+qFig. 4.
+Confusion matrix analysis of cross-category generalizations using the Willow ObjectClass dataset. The y-axis repre-
+sents categories of images used for training the model, and the x-axis represents categories of samples used to test. Among
+them, (a), (b), and (c) are the comparison of the generalization ability of the aforementioned baseline in matching accuracy. (d)
+represents the method proposed in this paper.
+Table 1. Matching accuracy (%) on the Willow ObjectClass.
+Method
+Car
+Duck
+Face
+M-bike
+W-bottle
+Mean
+GMN[17]
+67.90
+76.70
+99.80
+69.20
+83.10
+79.34
+NHGM[18]
+86.50
+72.20
+99.90
+79.30
+89.40
+85.50
+CIE-H[19]
+82.20
+81.20
+100.00
+90.00
+97.60
+90.20
+PIA-GM[8]
+88.60
+87.00
+100.00
+70.30
+87.80
+86.74
+PCA-GM[8]
+87.60
+83.60
+100.00
+77.60
+88.40
+87.44
+IPCA-GM[16]
+90.40
+88.60
+100.00
+83.00
+88.30
+90.06
+PA-GM (ours)
+92.70
+91.30
+100.00
+84.50
+93.80
+92.46
+Table 2. Matching accuracy (%) on the IMC-PT-SparseGM.
+Method
+Reichstag
+Sacre coeur
+St peters square
+Mean
+CIE-H[19]
+42.24
+28.47
+30.78
+33.83
+PIA-GM[8]
+71.46
+41.31
+42.64
+51.80
+PCA-GM[8]
+69.38
+39.86
+42.10
+50.40
+IPCA-GM[16]
+72.96
+43.80
+44.93
+53.89
+GANN-GM[20]
+76.02
+44.15
+50.49
+56.89
+PA-GM (ours)
+96.28
+75.93
+81.66
+84.63
+optimizer to train our models with a learning rate of 1×10−4.
+To overcome the over-smoothing problem common to graph
+neural network models, we adopt a two-layer graph embed-
+ding and restrict the degree of smoothing in our experiments.
+Metrics. We evaluate the graph matching capacity of the
+proposed method using the matching accuracy, which is de-
+fined by accuracy = � AND(Si,j, Sgt
+i,j)/N from [8, 16].
+Note that Si,j is the element of the predicted permutation ma-
+trix representing the correspondence matching of node i and
+node j from two different graphs. Similarly, Sgt
+i,j is the corre-
+spondence of the ground truth between two nodes, and N is
+the number of matching node pairs.
+3.2. Graph Matching Results on Real-world Datasets
+Table 1 and Table 2 report the overall comparison of the per-
+formance results for graph matching accuracy. The CIE-H
+method excels on the M-bike and W-bottle classes of the
+Willow ObjectClass dataset, while our method scores highest
+in the two categories of accuracy, and the average accuracy.
+By successfully incorporating the position information, our
+model achieves excellent results in the average accuracy of
+Willow ObjectClass and IMC-PT-SparseGM datasets. Based
+on these results, it can be concluded that our method performs
+well on graph matching compared with existing methods.
+3.3. Cross-category Generalization Study
+To assess the robustness and generalization ability of our
+method on the different object categories, we further conduct
+the cross-category generalization study. Specifically, we train
+each of the five classes separately, and then test the general-
+ization ability of the validation model for all five classes with
+the separate training classes. Comparing the elements of Fig.
+4, it is clear that the generalization ability of the model frame-
+work that incorporates position information is superior to the
+alternative methods. In addition, the matching accuracy for
+face data is generally rather large. This may be related to the
+relatively simple background of the data and less interference
+from noise.
+4. CONCLUSION
+In this paper, we propose a novel deep learning method for
+graph matching based on node-wise embeddings between
+graphs that combine position-aware node embeddings. Dur-
+ing the experiments, the proposed method is compared with
+alternative methods to demonstrate its robustness and effec-
+tiveness. Comparing our methodology to existing methods
+on real-world datasets demonstrates its state-of-the-art per-
+formance. Further improvements of graph matching will be
+achieved with the use of different relative positions strategies
+in future work.
+
+PIA-GM Accuracy (diag:0.8580, all:0.5800)
+PCA-GM Accuracy (diag:0.8580, all:0.5800)
+IPCA-GM Accuracy (diag:0.8580, all:0.5800)
+PA-GM Accuracy (diag:0.8580, all:0.5800)
+1.0 
+0.814
+0.374
+0.983
+0.423
+0.489
+0.832
+0.512
+0.706
+0.426
+0.425
+0.947
+0.608
+0.949
+0.509
+0.538
+0.93
+0.72
+0.996
+0.561
+0.849
+0.9
+ Duck
+Duck
+0.496
+0.458
+0.591
+0.864
+0.795
+0.365
+0.498
+0.552
+0.902
+0.908
+0.417
+0.52
+ Duck
+0.802
+0.8537
+0.445
+0.822
+0.902
+0.996
+0.665
+0.773
+0.8
+Face
+Face
+Face
+ 0.7
+0.236
+0.132
+0.998
+0.252
+0.165
+0.482
+0.424
+1.0
+0.358
+0.328
+0.618
+0.626
+1.0
+0.468
+0.474
+0.466
+0.32
+1.0
+0.437
+0.309
+Winebottle Motorbike
+ Motorbike
+: Motorbike
+: Motorbike
+0.6
+0.487
+0.47
+0.995
+0.762
+0.459
+0.383
+0.259
+0.75
+0.69
+0.283
+0.594
+0.549
+0.9
+0.792
+0.425
+0.779
+0.682
+1.0
+0.838
+0.696
+ 0.5
+Winebottle
+Winebottle 
+Winebotle 
+0.507
+0.53
+0.9699
+0.486
+0.914
+0.543
+0.606
+0.766
+0.37
+0.877
+0.64
+0.678
+0.96
+0.476
+0.92
+0.716
+0.625
+0.9279
+0.546
+0.934
+ 0.4
+Winebottle Motorbike Face
+Duck
+Car
+Winebottle Motorbike Face
+Duck
+Car
+Winebottle Motorbike Face
+Duck
+Car
+Winebottle Motorbike Face
+Duck
+Car
+(a)
+(b)
+(c)
+(d)5. REFERENCES
+[1] Steven Gold and Anand Rangarajan, “A graduated as-
+signment algorithm for graph matching,” IEEE Transac-
+tions on pattern analysis and machine intelligence, vol.
+18, no. 4, pp. 377–388, 1996.
+[2] Kexin Deng, Jie Tian, Jian Zheng, Xing Zhang, Xiao-
+qian Dai, and Min Xu,
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+[4] Miao Wang, Yu-Kun Lai, Yuan Liang, Ralph R Martin,
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+[13] Minsu Cho, Karteek Alahari, and Jean Ponce, “Learn-
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+Matas, Pascal Fua, Kwang Moo Yi, and Eduard Trulls,
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+practice,” International Journal of Computer Vision, pp.
+517–547, 2021.
+[15] Diederik P Kingma and Jimmy Ba, “Adam: A method
+for stochastic optimization,” International Conference
+on Learning Representations, 2015.
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+“Combinatorial learning of robust deep graph matching:
+an embedding based approach,” IEEE Transactions on
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+matching,” in IEEE Conference on Computer Vision and
+Pattern Recognition, 2018, pp. 2684–2693.
+[18] Runzhong Wang, Junchi Yan, and Xiaokang Yang,
+“Neural graph matching network:
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+quadratic assignment problem with extension to hy-
+pergraph and multiple-graph matching,” IEEE Trans-
+actions on Pattern Analysis and Machine Intelligence,
+2021.
+[19] Tianshu Yu, Runzhong Wang, Junchi Yan, and Baoxin
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+“Graduated assignment for joint multi-graph matching
+and clustering with application to unsupervised graph
+matching network learning.,” in NeurIPS, 2020.
+

69AzT4oBgHgl3EQf-P6_/content/tmp_files/load_file.txt

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+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQf-P6_/content/2301.01932v1.pdf,len=370
+page_content='PA-GM: POSITION-AWARE LEARNING OF EMBEDDING NETWORKS FOR DEEP GRAPH  MATCHING  Dongdong Chen1, Yuxing Dai2, Lichi Zhang1, Zhihong Zhang2, Edwin R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQf-P6_/content/2301.01932v1.pdf'}
+page_content=' Hancock3  1 Shanghai Jiao Tong University  2 Xiamen University  3 University of York  ABSTRACT  Graph matching can be formalized as a combinatorial opti-  mization problem, where there are corresponding relation-  ships between pairs of nodes that can be represented as edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQf-P6_/content/2301.01932v1.pdf'}
+page_content='  This problem becomes challenging when there are potential  ambiguities present due to nodes and edges with high simi-  larity, and there is a need to find accurate results for similar  content matching.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQf-P6_/content/2301.01932v1.pdf'}
+page_content=' In this paper, we introduce a novel end-to-  end neural network that can map the linear assignment prob-  lem into a high-dimensional space augmented with node-level  relative position information, which is crucial for improving  the method’s performance for similar content matching.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQf-P6_/content/2301.01932v1.pdf'}
+page_content=' Our  model constructs the anchor set for the relative position of  nodes and then aggregates the feature information of the tar-  get node and each anchor node based on a measure of rela-  tive position.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQf-P6_/content/2301.01932v1.pdf'}
+page_content=' It then learns the node feature representation by  integrating the topological structure and the relative position  information, thus realizing the linear assignment between the  two graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQf-P6_/content/2301.01932v1.pdf'}
+page_content=' To verify the effectiveness and generalizability of  our method, we conduct graph matching experiments, includ-  ing cross-category matching, on different real-world datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQf-P6_/content/2301.01932v1.pdf'}
+page_content='  Comparisons with different baselines demonstrate the supe-  riority of our method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQf-P6_/content/2301.01932v1.pdf'}
+page_content=' Our source code is available under  https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQf-P6_/content/2301.01932v1.pdf'}
+page_content='com/anonymous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQf-P6_/content/2301.01932v1.pdf'}
+page_content='  Index Terms— Graph Matching, Graph Embedding,  Deep Neural Network  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQf-P6_/content/2301.01932v1.pdf'}
+page_content=' INTRODUCTION  Graph matching aims to establish the relationship between  two or more graphs based on information derived from their  nodes and edges [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQf-P6_/content/2301.01932v1.pdf'}
+page_content='  Due to its ability to better express  and encode data relationships, graph matching has gained  considerable traction in computer vision and related fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQf-P6_/content/2301.01932v1.pdf'}
+page_content='  Applications of graph matching techniques include image  registration in medical image analysis [2], link analysis in  social networks [3] and image extrapolation in computer  vision[4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQf-P6_/content/2301.01932v1.pdf'}
+page_content=' Current methods for solving the graph matching  problem can be divided into two main classes: a) learning-  free or b) learning-based [5, 6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQf-P6_/content/2301.01932v1.pdf'}
+page_content=' In the first case, the critical  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQf-P6_/content/2301.01932v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQf-P6_/content/2301.01932v1.pdf'}
+page_content=' A failed example of mismatching the position of the  animal dog’s ears in the two images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQf-P6_/content/2301.01932v1.pdf'}
+page_content='  element is the mathematical method used for obtaining an  approximate solution to an intrinsically NP-hard problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQf-P6_/content/2301.01932v1.pdf'}
+page_content='  On the other hand, learning-based methods aim to improve  the solver with a number of methods including deep neural  networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQf-P6_/content/2301.01932v1.pdf'}
+page_content='  However, these methods can only compute a similarity  score for the whole graph, or rely on inefficient global match-  ing procedures[7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQf-P6_/content/2301.01932v1.pdf'}
+page_content=' For example, [8] only considers the em-  bedding of local information for nodes in the graph, leading  to a tendency to inconsistently match similar nodes from dif-  ferent regions of the graph, and thus resulted to have ambigu-  ities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQf-P6_/content/2301.01932v1.pdf'}
+page_content=' As is illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQf-P6_/content/2301.01932v1.pdf'}
+page_content=' 1, it shows that the node em-  bedding representation, which relies only on local structural  information and semantic node information, lacks sufficient  discrimination to effectively resolve these ambiguities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQf-P6_/content/2301.01932v1.pdf'}
+page_content=' As a  result, it is difficult to distinguish the left and right ears of  the dog in the example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQf-P6_/content/2301.01932v1.pdf'}
+page_content=' From this, relative positional infor-  mation is the key to the matching of diagrams, especially in  cases where the graphs have similar semantic and structural  content.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQf-P6_/content/2301.01932v1.pdf'}
+page_content='  In this paper, we introduce the idea of position-awareness  to solve the above-mentioned problem in graph matching.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQf-P6_/content/2301.01932v1.pdf'}
+page_content=' We  first construct the graph as input to the model with node fea-  tures extracted from an image, and then extract a collection  of anchor points as reference coordinates for each node in the  graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQf-P6_/content/2301.01932v1.pdf'}
+page_content=' We further propose a corresponding position-aware  node embedding algorithm to capture the relative positional  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQf-P6_/content/2301.01932v1.pdf'}
+page_content='01932v1  [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQf-P6_/content/2301.01932v1.pdf'}
+page_content='CV]  5 Jan 2023  Xinformation of the nodes in the graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQf-P6_/content/2301.01932v1.pdf'}
+page_content=' With the node-wise  graph embedding to hand, we identify a node permutation for  node-to-node correspondence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQf-P6_/content/2301.01932v1.pdf'}
+page_content='  The main contributions of this paper are as follows:  We propose a novel Position-Aware Learning of Em-  bedding Network for Deep Graph Matching (PA-GM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQf-P6_/content/2301.01932v1.pdf'}
+page_content='  To the best of our knowledge, there are no methods that  consider the learning of an embedding augmented with  relative positional information in graph matching tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQf-P6_/content/2301.01932v1.pdf'}
+page_content='  This restriction hampers their applications in terms of  matching accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQf-P6_/content/2301.01932v1.pdf'}
+page_content='  We extract the relative positional information for the  nodes in constructing the node embedding instead of  the traditional neighborhood aggregation approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQf-P6_/content/2301.01932v1.pdf'}
+page_content=' We  fully consider the positional information relevant to the  matching of keypoints, which is proved to be effective  in our experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQf-P6_/content/2301.01932v1.pdf'}
+page_content='  The proposed framework is both scalable and flexible,  and benefits from the use of a relative position coef-  ficient together with an alignment loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQf-P6_/content/2301.01932v1.pdf'}
+page_content=' Experiments  show that the model achieves state-of-the-art results on  real-world datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQf-P6_/content/2301.01932v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQf-P6_/content/2301.01932v1.pdf'}
+page_content=' METHODS  In this paper, we intend to resolve the graph matching prob-  lem based on the supervised matching of graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQf-P6_/content/2301.01932v1.pdf'}
+page_content=' Specifi-  cally, we aim to learn an end-to-end model which can extract  graph information and their matches through given pair-wise  ground-truth correspondences for a set of graphs, and which  can be further generalized to unseen graph pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQf-P6_/content/2301.01932v1.pdf'}
+page_content=' The overall  pipeline for graph matching using the position-aware embed-  ding network is presented in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQf-P6_/content/2301.01932v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQf-P6_/content/2301.01932v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQf-P6_/content/2301.01932v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQf-P6_/content/2301.01932v1.pdf'}
+page_content=' Problem Definition  In this paper, we denote a graph by the triple G = (V, A, X)  which consists of a finite set of nodes V , an adjacency ma-  trix A, and a set of node attributes X extracted from im-  ages using CNN-based models[9, 10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQf-P6_/content/2301.01932v1.pdf'}
+page_content=' We construct a vec-  tor v ∈ {0, 1}nm×1 to indicate the match of vertices in the  source graph Gs = (Vs, As, Xs) and the target graph Gt =  (Vt, At, Xt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQf-P6_/content/2301.01932v1.pdf'}
+page_content='  The vector has elements vi,j = 1 if vertex  i ∈ Vs is matched to vertex j ∈ Vt, and vi,j = 0 if other-  wise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQf-P6_/content/2301.01932v1.pdf'}
+page_content=' It is worth noting that all the vertex matches are subject  to one-to-one mapping constraints �  j∈Vt vi,j = 1 ∀i ∈ Vs  and �  i∈Vs vi,j ≤ 1 ∀j ∈ Vt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQf-P6_/content/2301.01932v1.pdf'}
+page_content=' Furthermore, we construct a  square symmetric positive matrix M ∈ Rnm×nm as the affin-  ity matrix, to encode the edge-to-edge affinity between two  graphs in the off-diagonal elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQf-P6_/content/2301.01932v1.pdf'}
+page_content=' In this way, the two-  graph matching between Gs and Gt can be formulated as an  edge-preserving, quadratic assignment programming (QAP)  problem [11]:  argmax  v  v⊤Mv  s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQf-P6_/content/2301.01932v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQf-P6_/content/2301.01932v1.pdf'}
+page_content='  �  j∈Vt  vi,j = 1 ∀i ∈ Vs,  �  i∈Vs  vi,j ≤ 1 ∀j ∈ Vt,  (1)  where v ∈ {0, 1}nm×1 and M ∈ Rnm×nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQf-P6_/content/2301.01932v1.pdf'}
+page_content=' The goal of  graph matching is to establish a correspondence between two  attributed graphs, which minimizes the sum of local and ge-  ometric costs of assignment between the vertices of the two  graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQf-P6_/content/2301.01932v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQf-P6_/content/2301.01932v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQf-P6_/content/2301.01932v1.pdf'}
+page_content=' Position-Aware Node Embedding  Pixels at neighboring positions in the image convey similar  semantic information, therefore we need to distinguish them  to resolve matching ambiguities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQf-P6_/content/2301.01932v1.pdf'}
+page_content=' Here, we refer to the nodes  used as reference positions named as anchors and propose an  effective strategy to construct an anchor set P consisting of  all nodes in the graph, denoted by P = V , serving as a sta-  ble reference for all nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQf-P6_/content/2301.01932v1.pdf'}
+page_content=' To combine information from the  nodes and the anchors, we design the information aggregation  mechanism as is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQf-P6_/content/2301.01932v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQf-P6_/content/2301.01932v1.pdf'}
+page_content='  Considering that the amount of effective information  transmission to the nodes by the anchors with different rela-  tive distances is different, we first compute a relative position  coefficient qv,u between a pair of nodes v and u as:  qv,u =  e−dv,u  �  i∈V e−dv,i ,  (2)  where dv,u is the shortest path distance between the two  nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQf-P6_/content/2301.01932v1.pdf'}
+page_content=' In practice, to effectively reduce the time complexity  of calculating the shortest path and also to reduce interfer-  ence from those anchor points that are distant from the node  under consideration, we demand that the maximum shortest  path distance does not exceed the ceiling value r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQf-P6_/content/2301.01932v1.pdf'}
+page_content=' Otherwise,  the value of the coefficient is set to infinity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQf-P6_/content/2301.01932v1.pdf'}
+page_content=' We continue to  compute the information aggregation function I(v, u)(l) for  the l-th layer between node v and u as:  I(v, u)(l) = qv,uCONCAT(h(l−1)  v  , h(l−1)  u  ),  (3)  where h(l−1)  v  and h(l−1)  u  are the feature representation and po-  sition representation in the (l − 1)-th layer, and are combined  through the message aggregation function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQf-P6_/content/2301.01932v1.pdf'}
+page_content=' We further aggre-  gate the information from all node-anchor pairs to obtaining a  new representation in a high dimensional space using the non-  linear variation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQf-P6_/content/2301.01932v1.pdf'}
+page_content=' This hidden representation can be computed  using the update  h(l)  v  = σ(AGG(I(v, u)(l)| ∀u ∈ V )W (l)),  (4)  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQf-P6_/content/2301.01932v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQf-P6_/content/2301.01932v1.pdf'}
+page_content=' Overview of the end-to-end position embedding networks for deep graph matching.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQf-P6_/content/2301.01932v1.pdf'}
+page_content=' The blue source graph Gs and the  green target graph Gt are extracted with the node-wise graph feature representation in high-level through the two frameworks  of position-aware node embedding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQf-P6_/content/2301.01932v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQf-P6_/content/2301.01932v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQf-P6_/content/2301.01932v1.pdf'}
+page_content=' The position-aware embedding first aggregates the  message of each target node hv and each anchor point hui in  the anchor set via the aggregation function I(v, u) based on  the relative position coefficients q(v, u).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQf-P6_/content/2301.01932v1.pdf'}
+page_content=' The representation  matrix is then further aggregated using the learnable function  AGG and is finally transformed in a non-linear way to obtain  the feature output hv for the target node in the current layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQf-P6_/content/2301.01932v1.pdf'}
+page_content='  where AGG is typically a permutation-invariant function  (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQf-P6_/content/2301.01932v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQf-P6_/content/2301.01932v1.pdf'}
+page_content=', sum), and W (l) is a learnable weight vector for the l-th  layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQf-P6_/content/2301.01932v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQf-P6_/content/2301.01932v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQf-P6_/content/2301.01932v1.pdf'}
+page_content=' Graph Feature Matching  Utilizing the proposed position-aware node embedding, the  proposed scheme encodes each node with position informa-  tion into a high-level embedding space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQf-P6_/content/2301.01932v1.pdf'}
+page_content=' In this way, we can  simplify the second-order affinity matrix of paired graphs to  a be a linear one with position-aware learning of the embed-  ding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQf-P6_/content/2301.01932v1.pdf'}
+page_content=' With the final hidden representation of the source graph  Hs ∈ Rn∗F ′ and the target graph Ht ∈ Rm∗F ′ to hand, we  can obtain a soft correspondence between these graphs with  a node-wise affinity matrix through the inner product of the  embeddings of the two graphs being matched.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQf-P6_/content/2301.01932v1.pdf'}
+page_content=' Specifically, to  satisfy the condition that the original graph maps injectively  onto the target graph, we apply the Sinkhorn normalization  [12] to obtain rectangular doubly-stochastic correspondence  matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQf-P6_/content/2301.01932v1.pdf'}
+page_content='  S = sinkhorn(HsHT  t ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQf-P6_/content/2301.01932v1.pdf'}
+page_content='  (5)  We further employ the cross-entropy loss function as the  permutation loss between the predicted permutation matrix  and the ground truth:  L = −  �  i∈Vs,j∈Vt  �  Sgt  i,j log Si,j +  �  1 − Sgt  i,j  �  log (1 − Si,j)  �  ,  (6)  where Sgt is the ground truth permutation matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQf-P6_/content/2301.01932v1.pdf'}
+page_content=' Conse-  quently, the cross-entropy loss based on linear allocations can  be learned end-to-end no matter the number of nodes and  edges in the graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQf-P6_/content/2301.01932v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQf-P6_/content/2301.01932v1.pdf'}
+page_content=' EXPERIMENTS  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQf-P6_/content/2301.01932v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQf-P6_/content/2301.01932v1.pdf'}
+page_content=' Experimental Setting  Dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQf-P6_/content/2301.01932v1.pdf'}
+page_content=' We use the Willow Object Class [13], consisting  of 9963 images in total and is used as a benchmark by many  methods to measure the accuracy of image classification  and recognition algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQf-P6_/content/2301.01932v1.pdf'}
+page_content='  And IMC-PT-SparseGM [14]  datasets, containing 16 object categories and 25061 images,  which gather from 16 tourist attractions around the world.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQf-P6_/content/2301.01932v1.pdf'}
+page_content='  https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQf-P6_/content/2301.01932v1.pdf'}
+page_content='di.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQf-P6_/content/2301.01932v1.pdf'}
+page_content='ens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQf-P6_/content/2301.01932v1.pdf'}
+page_content='fr/willow/research/graphlearning/.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQf-P6_/content/2301.01932v1.pdf'}
+page_content='  Implementation Details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQf-P6_/content/2301.01932v1.pdf'}
+page_content=' The experiments are conducted  using two GeForce GTX 1080 Ti GPUs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQf-P6_/content/2301.01932v1.pdf'}
+page_content=' We employ a batch  size of 8 in training and evaluate the results of our method af-  ter 2000 epochs for each iteration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQf-P6_/content/2301.01932v1.pdf'}
+page_content=' We employ the Adam [15]  hs: hidden representation of source graph  ht: hidden representation of target graph  Position-Aware  Node embedding  Iteration  3  Ground truth  ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQf-P6_/content/2301.01932v1.pdf'}
+page_content='  (6)  Update  ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQf-P6_/content/2301.01932v1.pdf'}
+page_content='  ①  Invariant  {1,2,3, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQf-P6_/content/2301.01932v1.pdf'}
+page_content=', 8]  Gs  Anchor Matrix  Anchor-set  Embedding layer  Hs  HT  ZAND  Accuracy  Position-Aware  Node embedding  Graph Feature Matching  ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQf-P6_/content/2301.01932v1.pdf'}
+page_content='  Iteration  4-3  ①2  4-3  6  ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQf-P6_/content/2301.01932v1.pdf'}
+page_content='  ①  Update  Invariant  {1,2,3, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQf-P6_/content/2301.01932v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQf-P6_/content/2301.01932v1.pdf'}
+page_content=',8]  Gt  :  Anchor-set  Anchor Matrix  Embedding layerq(v,  hu?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQf-P6_/content/2301.01932v1.pdf'}
+page_content='  AGG  q(v, u3)  W  hv  v,u1~p  ,up)  (V,  qFig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQf-P6_/content/2301.01932v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQf-P6_/content/2301.01932v1.pdf'}
+page_content='  Confusion matrix analysis of cross-category generalizations using the Willow ObjectClass dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQf-P6_/content/2301.01932v1.pdf'}
+page_content=' The y-axis repre-  sents categories of images used for training the model, and the x-axis represents categories of samples used to test.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQf-P6_/content/2301.01932v1.pdf'}
+page_content=' Among  them, (a), (b), and (c) are the comparison of the generalization ability of the aforementioned baseline in matching accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQf-P6_/content/2301.01932v1.pdf'}
+page_content=' (d)  represents the method proposed in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQf-P6_/content/2301.01932v1.pdf'}
+page_content='  Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQf-P6_/content/2301.01932v1.pdf'}
+page_content=' Matching accuracy (%) on the Willow ObjectClass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQf-P6_/content/2301.01932v1.pdf'}
+page_content='  Method  Car  Duck  Face  M-bike  W-bottle  Mean  GMN[17]  67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQf-P6_/content/2301.01932v1.pdf'}
+page_content='90  76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQf-P6_/content/2301.01932v1.pdf'}
+page_content='70  99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQf-P6_/content/2301.01932v1.pdf'}
+page_content='80  69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQf-P6_/content/2301.01932v1.pdf'}
+page_content='20  83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQf-P6_/content/2301.01932v1.pdf'}
+page_content='10  79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQf-P6_/content/2301.01932v1.pdf'}
+page_content='34  NHGM[18]  86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQf-P6_/content/2301.01932v1.pdf'}
+page_content='50  72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQf-P6_/content/2301.01932v1.pdf'}
+page_content='20  99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQf-P6_/content/2301.01932v1.pdf'}
+page_content='90  79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQf-P6_/content/2301.01932v1.pdf'}
+page_content='30  89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQf-P6_/content/2301.01932v1.pdf'}
+page_content='40  85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQf-P6_/content/2301.01932v1.pdf'}
+page_content='50  CIE-H[19]  82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQf-P6_/content/2301.01932v1.pdf'}
+page_content='20  81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQf-P6_/content/2301.01932v1.pdf'}
+page_content='20  100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQf-P6_/content/2301.01932v1.pdf'}
+page_content='00  90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQf-P6_/content/2301.01932v1.pdf'}
+page_content='00  97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQf-P6_/content/2301.01932v1.pdf'}
+page_content='60  90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQf-P6_/content/2301.01932v1.pdf'}
+page_content='20  PIA-GM[8]  88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQf-P6_/content/2301.01932v1.pdf'}
+page_content='60  87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQf-P6_/content/2301.01932v1.pdf'}
+page_content='00  100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQf-P6_/content/2301.01932v1.pdf'}
+page_content='00  70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQf-P6_/content/2301.01932v1.pdf'}
+page_content='30  87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQf-P6_/content/2301.01932v1.pdf'}
+page_content='80  86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQf-P6_/content/2301.01932v1.pdf'}
+page_content='74  PCA-GM[8]  87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQf-P6_/content/2301.01932v1.pdf'}
+page_content='60  83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQf-P6_/content/2301.01932v1.pdf'}
+page_content='60  100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQf-P6_/content/2301.01932v1.pdf'}
+page_content='00  77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQf-P6_/content/2301.01932v1.pdf'}
+page_content='60  88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQf-P6_/content/2301.01932v1.pdf'}
+page_content='40  87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQf-P6_/content/2301.01932v1.pdf'}
+page_content='44  IPCA-GM[16]  90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQf-P6_/content/2301.01932v1.pdf'}
+page_content='40  88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQf-P6_/content/2301.01932v1.pdf'}
+page_content='60  100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQf-P6_/content/2301.01932v1.pdf'}
+page_content='00  83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQf-P6_/content/2301.01932v1.pdf'}
+page_content='00  88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQf-P6_/content/2301.01932v1.pdf'}
+page_content='30  90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQf-P6_/content/2301.01932v1.pdf'}
+page_content='06  PA-GM (ours)  92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQf-P6_/content/2301.01932v1.pdf'}
+page_content='70  91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQf-P6_/content/2301.01932v1.pdf'}
+page_content='30  100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQf-P6_/content/2301.01932v1.pdf'}
+page_content='00  84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQf-P6_/content/2301.01932v1.pdf'}
+page_content='50  93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQf-P6_/content/2301.01932v1.pdf'}
+page_content='80  92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQf-P6_/content/2301.01932v1.pdf'}
+page_content='46  Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQf-P6_/content/2301.01932v1.pdf'}
+page_content=' Matching accuracy (%) on the IMC-PT-SparseGM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQf-P6_/content/2301.01932v1.pdf'}
+page_content='  Method  Reichstag  Sacre coeur  St peters square  Mean  CIE-H[19]  42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQf-P6_/content/2301.01932v1.pdf'}
+page_content='24  28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQf-P6_/content/2301.01932v1.pdf'}
+page_content='47  30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQf-P6_/content/2301.01932v1.pdf'}
+page_content='78  33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQf-P6_/content/2301.01932v1.pdf'}
+page_content='83  PIA-GM[8]  71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQf-P6_/content/2301.01932v1.pdf'}
+page_content='46  41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQf-P6_/content/2301.01932v1.pdf'}
+page_content='31  42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQf-P6_/content/2301.01932v1.pdf'}
+page_content='64  51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQf-P6_/content/2301.01932v1.pdf'}
+page_content='80  PCA-GM[8]  69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQf-P6_/content/2301.01932v1.pdf'}
+page_content='38  39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQf-P6_/content/2301.01932v1.pdf'}
+page_content='86  42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQf-P6_/content/2301.01932v1.pdf'}
+page_content='10  50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQf-P6_/content/2301.01932v1.pdf'}
+page_content='40  IPCA-GM[16]  72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQf-P6_/content/2301.01932v1.pdf'}
+page_content='96  43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQf-P6_/content/2301.01932v1.pdf'}
+page_content='80  44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQf-P6_/content/2301.01932v1.pdf'}
+page_content='93  53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQf-P6_/content/2301.01932v1.pdf'}
+page_content='89  GANN-GM[20]  76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQf-P6_/content/2301.01932v1.pdf'}
+page_content='02  44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQf-P6_/content/2301.01932v1.pdf'}
+page_content='15  50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQf-P6_/content/2301.01932v1.pdf'}
+page_content='49  56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQf-P6_/content/2301.01932v1.pdf'}
+page_content='89  PA-GM (ours)  96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQf-P6_/content/2301.01932v1.pdf'}
+page_content='28  75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQf-P6_/content/2301.01932v1.pdf'}
+page_content='93  81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQf-P6_/content/2301.01932v1.pdf'}
+page_content='66  84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQf-P6_/content/2301.01932v1.pdf'}
+page_content='63  optimizer to train our models with a learning rate of 1×10−4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQf-P6_/content/2301.01932v1.pdf'}
+page_content='  To overcome the over-smoothing problem common to graph  neural network models, we adopt a two-layer graph embed-  ding and restrict the degree of smoothing in our experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQf-P6_/content/2301.01932v1.pdf'}
+page_content='  Metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQf-P6_/content/2301.01932v1.pdf'}
+page_content=' We evaluate the graph matching capacity of the  proposed method using the matching accuracy, which is de-  fined by accuracy = � AND(Si,j, Sgt  i,j)/N from [8, 16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQf-P6_/content/2301.01932v1.pdf'}
+page_content='  Note that Si,j is the element of the predicted permutation ma-  trix representing the correspondence matching of node i and  node j from two different graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQf-P6_/content/2301.01932v1.pdf'}
+page_content=' Similarly, Sgt  i,j is the corre-  spondence of the ground truth between two nodes, and N is  the number of matching node pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQf-P6_/content/2301.01932v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQf-P6_/content/2301.01932v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQf-P6_/content/2301.01932v1.pdf'}
+page_content=' Graph Matching Results on Real-world Datasets  Table 1 and Table 2 report the overall comparison of the per-  formance results for graph matching accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQf-P6_/content/2301.01932v1.pdf'}
+page_content=' The CIE-H  method excels on the M-bike and W-bottle classes of the  Willow ObjectClass dataset, while our method scores highest  in the two categories of accuracy, and the average accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQf-P6_/content/2301.01932v1.pdf'}
+page_content='  By successfully incorporating the position information, our  model achieves excellent results in the average accuracy of  Willow ObjectClass and IMC-PT-SparseGM datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQf-P6_/content/2301.01932v1.pdf'}
+page_content=' Based  on these results, it can be concluded that our method performs  well on graph matching compared with existing methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQf-P6_/content/2301.01932v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQf-P6_/content/2301.01932v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQf-P6_/content/2301.01932v1.pdf'}
+page_content=' Cross-category Generalization Study  To assess the robustness and generalization ability of our  method on the different object categories, we further conduct  the cross-category generalization study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQf-P6_/content/2301.01932v1.pdf'}
+page_content=' Specifically, we train  each of the five classes separately, and then test the general-  ization ability of the validation model for all five classes with  the separate training classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQf-P6_/content/2301.01932v1.pdf'}
+page_content=' Comparing the elements of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQf-P6_/content/2301.01932v1.pdf'}
+page_content='  4, it is clear that the generalization ability of the model frame-  work that incorporates position information is superior to the  alternative methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQf-P6_/content/2301.01932v1.pdf'}
+page_content=' In addition, the matching accuracy for  face data is generally rather large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQf-P6_/content/2301.01932v1.pdf'}
+page_content=' This may be related to the  relatively simple background of the data and less interference  from noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQf-P6_/content/2301.01932v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQf-P6_/content/2301.01932v1.pdf'}
+page_content=' CONCLUSION  In this paper, we propose a novel deep learning method for  graph matching based on node-wise embeddings between  graphs that combine position-aware node embeddings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQf-P6_/content/2301.01932v1.pdf'}
+page_content=' Dur-  ing the experiments, the proposed method is compared with  alternative methods to demonstrate its robustness and effec-  tiveness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQf-P6_/content/2301.01932v1.pdf'}
+page_content=' Comparing our methodology to existing methods  on real-world datasets demonstrates its state-of-the-art per-  formance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQf-P6_/content/2301.01932v1.pdf'}
+page_content=' Further improvements of graph matching will be  achieved with the use of different relative positions strategies  in future work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQf-P6_/content/2301.01932v1.pdf'}
+page_content='  PIA-GM Accuracy (diag:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQf-P6_/content/2301.01932v1.pdf'}
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+arXiv:2301.03080v1  [math.NA]  8 Jan 2023
+Perron-Frobenius operator filter for stochastic dynamical systems
+Ningxin Liu∗
+Lijian Jiang†
+Abstract
+The filtering problems are derived from a sequential minimization of a quadratic function
+representing a compromise between model and data.
+In this paper, we use the Perron-
+Frobenius operator in stochastic process to develop a Perron-Frobenius operator filter. The
+proposed method belongs to Bayesian filtering and works for non-Gaussian distributions for
+nonlinear stochastic dynamical systems. The recursion of the filtering can be characterized
+by the composition of Perron-Frobenius operator and likelihood operator.
+This gives a
+significant connection between the Perron-Frobenius operator and Bayesian filtering. We
+numerically fulfil the recursion through approximating the Perron-Frobenius operator by
+Ulam’s method. In this way, the posterior measure is represented by a convex combination
+of the indicator functions in Ulam’s method.
+To get a low rank approximation for the
+Perron-Frobenius operator filter, we take a spectral decomposition for the posterior measure
+by using the eigenfunctions of the discretized Perron-Frobenius operator. A convergence
+analysis is carried out and shows that the Perron-Frobenius operator filter achieves a higher
+convergence rate than the particle filter, which uses Dirac measures for the posterior. The
+proposed method is explored for the data assimilation of the stochastic dynamical systems.
+A few numerical examples are presented to illustrate the advantage of the Perron-Frobenius
+operator filter over particle filter and extend Kalman filter.
+keywords: Perron-Frobenius operator, Bayesian filtering, stochastic dynamical systems,
+particle filter
+1
+Introduction
+In recent years, the operator-based approach has been extensively exploited to analyze dy-
+namical systems. The two primary candidates of the approach are Perron-Frobenius operator
+and its dual operator, Koopman operator. Many data-driven methods have been developed
+for numerical approximation of these operators. The two operators are motivated to approxi-
+mate the dynamical system’s behavior from different perspectives. The Koopman operator is
+used to study the evolution of observations, while Perron-Frobenius operator (PFO) charac-
+terizes the transition of densities. Therefore, the PFO deals with the system’s uncertainties
+∗School of Mathematical Sciences, Tongji University, Shanghai 200092, China. ([email protected]).
+†School
+of
+Mathematical
+Sciences,
+Tongji
+University,
+Shanghai
+200092,
+China.
+([email protected]).
+1
+
+in the form of probability density functions of the state. In practice, it determines an abso-
+lutely continuous probability measure preserved by a given measurable transformation on a
+measure space.
+The Perron-Frobenius operator has been widely used to characterize the global asymp-
+totic behavior of dynamical systems derived from many different domains such as fluid dy-
+namics [1], molecular dynamics [2], meteorology and atmospheric sciences [3], and to estimate
+invariant sets or metastable sets with a toolbox like in [4]. It is of great interest to study the
+invariant density of Perron-Frobenius operator [5] and design efficient numerical approaches.
+Then one can apply ergodic theorems to the statistical properties of deterministic dynamical
+systems.
+Since PFO is able to transport density of a Markov process, its approximation is necessary
+for numerical model transition probability of the Markov process. Many different numerical
+methods [6], such as Ulam’s method and Petrov-Galerkin method, are proposed for approx-
+imation of the Perron-Frobenius operator.
+As the PFO operates on infinite-dimensional
+spaces, it is natural to project it onto a finite-dimensional subspace spanned by suitable
+basis functions. The projection is usually accomplished by Galerkin methods with weak ap-
+proximation. It was originally proposed by Ulam [7], who suggested that one can study the
+discrete Perron-Frobenius operator on the finite-dimensional subspace L1 of indicator func-
+tions according to a finite partition of the region. Convergence analysis of Ulam’s method is
+discussed in many literatures [8, 9].
+The classical filtering problems in stochastic processes are investigated in [10, 11]. In the
+paper, the models of filtering problems are considered with discrete-time and continuous-
+time stochastic processes defined by the solutions of SDEs, which can model a majority of
+stochastic dynamical systems in the real world. The filtering methods have been widely used
+for geophysical applications, such as oceanography [12], oil recovery [13], atmospheric science,
+and weather forecast [14]. Remarkably, as one of the filtering methods, Kalman filter [15] has
+been well-known for low-dimensional engineering applications in linear Gaussian models, and
+it has been also developed and utilized in many other fields [16–18]. For nonlinear problems,
+the classical filters, such as 3DVAR [19], Extended Kalman filter [10] and Ensemble Kalman
+filter [20], usually invoke a Gaussian ansatz.
+They are often used in the scenarios with
+small noisy observation and high dimensional spaces. However, these extensions rely on the
+invocations of Gaussian assumption. As a sequential Monte Carlo method, particle filter [21]
+is able to work well for the nonlinear and non-Gaussian filtering problems. It can be proved
+to estimate true posterior filtering problems in the limit of large number of particles.
+Although the particle filter (PF) can treat the nonlinear and non-Gaussian filtering prob-
+lems, it has some limitations in practice. First of all, particle filter handles well in low-
+dimensional systems, but it may occur degeneracy [22] when the systems have very large
+scale. It means that the maximum of the weights associated with the sample ensemble con-
+verges to one as the system dimension tends to infinity. To avoid degeneracy, it requires
+a great number of particles that scales exponentially with the system dimension. This is a
+manifestation of the curse of dimensionality. Resampling, adding jitter and localisation are
+introduced to circumvent this curse of dimensionality and get the accurate estimation of high-
+dimensional probability density functions (PDFs). The particle filter also does not perform
+well in geophysical applications of data assimilation, because the data in these application
+have strongly constraints on particle location [23]. This impacts on the filtering performance.
+2
+
+Besides, the prior knowledge of the transition probability density functions in particle filter
+is necessary to be known, and the efficient sampling methods such as acceptance-rejection
+method and Markov chain Monte Carlo, need to be used for complicated density functions.
+The sampling is particularly a challenge in high dimensional spaces. To overcome these diffi-
+culties, we propose a Perron-Frobenius operator filter (PFOF), which does not use particles
+and any sampling methods. The information of prior probability density is not required in
+the method, which needs data information instead. The method works well in nonlinear and
+non-Gaussian models.
+In this paper, we propose PFOF to treat nonlinear stochastic filtering problems. The
+method transfer filtering distribution with the Perron-Frobenius operator. For filtering prob-
+lems, the update of filtering distribution involves two steps: predication and analysis. In
+prediction, the density is transported with a transition density function given by a Markov
+chain of the underlying system. In analysis, the density is corrected by Bayes’ rule under
+the likelihood function given by observations. Thus, the update of filtering distribution can
+be expressed as a composition of PFO and likelihood functions. In the simulation process,
+Ulam’ method is used to discretize the PFO and project it onto a finite-dimensional space
+of indicator functions. Hence the filtering density is also projected onto the subspace and
+is represented by a linear convex combination of the indicator functions. The recursion of
+filtering distribution is then expressed by a linear map of weights vectors associated with
+the basis functions via the PFO and likelihood function. For the high dimensional problems,
+we propose a low-rank PFOF (lr-PFOF) using a spectral decomposition. To this end, we
+first use the eigenfunctions of the discretized PFO to represent the spectral decomposition of
+the density functions. Then we make a truncation of the decomposition and use the eigen-
+functions corresponding to the first few dominant eigenvalues. This can improve the online
+assimilation efficiency. The idea of PFO is extended to the continuous-time filtering prob-
+lems. In these problems, Zakai equation characterizes the transition of filtering density. We
+utilize the approximation of the Perron-Frobenius operator to compute the Zaikai equation
+and obtain the posterior density functions of the continuous-time filtering problems.
+We compare the proposed method with the particle filter. For PFOF, we give an error
+estimate in the total-variance distance between the approximated posterior measure and the
+truth posterior. The estimate implies that PFOF achieves a convergence rate O( 1
+N ), which
+is faster than particle filters with the same number N of basis functions. Our numerical
+results show that PFOF also renders better accuracy than that of extended Kalman filter.
+The rest of the paper is organized as follows. In Section 2, we express the Bayesian
+filter in terms of the Perron-Frobenius operator. In Section 3, we present the recursion of
+the filtering empirical measure with an approximated Perron-Frobenius operator. Then we
+derive PFOF as well as lr-PFOF, and analyze an error estimate subsequently.
+PFOF is
+also extended to the continuous-time filtering problems. A comprehensive comparison with
+particle filter is give in Section 4. A few numerical results of stochastic filtering problems
+are given in Section 5. Section 6 concludes the paper in a summary.
+3
+
+2
+Preliminaries
+We give a background review on Perron-Frobenius operator [24] (PFO) and Bayesian filter
+in this section. The Perron-Frobenius operator transports the distributions over state space
+and describes the stochastic behavior of the dynamical systems. The framework of Bayesian
+filter is introduced and summarized as a recursive formula with PFO.
+2.1
+Perron-Frobenius operator
+Let X be a metric space, B the Borel-σ-algebra on X, and Φ : X → X a nonsingular
+transformation. Let M denote the space of all finite measures on (X, B) and µ is a finite
+measure. The phase space is defined on a measure space (X, B, µ). The Perron-Frobenius
+operator P : M → M is a linear and infinite-dimensional operator defined by
+Pµ(A) = µ(Φ−1(A)),
+∀A ∈ B.
+(2.1)
+The PFO is a linear, positive and non-expansive operator, and hence a Markov operator.
+We can also track the action on distributions in the function space. In the paper, we denote
+L1(X) := L1(X, B, µ). Let f ∈ L1(X) be the probability density function (PDF) of a X-
+valued random variable x. Since Φ is a nonsingular with respect to µ, there is a g ∈ L1(X)
+satisfying
+�
+Φ−1(A) fdµ =
+�
+A gdµ for all A ∈ B. Then g is the function characterizing the
+distribution of Φ(x). The mapping f �→ g, defined uniquely by a linear operator P : L1(X) →
+L1(X) :
+�
+A
+Pf dµ =
+�
+Φ−1(A)
+f dµ,
+∀A ∈ B.
+(2.2)
+The operator P is called the Perron-Frobenius operator. With the definition (2.1) and (2.2),
+we make the connection between probability density function and the measure associated
+with the PFO. When f is a probability density function with respect to an absolutely con-
+tinuous probability measure µ ∈ M(X), g is another PDF with respect to the absolutely
+continuous probability measure µ ◦ Φ−1. In addition, the measure µ ∈ M(X) is an invariant
+measure of P when Pµ = µ holds.
+Let Ψ : R+ × X → X be a nonsingular flow map for a deterministic continuous-time
+dynamical system. Then Ψτ : X → X is nonsingular for each τ ∈ R+. The transfer operator
+Pτ : L1(X) → L1(X) is time-dependent and has an analogous definition to (2.2), such that
+�
+A
+Pτf dµ =
+�
+Ψ−1
+τ
+(A)
+f dµ.
+The {Pτ}τ≥0 forms a semigroup of the Perron-Frobenius operators. We note that {Pτ}τ≥0
+has an infinitesimal generator AP F by Hille-Yosida Theorem.
+Let us consider the Perron-Frobenius operator in stochastic dynamic systems and the
+infinitesimal generator of PFO associated to the stochastic solution semiflow induced by
+a stochastic dynamical equation (SDE). Let b : X → X and σ : X → X be smooth
+time-invariant functions. Suppose that a stochastic process xt is the solution to the time-
+homogeneous stochastic differential equation:
+dxt = b(xt)dt + σ(xt)dWt,
+x(t0) ∼ ρ0,
+(2.3)
+4
+
+where Wt is a standard Brownian motion. In this case, the distribution of the stochastic
+process xt can be described by a semigroup of Perron-Frobenius operators {Pτ}τ>0 on L1(X).
+The generator of {Pτ}τ>0 is a second-order differential operator on X. The PDE defined by
+the generator describes the evolution of the probability density of xt.
+Suppose that Φ is the mapping of the stochastic dynamical system (2.3) and Φ(x) is an
+X-valued random variable over the probability space (X, B, µ). Given a stochastic transition
+function pτ : X × B → [0, 1] induced by Φ, we consider probability measure µ translated
+with a linear operator defined in terms of the transition function pτ(x, ·). The stochastic
+PFO [25] Pτ : M → M is defined by
+Pτµ(A) =
+�
+X
+pτ(x, A) dµ(x),
+∀A ∈ B.
+(2.4)
+If pτ(x, ·) is absolutely continuous to µ for all x ∈ X, there exists a nonnegative transition
+density function qτ : X × X → R with qτ(x, ·) ∈ L1(X) and
+P(xt+τ ∈ A|xt = x) =
+�
+A
+qτ(x, y)dµ(y),
+∀A ∈ B.
+The transition density function is the infinite-dimensional counterpart of the transition ma-
+trix for a Markov chain. Now we define the stochastic PFO associated with transition density.
+If f ∈ L1(X) is a probability density function, the Perron-Frobenius semigroup of operators
+Pτ : L1(X) → L1(X), τ > 0, is defined by
+Pτf(y) =
+�
+X
+qτ(x, y)f(x) dµ(x).
+The PFO Pτ defined here translates the probability density function of xt with time. Let ρ
+be a probability density. The infinitesimal generator AP F of Pτ is given by
+AP Fρ = −∇ · (bρ) + 1
+2∇ · ∇ · (σσTρ).
+We assume that �ρ : [0, ∞) × X → [0, ∞) is the probability density function of the solution
+xt in (2.3) and ρ0 is the density function of the initial condition x0.
+Then �ρ solves the
+Fokker-Planck equation,
+
+
+
+∂�ρ
+∂t = −∇ · (b�ρ) + 1
+2∇ · ∇ · (σσT �ρ),
+(t, x) ∈ (0, ∞) × X,
+�ρ(0, x) = ρ0(x).
+If the phase space X is compact and b ∈ C3(X, X), the equation has a unique solution,
+which is given by
+�ρ(t, x) = Ptρ0(x).
+2.2
+Bayesian filter
+In this section, we present the framework of Bayesian filter in discrete time from the per-
+spective of Bayes’ rule.
+In filtering problems, a state model and an observation model
+5
+
+are combined to estimate the posterior distribution, which is a conditional distribution
+of the state given by observation. Let us consider the dynamical model governed by the
+flow Ψ ∈ C(Rn, Rn) with noisy observations y = {yj}j∈Z+ depending on the function
+h(x) : Rn → Rp:
+�
+xj+1 = Ψ(xj) + ξj, j ∈ N, x0 ∼ ρ0,
+yj+1 = h(xj+1) + ηj+1, j ∈ N,
+(2.5)
+where ξ := {ξj}j∈N is an i.i.d. sequence with ξj ∼ N(0, Σ) and η := {ηj}j∈Z+ is an i.i.d.
+sequence with ηj ∼ N(0, R). Let Yj = {yl}j
+l=1 denote the data up to time tj. The filtering
+problem aims to determine the posterior PDF p(xj|Yj) of the random variable xj|Yj and the
+sequential updating of the PDF as the data increases. The Bayesian filtering involves two
+steps: prediction and analysis. It provides a derivation of p(xj+1|Yj+1) from p(xj|Yj). The
+prediction is concerned with the map p(xj|Yj) �→ p(xj+1|Yj) and the analysis derives the map
+p(xj+1|Yj) �→ p(xj+1|Yj+1) by Bayes’s formula.
+Prediction
+p(xj+1|Yj) =
+�
+Rn p(xj+1|Yj, xj)p(xj|Yj)dxj
+=
+�
+Rn p(xj+1|xj)p(xj|Yj)dxj.
+(2.6)
+Note that p(xj+1|Yj, xj) = p(xj+1|xj), because Yj provides indirect information about deter-
+mining xj+1. Since p(xj+1|xj) is specified by the underlying model (2.5) and
+p(xj+1|xj) ∝ exp(−1
+2|Σ− 1
+2(xj+1 − Ψ(xj))|2),
+(2.7)
+the prediction provides the map from p(xj|Yj) to p(xj+1|Yj). Let �µjbe the prior probability
+measure corresponding to the density p(xj|Yj−1) and µj be the posterior probability measure
+on corresponding to the density p(xj|Yj). The stochastic process {xj, j ∈ N} of (2.5) is a
+Markov chain with the transition kernel p(·, ·) determined by p(xj, xj+1) = p(xj+1|xj). Then
+we can rewrite (2.6) as
+�µj+1(·) = (Pµj)(·) :=
+�
+Rn p(xj, ·)µj(dxj) =
+�
+Rn p(xj, ·)dµ(xj)1.
+(2.8)
+In particular, the operator P coincides with the Perron-Frobenius operator defined in (2.4).
+Analysis
+p(xj+1|Yj+1) = p(xj+1|Yj, yj+1)
+= p(yj+1|xj+1, Yj)p(xj+1|Yj)
+p(yj+1|Yj)
+= p(yj+1|xj+1)p(xj+1|Yj)
+p(yj+1|Yj)
+.
+(2.9)
+Note that p(yj+1|xj+1, Yj) = p(yj+1|xj+1) and Bayes’s formula is used in the second equality.
+The likelihood function p(yj+1|xj+1) is determined by the observation model: p(yj+1|xj+1) ∝
+1Refer to [26], if the function f ∈ L1(X) on a measure space (X, B, µ) is said to be µ integrable, we have
+�
+fdµ =
+�
+f(x)µ(dx) =
+�
+f(x)dµ(x).
+6
+
+exp(−1
+2|R− 1
+2(yj+1 − h(xj+1))|2). Let
+gj(xj+1) := exp(−1
+2|R− 1
+2(yj+1 − h(xj+1))|2).
+(2.10)
+The analysis provides a map from p(xj+1|Yj) to p(xj+1|Yj+1), so we can represent the update
+of the measure µj+1(·) by
+µj+1(·) = (Lj�µj+1)(·) :=
+gj(xj+1)�µj+1(·)
+�
+Rn gj(xj+1)�µj+1(·),
+(2.11)
+where the likelihood operator Lj is defined by
+(Ljµ)(dx) =
+gj(x)µ(dx)
+�
+Rn gj(x)µ(dx).
+(2.12)
+In general, the prediction and analysis provide the mapping from µj to µj+1. The prediction
+maps µj to �µj+1 through the Perron-Frobenius operator P, while the analysis maps �µj+1 to
+µj+1 through the likelihood operator Lj. Then we represent the µj+1 using formulas (2.8)
+and (2.11), and summarize Bayesian filtering as
+µj+1 = LjPµj.
+(2.13)
+The µ0 is assumed to be a known initial probability measure. We note that P does not
+depend on j, because the prediction step is governed by the same Markov process at each
+j. However, Lj depends on j because the different observations are used to compute the
+likelihood at each j. In this way, the evolution of µj processes through a linear operator P
+and a nonlinear operator Lj. The approximation of µj can be achieved by the numerical
+iteration of (2.13).
+3
+Bayesian filter in terms of Perron-Frobenius opera-
+tor
+It is noted that the Perron-Frobenius operator translates a probability density function
+with time according to the flow of the dynamics. We extend the idea to filtering problems
+to represent the transition of the posterior probability density function, i.e., the filtering
+distribution.
+Therefore, we propose a filtering method: Perron-Frobenius operator filter
+(PFOF). In the proposed method, the density function is projected onto an approximation
+subspace spanned by indicator functions. The fluctuation of the density function, which is
+approximated by weights vector, is transferred by PFO and likelihood operator. Moreover,
+we present a low-rank Perron-Frobenius operator filter (lr-PFOF), which is a modified version
+of the PFOF.
+3.1
+Perron-Frobenius operator filter
+The iteration (2.13) is helpful to design a filter method. According to definition (2.8), the
+operator P in the iteration is Perron-Frobenius operator corresponding to the flow Ψ of the
+7
+
+model (2.5). Based on the idea, we propose a Perron-Frobenius operator filter, which utilizes
+Ulam’s method [7] to approximate operator P in the iteration. In PFOF, we simply use P
+for Pτ because the discrete time steps of the state model keep the same. In this manner, the
+iteration of filtering distribution of PFOF becomes
+µN
+j+1 = LjPNµN
+j ,
+µN
+0 = µ0,
+(3.14)
+where PN calculated by the Ulam’s method is an approximation of P. Ulam’s method is
+a Galerkin projection method to discretize the Perron-Frobenius operator. We first give a
+discretisation of the phase space. Let B = {B1, · · · , BN} ⊂ B be a finite number of measure
+boxes and a disjoint partition of phase space X.
+The indicator function is a piecewise
+constant function and is defined by
+1Bi(x) =
+�
+1,
+if x ∈ Bi,
+0,
+otherwise.
+(3.15)
+Ulam proposed to use the space of a family of indicator functions {1B1, · · · , 1BN} as the
+approximation space for the PFO. We define the projection space VN := span{1B1, · · · , 1BN}.
+The VN ∈ L1(X) is regarded as an approximation subspace of L1(X). For each ρ ≥ 0 in
+L1(X), we define the operator πN : L1(X) → VN by
+πNρ =
+N
+�
+i=1
+ω(i)1Bi,
+where
+ω(i) :=
+�
+Bi ρ dµ
+µ(Bi) .
+(3.16)
+Then πN is the projection onto VN. Due to the projection, we define the discretized PFO
+PN as
+PN = πN ◦ P.
+We can represent the linear map PN|V 1
+N : V 1
+N → V 1
+N, where V 1
+N :=
+�
+f ∈ VN :
+�
+|f|dµ = 1
+�
+by
+a matrix P N = (P N
+ij ) ∈ RN×N whose entries P N
+ij =
+1
+µ(Bi)
+�
+Bi P1Bjdµ. The entries characterizes
+the transition probability from the box Bi to box Bj under the flow map Ψ. We can show
+P N
+ij = µ(Bi ∩ Ψ−1(Bj))
+µ(Bi)
+.
+(3.17)
+A Markov chain for Ψ arises as the discretization PN of the PFO, and the Markov chain has
+transition matrix P N. So the Ulam’s method can be described either in terms of the operator
+PN or the matrix P N. By the projection πN, the density ρ can be expressed as a vector
+W = [ω(1), · · · , ω(N)], where ω(i) is the weight of the basis function 1Bi. Since the entries P N
+ij
+represent the transition probability from Bi to Bj, they can be estimated by a Monte-Carlo
+method, which gives a numerical realization of Ulam’s method [6]. We randomly choose a
+large number of points {xl
+i}n
+l=1 in each Bi and count the number of times Ψ(xl
+i) contained in
+box Bj. Then P N
+ij is calculated by
+P N
+ij ≈ P N
+n,ij = 1
+n
+n
+�
+l=1
+1Bj(Ψ(xl
+i)).
+(3.18)
+8
+
+The Monte-Carlo method is used as an approximation to the integrals (3.17). The conver-
+gence of the Ulam’s method depends on the choice of the partition of the region and the
+number of points. Based on indicator basis functions, we denote the the empirical density
+in the PFOF with respect to the measure µN
+j as
+ρN
+j (x) =
+N
+�
+i=1
+ω(i)
+j 1Bi(x),
+(3.19)
+where 1Bi(·) is the indicator function defined by (3.15) and j represents the index of time
+sequence.
+In this way, the density ρN
+j
+can be represented by the vector of the weights
+Wj = [ω(1)
+j , · · · , ω(N)
+j
+]. Suppose that Wj = [ω(1)
+j , · · · , ω(N)
+j
+] and Wj+1 = [ω(1)
+j+1, · · · , ω(N)
+j+1] are
+separately the weights of πNρj and πNρj+1 := πNPρj. When the region is evenly divided,
+the evolution of density functions becomes a Markov transition equation of the weights:
+Wj+1 = WjP N,
+(3.20)
+where P N is the matrix form of discretized PFO. We consider the projection of the Pρj, i.e.,
+πNPρj =
+N
+�
+i=1
+ω(i)
+j+11Bi.
+(3.21)
+In addition, note that
+πNPρj = πNP
+N
+�
+i=1
+ω(i)
+j 1Bi =
+N
+�
+i=1
+ω(i)
+j πN(P1Bi).
+We denote πN(P1Bi) = �N
+k=1 cik1Bk, where
+cik =
+�
+X P(1Bi)1Bkdx
+µ(Bk)
+=
+�
+Bk P(1Bi)dx
+µ(Bk)
+=
+�
+Ψ−1(Bk) 1Bidx
+µ(Bk)
+= µ(Bi ∩ Ψ−1(Bk))
+µ(Bk)
+.
+Then we have
+πNPρj =
+N
+�
+i=1
+ω(i)
+j
+N
+�
+k=1
+cik1Bk =
+N
+�
+k=1
+N
+�
+i=1
+ω(i)
+j cik1Bk
+=
+N
+�
+k=1
+N
+�
+i=1
+µ(Bi)
+µ(Bk)ω(i)
+j P N
+ik 1Bk.
+Comparing to (3.21), we get
+ω(k)
+j+1 =
+N
+�
+i=1
+µ(Bi)
+µ(Bk)ω(i)
+j P N
+ik .
+(3.22)
+9
+
+Thus, if we give a uniform partition of the X, i.e., µ(Bi) = µ(Bj), ∀i, j ∈ N, then we get
+the result (3.20). With the expression, we design the following prediction step and analysis
+step to approximate the posterior distribution p(xj+1|Yj+1).
+Prediction In this step, we give a set of boxes {B1, · · · , BN} ⊂ B, which is a uniform
+partition of X, and denote the mass point of each box as x(i), i = 1, · · · , N. Define �
+Wj =
+[�ω(1)
+j , · · · , �ω(N)
+j
+] as the prior weight vector and Wj = [ω(1)
+j , · · · , ω(N)
+j
+] as the posterior weight
+vector. In equation (2.8), we note that the prior density p(xj+1|Yj) is computed under the
+linear operator P. To discretize the formula �µj+1 = Pµj, we build a map between the weights
+of the density function,
+�
+Wj+1 = WjP N.
+The formula contains the prior information of the underlying system (2.5) because the PFO
+in the formula is defined by the transition kernel p of the system. With the basis functions
+1Bi(·), the empirical prior measure is given by
+�µN
+j+1 =
+N
+�
+i=1
+�ω(i)
+j+11Bi(dx).
+Analysis In this step, we derive the posterior measure µN
+j+1. To achieve this, we apply
+Bayes’s formula (2.9) on weights and update the weights by
+ω(i)
+j+1 = �ω(i)
+j+1/(
+N
+�
+n=1
+�ω(n)
+j+1),
+�ω(i)
+j+1 = gj(x(i))�ω(i)
+j+1,
+(3.23)
+where gj(x) given by (2.10) denotes the likelihood function as before. Then the µN
+j+1 approx-
+imated by the indicator measure is given by
+µN
+j+1 =
+N
+�
+i=1
+ω(i)
+j+11Bi(dx).
+Note that we choose the mass point x(i) of each box Bi to calculate gj(x), i.e., the likelihood
+function. It is a reasonable choice to approximate the likelihood function of the points in
+the Bi. In both prediction step and analysis step, they only evolve weights {ω(i)
+j }N
+i=1 into
+{ω(i)
+j+1}N
+i=1 via {�ω(i)
+j+1}N
+i=1, and provide a transform from µN
+j to µN
+j+1. The complete procedure is
+summarized in Algorithm 2, named Perron-Frobenius operator filter. The algorithm consists
+of two phases: offline phase to compute P N by Ulam’s method, and online phase to update
+the approximation of the posterior measure. Besides, the standard Ulam’s method becomes
+inefficient in high-dimensional dynamical systems due to the curse of dimensionality. For
+this case, we may use the sparse Ulam method [27] instead. It constructs an optimal ap-
+proximation subspace and costs less computational effort than the standard Ulam’s method
+when a certain accuracy is imposed.
+10
+
+Algorithm 1 Perron-Frobenius operator filter
+Offline:
+Compute P N by Ulam’s method
+Online:
+1: Set j = 0 and µN
+0 (dx0) = µ0(dx0), compute ω(i)
+0 =
+�
+Bi µ0dx0
+µ(Bi)
+2: Let Wj = [ω(1)
+j , · · · , ω(N)
+j
+], compute �
+Wj+1 = WjP N
+3: Define �µN
+j+1 = �N
+i=1 �ω(i)
+j+11Bi(x)
+4: Denote ω(i)
+j+1 by (3.23), define µN
+j+1 = �N
+i=1 ω(i)
+j+11Bi(x)
+5: j+1→ j
+6: Go to step 2
+3.2
+Analysis of error estimate
+We analyze the error estimate of the Perron-Frobenius operator filter in this section to
+explore the factors, which determine convergence of the algorithm. Define a total-variation
+distance d(·, ·) between two probability measures µ and ν as follows:
+d(µ, ν) = 1
+2sup|f|∞≤1|Eµ(f) − Eν(f)|,
+where Eµ(f) =
+�
+X f(x)µ(dx) for f ∈ L1(X) and |f|∞ = supx|f(x)|. The distance d(·, ·) can
+also be characterized by the L1 norm of the difference between the two PDFs ρ and ρ′, which
+correspond to the measure µ and ν, respectively, i.e.,
+d(µ, ν) = 1
+2
+�
+X
+|ρ(x) − ρ′(x)|dx.
+(3.24)
+The distance induces a metric. To estimate the error, we recall the iteration (3.14) and see
+that the approximation error of the probability comes from the operator PN. To do this, we
+need the following lemmas.
+Lemma 3.1. (Theorem 4.8 in [23]) Suppose that P is the Perron-Frobenius operator defined
+in (2.8). Let µ and ν be two arbitrary probability measures. Then
+d(Pµ, Pν) ≤ d(µ, ν).
+Lemma 3.2. (Lemma 4.9 in [23]) Let gj be the likelihood function defined by (2.10) and the
+operator Lj defined by (2.12). Assume that there exists κ ∈ (0, 1] such that for all x ∈ X
+and j ∈ N,
+κ ≤ gj(x) ≤ κ−1.
+(3.25)
+Then we have
+d(Ljµ, Ljν) ≤ 2κ−2d(µ, ν).
+Lemma 3.3. (Theorem 2.4.1 in [28]) Let Ca be discrete Lipschitz cone defined as
+Ca = {φ : φ(x)
+φ(y) ≤ ea|x−y|, ∀x, y ∈ R}.
+11
+
+For each N > 0, the πN given by (3.16) denotes the projection of L1 onto VN. Then for any
+function f ∈ Ca,
+∥f − πNf∥L1 ≤ (ea/N − 1)∥f∥L1.
+By the above three lemmas, we analyze total-variance distance between the approximate
+measure µN
+J and the true measure µJ and give the following theorem.
+Theorem 3.4. If gj(x) satisfies the condition (3.25) and the probability density ρN
+j of the
+measure µN
+j satisfies PρN
+j ∈ Ca, ∀j ∈ N, then
+d(µN
+J , µJ) ≤
+J
+�
+j=1
+(2κ−2)j ea − 1
+2N
+.
+Proof. From the formula (2.13) and (3.14), we apply the triangle inequality to the distance
+d(µN
+j+1, µj+1) and get
+d(µN
+j+1, µj+1) = d(LjPNµN
+j , LjPµj)
+≤ d(LjPNµN
+j , LjPµN
+j ) + d(LjPµN
+j , LjPµj).
+According to Lemma 3.2 and Lemma 3.1, it follows that
+d(µN
+j+1, µj+1) ≤ 2k−2�
+d(PNµN
+j , PµN
+j ) + d(PµN
+j , Pµj)
+�
+≤ 2k−2�
+d(PNµN
+j , PµN
+j ) + d(µN
+j , µj)
+�
+,
+(3.26)
+Let us consider d(PNµN
+j , PµN
+j ). Suppose that ρ′
+j+1 is density function associated with the
+measure PµN
+j . Let ρN
+j and ρN
+j+1 be the density functions of µN
+j and µN
+j+1, respectively. By
+the definition of total-variance distance in (3.24), we have
+d(PNµN
+j , PµN
+j ) = 1
+2
+�
+X
+|ρN
+j+1(x) − ρ′
+j+1(x)|dx
+= 1
+2
+�
+X
+|PNρN
+j (x) − PρN
+j (x)|dx
+= 1
+2
+�
+X
+|πN ◦ PρN
+j (x) − PρN
+j (x)|dx,
+where we have used the equation PN = πN ◦ P in the last equality. Since PρN
+j (x) ∈ Ca, we
+use Lemma 3.3 and get
+d(PNµN
+j , PµN
+j ) ≤ 1
+2(ea/N − 1)
+≤
+1
+2N (ea − 1).
+(3.27)
+With the fact that µN
+0
+= µ0, we combine (3.27) with (3.26) and repeat the iterating to
+complete the proof.
+Theorem 3.4 estimates the online error of the PFOF algorithm.
+Since the Perron-
+Frobenious operator is numerically approximated offline by the matrix form P N
+n given by
+(3.18), we will analyze the offline error generated by the approximation. Each coefficient of
+P N
+n is computed by the Monte-Carlo approximation of (3.17) using a set of the sampling
+points {xl
+i}n
+l=1. We show that the matrix P N
+n converge to the matrix P N.
+12
+
+Proposition 3.5. If the matrix P N
+n is defined by (3.18) and P N is defined by (3.17), then
+the following convergence in distribution holds,
+√n((P N
+n )ij − (P N)ij)
+D
+−−−→
+n→∞ N (0, σn,N
+ij
+),
+(3.28)
+where
+(σn,N
+ij
+)2 =
+�
+X
+(1Ψ−1
+τ (Bj) · 1Bi)2dµ − (
+�
+X
+1Ψ−1
+τ (Bj) · 1Bidµ)2,
+(3.29)
+and N (0, σn,N
+ij
+) is the normal distribution with the mean 0 and standard deviation σn,N
+i,j .
+Proof. Note that the entries of P N
+n are given by
+P N
+n,ij = 1
+n
+n
+�
+l=1
+1Bj(Ψτ(xl
+i)),
+which is the Monte-Carlo approximation of
+P N
+ij =
+�
+X 1Ψ−1
+τ (Bj) · 1Bidµ
+�
+X 1Bidµ
+,
+with sampling points xl
+i drawn independently and uniformly from the box Bi. The denomi-
+nator
+�
+X 1Bidµ normalizes the entries P N
+ij so that P N becomes a right stochastic matrix, with
+each row summing to 1. The convergence result (3.28) follows directly from the convergence
+of Monte-Carlo integration [29].
+Proposition 3.5 indicates that there exits a constant CN(Ψτ, α) determined by the stan-
+dard deviation σn,N
+ij
+with a given confidence rate α ∈ [0, 1) such that for m large enough,
+the following estimate holds with probability at least α:
+∥ (P N
+n )ij − (P N)ij ∥∞≤ CN(Ψτ, α)n− 1
+2.
+(3.30)
+The result shows that the convergence of P N
+n to P N is in O(n− 1
+2).
+3.3
+A low-rank Perron-Frobenius operator filter
+In PFOF, we note that the number of blocks increases exponentially with respect to dimen-
+sions, resulting in the number of basis functions growing rapidly. Therefore, we propose
+a low-rank approximation, formed by eigenfunctions of the Perron-Frobenius operator, to
+represent the density. This approach can effectively reduce the number of the required basis
+functions. Let ρ still be a probability density function of the dynamical system governed by
+Ψ. Then it can be written as a linear combination of the independent eigenfunctions ϕi of
+P. So
+ρ(x, t) =
+∞
+�
+i=1
+aiϕi(x),
+ai ∈ C.
+13
+
+Suppose that λi is the eigenvalue corresponding to the eigenfunction ϕi of P, then
+Pρ(x, t) =
+∞
+�
+i=1
+λiaiϕi(x).
+Actually, the eigenfunction of the discretized PFO can be determined by the following propo-
+sition.
+Proposition 3.6. Let B = {B1, · · · , BN} ⊂ B be a uniform partition of the phase space
+X. If ξ is the left eigenvector of P N corresponding to the eigenvalue λ, then λ is also the
+eigenvalue of the restricted operator πNP with the eigenfunction ϕ ≜ ξTU, where U =
+[1B1(x), · · · , 1BN(x)]T.
+Proof. Let ϕ = �
+i ξ(i)1Bi. From Eq. (3.21) and Eq. (3.22),
+πNPϕ =
+N
+�
+j
+N
+�
+i
+ξ(i)P N
+ij 1Bj.
+Since ξP N = λP N, i.e., λξ(j) = �
+i ξ(i)P N
+ij , ∀j ∈ N, we get
+πNPϕ =
+N
+�
+j
+λξ(j)1Bj = λϕ.
+Thus, λ is also an eigenvalue of the restricted operator πNP with eigenfunction ϕ.
+In order to obtain the spectral expansion of the density function ρj := ρ(x, tj), we de-
+fine the matrix ϕ = [ϕ1, · · · , ϕN]T, where {ϕi}N
+i=1 are the eigenfunctions with respect to
+eigenvalues {λi}N
+i=1, with |λ1| = 1 ≥ |λ2| ≥ · · · ≥ |λN| ≥ 0. Let ρ0 = W0U and
+Ξ =
+
+
+ξT
+1
+ξT
+2...
+ξT
+N
+
+ .
+Then the eigenfunction is denoted as ϕ = ΞU and the density function of ρ1 is given by
+ρ1 = πNPρ0 = πNPW0U = πNPW0Ξ−1ϕ = ΛW0Ξ−1ϕ =
+N
+�
+i=1
+λiϕivi,
+(3.31)
+where Λ is a diagonal eigenvalue matrix for πNP and vi is the column vector of the matrix
+V = W0Ξ−1. The first r major eigenvalues and their corresponding eigenfunctions are used
+to approximate the density function. If the formula (3.31) is truncated by r < N, then the
+low-rank model of ρ1 has the form of
+ρ1 =
+r
+�
+i=1
+λiϕivi.
+14
+
+In this way, the low-rank model of density functions at time tj is given by
+ρj =
+r
+�
+i=1
+λj
+iϕivi,
+j ∈ N.
+Let us denote the low-rank approximation of Perron-Frobenius operator as ρj = �Pρj−1 ≜
+�r
+i=1 λiϕivj−1,i, where vj−1,i is the column vector of the matrix Vj−1 = Wj−1Ξ−1. We apply
+�P in the Bayesian filter to obtain the low-rank Perron-Frobenius operator filter (lr-PFOF),
+in which the probability measure satisfies the recursive formula
+µN
+j+1 = Lj �PµN
+j ,
+µN
+0 = µ0.
+To describe the following prediction and analysis steps, we first calculate the weak approxi-
+mation P N and get the eigenvalues Λ and left eigenvectors Ξ of P N.
+Prediction In this step, we give a model decomposition of the prior density p(xj+1|Yj).
+First, the Wj satisfying p(xj|Yj) = WjU is obtained from the previous analysis step. Next,
+compute the matrix Vj = WjΞ−1 and
+�ρj+1 = p(xj+1|Yj) =
+r
+�
+i=1
+λiϕivj,i.
+Analysis In this step, we derive the posterior density p(xj+1|Yj+1) via Bayes’s formula.
+Multiply p(xj+1|Yj) by likelihood function gj and have
+ρj+1 = p(xj+1|Yj+1) ∝
+r
+�
+i=1
+λiϕivj,igj.
+To normalize ρj+1, we rewrite the �ρj+1. Since ϕi = ξT
+i U = �N
+k=1 ξ(k)
+i 1Bk, we get
+�ρj+1 =
+r
+�
+i=1
+λi
+N
+�
+k=1
+ξ(k)
+i 1Bkvj,i =
+N
+�
+k=1
+�
+r
+�
+i=1
+λiξ(k)
+i vj,i
+�
+1Bk.
+Then we multiply by gj(x) and make a normalization to the weights of 1Bk, such that
+ω(k)
+j+1 = �ω(k)
+j+1/(
+N
+�
+n=1
+�ω(n)
+j+1),
+�ω(k)
+j+1 =
+r
+�
+i=1
+λiξ(k)
+i vj,igj(x(k)),
+(3.32)
+where x(k) is still the mass point of each box Bk. The posterior density becomes
+ρj+1 =
+N
+�
+k=1
+ω(k)
+j+11Bk = Wj+1U.
+Remark 3.1. Note that the complex eigenvalues and eigenvectors may appear in the eigen-
+decomposition of the matrix P N. When the stationary distribution �π of the system satisfies
+detailed balance, a symmetrization method is designed in [30] to solve the problem. Since
+15
+
+Algorithm 2 low-rank Perron-Frobenius operator filter
+Offline:
+Compute P N and its eigenvalue Λ and left eigenvector Ξ. Give the eigenfunction ϕ = ΞU.
+Online:
+1: Set j = 0 and ρ0 = W0U, compute ω(i)
+0 =
+�
+Bi µ0dx0
+µ(Bi)
+2: Denote Vj = WjΞ−1, compute �ρj+1 = �r
+i=1 λiϕivj,i
+3: Define gj by (2.10), give ρj+1 ∝ �r
+i=1 λiϕivj,igj
+4: Normalize weights by (3.32) and obtain Wj+1, let ρj+1 = �N
+k=1 ω(k)
+j+11Bk.
+5: j+1→ j
+6: Go to step 2
+ρ0, ρ1, · · · can be seen as a Markov chain with transition matrix P N, we suppose that P N
+satisfies detailed balance with respect to �π, i.e.,
+�πiP N
+ij = �πjP N
+ji ,
+∀i, j ∈ N.
+Then P N can be symmetrized by a similarity transformation
+S = �ΛP N �Λ−1,
+where �Λ =
+
+
+√�π1
+√�π2
+...
+√�πN
+
+ .
+Here the S is a symmetric matrix and this can be easily checked by detailed balance equation.
+It is known that S has a full set of real eigenvalues αj ∈ R and an orthogonal set of
+eigenvectors wj. Therefore, P N has the same eigenvalues αj and real left eigenvectors
+ψj = �Λwj.
+3.4
+Extension to continuous-time filtering problems
+In this subsection, we consider a continuous-time filtering problem, where the state model
+and observation are by the following SDEs,
+dx
+dt = f(x) +
+�
+Σc
+dWt
+dt ,
+x(t0) ∼ N (m0, C0),
+(3.33)
+dz
+dt = h(x) +
+�
+Rc
+dWt
+dt ,
+z(0) = 0.
+(3.34)
+Here Σc is the covariance of model error and Rc is the covariance of observation error. Sup-
+pose that the posterior measure µt governed by the continuous-time problem has Lebesgue
+density ρ(·, t) : Rn �→ R+ for a fixed t. Let ρ(x, t) = r(x, t)/
+�
+Rn r(x, t)dx, where r is the
+unnormalized density. For a positive definite symmetric matrix A ∈ Rp×p, we define the
+weighted inner product ⟨·, ·⟩A = ⟨A− 1
+2·, A− 1
+2·⟩ on the space L2([0, T]; Rp).
+The resulting
+16
+
+norm | · |A = |A− 1
+2 · |. In the continuous filtering problem, our interest is to find the dis-
+tribution of the random variable x(t)|{z(s)}s∈[0,t] as the time t increases. Zakai stochastic
+partial differential equation (SPDE) is a well-known equation whose solution characterizes
+the unnormalized density of posterior distribution [35]. The Zaikai equation has the form of
+∂r
+∂t = AP Fr + r
+�
+h, dz
+dt
+�
+Rc
+.
+(3.35)
+The partial differential operator AP F generates a continuous Perron-Frobenius semigroup
+{Pt, t ≥ 0}. Let {Qt
+s, 0 ≤ s ≤ t} be the stochastic semigroup [31] associated with with the
+following SDE
+dr′
+dt = r′
+�
+h, dz
+dt
+�
+Rc
+.
+(3.36)
+Then the Zakai equation (3.35) can be approximated by the following Trotter-like product
+formula
+rj+1 = Q
+tj+1
+tj
+Pτrj,
+(3.37)
+where τ = tj+1 − tj, ∀j ∈ N. For the fixed τ, Pτ is still denoted by P. By the reference [31],
+the Q
+tj+1
+tj
+describes the solution of the equation (3.36), i.e.,
+Q
+tj+1
+tj
+r(x) = exp
+�
+⟨h(x), zj+1 − zj⟩Rc − τ
+2|h(x)|2
+Rc
+�
+r(x).
+With the discrete scheme (3.37), we utilize the Perron-Frobenius operator to solve Zakai
+equation, rather than using Fokker-Planck operator AP F. Thus, we discretize P by Ulam’s
+method and project the density function onto VN. Let P N be the discretization of P. Let
+Wj and Wj+1 be the weights vectors with respect to πNrj and πNrj+1. Denote gc
+j(x) =
+exp
+�
+⟨h(x), zj+1 − zj⟩Rc − τ
+2|h(x)|2
+Rc
+�
+and
+Gj =
+
+
+gc
+j(x(1))
+gc
+j(x(2))
+...
+gc
+j(x(N))
+
+ ,
+where x(i) is the mass point of Bi. Then the transition of density functions turns into a map
+of the weights,
+Wj+1 = Gj ⊙
+�
+WjP N�
+.
+Here ⊙ denotes Hadamard product. In this case, the PFO is extended to the continuous-time
+filtering problem to estimate the posterior density function.
+4
+Comparison with particle filter
+Particle filter (PF) [32, 33] is an important filtering method to sequentially approximate the
+true posterior filtering distribution p(xj|Yj) in the limit of a large number of particles. In
+17
+
+practice, we approximate the probability density by a combination of locations of particles
+and weights associated with Dirac functions. Particle filter proceeds by varying the weights
+and determining the particle Dirac measures. It is able to take care of non-Gaussian and non-
+linear models. In this section, we will compare the computational accuracy and differences
+between PFOF and PF.
+Accordingly, we define µN
+j as the posterior empirical measure on RN approximating truth
+posterior probability measure µj and define �µN
+j
+on RN as the approximation of the prior
+probability measure �µj. Let
+µj ≈ µN
+j :=
+N
+�
+n=1
+ω(n)
+j δx(n)
+j ,
+�µj+1 ≈ �µN
+j+1 :=
+N
+�
+n=1
+�ω(n)
+j+1δ�x(n)
+j+1,
+where x(n)
+j
+and �x(n)
+j+1 are particle positions, and ω(n)
+j
+> 0, �ω(n)
+j+1 > 0 are the associated
+weights satisfying �N
+n=1 ω(n)
+j
+= 1, �N
+n=1 �ω(n)
+j+1 = 1. The empirical distribution is completely
+determined by particle positions and weights. The objective of particle filter is to calculate
+the update {x(n)
+j , ω(n)
+j } → {�x(n)
+j+1, �ω(n)
+j+1} and {�x(n)
+j+1, �ω(n)
+j+1} → {x(n)
+j+1, ω(n)
+j+1}, which define the
+prediction step and analysis step, respectively. Monte-Carlo sampling is used to determine
+particle positions in the prediction and Bayesian rule is used to update of the weights in the
+analysis.
+Prediction In this step, the prediction phase is approximated by the Markov chain
+{Ψ(xj)}j∈N with transition kernel p(xj, xj+1) = p(xj+1|xj). We draw �x(n)
+j+1 from the kernel p
+started from x(n)
+j , i.e., �x(n)
+j+1 ∼ p(x(n)
+j , ·). We keep the weights unchanged so that �ω(n)
+j+1 = ω(n)
+j ,
+and obtain the prior probability measure
+�µN
+j+1 =
+N
+�
+n=1
+ω(n)
+j δ�x(n)
+j+1.
+Analysis In this step, we apply Bayes’s formula to approximate the posterior probability
+measure. To do this, we fix the position of the particles and update the weights. With the
+definition of gj(x) in (2.10), we have the empirical posterior distribution
+µN
+j+1 =
+N
+�
+n=1
+ω(n)
+j+1δ�x(n)
+j+1,
+where
+ω(n)
+j+1 = �ω(n)
+j+1/(
+N
+�
+n=1
+�ω(n)
+j+1),
+�ω(n)
+j+1 = gj(�x(n)
+j+1)ωn
+j .
+(4.38)
+The first equation in (4.38) is a normalization. Sequential Importance Resampling (SIR)
+particle filter is a basic particle filter and shown in Algorithm 1.
+A resampling step is
+introduced in the algorithm. In this way, we can deal with the initial measure µ0 when it
+is not a combination of Dirac functions. We can also deal with the case when some of the
+particle weights are close to 1. The algorithm shows that each particle moves according
+18
+
+to the underlying model and is reweighted according to the likelihood. By the iteration of
+Bayesian filtering , we rewrite the particle filter approximated by the form
+µN
+j+1 = LjSNPµN
+j ,
+µN
+0 = µ0,
+(4.39)
+where the operator SN is defined as follows:
+(SNµ)(dx) = 1
+N
+N
+�
+n=1
+δx(n)(dx),
+x(n) ∼ µ
+i.i.d..
+Algorithm 3 Sequential Importance Resampling particle filter
+1: Set j = 0 and µN
+0 (dx0) = µ0(dx0)
+2: Draw x(n)
+j
+∼ µN
+j , n = 1, · · · , N
+3: Set ω(n)
+j
+= 1/N, n = 1, · · · , N, redefine µN
+j := �N
+n=1 ω(n)
+j δx(n)
+j
+4: Draw �x(n)
+j+1 ∼ p(x(n)
+j , ·)
+5: Define ω(n)
+j+1 by (3.23) and µN
+j+1 = �N
+n=1 ω(n)
+j+1δ�x(n)
+j+1
+6: j+1→ j
+7: Go to step 2
+By (4.39), we find that the randomness for the probability measure is caused by the sam-
+pling operator SN and the convergence of particle filter depends on the number of particles.
+The particle filter does recover the truth posterior distribution as the number of particles
+tends to infinity [34]. The following theorem gives a convergence result for PF.
+Theorem 4.1. (Theorem 4.5 in [23]) Let m be the number of particles and µm
+j the approx-
+imation measure in SIR particle filter. Assume that κ ∈ (0, 1] is the constant defined in
+Lemma 3.2, then the total-variance distance between µm
+J and µJ is estimated by
+d(µm
+J , µJ) ≤
+J
+�
+j=1
+(2κ−2)j 1
+√m.
+(4.40)
+Let J be fixed in Theorem 3.4 and Theorem 4.1. We find that the convergence rate of
+particle filter depends on the number of particles m. Similarly, the convergence of PFOF
+is determined by the number of blocks N used in the Ulam’s method. When N = m, i.e.,
+the same number of basis functions in the two methods, the rate of convergence is O( 1
+N )
+in PFOF and O(
+1
+√m) in SIR particle filter. The analysis shows that PFOF converges faster
+than the particle filter.
+Sampling from high-dimensional and complex transition kernels is difficult to realize in
+PF. The PFOF avoids the sampling and uses a data-driven approximation instead, which
+requires short-term path simulations rather than the form of transition density. Particle
+degeneracy is also a significant issue. As the number of effective particles decreases gradually,
+the efficiency of the particle filter becomes worse.
+It is known that particle filter is inefficient for high-dimensional models because of degen-
+eracy. So the accurate estimate of posterior PDF requires a great number of particles that
+19
+
+scales exponentially with the size of the system. In addition to resampling, adding jitter and
+localisation are effective modifications to solve the problem. The PFOF also has the “curse
+of dimensionality” problem in high dimensions as the partition scale expansion. One solu-
+tion to circumvent this problem is the sparse Ulam method. The low-rank Perron-Frobenius
+operator filter can enhance the efficiency of filtering problems.
+5
+Numerical results
+In this section, we present some numerical examples for filtering problems using the pro-
+posed PFOF. The system dynamics is unknown and some observations are given in the
+filtering problems. The PFOF and lr-PFOF are implemented to estimate posterior PDFs
+of the stochastic filtering problems. In Section 5.1, we consider an Ornstein-Uhlenbeck (O-
+U) process to identify the Gaussian PDF of the system and estimate its posterior PDFs
+with observations known.
+In Section 5.2, we consider a nonlinear filtering problem gov-
+erned by Bene˘s SDE, and estimate the non-Gaussian posterior PDFs. In Section 5.3, we
+consider a continuous-time filtering problem, which is a classical chaotic system Lorenz’63
+model with observations, to model posterior density of the state. We compare the proposed
+PFOF/lr-PFOF with particle filter and Extended Kalmn filter (ExKF). Numerical results
+show that PFOF achieves a better posterior PDF estimates than PF, and a more accurate
+state estimates than ExKF.
+5.1
+O-U process
+Let us consider an O-U process, which is a one-dimensional linear dynamical system,
+dxt = −λxtdt + dWt,
+x(t0) ∼ N (m0, C0),
+where λ > 0 and Wt is a standard Brownian motion. We now consider a state-space model
+formed by a discretization of the O-U process and the discrete observations of the state as
+follows,
+�
+x(tk+1) = exp(−λ∆tk)x(tk) + qk,
+qk ∼ N (0, Σk),
+y(tk) = Hx(tk) + rk,
+rk ∼ N (0, R),
+(5.41)
+where Σk = exp(−2λ∆tk), H = I and R = σ2. The parameters are given by λ = 1/2,
+m0 = 2, C0 = 0.1 and σ = 1. To apply PFOF, we compute Perron-Frobenius operator Pτ
+using Ulam’s method with time step τ = 0.1 and obtain an approximation form P N
+τ ∈ RN×N
+of Pτ. We take the phase space of xt is [−6, 6] and divide it into N = 100 grids, and each
+interval [zk, zk+1], k = 0, · · · , N − 1, defines a box Bk. We define an indicator function
+1Bk(x) on each Bk and randomly choose n = 100 sample points in the box to calculate P N
+τ .
+Given initial Gaussian distribution N (2, 0.1), we rewrite µ0 as a vector W0, which denotes
+the coefficients of µN
+0 . The P N
+τ
+acts on the weight vector to estimate probability value of xt
+on each Bk, i.e., P(xt ∈ Bk), t = qτ, q = 0, 1, 2 · · ·. Thus, we get the discrete probability
+density function (PDF) of xt at t. The simulation PDFs at different times are shown in the
+left column of Figure 5.1. By the figure, we see that the PDFs estimated by PFO are close
+20
+
+-6
+-4
+-2
+0
+2
+4
+6
+t=2.5
+0
+0.5
+1
+1.5prior PDF by P-F operator
+Truth
+PDF by PFO
+-6
+-4
+-2
+0
+2
+4
+6
+0
+0.5
+1
+1.5 empirical posterior PDF
+Truth
+PFOF
+-6
+-4
+-2
+0
+2
+4
+6
+0
+0.5
+1
+1.5histogram of posterior PDF
+Truth
+Particle filter
+-6
+-4
+-2
+0
+2
+4
+6
+t=5
+0
+0.5
+1
+1.5
+Truth
+PDF by PFO
+-6
+-4
+-2
+0
+2
+4
+6
+0
+0.5
+1
+1.5
+Truth
+PFOF
+-6
+-4
+-2
+0
+2
+4
+6
+0
+0.5
+1
+1.5
+Truth
+Particle filter
+-6
+-4
+-2
+0
+2
+4
+6
+t=10
+0
+0.5
+1
+1.5
+Truth
+PDF by PFO
+-6
+-4
+-2
+0
+2
+4
+6
+0
+0.5
+1
+1.5
+Truth
+PFOF
+-6
+-4
+-2
+0
+2
+4
+6
+0
+0.5
+1
+1.5
+Truth
+Particle filter
+Figure 5.1: The prior PDF estimated by PFO (left column), posterior PDF by PFOF (middle column)
+and posterior PDF by particle filter (right column) at different times.
+to the truth. By this way, the PDF is computed without solving Fokker-Planck equation
+and the estimation of PDF is actually the prior density in the model (5.41).
+Then we compute posterior probability density of the state-space model (5.41). We set
+N = 500 and n = 100. The posterior probability density function is estimated by Algorithm
+1 and the results are displayed in the middle column of Figure 5.1. From Figure 5.1, we find
+that the empirical posterior PDFs estimated by PFOF are close to the Gaussian posterior
+densities. To make comparison with PFOF, the particle filter is also used for the filtering
+problem. In the particle filter, 500 particles are drawn randomly to generate Dirac measure
+and construct empirical measure. Thus, the number of basis functions is equal to each other
+in the two methods. Figure 5.1 clearly shows that the empirical PDF calculated by PFOF
+is more accurate than that by PF. The numerical results support Theorem 3.4 and Theorem
+4.1.
+5.2
+Bene˘s-Daum filter
+In this subsection, we apply PFOF to a nonlinear filtering problem, whose state-space model
+is defined by the Bene˘s stochastic difference equation,
+dxt = tanh(xt)dt + dWt,
+(5.42)
+21
+
+with initial condition x0 = 0. Refer to [36], the probability density function of the equation
+(5.42) is given by
+p(x(t)) =
+1
+√
+2πt
+cosh(x(t))
+cosh(x0) exp
+�
+− t
+2
+�
+exp
+�
+− 1
+2t(x(t) − x0)
+�
+.
+We take the phase space [−15, 15] and uniformly divide it into 100 (N = 100) grids [zk, zk+1], k =
+0, ..., N − 1, each of which corresponds to a box Bk. The Ulam’s method is used to approxi-
+mate PFO. The time step is set as τ = 0.5 and the number of random sample points m = 400.
+The predicted PDF of xt at t = 1, t = 2.5 and t = 5 are shown in Figure 5.2. The PDFs are
+separately estimated by discretized PFO matrix P N ∈ R100×100 and low-rank approximation
+of PFO with truncation r = 30. We see that PDFs at t = 2.5 and t = 5 have two modes and
+the PFO can fairly approximate the two modes.
+-20
+-10
+0
+10
+20
+0
+0.05
+0.1
+0.15
+0.2
+0.25
+0.3
+0.35
+0.4
+t=1
+true PDF
+PDF by PFO
+PDF by lr-PFO, r=30
+-20
+-10
+0
+10
+20
+0
+0.02
+0.04
+0.06
+0.08
+0.1
+0.12
+0.14
+0.16
+0.18
+0.2
+t=2.5
+true PDF
+PDF by PFO
+PDF by lr-PFO, r=30
+-20
+-10
+0
+10
+20
+0
+0.05
+0.1
+0.15
+t=5
+true PDF
+PDF by PFO
+PDF by lr-PFO, r=30
+Figure 5.2: The PDF estimated by PFO and low-rank model at different times.
+First we want to calculate the truth posterior filtering distribution of the model (5.42)
+subject to observation. In this example, the observation model satisfies
+p(yk|x(tk)) = N (yk|x(tk), σ2).
+(5.43)
+According to [36] (Chapter 10.5), the transition density of the Bene˘s SDE is given by
+p(x(tk)|x(tk−1)) =
+1
+√2π∆tk−1
+cosh(x(tk))
+cosh(x(tk−1))exp(−1
+2∆tk−1)×exp
+�
+−
+1
+2∆tk−1
+(x(tk)−x(tk−1))2
+�
+,
+where ∆tk−1 = tk − tk−1. If we assume that the filtering solution at time tk−1 is of the form
+p(x(tk−1)|y1:k−1) ∝ cosh(x(tk−1))exp
+�
+−
+1
+2Pk−1
+(x(tk−1) − mk−1)2
+�
+for given mk−1 and Pk−1. Then we use the Chapman-Kolmogorov equation and give the
+prior density
+p
+�
+x(tk)|y1:k−1
+�
+∝ cosh
+�
+x(tk)
+�
+exp
+�
+−
+1
+2P −
+k
+(x(tk) − m−
+k )2
+�
+,
+22
+
+where
+m−
+k = mk−1,
+P −
+k = Pk−1 + ∆tk−1.
+The m−
+k and P −
+k
+are sufficient statistics representing prior density functions.
+By Bayes’
+formula, the posterior density of x(tk) is given by
+p
+�
+x(tk)|y1:k
+�
+∝ cosh
+�
+x(tk)
+�
+exp
+�
+−
+1
+2Pk
+�
+x(tk) − mk
+�2
+�
+,
+(5.44)
+where the equations of parameters mk and Pk in the posterior density satisfy
+mk = m−
+k +
+�
+P −
+k
+P −
+k + σ2
+�
+(yk − m−
+k ),
+P −
+k = Pk−1 + ∆tk−1.
+Thus, the reference posterior distribution is defined by (5.44).
+To apply PFOF to the nonlinear filtering problem, we make a finer division of the phase
+interval [−15, 15] to obtain 400 boxes. Besides, we choose enough sample points in Ulam’s
+method to reduce error of Monte-Carlo as much as possible. The observations yk are arti-
+ficially obtained by simulating the underlying model (5.42) and adding noise according to
+(5.43), where σ = 1. The observable interval is [0, 5] with a time step ∆tk = 0.1. The
+initial distribution for the filtering process is chosen to be m0 = 0, P0 = 2. Particularly, we
+also use the particle filter as a comparison. In the prediction, we are not allowed to draw
+sample points directly because of a complex transition probability density function. We use
+Acceptance-Rejection method to resolve the issue. We first show the results of posterior
+mean estimated by PFOF and lr-PFOF (r=40) in Figure 5.3, together with truth and ob-
+servations. The mean is obtained by averaging the posterior distribution of PFOF/lr-PFOF
+and it is close to the truth as the figure shows.
+The posterior densities estimated by PFOF, lr-PFOF and particle filter are shown in
+Figure 5.4 together with the truth. The truncation parameters in lr-PFOF are separately
+set as r = 10, r = 20 and r = 40. The estimation accuracy of lr-PFOF gradually improves as
+the number of truncation basis functions increases, and achieves almost the same as PFOF
+when r = 40 < N = 400. Although the number of basis functions is the same in both
+PFOF and particle filter, there exit clear difference between the two methods. The results
+show that the accuracy of PFOF is higher than that of particle filter in the non-Gaussian
+and nonlinear filtering problem. This further confirms the theoretical analysis in Section
+3. As shown in Table 1, both PFOF and lr-PFOF use less CPU-time than SIR particle
+filter does. Actually, the CPU-time in particle filter is mainly from Acceptance-Rejection
+sampling. From the table, it can be seen that lr-PFOF can reduce online computation time
+comparing to PFOF.
+23
+
+0
+0.5
+1
+1.5
+2
+2.5
+3
+3.5
+4
+4.5
+5
+-3
+-2
+-1
+0
+1
+2
+3
+4
+5
+6
+Truth
+PFOF-mean
+lr-PFOF-mean,r=40
+observation
+Figure 5.3: The mean estimated by PFOF and lr-PFOF.
+-15
+-10
+-5
+0
+5
+10
+15
+t=1
+0
+0.2
+0.4
+0.6
+0.8
+1
+1.2
+empirical posterior PDF
+Truth
+lr-PFOF,r=10
+r=20
+r=40
+-15
+-10
+-5
+0
+5
+10
+15
+0
+0.2
+0.4
+0.6
+0.8
+1
+1.2
+empirical posterior PDF
+Truth
+PFOF
+-15
+-10
+-5
+0
+5
+10
+15
+0
+0.2
+0.4
+0.6
+0.8
+1
+1.2
+histogram of posterior PDF
+Truth
+Particle filter
+-15
+-10
+-5
+0
+5
+10
+15
+t=2.5
+0
+0.2
+0.4
+0.6
+0.8
+1
+1.2
+Truth
+lr-PFOF,r=10
+r=20
+r=40
+-15
+-10
+-5
+0
+5
+10
+15
+0
+0.2
+0.4
+0.6
+0.8
+1
+1.2
+Truth
+PFOF
+-15
+-10
+-5
+0
+5
+10
+15
+0
+0.2
+0.4
+0.6
+0.8
+1
+1.2
+Truth
+Particle filter
+-15
+-10
+-5
+0
+5
+10
+15
+t=5
+0
+0.2
+0.4
+0.6
+0.8
+1
+1.2
+Truth
+lr-PFOF,r=10
+r=20
+r=40
+-15
+-10
+-5
+0
+5
+10
+15
+0
+0.2
+0.4
+0.6
+0.8
+1
+1.2
+Truth
+PFOF
+-15
+-10
+-5
+0
+5
+10
+15
+0
+0.2
+0.4
+0.6
+0.8
+1
+1.2
+Truth
+Particle filter
+Figure 5.4: The posterior PDF by lr-PFOF (left column), PFOF (middle column) and particle filter (right
+column) at different times.
+Table 1: CPU-time (seconds) for posterior PDF with different methods.
+Methods
+PFOF
+lr-PFOF (r=10)
+lr-PFOF (r=20)
+lr-PFOF (r=40)
+particle filter
+offline
+0.1599
+0.2536
+0.2649
+0.2689
+6906.9486
+online
+0.0673
+0.0150
+0.0431
+0.0641
+24
+
+5.3
+Lorenz’63 model
+Lorenz developed a mathematical model for atmospheric convection in 1963. The Lorenz’63
+model is the simplest continuous-time system to exhibit sensitivity to initial conditions and
+chaos, and it is popular example used for data assimilation. For some parameters and initial
+conditions, the system may perform a chaotic behaviour. The model consists of three coupled
+nonlinear ordinary differential equations with the solution v = (v1, v2, v3) ∈ R3. We consider
+the Lorenz’63 model with additive white noise,
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+dv1
+dt = a(v2 − v1) + σ1
+dW1
+dt
+dv2
+dt = −av1 − v2 − v1v3 + σ2
+dW2
+dt
+dv3
+dt = v1v2 − bv3 − b(r + a) + σ3
+dW3
+dt
+v(0) ∼ N (m0, C0),
+where Wj are Brownian motions assumed to be independent. We use the classical parameter
+values (a, b, r) = (10, 8
+3, 28) and set σ1 = σ2 = σ3 = 2. The initial mean m0 is given by
+(0, 0, 0) and covariance matrix is an identity matrix I3 ∈ R3×3. We give the continuous
+observation z(t), which is governed by a SDE
+
+
+
+dz
+dt = h(v) + γ dWz
+dt
+z(0) = 0,
+with γ = 0.2.
+The purpose of this example is to explore the performance of PFOF in
+continuous-time filtering problems. We compare the assimilation results based on Perron-
+Frobenius operator and continuous-time Extended Kalman filter. The posterior means es-
+timated by the two methods are shown in Figure 5.5 and Figure 5.7.
+The two figures
+are corresponding to different observations h(v) = Hv, where the former is determined by
+H = [0, 1, 0] and the latter is determined by H = [0, 0, 1]. In particular, we find that the
+choice of observations in Lorenz models is quite influential, especially for ExKF. The stability
+of ExKF significantly depends on the observation. Because the insufficient observations may
+keep the filter away from the truth and cause significant model error, and it may easily lead
+to the numerical instability once the deviation occurs. However, the results of reconstruc-
+tion by PFOF much less affected by observation model, so the method shows much better
+robustness than ExKF.
+Figure 5.6 shows the consequence of mean-square error with v2 or v3 as the different
+observation. For ExKF, we find that there is a large error in estimating mean by ExKF
+when the third component v3 is observed. To better visualize the results, we compare the
+trajectories of mean obtained by PFOF and ExKF in Figure 5.8 together with truth. We
+find the trajectory mean of PFOF agrees with the truth more than the the trajectory mean
+of ExKF.
+For v3 as an observation, the one-dimensional and two-dimensional marginal probability
+distributions are displayed in Figure 5.9. The figure aims to intuitively describe distribution
+of the single value and correlation of the different components.
+As shown in the figure,
+25
+
+t
+0
+0.2
+0.4
+0.6
+0.8
+v1
+-15
+-10
+-5
+0
+5
+10
+15
+20
+25
+truth
+PFOF-mean
+ExKF-mean
+standard deviation of ExKF
+t
+0
+0.2
+0.4
+0.6
+0.8
+v2
+-20
+-15
+-10
+-5
+0
+5
+10
+15
+20
+25
+30
+truth
+PFOF-mean
+ExKF-mean
+standard deviation of ExKF
+t
+0
+0.2
+0.4
+0.6
+0.8
+v3
+-30
+-25
+-20
+-15
+-10
+-5
+0
+5
+10
+truth
+PFOF-mean
+ExKF-mean
+standard deviation of ExKF
+Figure 5.5: The posterior mean of each component by ExKF and PFOF in Lorenz’63 model with continuous
+observation. The component v2 is observed.
+t
+0
+0.2
+0.4
+0.6
+0.8
+MSE
+0
+50
+100
+150
+200
+250
+ExKF, v2 observed
+t
+0
+0.2
+0.4
+0.6
+0.8
+MSE
+0
+10
+20
+30
+40
+50
+60
+70
+80
+90
+100
+PFOF, v2 observed
+t
+0
+0.2
+0.4
+0.6
+0.8
+MSE
+0
+200
+400
+600
+800
+1000
+1200
+1400
+1600
+1800
+ExKF, v3 observed
+t
+0
+0.2
+0.4
+0.6
+0.8
+MSE
+0
+10
+20
+30
+40
+50
+60
+70
+80
+90
+100
+PFOF, v3 observed
+Figure 5.6: The mean-square error ∥v(t) − m(t)∥2
+2 of filters.
+26
+
+t
+0
+0.2
+0.4
+0.6
+0.8
+v1
+-15
+-10
+-5
+0
+5
+10
+15
+20
+25
+truth
+PFOF-mean
+ExKF-mean
+standard deviation of ExKF
+t
+0
+0.2
+0.4
+0.6
+0.8
+v2
+-20
+-15
+-10
+-5
+0
+5
+10
+15
+20
+25
+30
+truth
+PFOF-mean
+ExKF-mean
+standard deviation of ExKF
+t
+0
+0.2
+0.4
+0.6
+0.8
+v3
+-30
+-25
+-20
+-15
+-10
+-5
+0
+5
+10
+truth
+PFOF-mean
+ExKF-mean
+standard deviation of ExKF
+Figure 5.7: The posterior mean of each component by ExKF and PFOF in Lorenz’63 model with continuous
+observation. The component v3 is observed.
+-20
+-15
+-10
+-5
+0
+v2
+5
+10
+15
+20
+-15
+-10
+v1
+-5
+0
+5
+10
+15
+-10
+-5
+0
+-20
+-25
+-15
+v3
+truth
+PFOF-mean
+ExKF-mean
+Figure 5.8: The trajectories of mean by PFOF and ExKF.
+27
+
+one-dimensional marginal distributions of the observed component are closer to Gaussian
+distributions than the other two components. This phenomenon reflects that when a com-
+ponent is used as an observation, its mean estimates will be more accurate than the other
+unobserved components.
+The results above show that PFOF has a higher accuracy for state estimates than ExKF
+in this chaotic nonlinear system. The former can also give estimates of probability density
+functions to gain more information of the state in the probabilistic sense.
+6
+Conclusions
+A new filtering method was proposed to estimate filtering distribution of the state under
+the framework of Perron-Frobenius operator. We formulated filtering problems for discrete
+and continuous stochastic dynamical systems and applied the Perron-Frobenius operator to
+propagation of the posterior probability density function. The finite-dimensional approxi-
+mation of the PFO was realized by Ulam’s method, which provides a Galerkin projection
+space spanned by indicator functions. With Ulam’s method, the posterior PDF was dis-
+cretized and expressed by the weights of basis functions. Then the evolution of PDF became
+the transition of the weights vectors, which were iterated by PFO and likelihood function.
+This procedure was called Perron Frobenius operator filter. Thus, the empirical PDF was
+determined by a convex combination of indicator functions. We gave an error estimate of
+the proposed method and proved that its accuracy is higher than that of particle filters. Fur-
+thermore, a low-rank Perron-Frobenius operator filter was presented to approximate density
+functions via spectral decomposition. The decomposition was realized by eigendecomposi-
+tion of discretized PFO. Finally, the proposed method was implemented for three stochastic
+filtering problems, which included a linear discrete system, a nonlinear discrete system and
+a nonlinear continuous chaotic system.
+The numerical results showed that the proposed
+method has better accuracy and better robustness compared with particle filters and ExKF.
+Acknowledgement: L. Jiang acknowledges the support of NSFC 12271408 and the
+Fundamental Research Funds for the Central Universities.
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+Scheduling Inference Workloads on Distributed Edge Clusters with
+Reinforcement Learning
+Gabriele Castellano†∗, Juan-Jos´e Nieto ‡§, Jordi Luque ‡, Ferr´an Diego ‡, Carlos Segura ‡
+Diego Perino ‡, Flavio Esposito†, Fulvio Risso∗, Aravindh Raman ‡
+†Computer Science, Saint Louis University, USA
+∗Computer and Control Engineering, Politecnico di Torino, Italy
+§Universitat Polit`ecnica de Catalunya, Spain
+‡Telef´onica Research, Spain
+Email: †{gabriele.castellano}@slu.edu, ‡{jordi.luque}@telefonica.com
+Abstract—Many real-time applications (e.g., Augmented/Vir-
+tual Reality, cognitive assistance) rely on Deep Neural Networks
+(DNNs) to process inference tasks. Edge computing is considered
+a key infrastructure to deploy such applications, as moving
+computation close to the data sources enables us to meet stringent
+latency and throughput requirements. However, the constrained
+nature of edge networks poses several additional challenges to
+the management of inference workloads: edge clusters can not
+provide unlimited processing power to DNN models, and often
+a trade-off between network and processing time should be
+considered when it comes to end-to-end delay requirements. In
+this paper, we focus on the problem of scheduling inference
+queries on DNN models in edge networks at short timescales
+(i.e., few milliseconds). By means of simulations, we analyze
+several policies in the realistic network settings and workloads
+of a large ISP, highlighting the need for a dynamic scheduling
+policy that can adapt to network conditions and workloads.
+We therefore design ASET, a Reinforcement Learning based
+scheduling algorithm able to adapt its decisions according to
+the system conditions. Our results show that ASET effectively
+provides the best performance compared to static policies when
+scheduling over a distributed pool of edge resources.
+I. INTRODUCTION
+In the last years, we have witnessed the growing popularity
+of applications leveraging Deep Neural Networks (DNNs),
+from Augmented/Virtual Reality (AR/VR) to cognitive assis-
+tance or video surveillance. The DNN model training process
+typically does not have strict latency constraints and it is
+performed offline in well-provisioned centralized data-centers
+or in a distributed fashion via, e.g., federated learning [1].
+Differently, the DNN inference task is usually performed
+online with constraints in terms of accuracy, throughput, and
+latency, which may significantly differ across applications. For
+instance, services like cognitive assistance require high accu-
+racy but may tolerate few hundreds of milliseconds latency,
+while others, like self-driving cars, have more stringent latency
+needs (i.e., tens of milliseconds).
+Providing an inference service requires to address several
+challenges to meet this diverse set of application constraints,
+e.g., the selection of the appropriate variant of the model
+to be used (programming framework, compiler optimization,
+batching size, etc.), the processing unit to leverage for the
+* Gabriele Castellano and Juan-Jos´e Nieto contributed equally to this work
+during his internship at the Telef´onica Research team, in Spring 2020.
+inference (e.g., GPU, CPU, TPU), and the nodes and resources
+(e.g., memory, computing) to be allocated to every application.
+This requires management at different timescale. On a short
+timescale (i.e, milliseconds), a scheduler is in charge of
+selecting the appropriate computing instance for every new
+incoming request to meet its application requirements. This in-
+cludes not only the selection of the computation node but also
+the appropriate model variant and computation technology. On
+a longer timescale (i.e., seconds, minutes), an orchestrator
+selects the proper model variants to deploy, optimizes their
+placement across the nodes, and allocates the appropriate
+resources to them. Recent work [2]–[5] focused on data cen-
+ters and proposed DNN inference workload management for
+such environments. Further, commercial solutions have been
+deployed in recent years [6]–[8] by major cloud providers.
+Edge computing is considered a key enabler to deploy
+DNN-based applications with stringent delay or bandwidth
+requirements, as it moves computation capabilities closer to
+end-users with respect to centralized cloud platforms. This
+is especially the case for users connected via mobile access
+(e.g. 5G). However, realizing DNN inference at the edge
+poses several additional challenges. Edge infrastructures are
+indeed complex networks composed of several layers with
+heterogeneous limited resources and different latencies to end
+users [9]. Due to the less availability of resources at edge,
+multiple inference models of different capacities should be
+considered, and end-to-end delay requirements may lead to
+considering a trade-off between network delay and processing
+time. This differs from centralized cloud platforms, which
+usually feature large pools of uniform hardware available
+in a single location where DNN models can be scaled up
+almost indefinitely. For these reasons, the optimal selection
+of inference models while scheduling real-time requests at
+Edge is still a challenging task. Recent work combined edge
+computing and deep learning [10], with a focus on scheduling
+requests to minimize end-to-end delay [11] or maximize
+accuracy [12]. However, none of the existing work analyzes
+inference workload optimization taking into account different
+application constraints in realistic edge network settings.
+In this paper, we focus on the problem of scheduling DNN
+inference requests taking into account not only accuracy (i.e.,
+model selection) but also throughput and latency constraints
+arXiv:2301.13618v1  [cs.LG]  31 Jan 2023
+
+under realistic edge deployment settings. First, we model our
+distributed edge inference system and provide a definition of
+the scheduling problem (Section III), also proposing several
+baseline static scheduling policies both original and from
+literature. From evaluating static policies on a realistic network
+topology, we observe that a policy that always performs better
+does not exist, as different applications may benefit differently
+from each scheduling strategy. Based on the insights derived
+by this analysis we propose ASET1 (Adaptive Scheduling
+of Edge Tasks), an adaptive scheduling algorithm based on
+Reinforcement Learning (Section IV), which dynamically fol-
+lows system conditions and apps requirements optimizing its
+decisions accordingly. We evaluate ASET simulating three
+topologies based on the realistic network of a large ISP
+and using a pool of reference edge applications (Section V).
+Our findings show that, while some static policies are well
+suited to optimize workloads on cloud-based topologies, ASET
+improves performance over any static policy when resources
+are distributed across the edge network, effectively increasing
+the percentage of successfully handled queries.
+II. RELATED WORK
+The provisioning of on-demand inference services has been
+investigated in several recent works.
+Inference scheduling in data centers. Most of the existing so-
+lutions address the common scenario where inference queries
+have to be scheduled over the resources of a Data Center. Some
+of the main production systems are Tensorflow Serving [6],
+Azure ML [7], and Cloud ML [8]. Most scientific works
+focused on proposing algorithms and strategies to improve
+the performance and ease of use of such cloud inference sys-
+tems. [2] and [3] address the problem of scheduling Directed
+Acyclic Graph (DAGs) tasks with the objective of improving
+the throughput; GrandSLAm [2] relies on a prediction model
+that estimates job duration, while [3] proposes an efficient
+RL approach to select the number of servers to allocate
+for a given job. Being oriented to a Cloud infrastructure,
+none of them takes into account network latency between
+the servers and their heterogeneity. In [13] a Model Master
+manages the dynamic allocation of DNN models across the
+servers of a heterogeneous data center based on Azure ML,
+and proposes a protocol among servers to forward queries to
+the correct destination. Clipper [4] provides a generalization
+of TensorFlow Serving [6] to enable the usage of different
+frameworks. One of the most complete solutions is provided
+by INFaaS [5], which focuses on ease of use, providing
+transparent scheduling of incoming queries over available
+model variants, and autoscaling of deployed models based
+on load thresholds. However, all the previous works address
+the scheduling problem only from the boundaries of a data
+center, considering neither (i) network latency, thus becoming
+no suitable in scenarios with real-time constraints, nor (ii)
+resource constrained clusters, thus failing to address situations
+where workers cannot be indefinitely scaled up/out.
+1In ancient Egyptian mythology, Aset was a major goddess said to have
+power over fate itself.
+Inference offloading. Another related set of works concerns
+offloading, with a focus on the end-devices. While offloading
+has been widely studied in the literature [14], [15], the specific
+use case of DNN workload introduces additional degrees of
+freedom (e.g., model variant selection and configuration) that
+can be exploited for improving optimization over the mere
+selection of the task placement. Some recent works [16]–
+[18] provides intelligent offloading techniques for DNN tasks.
+DeepDecision [17] addresses the problem in the particular case
+of a single device running a single application; queries are
+scheduled among a series of local small models providing
+different performance/requirements trade-off, and one remote
+model, which provides the best performance. On the other
+hand, LinkShare [18] focuses on the orthogonal problem of
+ordering the offloaded requests from multiple apps on the
+same device, with the main constraint of network bandwidth.
+MCDNN [16] proposes a scheduler to handle queries from
+multiple applications on the same device, deciding (i) the
+model variant to be used and (ii) whether to offload the
+inference task or not, seeking average accuracy maximization.
+Such decisions are taken considering constraints such as
+latency requirements, device energy, cloud monetary budget.
+Inference and edge computing. Fewer and more recent are
+the trends that combine DNN with edge computing [10], with
+the aim of overcoming scalability and latency limitations of
+cloud computing. The use of edge computing brings additional
+challenges deriving from the high resource requirements of
+DNN based tasks on less powerful edge compute resources.
+Despite some issues have been addressed in recent works [11],
+[12], [19], [20], edge-oriented solutions for inference systems
+are still largely embryonic compared to data center solutions,
+with many open challenges. CloudPath [19] focuses on the
+problem of data distribution on a hierarchical continuum of
+computing resources between edge and cloud. In [20], authors
+propose an approach to schedule DAGs across multiple edge
+servers, seeking minimization of end-to-end latency. However,
+the proposed algorithm assumes the possibility to indefinitely
+allocate new edge servers when needed, with no geographical
+restrictions, thus not addressing the problem of constrained
+resources at the edge. Other works [11], [12] study the problem
+of processing data streams from scattered devices, exploiting
+the geographically distributed edge/cloud clusters. In particu-
+lar, VideoEdge [12] assumes a deployment of cameras gener-
+ating a known set of video streams, on which various DNN
+tasks should be performed. The proposed approach decides
+globally the cluster where each stream should be processed,
+as well as the model variant to employ and its configuration,
+considering computation and network bandwidth as constraints
+and seeking accuracy maximization. However, neither pro-
+cessing nor network latencies are taken as constraints, thus
+making this approach not suitable for interactive or critical
+scenarios (e.g., virtual reality, autonomous driving, and more).
+A similar use case is analyzed in [11], which focuses on
+minimizing the end-to-end latency processing data flowing
+from the edge to the cloud. However, it only considers the
+problem of task allocation, missing the possibility to optimize
+
+properly selecting model variants and their configurations.
+To the best of our knowledge, none of the existing works
+on inference serving systems addresses the problem simul-
+taneously considering (i) end-to-end latency, accuracy, and
+throughput constraints, (ii) edge-cloud computing and multi-
+cluster deployment, (iii) real-time job dispatching, (iv) opti-
+mization on model variant selection.
+III. SCHEDULING IN EDGE-CLOUD INFRASTRUCTURE
+In this section, we formally define the problem of schedul-
+ing inference tasks on a distributed edge-cloud infrastructure.
+Additionally, we describe a set of static scheduling policies
+(both original and from literature), that we then use in Sec-
+tion IV as a baseline for our dynamic scheduling approach.
+A. System modeling
+Applications and data-streaming sources. We consider a set
+of sources (e.g., end users, IoT devices, vehicles) running
+a variety of applications (e.g., virtual reality, autonomous
+driving) each relying on one or more DNN inference tasks.
+Every application generates queries to be processed, i.e., each
+query represents the request to perform a specific inference
+task j ∈ J (e.g., object detection, speech recognition) on
+a given input (e.g., a video frame), where J is the set of
+inference tasks supported by the system. Since applications
+often require more than one query to be processed, we treat
+sequential queries as streams (e.g., all the frames captured by
+an AR headset). Therefore, each query q belongs to a stream
+i ∈ I, being I the entire set of streams currently served by
+the system. Every query of a stream has a set of requirements
+such as a maximum end-to-end delay Di, and a minimum
+required accuracy Ai. Additionally, every stream i has a data
+rate ρi, that is the number of queries submitted each second
+(e.g., frame rate), and every query of stream i has an input of
+size ζi (e.g., frame size). Note that all queries of a stream are
+for the same task j ∈ J with the same requirements.
+DNN Models and Variants. Every inference task j can be
+served using a Deep Neural Network model m among the
+set of M j models that are trained for task j. Therefore, the
+system provides a total of Nm = �
+j∈J |M j| DNN models.
+Take object detection as an example application. A model m
+represents a particular Neural Network architecture with pre-
+trained weights (e.g., yolo-v3, ssd-mobilenet-v1), and features
+a given accuracy Am (mean average precision - mAP). A
+model m can be deployed and run through different setups
+and underlying hardware (e.g., SSD Mobilenet v1 on (i)
+Tensorflow-GPU with batch size 8, or on (ii) Opencv-CPU
+batch size 1 and 2 replicas, and more), thus obtaining a set
+V m of different model variants. A model variant v features
+a given processing delay Dv, throughput capacity Cv (i.e.,
+the maximum number of queries it can process per second),
+and resource usage rv ∈ Rk
++ (e.g., in terms of CPU, system
+memory and GPU memory). Note that the processing delay
+may vary based on the size ζi ∈ R+ of the input data, thus it
+is a function Dv : R+ → R+; with Dv we refer to the time
+____
+____
+____
+____
+__
+____
+____
+____
+____
+__
+____
+____
+____
+____
+__
+Tasks Scheduler
+____
+____
+____
+____
+__
+____
+____
+____
+____
+__
+____
+____
+____
+____
+__
+___
+___
+Queries
+Task: Obj. det.
+Constraints:
+•
+Latency
+•
+Accuracy
+Stream:
+•
+Size
+•
+Rate
+Tasks Scheduler
+____
+____
+____
+____
+__
+____
+____
+____
+____
+__
+Model-Variant
+Task: obj. det. 
+Specs:
+•
+mAP
+•
+Processing Time
+•
+Load Capacity
+Load: current QPS
+Fig. 1: The scheduler dispatches streams of queries on available model variants
+based on their constraints and geographical position of clusters.
+needed to process the maximum input size supported by the
+model (analogous considerations hold for the capacity Cv).
+Network topology and computing clusters. We consider a
+geographically distributed cloud-edge infrastructure composed
+by Nν computing clusters (e.g., a centralized data center, a
+telco regional cloud, an eNodeB) typically organized in a
+hierarchical topology. Each cluster potentially provides dif-
+ferent resources. We denote cn ∈ Rk
++ the overall capacity of
+cluster n, with cnk representing the amount of resource k ∈ N
+available on cluster n. Examples of resources include CPU,
+system memory and GPU memory.
+Model variants are deployed at different computing clus-
+ters consuming a different amount of resources. On a long
+timescale (i.e., seconds, minutes), an orchestrator selects the
+appropriate set of model variants to deploy, optimizes their
+placement across the clusters, and allocates the appropriate
+resources. Finally, stream sources are connected to a small
+cluster at the lower layer of the hierarchy. This can be either
+the antenna/eNodeB in case of cellular communication or the
+home gateway in the fixed access case. Queries need to be
+scheduled for processing across model variants available at
+different computing clusters to meet application requirements
+on a short timescale (i.e., tens of milliseconds). In the fol-
+lowing, we provide a definition of the scheduling problem we
+tackle in this paper.
+B. Scheduling problem definition
+We assume a scheduler is located at the nearest compute
+cluster available to existing stream sources, i.e., antenna/eN-
+odeB or the home gateway/central office in the fixed access
+case. It follows every stream source is served by a scheduler
+s among Ns different ones (one per each lower layer cluster).
+Each scheduler s has a given average network delay ds
+n
+towards each cluster n; we also model the associated delay
+deviation as σs
+n. Note that an additional access delay from the
+stream source to the scheduler has to be taken into account
+(e.g, the radio latency between a device and the nearest 5G
+antenna). We denote δi the additional access delay that affects
+stream i. Every scheduler is aware of each model variant v
+currently available on each cluster n, each with its current
+load Lvn(t) (measured in terms of incoming queries per
+second2). Based on the current conditions of the available
+2Each stream i contributes to the total load as a fraction ηi
+v of its data
+rate ρi (named fractional load), based on the stream data size ζi.
+
+88880
+488 (0)model variants, for every stream i it serves, a scheduler s
+decides which model variant v on which cluster n should be
+used to process stream i.
+When scheduling a stream i to the proper model variant/-
+cluster, the scheduler takes into account application require-
+ments. Specifically, it considers the stream data size ζi, its
+data rate ρi, its bit rate bi, the maximum tolerated end-to-end
+delay Di and the minimum required accuracy Ai, satisfying
+the following constraints:
+(i) the selected model variant v is a valid implementation of
+task j required by i,
+v ∈ V m ∧ m ∈ M j;
+(1)
+(ii) the load capacity of the chosen model variant is not
+exceeded,
+Lvn(t) + ηi
+vρi ≤ Cv,
+(2)
+being ηi
+v the fractional load of stream i for model variant v;
+(iii) the sum of expected network delay and processing time
+does not exceed the maximum tolerated delay,
+2(δi + ds
+n + 2σs
+n) + biζi + Dv(ζi) ≤ Di,
+(3)
+where the first addendum is the round-trip propagation time,
+the second is the transmission delay for one query and the
+third is the time needed to process the query;
+(iv) the selected model provides an adequate accuracy
+Am ≥ Ai.
+(4)
+A graphical representation of the scheduling problem is
+depicted in Figure 1, while a scheduling policy can be formally
+defined as follows.
+Definition 1. (scheduling policy). Let us consider a stream i to
+be processed through a task j on an edge-cloud infrastructure
+that features a set of V m compatible model variants over Nν
+clusters (|N| = Nν). A scheduling policy is any function
+β : I → V m, N
+(5)
+that binds stream i to a feasible model variant v ∈ V m de-
+ployed on cluster n ∈ N, so that constraints at Equations (1),
+(2), (3), and (4) are satisfied.
+Note that, as requests are handled in real-time, scheduler
+decisions should be taken in an amount of time that is
+negligible compared to the stream latency requirements.
+Scheduling performance metrics and objectives. Based on
+the scheduling decisions, in a given time instant t the stream
+i will feature a reject ratio qR
+i (t) ∈ [0, 1], i.e., the fraction
+of queries from stream i that have not been processed by the
+system because of resource unavailability, and a failure ratio
+qF
+i (t) ∈ [0, 1], i.e. the fraction of queries that have been served
+violating one or more application requirements (i.e., delivered
+out of maximum tolerated delay).
+The goal of the scheduler is typically to maximize, over
+time, the fraction of queries that are served successfully, i.e.,
+to minimize the sum of reject ratio and failure ratio.
+C. Static scheduling policies
+Several policies have been proposed for static scheduling
+of inference tasks on edge clusters [9], [21]. In this work we
+consider the following ones (both original and from literature):
+1) closest: bind stream i to any feasible model variant v∗ lo-
+cated on the cluster n∗ that features the lower network latency
+to serving scheduler s, i.e., n∗ = arg minn∈N (ds
+n + 2σs
+n).
+This policy may lead to the early saturation of smaller clusters
+at the very edge, as they are always preferred [22].
+2) load balancing: bind the input stream to model variant v∗
+on cluster n∗ such that (v∗, n∗) = arg minv,n∈V m×N Lvn(t).
+This policy can bring huge performance gains compared to
+closest [22]; however, it may lead to unfair allocation when
+latency-sensitive applications are in the minority.
+3) farthest: bind stream i to any feasible model variant v∗
+located on the cluster n∗ with the highest (still feasible) net-
+work latency, i.e. n∗ = arg maxv∈N (ds
+n + 2σs
+n). As opposed
+to closest, this policy preserves smaller clusters at the very
+edge for those apps that really need them [23]; however, it is
+highly affected by the unreliability of network delay for long
+distance communications.
+4) cheaper: bind stream i to model variant v∗ on clus-
+ter n∗ such that the expected end-to-end delay (round-
+trip and processing time) is maximized, i.e., (v∗, n∗) =
+arg minv,n∈V m×N (2(ds
+n + 2σs
+n) + Dv(ζi)). We designed this
+policy as an improvement over farthest, as it additionally tries
+to preserve the most performing model variants.
+5) random-proportional latency: bind stream i to model
+variant v on cluster n with probability 1/(2(ds
+n + 2σs
+n) +
+Dv(ζi)). This guarantees that, on a large enough number of
+streams, bindings are proportionate to end-to-end delays [21].
+6) random-proportional load: bind stream i to model variant
+v on cluster n with probability Cv/Lvn(t). This guarantees
+that, on a large enough number of streams, bindings are
+proportional to the capacity of each model variant.
+7) least impedance: bind stream i to model variant v∗ on
+cluster n∗ such that end-to-end latency to s is minimized, i.e.,
+(v∗, n∗) = arg minv,n∈V m×N (2(ds
+n + 2σs
+n) + Dv(ζi)) [21].
+This greedy policy leads to the best performance when the
+overall load is low, but may suffer from a high rejection rate
+once the closest and fastest model variants are saturated.
+Our experiments (Section V) show that, for a heterogeneous
+pool of applications, a policy that always performs better
+than the others does not exists: different applications may
+benefit differently from each scheduling strategy, and also the
+physical topology and the particular streams arrivals can be
+determinant. Based on these findings, in the next section we
+propose ASET, an algorithm for Adaptive Scheduling of Edge
+Tasks that leverages Reinforcement Learning to optimize its
+decisions dynamically based on the current system conditions.
+IV. ASET SCHEDULING ALGORITHM
+Our adaptive scheduling approach aims to learn the optimal
+policy depending on current system conditions, e.g, current
+applications, network topology, and stream arrivals that vary
+over time. Due to the lack of labeled data, the optimal
+
+____
+____
+____
+____
+ACTION
+REWARD
+ASET	RL	Agent
+STATE
+3
+1
+Incoming	Workloads
+____
+____
+Cloud-Edge	Infrastructure
+2
+4
+5
+Fig. 2: Algorithm overview. State St, sampled from the environment, is
+forwarded through the agent DNN, which outputs action At; performing At
+on the environment contributes to reward rt+1 obtained at the next step.
+policy learning is formulated as a Reinforcement Learning
+(RL) problem; hence, an intelligent agent tries to learn the
+optimal policy selection strategy according to the observed
+state of the environment. This is accomplished by an RL
+policy that estimates a probability distribution of each possible
+action (policy selection) that cumulatively maximizes a reward
+(typically maximizing the fraction of queries that are served
+successfully), as shown in Figure 2.
+Let us consider a learner and decision-maker called the
+agent, and an environment that is the external world that the
+agent interacts with at discrete time steps t. Given St ∈ S,
+where S is the set of possible states of the environment, the
+agent can select an action At ∈ A(St), standing for the set of
+available actions in state St. The agent receives an observation
+of the environment St at time t and, one step later, a numerical
+reward, rt+1 ∈ R ⊂ R, and it jointly determines the action
+At to perform, which, in part, yields to next state St+1.
+Definition 2. (stochastic reinforcement learning policy). An
+RL policy πφ, where φ ∈ Rd denotes policy parameters, is
+any function or algorithm that determines and maps the next
+action to take by an agent. A stochastic RL policy, additionally,
+estimates a probability distribution over actions that an agent
+can take at a given state:
+πφ : A x S → [0, 1],
+(6)
+πφ(a|s)
+def
+== P(take action a|given state s).
+Overall, the goal of the proposed adaptive scheduling is to
+learn an optimal sequence of static network scheduling policies
+that maximizes the percentage of successfully dispatched
+streams. At a T seconds rate, the RL-based scheduler sam-
+ples the environment by collecting a variety of observations
+from the edge-cloud infrastructure, e.g., responses and loads,
+building up the current state St of the environment. Then,
+the agent evaluates a discrete set A of actions and chooses
+an action At ∈ A, where A stands in this work for the set
+of available network scheduling policies β. Note that the set
+of actions does not depend on the state itself, thus the sets
+A(St) = A are the same (Section III-C). Therefore, every time
+that the agent takes an action At, the state of the environment
+St is observed and a reward score rt+1 is used as feedback
+information to improve the policy selection, see Figure 2. In
+this work, these rewards are defined as a linear combination of
+the ratio of “failed” queries and the ratio of queries that have
+0
+100
+200
+300
+400
+time (s)
+30
+40
+50
+60
+70
+80
+90
+100
+% of success queries
+RP-LOAD
+CHEAPER
+FARTHEST
+CHEAPER
+load-balancing
+rp-load
+least-impedance
+closest
+farthest
+cheaper
+rp-latency
+random
+ASET
+(a) Cloud-based topology.
+0
+100
+200
+300
+400
+time (s)
+40
+50
+60
+70
+80
+90
+100
+% of success queries
+CHEAPER
+FARTHEST
+CHEAPER
+FARTHEST
+CHEAPER
+load-balancing
+rp-load
+least-impedance
+closest
+farthest
+cheaper
+rp-latency
+random
+ASET
+(b) Edge-based topology.
+Fig. 3: The ASET RL agent infers the optimal policy sequence based on the
+system conditions, seeking an optimal binding between workloads and model
+variants that maximizes the percentage of success queries. Plots show two
+runs on a cloud-based topology and on an edge-based one (see Section V).
+been “rejected” for lack of available resources (Section IV-C).
+The particular policy βt, selected by the agent at time t, is
+used to dispatch all incoming streams during the subsequent
+time window [t, t + T]. Therefore, given the corresponding
+states sequence S = [S0, ST , S2T , ..., SkT ] with k ∈ N, the re-
+sulting overall scheduling policy β(S) = [β0, βT , β2T , ..., βkT ]
+dynamically maps, with the corresponding baseline policies βt,
+a stream i to a model variant v and its deployment on cluster
+n. From now, and for the sake of simplicity, we will refer as π
+to the policy learned by the ASET agent (Definition 2), which
+leads to a particular static policy sequence β(S). It corresponds
+to any function employed to estimate the optimal sequence of
+actions that the agent should perform at each time window
+[t, t+T] and given a state St, β(S) = [A0, AT , A2T , ..., AkT ].
+The intuition of this behavior is provided in Figure 3. Note that
+each of the static scheduling policies from Section III-C cor-
+responds to a deterministic agent that always returns the same
+action At independently of the system state; whereas the pol-
+icy π learned by the ASET agent can be seen as a meta-policy
+(or as a policy of baseline scheduling strategies) that also
+satisfies the constraints from Equations (1), (2), (3), and (4).
+A. Deep Q-Learning policy optimization
+Our RL agent has to cope with a discrete set of actions, with
+A ⊂ N. This is often modeled in literature as a stochastic
+process with no memory, which is a Markov Decision Pro-
+cess [24] (MDP). In this work, our MDP defined by tuples
+(S, A, T , R, γ) represents states comprised of partial obser-
+vations from the system. Nonetheless, the model parameters
+of such MDP are unknown, i.e., the transition probabilities
+T (s′|a, s) and the rewards R(s′|a, s) of taking the action
+At = a and moving from state St = s to state St+1 = s′.
+Note that the ASET agent should experience each transition
+among states at least once, or even multiple times to get a
+reliable estimation of both transition and cumulative rewards.
+At each step t = kT, with k ∈ N, the RL agent can choose
+one of several possible scheduling policy-actions, βt ≡ At.
+The transition probability T (s′|a, s) depends in part on the
+chosen action, and, additionally, from some positive or neg-
+ative reward that may be returned by every state transition,
+named return of actions. Overall, our objective is to find a
+strategy, i.e., a policy π mapping to a sequence β(S), that
+maximizes the expected return G(t) of rewards over time.
+
+Thus, G(t) is defined in terms of the cumulative weighted
+rewards along with states and given the corresponding optimal
+sequence of actions to take in the future:
+G(t) =
+H
+�
+τ=0
+γτrτ
+γ ∈ [0, 1],
+(7)
+where rτ = R(s′|a, s) is the reward at time step τ due to
+corresponding state transition (s, s′), γ is a weighting factor
+that reduces the contribution of long-term rewards, usually
+known as the discount factor, and time H is the last time
+step within a training episode (see Section IV-C for further
+details). Therefore, the RL agent’s target policy is
+π∗(a|s) = arg max
+πφ
+Et∗∼πφ {G(t)} ,
+(8)
+which translates the scheduler state into a distribution over
+actions, see Definition 2. Note that the expectation is computed
+over the distribution of trajectories t∗ = (s0, a0, s1, ...).
+In Q-Learning, the optimal pair values (s, a), i.e., those
+yielding to the sequence of optimal actions, are generally
+called Quality-Values (Q-Values) and noted as Q∗(s, a) [25].
+They correspond to the sum of weighted rewards that the RL
+agent can expect on average after performing action a on state
+s. It is also known as the expected return of actions,
+Q(s, a) = Et∗∼πφ {Gt|St = s, At = a} .
+(9)
+Bellman [24] showed that if an agent’s trajectory follows
+the highest Q-Values, then its policy is optimal and leads
+to the highest G(t) as well. Bellman also reported that an
+accurate estimate of Q-Values can be found recursively by
+using the Bellman Optimality Equation, also known as the
+Value Iteration algorithm. In fact, Q-Learning is an adaptation
+of Bellman’s value iteration algorithm, where a policy is
+implicitly, or off-line, learned by following trajectories yield-
+ing to the highest Q-Values [25]. It is usually computed by
+dynamic programming and assumes that the optimal value of
+state St = s is equal to the reward it will get on average,
+after taking one optimal action a and adding the expected
+optimal value of all possible next states along the future path of
+decisions, that is Q(s, a) = Eπ {r + γ maxa′ Q(s′, a′)|s, a}.
+Equation (9) turns out into the following iteration algorithm,
+which converges to the optimal Q∗(s, a),
+Qk+1(s, a) ←
+�
+s′
+T (s, a, s′)
+�
+r + γ max
+a′ Qk(s′, a′)
+�
+,
+(10)
+for all s′ ∈ S, a′ ∈ A and k ∈ N as iteration step. For
+simplicity, we set the transition probability matrix T to all
+elements equal to 1, allowing initial transitions among all seen
+states. Once Q-Values are estimated, the optimal policy π∗
+for the RL agent corresponds to chose the action that has the
+highest Q-Values: π∗(a|s) = arg maxπ Qπ(s, a), for all s ∈ S
+and a ∈ A ≡ β static policies in Section III-C.
+However, previous algorithm does not scale for large MDPs
+with a large number of states. A solution is to approximate
+the optimal Q∗(s, a) using a Deep Neural Network, named
+Deep Q-Network (DQN) [26], to get an estimate Q(s, a; φ) ≈
+Q∗(s, a), where φ stands for the parameters of the DQN
+model, see line 16 in Algorithm 1. The using of a DQN for
+approximate Q-Learning is known as Deep Q-Learning.
+B. State Encoding
+We model the state in a continuous fashion, representing
+the environment in a given time t as a set of some particular
+features sampled from the system and averaged along a time
+window of size T. Features are evaluated separately for each
+available worker w ∈ W,3 and are as follows: (i) the number
+|Iw| of streams currently served by worker w, being Iw =
+{i ∈ I| β(i) = (v, n)}; (ii) the current throughput Rw(t) of
+worker w, in terms of responses delivered at the time instant
+t; (iii) the current load Lw(t), measured in terms queries per
+second normalized on input size (as defined in Section III-B);
+(iv) number of incoming instant queries grouped by stream
+characteristics, e.g., queries of all streams that require end-to-
+end delay within a given range [δ1, δ2[ and features a data rate
+in the interval [ρ4, +∞[, i.e., �
+i∈I1,4 ρi, where I1,4 = {i ∈
+I| Di ∈ [δ1, δ2[∧ρi ∈ [ρ4, +∞[}. In particular, we consider
+a partition 0 = δ0 < δ1 < δ2 < ... < δNδ−1 of R+ with Nδ
+delay intervals, and a second partition 0 = ρ0 < ρ1 < ρ2 <
+... < ρNρ−1 of R+ with Nρ input-rate intervals, evaluating
+Nδ · Nρ different sum of instant queries, that is one feature
+for each combination of the two partitioning sets. The features
+defined so far constitute a vector as,
+Sw,t =
+�
+|Iw|, Rw(t), Lw(t),
+�
+i∈I0,0
+ρi, . . . ,
+�
+i∈INδ−1,Nρ−1
+ρi
+�
+(11)
+where Sw,t ∈ R3+Nδ·Nρ
++
+. Therefore, the complete state S is
+modeled as a three-dimensional vector in R(3+Nδ·Nρ)×|W |×T
++
+,
+that is, each feature in (11) is first evaluated for each available
+worker (each model variant on each node), and then for
+each time instant within the considered time window. For
+instance, vector Sw stores, for worker w, every features in
+(11) evaluated at every time instant t−T +1, t−T +2, ..., t
+within time window [t − T + 1, t]. From now, we refer to the
+state vector encoding as simply s or st for a generally speaking
+state or a state referred to a time window, respectively.
+C. Training
+The proposed RL scheduling agent is trained over a series
+of episodes that resemble various scenarios. Each episode
+corresponds to a different workload execution with given
+parameters, e.g. requirements from tasks, number of clients
+per minute (λ) or the seed value (ζ) for random number
+generation (RNG), and is concluded when the percentage
+of success queries, qS
+t , falls below a given threshold θ or
+when a timeout H is reached. This allows us to speed up
+the training by terminating unsuccessful or steady episodes
+quickly. At every time step t, a reward rt scores the rate of
+successful queries, see Algorithm 1 at lines 9-10, where qF
+t
+3A worker is a model variant instance v running on a particular cluster n,
+therefore we can assume index w = n · v + v.
+
+Algorithm 1 ASET training procedure
+1: initialize replay buffer D = ∅ ring of size Nr and φ0 DQN parameters
+2: initialize training RNG seeds ζ ∈ [0, ..., P − 1]
+3: for episode k ∈ [0, ..., M] do
+4:
+sample random seed ζ ∼ U(P)
+5:
+initialize πk(a|s) = ϵ U(β) + (1 − ϵ)δ(a = arg maxa Q(s, a; φk))
+6:
+for step t ∈ [0, ..., H] do
+7:
+sample at ∼ πk(a|s)
+8:
+sample st+1 state from edge-network and reward rt
+9:
+if ϕ > 1 − (qF
+t + qR
+t ) then rt = −(qF
+t + qR
+t )
+10:
+else rt = ψ
+11:
+D ← D ∪ {(st, at, st+1, rt)} with circular replacement
+12:
+if qS
+t ≤ θ then t ← H, ends this episode
+13:
+φk,0 ← φk
+14:
+for gradient descend step g ∈ [0, ..., G − 1] do
+15:
+sample batch B ⊂ D with idx ∼ U(Nr)
+16:
+compute Ct ← rt + γ arg maxat+1 Q(st+1, a(st+1); φk)
+17:
+compute loss L = �
+i(Ci − Q(si, ai; φk))2
+18:
+update replay network φk,g+1 ← φk,g − α∇φk,gL
+19:
+update DQN parameters φk+1 ← φk,G
+is the ratio of “failed” queries, i.e., those delivered violating
+one or more constraints (e.g., out of tolerated delay), and qR
+t
+is the ratio of queries “rejected” by the system for lack of
+resources, normalized by corresponding time window. ψ is
+a penalty inverse proportional to the episode active time. It
+ensures that both short and bad action trajectories do not reach
+higher returns than optimal ones. Note that DQN network
+is used to minimize the target loss L, see lines 14-18, by
+Adam optimizer and α learning rate. It takes gradient steps
+on the Bellman error objective L, see factor C at line 16 and
+Eq. (10), concurrently with data collection from the replay
+buffer [27] for an efficient off-line learning. This is a common
+hybrid approach to implement Q-Learning [25], [28], [29].
+Additionally, we employ an ϵ-greedy exploration policy, see
+line 5, with parameter ϵ dynamically updated. The architecture
+of our DQN consists of a stacking of convolutional layers
+that extracts temporal correlations from the state tensor S.
+Such feature extraction part is composed of three convolutional
+layers with 4x4 kernels along the time and feature dimensions,
+followed by Re-Lu activation and max-pooling. Finally, two
+linear layers squeeze the dimension from 256 to as many
+outputs as different static policies β.
+V. PERFORMANCE EVALUATION
+We evaluate ASET using a prototype implementation of an
+edge inference system that will be released upon acceptance
+of the paper. We first use our prototype to run small scale
+experiments with the aim of profiling some representative
+models and their variants (results not shown). Then we use
+such profiling data to run large scale experiments on a simu-
+lated setup, comparing the performance of ASET to those of
+static scheduling policies.
+A. Evaluation settings
+System Prototype. Our prototype implements the edge in-
+ference system functionalities described in Section III. On
+each cluster, a Master deploys workers and routes streams
+between them and remote clients; each Worker runs in a
+Docker container and implements a pipeline that processes
+queries in FIFO order from different streams, based on the
+model variant batch size; a Monitoring agent on each cluster
+collects stats from model variants usage and their performance,
+used (i) to build a catalog of model variants and (ii) to provide
+each Scheduler with aggregated observations on the system
+state. We use such a prototype to profile variants of pre-
+trained inference models with respect to their resource usage
+and performance (see below).
+Simulation Setup. To evaluate our approach on a large
+scale, we set up a simulated environment where each worker
+simulates the inference task based on the profiling information
+available for its model variant. Therefore, empty responses are
+generated for each batch of queries after simulating a pro-
+cessing delay (based on a normal distribution). Additionally,
+we simulate network delay between stream sources and des-
+tination clusters (see below for considerations on the network
+topologies), as well as the transmission delay. Apart from
+simulated workers, other system components are deployed
+using their prototype implementation. Therefore, the system
+operates on a realistic timescale.
+Network topology. We leverage the network topology of a
+large ISP to assess scheduling performance under realistic
+settings. Specifically, our simulated environment is a cloud-to-
+edge topology with clusters of different sizes deployed hierar-
+chically. To preserve ISP confidentiality, we only report a high-
+level summary of topology, latency, and hardware distribution
+characteristics. Similarly to the tiered topologies from [30],
+[31], our topology can provide clusters with computation
+capabilities at different layers: network access (e.g., antennas,
+home gateway), central offices (multiple payers), operator
+data center, and remote cloud (third parties). Specifically, we
+focus on three scenarios: (i) dc-cloud, where resources are
+deployed at ISP data center and remote cloud only; (ii) co-dc-
+cloud, where resources are deployed at central offices, operator
+data center and remote cloud; (iii) full-edge topology, where
+clusters are deployed at all layers previously mentioned. Note
+that we limit the simulations to the topology serving 1,000
+antennas from the full ISP topology, and appropriately scale
+resources (see below). For the evaluation, we assume a 5G
+radio access technology with antennas deployed similarly to
+LTE. Network/transmission delays range from few millisec-
+onds, to reach the eNodeBs behind the antennas, to the order
+of ten milliseconds for central offices and ISP data centers,
+and few tens of milliseconds for the remote cloud.
+Requests workload. Requests are generated following a Pois-
+son distribution. Each generator runs on average λ clients per
+minute querying the scheduler of a given geographical area
+(antenna). Once spawned, each client requests for processing
+a stream featuring randomized characteristics in terms of frame
+rate, required end-to-end latency, required model accuracy,
+frame sizes, stream duration. To capture realistic queries char-
+acteristics, we modeled metrics of generated streams according
+to the reference edge applications in Table I. In our settings,
+a generator with λ = 60 brings a load of almost 1000 queries
+per second on the serving antenna.
+Computing clusters and model variant. We assume a given
+
+TABLE I: Characteristics of reference applications [32], [33].
+Edge app
+Tolerated
+delay
+Frame
+rate
+Streams
+duration
+Required
+accuracy
+Pool
+95 ms
+5 FPS
+5-10 s
+10 mAP
+Workout Assistant
+300 ms
+2 FPS
+90 s
+10 mAP
+Ping-pong
+150 ms
+15-20 FPS
+20-40 s
+15 mAP
+Face Assistant
+370 ms
+5 FPS
+1-5 s
+30 mAP
+Lego/Draw/Sandwich
+600 ms
+10-15 FPS
+60 s
+25 mAP
+Gaming
+20-30 ms
+25 FPS
+10-30 m
+35 mAP
+Connected Cars
+150 ms
+10-15 FPS
+15-30 m
+40 mAP
+Tele-Robots
+25-35 ms
+10 FPS
+5 m
+40 mAP
+Remote-driving
+20-30 ms
+20 FPS
+15-30 m
+50 mAP
+Interactive AR/VR
+30-50 ms
+25 FPS
+30-60 s
+35 mAP
+pool
+workout
+pingpong
+face
+lego
+gaming
+cars
+robots
+driving
+ar/vr
+0
+20
+40
+60
+80
+100
+% of success queries
+load-balancing
+rp-load
+least-impedance
+closest
+farthest
+cheaper
+rp-latency
+ASET
+(a) Average values for clients rate λ = 60
+pool
+workout
+pingpong
+face
+lego
+gaming
+cars
+robots
+driving
+ar/vr
+0
+20
+40
+60
+80
+100
+% of success queries
+load-balancing
+rp-load
+least-impedance
+closest
+farthest
+cheaper
+rp-latency
+ASET
+(b) Average values for episodes with dynamic clients rates.
+Fig. 4: Success percentage for different apps on the full-edge topology.
+reference hardware distribution across clusters, with comput-
+ing capabilities increasing from the access network to the
+cloud. Specifically, the access network can be equipped with
+an 8-16 cores machine, 16 GB of memory and a small TPU,
+central offices can host in the order of tens of servers (32-
+64 CPUs, 128-256 GB, and few GPUs), ISP data centers
+can host hundreds of servers, while for the centralized cloud
+we assume unlimited resources. In our evaluation, we focus
+on DNN models for the object detection task, as it is one
+of the most challenging and computation-intensive inference
+service [34], [35]. Using our prototype we profile MobileNet-
+SSD, Yolo-v3, and Tinyyolo-v2 models [36], [37], with CPU
+and GPU variants on different batch sizes, scaling on allocated
+resources and number of replicas. Such a set of results is not
+shown for lack of space. We use profiled information to run
+our simulations on top of the three topologies described above.
+On each cluster, workers have been scaled on the number of
+replicas up to resource saturation.
+B. Experimental Results
+We compare the performance of the baseline policies de-
+scribed in Section III-C distinguishing results for different
+applications from Table I. As a performance metric we con-
+sider the percentage of queries that are successfully processed
+by the system satisfying the application QoS requirements.
+Figure 4a shows results of multiple runs with λ = 60. Results
+0
+100
+200
+300
+400
+time (s)
+40
+50
+60
+70
+80
+90
+100
+% of success queries
+load-balancing
+rp-load
+least-impedance
+closest
+farthest
+cheaper
+rp-latency
+random
+ASET
+(a) Time avg on dc-cloud for λ = 60.
+20
+40
+60
+100
+# of streams per minute (poisson λ)
+30
+40
+50
+60
+70
+80
+90
+100
+% of success queries
+load-balancing
+rp-load
+least-impedance
+closest
+farthest
+cheaper
+rp-latency
+random
+ASET
+(b) Different clients rate on dc-cloud.
+0
+100
+200
+300
+400
+time (s)
+40
+50
+60
+70
+80
+90
+100
+% of success queries
+load-balancing
+rp-load
+least-impedance
+closest
+farthest
+cheaper
+rp-latency
+random
+ASET
+(c) Time avg on co-dc-cloud for λ = 60.
+20
+40
+60
+100
+# of streams per minute (poisson λ)
+30
+40
+50
+60
+70
+80
+90
+100
+% of success queries
+load-balancing
+rp-load
+least-impedance
+closest
+farthest
+cheaper
+rp-latency
+random
+ASET
+(d) Different clients rate on co-dc-cloud.
+Fig. 5: Performance of ASET compared with static policies for (ab) the dc-
+cloud topology and (cd) the co-dc-cloud topology.
+suggest that there is no one-size-fits-all policy, as various
+applications may benefit differently from each policy. Varying
+the rate of stream requests on the antenna (Figure 4b) may
+further increase the uncertainty of relying on a single policy.
+In the following, we compare the performance of the ASET RL
+scheduling approach with the performance of static policies,
+evaluating the benefits it can introduce in the various scenarios.
+We trained three different versions of ASET (one for each
+topology). In particular, we sample the state using a time
+window T = 25 seconds, and we experimentally chose an
+episode timeout of 8 minutes to avoid steady states in the
+network. Despite we evaluate on multiple clients rate, our
+agent has been trained only on episodes with λ = 60.
+Cloud deployment. When all the available resources are
+located in a few centralized clusters, the various static policies
+have small differences in performance and a dynamic approach
+has little room for improvement. Results for the dc-cloud
+topology are shown in Figures 5ab. In particular, Figure 5a
+plots, for every moment of the simulation (time axis), the
+percentage of queries that are handled successfully, averaging
+multiple runs with different workloads. The graph shows that,
+for this topology, ASET does not improve over static policies,
+and it even performs worse for higher lambdas (Figure 5b).
+Figures 5cd shows that moving some resources to Central
+Offices (co-dc-cloud topology) makes a huge difference: in
+general, all the policies achieve a higher success ratio on this
+configuration (Figure 5c), as they can exploit the additional
+lower latency spots, and the higher level of distribution gives
+to ASET a certain margin of improvement. Figure 5d shows
+that ASET introduces some improvement over all the baselines
+for every lambda, despite being trained only for λ = 60.
+Edge deployment. The results so far suggest that a good
+distribution of computing resources is a key factor to improve
+against static scheduling policies. As shown in Figure 6, the
+benefits of using a dynamic scheduling approach become more
+
+0
+100
+200
+300
+400
+time (s)
+40
+50
+60
+70
+80
+90
+100
+% of success queries
+load-balancing
+rp-load
+least-impedance
+closest
+farthest
+cheaper
+rp-latency
+random
+ASET
+(a) Queries handled successfully.
+20
+40
+60
+100
+# of streams per minute (poisson λ)
+0
+20
+40
+60
+80
+100
+% of success queries
+load-balancing
+rp-load
+least-impedance
+closest
+farthest
+cheaper
+rp-latency
+random
+ASET
+(b) Different clients rate on full-edge.
+0
+100
+200
+300
+400
+time (s)
+0
+5
+10
+15
+20
+25
+30
+35
+% of failed queries
+load-balancing
+rp-load
+least-impedance
+closest
+farthest
+cheaper
+rp-latency
+random
+ASET
+(c) Queries delivered with QoS violations.
+0
+100
+200
+300
+400
+time (s)
+0
+5
+10
+15
+20
+25
+30
+% of rejected queries
+load-balancing
+rp-load
+least-impedance
+closest
+farthest
+cheaper
+rp-latency
+random
+ASET
+(d) Queries rejected for lack of resources.
+Fig. 6: Performance of ASET compared with static policies for the full-edge
+topology. (a) (c) and (d) show averages of multiple runs with λ = 60.
+concrete in a full-edge topology, where resources are better
+distributed on multiple smaller clusters in different locations.
+In fact, Figure 6a shows that the dynamic approach of ASET is
+able to achieve a constant improvement over any static policy,
+with a higher success ratio over time. In particular, Figures 6cd
+show that, while maintaining the same rejection rate as the best
+static-policy, ASET effectively reduces the number of queries
+that are handled violating one or more QoS requirements.
+Moreover, Figure 6b shows that an ASET agent trained only
+for λ = 60 can also generalize on different requests rate, even
+supporting a load of more than 1600 queries per second (λ =
+100) on a single antenna.
+Dynamic input rate. We have performed some additional
+experiments to evaluate how the system behaves in dynamic
+situations where the requests rate varies over time. For this
+purpose, we have set up some dynamic runs where the lambda
+value changes every 150 seconds: a first pattern simulates
+a particularly fast variation with values of 20, 60, and 100
+clients per minute (Figure 7a); a different pattern simulates
+a more steady scenario where the requests rate first moves
+from 60 to 40 clients per minute, then drops to 20, and finally
+slowly goes back to 60 (Figure 7b). Similar to previous plots,
+the outcomes for this set of experiments are shown averaging
+values over time for multiple runs (Figure 7). Results in both
+figures show that having a dynamic requests arrival even intro-
+duces a bigger margin for improvement that ASET effectively
+exploits reaching the highest percentage of queries handled
+successfully. This appears particularly evident in the case
+where the variation between client arrivals is faster and bigger
+(Figure 7a). This result suggests that, while some of the static
+policies may achieve decent performance when the system
+load is stable, they struggle on more dynamic scenarios. In
+such situations, an adaptive algorithm such as ASET is more
+suitable as it can learn how to best optimize the system under
+different conditions. Moreover, results suggest that ASET
+0
+100
+200
+300
+400
+time (s)
+40
+50
+60
+70
+80
+90
+100
+% of success queries
+load-balancing
+rp-load
+least-impedance
+closest
+farthest
+cheaper
+rp-latency
+random
+ASET
+(a) Burst variation of λ from 20 to 100.
+0
+100
+200
+300
+400
+time (s)
+40
+50
+60
+70
+80
+90
+100
+% of success queries
+load-balancing
+rp-load
+least-impedance
+closest
+farthest
+cheaper
+rp-latency
+random
+ASET
+(b) Steady variation of λ between 60 and 20.
+Fig. 7: Performance of ASET varying the requests rate over time with two
+different load variation patterns (full-edge topology).
+0
+2
+4
+6
+8
+time (s)
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+CDF
+λ = 60
+λ = 80
+λ = 100
+switching policy delay
+requests interval
+(a) Cumulative distribution of delay.
+0
+100
+200
+300
+400
+500
+600
+700
+training episodes
+0
+20
+40
+60
+80
+100
+testing % of success queries
+topology
+full-edge
+co-dc-cloud
+dc-cloud
+(b) Learning curve.
+Fig. 8: (a) Delay for switching policy compared with requests arrival intervals.
+(b) Learning curve while training ASET on different topologies.
+training generalizes enough as the algorithm performs well
+under previously unseen dynamic conditions.
+Training and applicability. Figure 8a shows the cumulative
+distribution of the time needed by ASET to infer a switching
+decision from the current policy to the one that is best suitable
+for the current system conditions (non-dashed line). The
+switching delay is compared with distributions of the intervals
+between subsequent requests for different lambdas. As shown,
+even for very large client loads (100 clients connected to
+the antenna), the time interval between two stream arrivals is
+typically in the scale of seconds or hundreds of milliseconds,
+while the delay for switching between policies is one order of
+magnitude smaller. Finally, Figure 8b shows the learning curve
+of ASET for different topologies on continuous stream request
+arrivals. The figure shows that ASET quickly reaches a certain
+level of performance in the first training iterations (before 100
+episodes) independently from the topology complexity, leaving
+room for extra improvements to the subsequent episodes based
+on the margin left by the topology itself.
+VI. CONCLUSIONS
+This paper proposes ASET, an adaptive algorithm based on
+Reinforcement Learning for scheduling inference workloads
+at the network edge. ASET solves the problem of exploiting
+scattered clusters of resources to serve inference queries from
+multiple edge applications (e.g., AR/VR, cognitive assistance).
+We model an edge inference system where queries from
+different access networks are processed across a multitude
+of distributed processing locations. The constrained nature
+of the edge network introduces a trade-off between network
+delay and processing time based on the various available
+DNN models. In such a scenario, ASET optimizes the binding
+between inference stream requests and available DL models
+
+across the network, maximizing the throughput and ensuring
+that any requirement in terms of inference accuracy and end-
+to-end delay is satisfied. We evaluated our approach over the
+realistic network topology of a large ISP and considering a
+heterogeneous pool of edge applications. Our findings show
+that ASET effectively improves the performance compared to
+static policies when resources are deployed across the whole
+edge-cloud infrastructure.
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+

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+Non-Invertible Symmetries in Supergravity
+Eduardo Garc´ıa-Valdecasas
+1Jefferson Physical Laboratory, Harvard University,
+Cambridge, MA 02138, USA
+2Universidad Autonoma de Madrid, Ciudad Universitaria de Cantoblanco,
+28049 Madrid, Spain
+E-mail: [email protected]
+Abstract: Non-invertible symmetries have been extensively studied in quantum field
+theories in recent years. In this note we initiate their study in supergravity. We find
+infinite families of non-invertible defects in 11d and 10d Type II supergravities. These
+operators display a rich action on different probe branes. We comment on how these
+symmetries are removed in the UV completion, M-theory and Type II String Theory
+and how their existence strengthens the link between the absence of global symmetries
+in Quantum Gravity and the Completeness Hypothesis.
+arXiv:2301.00777v1  [hep-th]  2 Jan 2023
+
+Contents
+1
+Introduction
+1
+2
+Review of QFT examples
+3
+3
+Non-invertible symmetries in Supergravity
+6
+3.1
+11d Supergravity
+6
+3.2
+Type IIA Supergravity
+9
+3.3
+Type IIB Supergravity
+12
+4
+A different approach
+13
+5
+Discussion
+15
+A The 7d ZN TQFT
+17
+B Explicit construction of the non-invertible defects by half gauging
+18
+1
+Introduction
+Symmetry is one of the basic principles in contemporary physics and quantum field
+theory in particular.
+A modern definition of symmetry, pioneered in [1], identifies
+symmetry with the topological sector of the theory.
+This is precisely what makes
+it a universal notion that helps classify quantum field theories and is robust under
+smooth deformations such as the RG flow. In more detail, the symmetry of a theory
+is given by the set of all topological operators and their fusion rules.
+In the most
+general definition one allows for topological operators of any codimension and fusion
+rules that may not obey a group law. The usual symmetries such as baryon number are
+generated by topological operators of codimension 1, Ug(Σd−1), hence acting on Hilbert
+space, that obey a group-law fusion Ug1(Σd−1)×Ug2(Σd−1) = Ug1·g2(Σd−1). Allowing for
+topological operators of different codimensions gives rise to higher-form symmetries. A
+prototypical example is the 1-form symmetry of free Maxwell theory acting on Wilson
+loops. Allowing for non group-laws brings non-invertible symmetries into the game.
+These are generated by topological operators that need not have an inverse, hence the
+– 1 –
+
+name. In this work we will be interested in p-form symmetries generated by codimension
+p + 1 operators with various p′s and obeying non-invertible fusion rules.
+While non-invertible symmetries may look exotic at first, in recent years it has
+been understood that they are essentially ubiquitous. In fact, theories as simple as free
+Maxwell in 4d are plagued with non-invertible symmetry operators. For a partial list of
+these recent developments see [2–38]. For a partial list of earlier results in 2d see [39–50].
+While these symmetries have been extensively studied in quantum field theory, they are
+almost uncharted territory in supergravity1. In this work we make a first, if superficial,
+approach to these new lands by studying some non-invertible defects in 11d, 10d Type
+IIA and Type IIB supergravity. The core idea is that Chern-Simons terms, which are
+usually believed to imply explicit breaking of the higher form symmetries of the gauge
+fields [51], are many times just a signal of their non-invertibility. Let us illustrate the
+argument, already formulated in [21] by building on earlier work [16, 17]. In the absence
+of Chern-Simons terms, gauge theories typically have equations of motion and Bianchi
+identities of the form,
+d ⋆ F = 0,
+dF = 0
+(1.1)
+These equations imply the existence of higher-form electric and magnetic symmetries.
+Adding Chern-Simons terms generically spoils these symmetries, particularly the elec-
+tric one, as the equation of motion now takes the form of a non-conservation equation,
+for instance:
+d ⋆ F = F ∧ F
+(1.2)
+However, as we discuss in more detail in the following section, in many cases the non-
+conservation is mild enough, since the right hand side vanishes in trivial topology, that
+one can still define topological operators implementing the symmetry. The price to pay
+is that one needs to dress the operator with some topological degrees of freedom that
+generate a non-invertible fusion rule. Given that supergravity theories are plagued with
+Chern-Simons terms2 one expects a very rich set of non-invertible symmetries! These
+symmetries will naturally act on probe branes.
+An important lesson from Quantum Gravity is that exact global symmetries are
+incompatible with it [52–59]. This implies that all the symmetries described in this
+work must be broken by the UV completion, be it M-theory or Type II String Theory.
+As we will see this is easily achieved by the presence of dynamical branes, in analogy to
+what happens with less exotic symmetries [60]. For further discussions on non-invertible
+symmetries and Quantum Gravity see [2, 3, 12].
+1See [38] for a noteworthy exception.
+2Many times these Chern-Simons terms are actually absorbed in modified Bianchi identities.
+– 2 –
+
+The remainder of this note is organised as follows. In section 2 we review non-
+invertible symmetries in quantum field theories with Chern-Simons terms. In section 3
+we present infinite families of non-invertible defects for 11d supergravity and the 10d
+Type II supergravities.
+We study their action on probe branes in some detail.
+In
+section 4 we elaborate in a different approach to the same question. We find a different
+set of topological operators and comment on their connection to the former ones. We
+conclude in section 5 with comments on our results, their implications and an outlook
+of future work. We leave for the Appendices A and B the explicit construction of the
+TQFT’s that are used in the main text to construct the general topological operators.
+2
+Review of QFT examples
+Chern-Simons terms typically turn the equations of motion for gauge fields into non-
+conservation equations for the currents they carry.
+Until recently, it was believed
+that these non-conservation equations implied the explicit breaking of the symmetry
+obtained by integrating the no longer conserved current.
+However, when the non-
+conservation is mild enough, one may construct topological operators for the symme-
+try by appropriately dressing them with gaped degrees of freedom. More concretely,
+consider a non-conservation equation for a U(1) current Jp of the form,
+d ⋆ Jp = Gd−p+1
+(2.1)
+where Gd−p+1 is a wedge product of gauge field strengths. If Gd−p+1 is locally exact
+Gd−p+1 = dKd−p and its integral vanishes in manifolds of trivial topology3, one can
+safely improve the current,
+d(⋆Jp − Kd−p) = 0
+(2.2)
+This improved current is conserved but may be non gauge invariant. The integrated
+charge for such a current is known as a Page charge [61] and we can use it to write an
+operator,
+e
+2πiα
+�
+Σd−p ⋆Jp−Kd−p
+(2.3)
+For simple enough spacetime manifolds (trivial topology) this operator is actually topo-
+logical and well defined. In fact, in most cases4 a new operator can be introduced with
+3With trivial topology we refer to manifolds homeomorphic to R4 or S4.
+4An exception is the case of Gd−p+1 being a single field strength, as in 3d Chern-Simons theory
+or 4d BF theory. For these theories however,
+�
+M2 Gd−p+1 =
+�
+M2 F ̸= 0, so our assumptions are not
+satisfied.
+– 3 –
+
+Kd−p modified to be also topological and gauge invariant in arbitrary manifolds. Let
+us consider a particularly simple example, 5d Chern-Simons Maxwell theory [21],
+S = 2π
+�
+−1
+2F2 ∧ ⋆F2 + 1
+6A1 ∧ F2 ∧ F2
+(2.4)
+Note that we have chosen to stick to the conventions in the supergravity literature.
+In particular, we have chosen field strengths to have integer periods,
+�
+F ∈ Z. In the
+absence of a Chern-Simons term, this theory would have U(1)(1)
+e
+× U(1)(2)
+m electric and
+magnetic symmetries with currents je = F2, jm = ⋆F2. These currents are conserved
+due to the equation of motion and the Bianchi identity, respectively. However, the
+Chern-Simons term seems to break the electric symmetry: the equation of motion
+becomes,
+d ⋆ F2 = 1
+2F2 ∧ F2
+(2.5)
+This equation implies that F2 is not conserved anymore, so the naive operator,
+Uα(Σ3) = exp
+�
+2πiα
+�
+Σ3
+⋆F2
+�
+(2.6)
+is no longer topological. However, F2 ∧ F2 is locally exact and, since there is no U(1)
+instanton number in S4, its integral vanishes in manifolds of trivial topology5. We may
+then improve the current,
+d
+�
+⋆F2 − 1
+2A1 ∧ F2
+�
+= 0
+(2.7)
+which is conserved but not gauge invariant. The corresponding topological operator,
+only valid for simple enough topology, is,
+˜Uα(Σ3) = exp
+�
+2πiα
+�
+Σ3
+⋆F2 − 1
+2A1 ∧ F2
+�
+(2.8)
+While this operator is not valid for arbitrary topology, if α ∈ [0, 1) is rational, one can
+define an improved operator which is topological and gauge invariant for arbitrary Σ3.
+For the particular case of α = 1/N, with N ∈ Z, the corresponding operator is,
+D1/N(Σ3) =
+�
+Dc1
+���
+Σ3exp
+�
+2πi
+� ⋆F2
+N + N
+2 c1 ∧ dc1 − c1 ∧ F2
+�
+(2.9)
+where c1 is a U(1) gauge field localized in the topological defect. That this operator is
+the gauge invariant version of 2.8 can be morally seen by integrating out c1 = A1/N.
+5This statement is sensitive to the UV completion of the quantum field theory at hand. For instance,
+there are stringy instantons in single D-branes in String Theory [62] and there are also U(1) instantons
+in non-commutative spacetimes [63]. We thank I˜naki Garc´ıa-Etxebarria for raising this point.
+– 4 –
+
+Figure 1: A deformation of the naive operator Uα(Σ3) generates an anomalous phase
+in the region swept by it.
+While this integration is not a correct equation for A1, c1 which are both U(1) gauge
+fields, it gives the correct result. For more details, including an explicit construction of
+the defect using higher half-space gauging see [21]. For a general α = p/N, p, N ∈ Z,
+a good topological operator can also be written in terms of the minimal An,p TQFT as
+in Equation (A.3). For more details on An,p, see [64]. What we have done is to dress
+the naive operator in 2.6 in such a way that the new degrees of freedom cancel the
+anomalous phase it generates as it is deformed,
+Uα(Σ′
+3) = Uα(Σ3)e
+2πiα
+2
+�
+Σ4 F2∧F2
+(2.10)
+For a pictorial representation see Figure 1. The upshot is that the rational α = p/N
+subgroup of the U(1) electric 1-form symmetry survives and is generated by A.3. The
+price to pay is that the symmetry becomes non-invertible, as can be seen by explicitly
+computing the fusion rules of the topological operators. These operators act on Wilson
+lines as the naive operators would have, but they also act on ’t Hooft surfaces by
+attaching a fractional flux to them. The action is completely analogous to what we
+will describe for 11d supergravity in section 3.1. We will denote these non-invertible
+symmetries with rational valued parameters as Γ(1)
+Q . The above construction generalizes
+to mixed Chern-Simons terms [21]. Consider for instance,
+S = 2π
+�
+−1
+2F2 ∧ ⋆F2 − 1
+2G2 ∧ ⋆G2 + 1
+2C1 ∧ F2 ∧ F2
+(2.11)
+Where F2 = dA1, G2 = dC1. In the absence of the Chern-Simons coupling, this theory
+has electric and magnetic symmetries
+U(1)(1)
+e,A × U(1)(2)
+m,A × U(1)(1)
+e,C × U(1)(2)
+m,C
+(2.12)
+Yet again, the Chern-Simons coupling spoils the conservation equations of the electric
+symmetries, which become,
+d ⋆ F2 = G2 ∧ F2,
+d ⋆ G2 = 1
+2F2 ∧ F2
+(2.13)
+– 5 –
+
+In the same vein as before, these two currents can be improved to Page currents.
+In turn, topological operators corresponding to the Page charge can be appropriately
+defined. For α = 1/N these operators can be written as,
+DG
+1/N(Σ3) =
+�
+Dc1
+���
+Σ3exp
+�
+2πi
+� ⋆G2
+N
++ N
+2 c1 ∧ dc1 − c1 ∧ F2
+�
+(2.14)
+DF
+1/N(Σ3) =
+�
+Dc1Dv1
+���
+Σ3exp
+�
+2πi
+�
+⋆F2
+N + Nv1 ∧ dc1 − c1 ∧ G2 − v1 ∧ F2
+�
+(2.15)
+These two constructions can once again be extended to any rational α = p/N by using
+the An,p theories. The upshot is that the electric symmetries are turned non-invertible
+by the Chern-Simons terms and the true symmetry of the theory is,
+Γ(1)
+Q,A × U(1)(2)
+m,A × Γ(1)
+Q,C × U(1)(2)
+m,C
+(2.16)
+In the remainder of this note we explore similar constructions in supergravity, where
+Chern-Simons terms are ubiquitous.
+3
+Non-invertible symmetries in Supergravity
+3.1
+11d Supergravity
+The construction of the non-invertible defects in the previous examples is only possible
+thanks to the existence of a particular 3d TQFT with the appropriate 1-form symmetry
+and anomaly. For would-be U(1) actions with phase α = 2π/N the full topological
+operator is 2.14 and the TQFT that is stacked on top of the naive operator 2.6 takes
+the form,
+AN,1
+3
+(Σ3) =
+�
+Dc1
+���
+Σ3exp
+�
+2πi
+�
+Σ3
+N
+2 c1 ∧ dc1 − c1 ∧ F2
+�
+(3.1)
+where c1 is an auxiliary U(1) gauge field living in Σ3 and F2 is the magnetic current
+in the bulk that needs to be gauged to obtain the defect. This TQFT, and its general-
+izations AN,p
+3
+(Σ3), are called fractional quantum Hall states, or FQHE states, and are
+particular to 3d. In 7d an analog FQHE construction con be made6, where one can
+write the following TQFT,
+AN,1
+7
+(Σ7) =
+�
+Dc3
+���
+Σ7exp
+�
+2πi
+�
+Σ7
+N
+2 c3 ∧ dc3 − c3 ∧ F4
+�
+(3.2)
+As we now argue, this TQFT precisely arises in 11d supergravity. Consider its action,
+S =
+1
+2κ2
+11
+�
+Σ11
+√−gR − 1
+2F4 ∧ ⋆F4 − 1
+6A3 ∧ F4 ∧ F4
+(3.3)
+6See, for instance [65].
+– 6 –
+
+Where 2κ2
+11 = (2π)−1(2πlp)9. If the Chern-Simons coupling is turned off, this theory
+has 2 symmetries U(1)(3)
+e
+× U(1)(6)
+m with currents je = F4, jm = ⋆F4 conserved thanks
+to the equation of motion and the Bianchi identity of F4. As discussed in [60] it also
+has a Chern-Weil symmetry with current F4 ∧ F4, which will not be relevant for our
+purposes. Once one includes the CS interaction the conservation equation of U(1)(3)
+e
+is
+modified to,
+d ⋆ F4 = 1
+2F4 ∧ F4
+(3.4)
+Hence the CS term explicitly breaks U(1)(3)
+e
+and gauges the Chern-Weil current. By
+now the game we must play is clear, this current fulfills the conditions to be improved
+to a U(1) Page current,
+d
+�
+⋆F4 − 1
+2A3 ∧ F4
+�
+= 0
+(3.5)
+which is conserved but not gauge invariant. The naive operator, which is only valid for
+trivial topology is,
+Uα(Σ7) = exp
+�
+2πiα
+�
+Σ7
+⋆11F4 − 1
+2A3 ∧ F4
+�
+(3.6)
+Writing the corresponding good topological operator is straightforward. For α = 1/N ∈
+U(1) we just need to stack the 7d FQHE theory. The resulting operator is,
+D1/N(Σ7) =
+�
+Dc3
+���
+Σ7exp
+�
+2πi
+�
+Σ7
+⋆11F4
+N
++ N
+2 c3 ∧ dc3 − c3 ∧ F4
+�
+(3.7)
+In fact, we can make use of the A(N,p)
+7
+[b4] theory defined in Appendix A to build a
+topological operator for any α ∈ [0, 1) of the form α = p/N, with p, N ∈ Z,
+Dp/N(Σ7) = exp
+�2πip
+N
+�
+Σ7
+⋆11F4
+�
+× A(N,p)
+7
+�F4
+N
+�
+(3.8)
+These defects can be explicitly constructed by higher gauging the magnetic U(1)(6)
+m
+symmetry, as detailed in appendix B. The upshot of the discussion above is that,
+contrary to expectations, the electric U(1)(3)
+e
+symmetry is not completely broken, but
+a non-invertible rational discrete subgroup remains. The symmetries associated to F4
+in M-theory are then,
+Γ(3)
+Q × U(1)(6)
+m
+(3.9)
+Of course we expect these two symmetries to be broken by the UV-completion of 11d
+supergravity. This is indeed the case in M-theory, as inclusion of dynamical M2- and
+M5-branes breaks them explicitly. In fact, given that one needs to gauge the magnetic
+– 7 –
+
+symmetry to build the electric defects, the presence of dynamical M5-branes is enough
+to break both symmetries. A consequence is that Γ(3)
+Q
+must be broken at an energy
+scale lower or equal than U(1)(6)
+m .
+Let us now study more carefully the action of the topological defect 3.8 on electric
+and magnetic probes, which we call probe M2- and M5-branes for obvious reasons. On
+probe M2-branes, which source ⋆F4, the action is invertible and given by the first term
+in 3.8. The action is indistinguishable from the one we expected for the electric 3-form
+symmetry. The second term in 3.8 can detect magnetic charges, sources for F4. In
+order to find the precise action consider the construction of the topological defect in Σ7
+via higher gauging of the magnetic symmetry as explained in Appendix B. Take 11d
+spacetime to be R11 with coordinates x1, ..., x11 such that Σ7 spans x1, ..., x7 and the
+gauging is defined for x8 > 0 such that Σ8 = R7 × R>0
+x8 . Denote by HF4
+m a 6-dimensional
+source of m units of F4 flux, which could be a bunch of probe M5-branes, spanning
+directions x5,6,7,9,10,11. If we displace HF4
+m from x8 < 0 to x8 > 0 it enters into the
+region where the magnetic symmetry is gauged and it stops being gauge invariant.
+In more detail, the 3-dimensional part of the probe M5 worldvolume that is inside the
+submanifold where the symmetry is gauged, Σ3 ≡ R3
+x5,x6,x7 = WV (M5)∩Σ8, transforms
+under b4 → b4 + dΛ3 gauge transformations by picking up a phase,
+HF4
+m → HF4
+m e2πim
+�
+Σ3 Λ3,
+for
+x8 > 0
+(3.10)
+Hence, in the x8 > 0 region the gauge invariant object is,
+HF4
+m e−2πim
+�
+Σ4 b4 = HF4
+m e
+2πipm
+N
+�
+Σ4 F4
+(3.11)
+where Σ4 is such that ∂Σ4 = Σ3. To write the right hand side we have used the equation
+of motion for b4 in B mod N. We conclude that the defect acts on the magnetic source
+by attaching a fractional F4 flux along Σ4, as depicted in Fig. 2,
+Dp/N(Σ7)HF4
+m = HF4
+m e
+2πipm
+N
+�
+Σ4 F4.
+(3.12)
+An important consequence of this action is that, if m ̸= 0 mod N, the symmetry defect
+annihilates the magnetic source, as argued in [66]. Consider again Figure 2. The idea
+is that, since Dp/N(Σ7) is topological, it may be shrunk to a point and removed, giving
+rise to a topological endpoint for V (Σ4) pm
+N ≡ e
+2πipm
+N
+�
+Σ4 F4. This endpoint is local and its
+existence implies that V (Σ4) pm
+N can develop a hole and disappear. However, V (Σ4) pm
+N
+is a generator of the magnetic symmetry U(1)(6)
+m which acts faithfully, so it better be
+that it can’t just go away! The only solution is that correlation functions with this
+– 8 –
+
+Figure 2: A magnetic source for F4 crosses the topological defect and picks up a
+fractional F4 flux attached to it.
+endpoint give zero. We conclude that the topological operator annihilates HF4
+m sources
+unless,
+pm
+N ∈ Z
+(3.13)
+However, if m = 0 mod N, V (Σ4) pm
+N becomes trivial except at the boundary of Σ4 and
+the action of the topological operator leaves an operator insertion in Σ3. We leave the
+study of this junction in detail for future work.
+3.2
+Type IIA Supergravity
+The existence of non-invertible symmetries in 11d supergravity suggests the existence
+of appropriate counterparts in Type IIA supergravity, which arises when dimensionally
+reducing on a circle. We will not attempt direct dimensional reduction of the symme-
+tries in this work. We study instead the non-invertible symmetries directly from the
+10 formulation. Consider the low energy action of type IIA string theory in 10d [67],
+SIIA =
+1
+2κ2
+�
+M10
+√−g
+�
+e−2Φ
+�
+R + 4|dΦ|2−1
+2|H3|2
+�
+− 1
+2|F2|2−1
+2| ˜F4|2
+�
+−
+− 1
+2κ2
+�
+M10
+1
+2B2 ∧ F4 ∧ F4
+(3.14)
+where ˜F4 = dA3 + A1 ∧ H3 is invariant under the gauge transformation (A1, A3) →
+(A1 + dλ0, A3 − λ0 ∧ H3). The equations of motion for F2, ˜F4 and H3 are,
+d ⋆ F2 = H3 ∧ ⋆ ˜F4
+(3.15)
+d ⋆ ˜F4 = H3 ∧ F4 = H3 ∧ ˜F4
+(3.16)
+d ⋆ H3 = 1
+2
+˜F4 ∧ ˜F4 + F2 ∧ ⋆ ˜F4
+(3.17)
+– 9 –
+
+The Bianchi Identities are,
+dH3 = 0,
+dF2 = 0,
+d ˜F4 = F2 ∧ H3
+(3.18)
+The Bianchi identities for F2, H3 imply that there are two conserved currents: ⋆H3, ⋆F2.
+These generate two magnetic symmetries U(1)(6)
+m × U(1)(7)
+m which are well understood,
+see for instance [60]. The remaining equations can be rewritten by introducing Hodge
+dual field strengths ˜Fp = ⋆ ˜F10−p as,
+d ˜F4 = F2 ∧ H3,
+d ˜F6 = ˜F4 ∧ H3,
+d ˜F8 = ˜F6 ∧ H3,
+(3.19)
+d ⋆ H3 = 1
+2
+˜F4 ∧ ˜F4 + F2 ∧ ˜F6
+(3.20)
+The newly introduced gauge invariant field strengths are defined as ˜Fp = dAp−1+Ap−3∧
+H3. From the equations above we see that there are three candidates for non-invertible
+symmetries. Indeed, the right hand side in equations 3.19 can be shown to be locally
+exact and its integral to vanish in manifolds with trivial topology, so we may write the
+following conserved but gauge variant Page currents,
+d
+�
+˜F4 − A1 ∧ H3
+�
+= 0
+(3.21)
+d
+�
+˜F6 − A3 ∧ H3
+�
+= 0
+(3.22)
+d
+�
+˜F8 − A5 ∧ H3
+�
+= 0
+(3.23)
+At this point the strategy seems clear: we can dress the would-be topological operators
+with some gaped degrees of freedom to obtain a good topological operator. However,
+for eqs. (3.22) and (3.23) things are not so simple. Let us focus first in eq. (3.21).
+We consider a U(1) transformation with α = 1/N and in analogy to 2.15 propose the
+following operator,
+DF4
+1/N(Σ4) =
+�
+Dc1Dv2
+���
+Σ4exp
+�
+2πi
+�
+Σ4
+˜F4
+N + Nv2 ∧ dc1 − c1 ∧ H3 − v2 ∧ dA1
+�
+(3.24)
+This operator is gauge invariant and can be built explicitly by gauging a Z(6)
+N × Z(7)
+N
+subgroup of the magnetic symmetry U(1)(6)
+m × U(1)(7)
+m with currents ⋆H3 and ⋆F2 in
+a manifold Σ5 such that ∂Σ5 = Σ4, as shown in Appendix B. In fact, as discussed
+there, the operator can be built for arbitrary α = p/N in a similar way, such that
+it is explicitly topological and gauge invariant. We hence conclude that the would-be
+magnetic U(1)(5)
+m symmetry of ˜F4, which is broken by the modified Bianchi identity,
+– 10 –
+
+becomes a non-invertible Γ(5)
+Q symmetry. Let us describe how the topological operator
+acts on the different probe branes. Since this discussion is a straightforward extension of
+the 11d supergravity we will be brief. A related in-depth discussion for the case of axion
+electrodynamics is presented in [16, 68]. The defect acts invertibly in probe D4-branes,
+as the naive magnetic symmetry of ˜F4 would have, but it also acts non-invertibly in
+probe NS5- and D6-branes. In particular, these probes are not gauge invariant if they
+intersect the auxiliary higher-gauging submanifold and a flux needs to be attached to
+them. The upshot is that the topological defect with α = p/N annihilates the probe
+NS5- or D6-branes if their charge m does not satisfy the following relation,
+pm
+N ∈ Z
+(3.25)
+If the relation above is satisfied, the action of the topological defect on the probe
+NS5- or D6-brane leaves behind an operator stuck in their worldvolume of dimension
+1 for the NS5-brane and 2 for the D6-brane. While the study of these junctions is
+very interesting, it goes beyond the scope of this work and we leave it for the future.
+We expect that anomaly inflow from the bulk will impose the existence of certain
+worldvolume degrees of freedom on the branes that will in turn be used to furnish the
+appropriate operators.
+Consider now the Page currents in eqs. (3.22) and (3.23). As we now explain, the
+would-be topological operator analogous to 3.24 can’t be built by higher gauging be-
+cause there is a higher anomaly. For 3.22, for instance, one could propose the following
+operator,
+DF6
+1/N(Σ6) =
+�
+Dc3Dv2
+���
+Σ6exp
+�
+2πi
+�
+Σ6
+˜F6
+N + Nv2 ∧ dc1 − c3 ∧ H3 − v2 ∧ dA3
+�
+(3.26)
+One immediately notices however, that the last term is not invariant under (A1, A3) →
+(A1 + dλ0, A3 − λ0 ∧ H3) gauge transformations. One can gain further insight into this
+issue by explicitly introducing ZN background gauge fields for ⋆H3 and ⋆dA3 in Σ7
+such that ∂Σ7 = Σ6. If correct, this higher gauging should generate 3.26. There is, at
+first sight, nothing wrong in gauging the currents ⋆H3 and ⋆dA3 since they are both
+conserved, the gauging is implemented by adding the following terms to the action7,
+δS = 2πi
+�
+Σ7
+Np′b3 ∧ b4 + Nb3 ∧ dˆc3 + Nb4 ∧ dˆv2 − b4 ∧ H3 − b3 ∧ dA3
+(3.27)
+7For more details on this procedure see appendix B.
+– 11 –
+
+The problem is that under an A1 gauge transformation the last term picks up an
+anomalous piece,
+2πi
+�
+Σ7
+b3 ∧ dλ0 ∧ H3
+(3.28)
+We conclude that there is a higher anomaly8 encoded by inflow as 2πi
+�
+b3 ∧ dA1 ∧ H3
+which precludes the higher gauging of the symmetry associated with the current ⋆dA3.
+A similar statement holds for ⋆dA5. A different way of phrasing the problem is by
+noticing that there is no magnetic symmetry associated to the would-be currents ⋆dA3
+or ⋆dA5. We do not have the necessary symmetries for the higher gauging procedure.
+An open possibility is to gauge the 5-form non-invertible magnetic symmetry for ˜F4 that
+we just built. This would amount to performing an explicit sum over the non-invertible
+defects in Σ7 instead of the coupling to the background field b3. We would in turn be
+able to do it for ˜F8 by using the newly found operators. It is not clear however how to
+perform this gauging, which we leave for future work. In section 4 we comment on how
+this problem seems to be avoided in the alternative approach presented in [36, 38].
+3.3
+Type IIB Supergravity
+The discussion above translates almost verbatim to Type IIB supergravity, which has
+the following action,
+SIIB =
+1
+2κ2
+�
+M10
+√−g
+�
+e−2Φ
+�
+R + 4|dΦ|2−1
+2|H3|2
+�
+− 1
+2|F1|2−1
+2| ˜F3|2−1
+2| ˜F5|2
+�
+−
+− 1
+4κ2
+�
+M10 A4 ∧ H3 ∧ F3
+(3.29)
+where gauge invariant field strengths ˜F3 = F3−A0∧H3 and ˜F5 = F5− 1
+2A2∧H3+ 1
+2B2∧F3
+have been introduced.
+This action needs to be supplemented with the self-duality
+condition for ˜F5: ˜F5 = ⋆ ˜F5. The equations of motion are:
+d ⋆ H3 = − ˜F5 ∧ ˜F3 + F1 ∧ ⋆ ˜F3
+(3.30)
+d ⋆ F1 = −H3 ∧ ⋆ ˜F3
+(3.31)
+d ⋆ ˜F3 = ˜F5 ∧ H3
+(3.32)
+d ⋆ ˜F5 = − ˜F3 ∧ H3
+(3.33)
+8This anomaly is in fact inherited from a standard anomaly in the bulk theory given by inflow as
+∼ B6 ∧ dA1 ∧ H3, B6 being the background coupled to the would-be current ⋆dA3.
+– 12 –
+
+And the Bianchi Identities are,
+dH3 = 0
+(3.34)
+dF1 = 0
+(3.35)
+d ˜F3 = −F1 ∧ H3
+(3.36)
+d ˜F5 = − ˜F3 ∧ H3
+(3.37)
+Note that 3.33 and 3.37 are the same equation. There are two conserved currents,
+⋆H3 and ⋆F1, which generate a U(1)(6)
+m × U(1)(8)
+m magnetic symmetry. Armed with
+the experience from the previous section we recognise that only the modified Bianchi
+identity ˜F3 = −F1 ∧H3 can give rise to a non-invertible symmetry9. The corresponding
+topological operator is,
+DF3
+1/N(Σ3) =
+�
+Dc0Dv2
+���
+Σ3exp
+�
+2πi
+�
+Σ3
+˜F3
+N − Nv2 ∧ dc0 + c0 ∧ H3 + v2 ∧ dA0
+�
+(3.38)
+which can be checked to correspond to the integral of the corresponding Page current
+upon naive integration of c0 = A0/N and v2 = −B2/N.
+Once again we conclude
+that the U(1)(5)
+m with non-conserved current ⋆ ˜F3 is not completely broken, but a non-
+invertible Γ(5)
+Q
+remains. The discussion regarding the action of this operator on the
+different probe objects in the theory is completely analogous as the one for Type IIA
+and we choose to free the reader from it. Let us just mention that it acts invertibly in
+D5-branes and non-invertibly in NS5- and D7-branes.
+4
+A different approach
+We have so far seen how to explicitly build non-invertible topological defects in 11d
+and 10d supergravity. In each of those cases, a would-be broken U(1) symmetry is still
+realized by more exotic topological operators. The price to pay is that the topological
+operators only exist for rational angles α = p/N and their fusion rules become non-
+invertible. As we have discussed, the existence of the topological operators is intimately
+tied to the existence of precise higher form symmetries, a discrete subgroup of which
+can be gauged in a particular way. In this section we compare our approach to the
+9Similar comments as in previous section apply here. One may wonder whether other modified
+Bianchi identities can give rise to non-invertible symmetries upon gauging the other non-invertible
+symmetries.
+– 13 –
+
+one pioneered in [36, 38] and applied to supergravity in [38]. The idea is elegant and
+simple, consider for instance the Page charge associated to 3.21,
+Uα(Σ4) = exp
+�
+2πiα
+�
+Σ4
+˜F4 − A1 ∧ H3
+�
+(4.1)
+This current is topological but not gauge invariant under A1 → A1 + dλ0.
+In the
+preceding sections we saw that a gauge invariant version of 4.1 can be written for
+α = p/N by stacking an appropriate TQFT. A simpler approach is to introduce a
+Stueckelberg-like field that restores gauge invariance. A compact scalar θ transforming
+under A1 gauge transformations as θ → θ + λ0 does the job,
+U ′
+α(Σ4) =
+�
+Dθ
+���
+Σ4exp
+�
+2πiα
+�
+Σ4
+˜F4 − (A1 − dθ) ∧ H3
+�
+(4.2)
+An immediate advantage of this approach is that α is not limited to be rational and
+the whole U(1) symmetry is unbroken. Still this U(1) can be seen to act non-invertibly
+on sources of H3 flux. Consider the following setup, Σ4 = S3 × S1 with m units of H3
+flux
+�
+S3 H3 = m, which could be sourced by m NS5-branes, for instance. We wish to
+evaluate the path integral with the insertion of the topological operator,
+⟨U ′
+α(S3 × S1)⟩ =
+��
+Dθ
+���
+Σ4exp
+�
+2πiα
+�
+Σ4
+˜F4 − (A1 − dθ) ∧ H3
+��
+(4.3)
+We can evaluate explicitly the term ∼ dθH3 by taking H3 to be a background:
+�
+Dθ exp
+�
+2πiα
+�
+S3×S1 dθ ∧ H3
+�
+=
+�
+ω∈z
+e2πiαmω
+(4.4)
+where we have traded the integral over θ by a sum over its periods in S1. The resulting
+sum is a delta function that is non-zero only for
+αm ∈ Z
+(4.5)
+An important remark is that this operator does not act on objects that only source
+dA1. Indeed, if we consider a similar setup as before but with Σ4 = S2 × S1 × S1 and
+m units of dA1 flux in S2, we find ⟨U ′
+α(S2 × S1 × S1)⟩ = 1. This is in sharp contrast
+with the topological operator that we built previously, 3.24, which acts non-invertibly
+both on sources of H3 and dA1. In order to better understand the connection between
+the two approaches, let us define yet another operator,
+ˆUα(Σ4) =
+�
+DθDc1
+���
+Σ4exp
+�
+2πiα
+�
+Σ4
+˜F4 − 1
+2(A1 − dθ) ∧ H3 − 1
+2(B2 − dc1) ∧ dA1
+�
+(4.6)
+– 14 –
+
+Where (B2, c1) → (B2 + dΛ1, c1 + Λ1).
+That this operator is gauge invariant and
+topological on a closed manifold can be checked by extending it to 5d10. This new
+operator acts non-invertibly in sectors with either
+�
+H3 = m or
+�
+dA1 = m fluxes.
+The argument is analog to the previous case and implies the vanishing of the partition
+function unless
+αm ∈ 2Z.
+We note that the action of this operator is hence not equivalent to 3.24 for which α
+has to satisfy a less stringent condition in terms of m: 3.2511. A further difference is
+that this operator admits a straightforward generalization to arbitrary Page charges.
+Let us write for completeness the one associated with the current in Eq. 3.22:
+ˆUα(Σ6) =
+�
+Dc1Dc2
+���
+Σ6exp
+�
+2πiα
+�
+Σ6
+˜F6 − 1
+2(A3 − dc2) ∧ H3 − 1
+2(B2 − dc1) ∧ dA3
+�
+(4.8)
+Gauge invariance is a bit trickier since dA3 transforms under A1 gauge transformations
+as dA3 → dA3 − dλ0 ∧ H3 but can be checked to hold in closed manifolds.
+5
+Discussion
+In this note we have started the exploration of non-invertible symmetries in supergrav-
+ity, the low energy limit of M-theory and String Theory. We have found that, in each
+case studied, a U(1) higher form symmetry that seemed broken by Chern-Simons (or
+modified Bianchi Identities) can be recovered in the form of a non-invertible topological
+operators for each rational value of α ∈ [0, 1). We have described how these operators
+can be constructed explicitly by higher gauging discrete subgroups of the invertible sym-
+metries of the theory. This construction automatically implies their topological nature
+and allowed us to deduce their action on the different brane probes of the theory. Fi-
+nally, we have constructed alternative topological operators by using Stueckelberg-like
+fields. This approach has the advantage of being defined for any irrational α. Another
+advantage is that it admits a straightforward generalization to any Page charge, which
+is unclear how to do for the first method. We conclude in this section by making some
+10For further arguments on its topological nature one can formulate similar considerations as the
+ones presented in Appendix A of [38].
+11For the interest of simplicity we have avoided discussing the 11d supergravity topological operator
+in this approach. In that case one must compare,
+U ′
+α(Σ7) =
+�
+Dc2
+���
+Σ7exp
+�
+2πiα
+�
+Σ7
+⋆11F4 − (A3 − dc2) ∧ F4
+�
+(4.7)
+and 3.8. Again a factor 2 appears which implies that the two operators are not completely equivalent.
+– 15 –
+
+comments on the applications of these symmetries, particularly in the context of the
+Swampland program [69, 70].
+An interesting application of non-invertible symmetries of this kind is that they
+require the existence of auxiliary symmetries that need to be gauged for the non-
+invertible topological operator to exist.
+This gives rise to a hierarchy between the
+scales of symmetry breaking of the different symmetries of the theory12.
+Consider
+the operator in 3.8 for instance. The existence of the 3-form non-invertible symmetry
+requires the existence of an exact 6-form magnetic symmetry. This implies a hierarchy
+between the energy at which these symmetries are broken in a UV-completion.
+E(Γ(3)
+Q ) ≤ E(U(1)(6)
+m )
+(5.1)
+In particular, if one assumes that Γ(3)
+Q , U(1)(6)
+m are broken explicitly by including dy-
+namical objects electrically charged under them, M2- and M5-branes, respectively, one
+concludes the following relation between their tension,
+T 1/3
+M2 ≲ T 1/6
+M5
+(5.2)
+which is true up to an order 1 factor with the values TM2 = (2π)−2l−3
+p , TM5 = (2π)−5l−6
+p
+13.
+Similar considerations apply to the operator in 3.24. In that case, to build Γ(5)
+Q defects
+we need to gauge a subgroup of U(1)(6)
+m × U(1)(7)
+m which implies,
+E(Γ(3)
+Q ) ≤ min
+�
+E(U(1)(6)
+m ), E(U(1)(7)
+m )
+�
+(5.3)
+If we again assume that the symmetries are only broken by the presence of the minimal
+branes charged under them we have,
+T 1/5
+D4 ≲ min
+�
+T 1/6
+NS5, T 1/7
+D6
+�
+(5.4)
+which again is fulfilled in Type IIA String Theory by the NS5-brane. A similar relation,
+which is again satisfied, applies to the topological operator of Type IIB String Theory.
+The assumption that the symmetries are only broken by their coupling to their minimal
+objects is not true in either M-theory or Type II String Theory, so these relations needed
+not hold. It is still amusing to see that they are indeed satisfied.
+Let us now elaborate a bit on the relation between these symmetries and the
+completeness hypothesis. The two simplest mechanisms by which a putative UV com-
+pletion breaks higher-form symmetries of the low energy theory is by the presence of
+12This is clear for the operators constructed in section 3, while it is less clear for the ones in section 4.
+13Note that a similar relation is enforced by the 2-group structure.
+– 16 –
+
+Chern-Simons couplings and the introduction of dynamical objects [3, 51, 60]. In this
+work we have argued that Chern-Simons terms are generically not enough to effectively
+break the symmetries, making the case for the need of adding dynamical objects. This
+makes more clear than ever the connection between having a complete spectrum (The
+Completeness Hypothesis) and the absence of generalized global symmetries.
+We finish with an outlook of the future work.
+• We have seen two different approaches to finding non-invertible symmetries for
+non-conservation equations. It would be very interesting to better understand
+the connection between the two approaches. In particular, it would help if we
+understood how to construct the operators in section 4 explicitly, maybe from
+higher gauging.
+• Relatedly, we have left for future work the understanding of whether more “ra-
+tional valued” non-invertible operators can be built in Type II supergravity by
+further gauging the non-invertible symmetries.
+• It would be very interesting to elaborate on the actions of the topological opera-
+tors in the different probe branes. We expect very rich fusion rules, in a similar
+spirit to [66].
+• We expect these results to have many applications and generalizations in string
+compactifications. In particular, it would be very interesting to see if our methods
+can be generalized to compactifications with generalized θ-terms, as studied in
+[71].
+Acknowledgments
+I am pleased to thank Jerem´ıas Aguilera, Riccardo Argurio, Shani Meynet, Miguel
+Montero and Damian van de Heisteeg for related and insightful discussions. I would
+also like to thank Shani Meynet, Antoine Pasternak, Valdo Tatitscheff and specially
+Riccardo Argurio and I˜naki Garc´ıa-Etxebarria, for taking time during Christmas to read
+this manuscript and share important comments. Finally, I want to thank Raquel for
+her support. This research is supported by a Margarita Salas award CA1/RSUE/2021-
+00738 from the Plan de Recuperaci´on, Transformaci´on y Resiliencia of the Ministerio
+de Universidades, UAM and Harvard.
+A
+The 7d ZN TQFT
+In this appendix we define the 7d TQFT with 3-form ZN symmmetry and anomaly as
+needed to render the 11d supergravity defects gauge invariant. We start by briefly re-
+– 17 –
+
+viewing the 3d An,p theory. Consider a smooth deformation of the would-be topological
+defect in 2.6 with α = p/N. The operators fails to be topological due to the equation
+of motion, which gives it a phase,
+exp
+�iπp
+N
+�
+M4
+F2 ∧ F2
+�
+(A.1)
+One looks for a TQFT that can cancel this phase. This is precisely what the AN,p[B2]
+theory does. Indeed it is defined to have 1-form symmetry Z(1)
+N and anomaly given by
+inflow as,
+S(N,p)
+3
+= −iπpN
+�
+M4
+B2 ∧ B2
+(A.2)
+Where B2 is a Z(1)
+N
+background gauge field with holonomies in Z/N. We may now
+identify the background B2 = F2/N to cancel the phase A.1 so that,
+U p
+N (Σ3) × A(N,p)
+�F2
+N
+�
+(A.3)
+is topological and gauge invariant, as reviewed in the main text. For further details on
+how to define the A(N,p)[B2] theory the reader may check [16, 64]. Consider now the
+case of 11d supergravity and the non-conservation equation in 3.4. The naive operator
+is not topological, as it picks up a phase
+exp
+�iπp
+N
+�
+M8
+F4 ∧ F4
+�
+(A.4)
+In analogy with the discussion above we define a 7d TQFT A(N,p)
+7
+[B4] with 3-form Z(3)
+N
+symmmetry and anomaly characterized by inflow as,
+S(N,p)
+7
+= −iπpN
+�
+M8
+B4 ∧ B4
+(A.5)
+Where B4 is a Z(3)
+N background gauge field with holonomies in Z/N.
+B
+Explicit construction of the non-invertible defects by half
+gauging
+In this appendix we construct the rational valued non-invertible defects introduced
+in the main text by using the technique of higher gauging [10].
+Consider first the
+non-invertible topological operator introduced for 11d supergravity 3.8. We choose an
+– 18 –
+
+auxiliary manifold Σ8 such that Σ7 = ∂Σ8 and gauge a Z(6)
+N
+subgroup of the mag-
+netic symmetry U(1)(6) with appropriate discrete torsion. This gauging is described by
+adding the following terms to the path integral,
+δS = 2πi
+�
+Σ8
+Nb4 ∧ dˆc3 + b4 ∧ F4 + Np′
+2 b4 ∧ b4
+(B.1)
+Where pp′ = 1 mod N. Let us unpack a bit the expression above. The second term
+describes the coupling of the U(1)(6) current to a a background b4 in Σ8. The first term
+is a coupling to a U(1) Lagrange multiplier gauge field ˆc3 whose job is to restrict the
+holonomy of b4 so that it is effectively a ZN gauge field. The third term is a discrete
+torsion that one may always add. The equation of motion for b4 is Ndˆc3+F4+Np′b4 = 0.
+Making b4 dynamical implements the gauging. Consider a closed Σ8. If we use the
+equation of motion and remove terms that are multiples of 2πi, we see that our gauging
+precisely cancels the phase in A.4, as needed. If we instead take p′ = 1 and Σ8 manifold
+with boundary ∂Σ8 = Σ7 we explicitly find,
+δS = −iπN
+�
+M8
+b4 ∧ b4 +
+�
+∂Σ7
+A(N,1)
+7
+[B4]
+(B.2)
+We thus conclude that the gauging above precisely generates A(N,p)
+7
+in Σ7 = ∂Σ8. This
+explicit construction of the defect is a further check of its topological nature.
+An
+important point for this gauging to work is that the theory must be self-dual under it,
+as emphasized in [66]. In particular, if one p-gauges a discrete q-form symmetry in a
+d-dimensional theory, for the symmetries to be the same before and after the gauging
+the following relation must hold,
+q = (d + p − 2)/2
+(B.3)
+In the case at hand, q = 6, p = 3 and d = 11, the relation is fulfilled.
+Consider now the topological defect introduced for type IIA supergravity 3.24. The
+classical phase that one whises to cancel is now,
+exp
+�2πip
+N
+�
+M5
+dA1 ∧ H3
+�
+(B.4)
+We have to gauge a Z(6)
+N ×Z(7)
+N subgroup of the magnetic symmetry U(1)(6)
+m ×U(1)(7)
+m in
+a 5-dimensional manifold Σ5, so we introduce two background fields b2 and b3, respec-
+tively. We also introduce Lagrange multipliers ˆc1, ˆv2 enforcing ZN holonomies and a
+discrete torsion term. The mixed gauging is hence implemented by adding the following
+term to the path integral,
+δS = 2πi
+�
+Σ5
+Np′b3 ∧ b2 + Nb3 ∧ dˆc1 + Nb2 ∧ dˆv2 − b3 ∧ dA1 − b2 ∧ H3
+(B.5)
+– 19 –
+
+Direct computation in a closed manifold using the equations of motion of b2, b3 shows
+that one precisely cancels the phase in B.4. In a manifold with boundary and with
+p = 1 one recovers the TQFT in 3.24 as expected. This gauging allows us to generalize
+the topological defect to arbitrary p. Note that in this gauging the quantum symmetries
+are interchanged, in the sense that the quantum symmetry from gauging Z(6)
+N becomes
+part of U(1)(7)
+m after gauging and viceversa. This ensures that the symmetry before and
+after the gauging is the same. A similar construction applies with minor modifications
+to the Type IIB defect in 3.38.
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+

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+1
+Enhancing NOMA Networks via Reconfigurable
+Multi-Functional Surface
+Ailing Zheng, Wanli Ni, Wen Wang, and Hui Tian
+Abstract—By flexibly manipulating the radio propagation en-
+vironment, reconfigurable intelligent surface (RIS) is a promising
+technique for future wireless communications. However, the
+single-side coverage and double-fading attenuation faced by con-
+ventional RISs largely restrict their applications. To address this
+issue, we propose a novel concept of multi-functional RIS (MF-
+RIS), which provides reflection, transmission, and amplification
+simultaneously for the incident signal. With the aim of enhancing
+the performance of a non-orthogonal multiple-access (NOMA)
+downlink multiuser network, we deploy an MF-RIS to maximize
+the sum rate by jointly optimizing the active beamforming and
+MF-RIS coefficients. Then, an alternating optimization algorithm
+is proposed to solve the formulated non-convex problem by
+exploiting successive convex approximation and penalty-based
+method. Numerical results show that the proposed MF-RIS
+outperforms conventional RISs under different settings.
+Index Terms—Multi-functional reconfigurable intelligent sur-
+face, non-orthogonal multiple access, rate maximization.
+I. INTRODUCTION
+Compared to orthogonal multiple access (OMA), non-
+orthogonal multiple access (NOMA) is capable of achieving
+high spectrum efficiency and massive connectivity [1]. Prior
+investigations have shown that the differences between users’
+channel conditions can be exploited to enhance NOMA per-
+formance [2]. However, users in large-scale networks may
+have poor or similar channel conditions, which hinders the
+application of successive interference cancellation (SIC) and
+the effective implementation of NOMA. Therefore, adjusting
+channel conditions and enhancing channel diversity are able
+to release the potential of NOMA in practical networks.
+Recently, with the ability to reshape the wireless propaga-
+tion environment, reconfigurable intelligent surface (RIS) has
+emerged as a key technique to improve the performance of
+NOMA networks [3]. By properly designing the reflection
+coefficients, RIS is able to smartly change the combined
+channels to enhance the differences among users, thus boosting
+the performance of NOMA in large-scale networks. Initial
+investigations on RIS-aided NOMA networks in [3]–[7] had
+verified the superiority of the integration of NOMA and RIS.
+Specifically, the authors of [3] and [4] performed comprehen-
+sive discussions of the main challenges and futuristic use cases
+regarding RIS-aided NOMA networks. Moreover, the works in
+[5]–[7] demonstrated the benefits brought by RISs to achieve
+performance trade-off among multiple NOMA users through
+This letter was supported by the Natural Science Foundation of Shandong
+Province under Grant No. ZR2021LZH010. The associate editor coordinating
+the review of this letter and approving it for publication was Lina Bariah.
+(Corresponding author: Hui Tian.)
+A. Zheng, W. Ni, W. Wang, and H. Tian are with the State Key Lab-
+oratory of Networking and Switching Technology, Beijing University of
+Posts and Telecommunications, Beijing 100876, China (e-mail: {ailing.zheng,
+charleswall, wen.wang, tianhui}@bupt.edu.cn).
+smartly adjusting the decoding order. However, the existing
+literature on RIS-aided NOMA networks mostly uses single
+functional RIS (SF-RIS) that only supports signal reflection
+or transmission/refraction. This implies that only users located
+in a single side can be served by the SF-RIS if no additional
+operations are performed.
+To overcome this limitation, the authors of [8] proposed the
+concept of dual-functional RIS (DF-RIS). Unlike SF-RIS, DF-
+RIS refers to the reconfigurable dual-functional surface that
+can conduct signal reflection and transmission simultaneously,
+such as simultaneous transmitting and reflecting RIS (STAR-
+RIS) [9] and intelligent omni-surface (IOS) [10]. Specifically,
+the coverage characterization of STAR-RIS-aided NOMA net-
+works was investigated in [9] by studying a coverage range
+maximization problem. The authors of [10] considered the
+average rate maximization problem in an IOS-aided NOMA
+networks with spatially correlated channels. Furthermore, the
+effective capacity and secrecy outage probability of STAR-
+RIS-aided NOMA networks were derived in [11] and [12],
+respectively. However, although the effective coverage can
+be enhanced by the existing DF-RIS, the signals relayed by
+the DF-RIS still suffer from channel fading twice due to
+the features of cascaded channels. This double-fading effect
+inevitably deteriorates the achievable performance of passive
+RIS-assisted wireless networks. Therefore, it is necessary to
+design new RIS architectures to mitigate the double-fading
+attenuation problem faced by the existing RISs.
+In this letter, a novel multi-functional RIS (MF-RIS) is
+proposed to address the issues aforementioned. Specifically,
+the proposed MF-RIS can not only divide the incident signal
+into transmission and reflection two parts based on the field
+equivalence principle, but also amplify the outgoing signal
+with the help of active loads. Thus, the MF-RIS is able
+to facilitate a full-space coverage and overcome the double-
+fading issue. Then, we investigate a sum rate maximization
+problem in an MF-RIS-aided NOMA network. Compared to
+the existing problems formulated in [8] and [9], the newly
+introduced MF-RIS constraints and highly coupled variables
+make the performance optimization more complicated. The
+main contributions of this letter are summarized as follows:
+1) We propose a new concept of MF-RIS by integrating the
+surface electric and magnetic impedances, and power amplifier
+into each element so that the incident signal can be reflected,
+refracted, and amplified simultaneously. 2) We formulate a
+non-convex optimization problem to maximize the throughout
+of an MF-RIS-aided NOMA network, where the MF-RIS is
+deployed to constructively enhance the channel condition by
+flexibly adjusting the radio propagation environment. 3) To
+solve the formulated non-convex problem, we propose an
+efficient iterative algorithm by alternatively optimizing the
+arXiv:2301.13630v1  [cs.IT]  31 Jan 2023
+
+2
+BS
+BS
+Transmission-only
+Reflection-only
+BS
+BS
+���
+���
+���
+���
+���
+���
+Reflected signal
+Transmitted signal
+Power supply
+Incident signal
+SF-RIS
+DF-RIS
+MF-RIS
+�
+Users in the reflection space
+Users in the transmission space
+Incident signal
+coverage
+Signal amplification
+Passive relay
+Amplifier
+Phase shifter
+Power supply
+Reflected phase
+Transmitted phase
+Patch
+Patch
+max
+�
+�
+Incident signal
+Reflected signal
+Transmitted signal
+Power splitting
+Reflected signal Transmitted signal
+/
+max
+max
+[0,
+]
+r t
+m
+r
+t
+m
+m
+�
+�
+�
+�
+�
+�
+�
+�
+360�
+1
+r
+t
+m
+m
+�
+�
+�
+�
+/
+/
+/
+r t
+m
+j
+r t
+r t
+m
+m
+m
+y
+e
+s
+�
+�
+�
+/
+/
+/
+r t
+m
+j
+r t
+r t
+m
+m
+m
+y
+e
+s
+�
+�
+�
+Fig. 1. Conventional RIS vs. the proposed MF-RIS-aided NOMA networks.
+active beamforming and MF-RIS coefficients based on the
+penalty-based method and successive convex approximation
+(SCA). 4) Simulation results show that the proposed MF-RIS-
+aided NOMA network can provide up to about 59% sum rate
+gain than the SF-RIS, and the MF-RIS prefers to be deployed
+at the user side for better performance.
+II. SYSTEM MODEL AND PROBLEM FORMULATION
+A. System Model
+We consider an MF-RIS-aided NOMA downlink network,
+where an N-antenna BS communicates with K single-antenna
+users with the aid of an MF-RIS comprising M elements, as
+shown in the right of Fig. 1. The sets of elements and users
+are denoted by M = {1, 2, . . . , M} and K = {1, 2, . . . , K},
+respectively. The channels of BS-user, BS-RIS, and RIS-
+user are denoted by hk
+∈
+CN×1, H
+∈
+CM×N, and
+gk ∈ CM×1, respectively. Furthermore, we define up =
+[
+�
+βp
+1ejθp
+1 ,
+�
+βp
+2ejθp
+2 , . . . ,
+�
+βp
+Mejθp
+M ]T
+∈ CM×1 as the
+transmission (p = t) or reflection (p = r) beamforming vector,
+where p ∈ {t, r} denotes the transmission and reflection
+spaces, βp
+m ∈ [0, βmax] and θp
+m ∈ [0, 2π) represent the
+amplitude and the phase shift response of the m-th element,
+respectively, with the maximum amplification factor βmax ≥ 1.
+Due to the law of energy conservation, we have βr
+m + βt
+m ≤
+βmax. If user k is located at the reflection space, the diagonal
+matrix of the MF-RIS for user k is given by Θk = diag(ur);
+otherwise Θk = diag(ut).
+We assume that the perfect channel state information (CSI)
+of all channels is available at the BS. Then the signal received
+at user k is expressed as
+yk = (hH
+k + gH
+k ΘkH)x + gH
+k Θkns + nk, ∀k,
+(1)
+where x = �
+k wksk denotes the transmit signal, wk and sk ∈
+CN(0, 1) represent the transmit precoder and the information
+symbol for user k, respectively. ns ∈ CN(0, σ2
+sIM) denotes
+the dynamic noise at the MF-RIS with each element’s noise
+power σ2
+s, and nk ∈ CN(0, σ2
+k) denotes the additive white
+Gaussian noise at user k with power σ2
+k.
+By employing SIC, the strong user can mitigate the inter-
+ference from weak users to improve the signal-to-interference-
+plus-noise ratio. Similar to [4]–[6], with the assistance of RIS
+to flexibly adjust channel conditions of multiple users, we
+assume that users’ indexes are ranked in an increasing order
+with respect to their channel gains, i.e.,
+∥�h1∥2 ≤ ∥�h2∥2 ≤ · · · ≤ ∥�hK∥2,
+(2)
+where �hk = hH
+k +gH
+k ΘkH is the equivalent combined channel.
+For the fixed decoding order, the corresponding achievable
+sum rate of user k is given by Rk = log2(1 + γk), where γk
+can be obtained by
+γk =
+|�hkwk|2
+�K
+i=k+1(|�hkwi|2) + |gH
+k Θkns|2 + σ2
+k
+, ∀k.
+(3)
+B. Problem Formulation
+In this letter, we aim to maximize the achievable sum rate
+of all users by jointly optimizing the active beamforming at
+the BS and the coefficients at the MF-RIS. Under the transmit
+and amplification power constraints, and the quality-of-service
+(QoS) requirement of users, the considered optimization prob-
+lem can be formulated as
+max
+wk,Θk
+�K
+k=1 Rk
+(4a)
+s.t.
+�K
+k=1 ∥wk∥2 ≤ Pmax,
+(4b)
+�K
+k=1(∥ΘkHwk∥2+∥ΘkIM∥2
+F σ2
+s)≤Po,
+(4c)
+βr
+m + βt
+m ≤ βmax, 0 ≤ βp
+m ≤ βmax, ∀m, ∀p, (4d)
+Rk ≥ Rmin
+k
+, θp
+m ∈ [0, 2π), (2), ∀k, ∀m, ∀p, (4e)
+where Pmax and Po denote the maximum transmit and am-
+plification power at the BS and MF-RIS, respectively. Rmin
+k
+represents the minimum rate requirement of user k. Specifi-
+cally, the constraints for transmit power, amplification power,
+the QoS requirements and the decoding order are given in
+(4b)-(4e), respectively. It can be observed that the formulated
+problem (4) is intractable due to the non-convex objective
+function and constraints. Besides, the active beamforming and
+MF-RIS coefficients are highly coupled, making it difficult
+to be solved directly. Thus, we aim to transform problem
+(4) into some tractable convex subproblems and solve them
+
+3
+separately and alternatively over iterations. In the next section,
+we adopt alternating optimization method to obtain the active
+beamforming and the MF-RIS coefficients efficiently.
+III. PROPOSED SOLUTION
+A. Active Beamforming Design
+Given the MF-RIS coefficients, the active beamforming
+optimization problem is still non-convex. To solve it, we first
+introduce an auxiliary variable set {Ak, Bk|k ∈ K}, where Ak
+and Bk are defined as
+Ak
+−1 = |�hkwk|2,
+(5)
+Bk =
+�K
+i=k+1(|�hkwi|2) + |gH
+k Θkns|2 + σ2
+k.
+(6)
+Thus, the achievable data rate can be rewritten as Rk =
+log2
+�
+1 + (AkBk)−1�
+.
+Then, the active beamforming optimization problem in (4)
+can be equivalently expressed as
+max
+wk,Ak,Bk,Rk
+�K
+k=1 Rk
+(7a)
+s.t.
+log2
+�
+1 + (AkBk)−1�
+≥ Rk, ∀k,
+(7b)
+Ak
+−1 ≤ |�hkwk|2, ∀k,
+(7c)
+Bk ≥
+K
+�
+i=k+1
+(|�hkwi|2)+|gH
+k Θkns|2+σ2
+k, ∀k,(7d)
+Rk ≥ Rmin
+k
+, (4b), (4c), ∀k.
+(7e)
+We further define �Hk = �hH
+k �hk, Dk = (HHΘk)(HHΘk)H
+and Wk = wkwH
+k , where Wk ⪰ 0, and rank(Wk) = 1.
+Then, we have
+|�hkwk|2 = Tr( �HkWk), ∥ΘkHwk∥2 = Tr(WkDk).
+(8)
+Therefore, problem (7) can be reformulated as
+max
+Wk,Ak,Bk,Rk
+�K
+k=1 Rk
+(9a)
+s.t.
+Ak
+−1 ≤ Tr( �HkWk), ∀k,
+(9b)
+Bk≥
+K
+�
+i=k+1
+Tr( �HkWi)+|gH
+k Θkns|2+σ2
+k, ∀k,(9c)
+�K
+k=1Tr(Wk) ≤ Pmax,
+(9d)
+�K
+k=1
+�
+Tr(WkDk)+∥ΘkIM∥2σ2
+s
+�
+≤Po,
+(9e)
+rank(Wk) = 1, ∀k,
+(9f)
+Wk ⪰ 0, Rk ≥ Rmin
+k
+, (7b), ∀k.
+(9g)
+In order to deal with the non-convex constraint (7b), we
+adopt the first-order Taylor expansion, and then we obtain the
+lower bound as follows:
+log2(1+
+1
+AkBk
+)≥log2(1+
+1
+A(τ1)
+k
+B(τ1)
+k
+)− log2 e(Ak−A(τ1)
+k
+)
+A(τ1)
+k
+(1+A(τ1)
+k
+B(τ1)
+k
+)
+−
+log2 e(Bk−B(τ1)
+k
+)
+B(τ1)
+k
+(1+A(τ1)
+k
+B(τ1)
+k
+)
+∆= Rk,
+(10)
+where A(τ1)
+k
+and B(τ1)
+k
+are feasible points of Ak and Bk in
+the τ1-th iteration, respectively.
+For the non-convex rank-one constraint in (9f), we assume
+to transform it to a penalty term in the objective function,
+which can be solved by SCA. Thus, we firstly introduce an
+equivalent equality:
+∥Wk∥∗ − ∥Wk∥2 = 0, ∀k,
+(11)
+where ∥Wk∥∗ = �
+i εi(Wk) and ∥Wk∥2 = ε1(Wk) denote
+the nuclear norm and the spectral norm of Wk, respectively.
+εi(Wk) is the i-th largest singular value of matrix Wk. Thus,
+when the matrix Wk is rank-one, equality (11) holds.
+Next, we employ the penalty method to solve problem
+(9) by adding (11) to the objective function (9a). Since the
+penalty term (11) makes the objective function not convex,
+we apply the first-order Taylor expansion to obtain a convex
+upper bound of (11) as follows:
+∥Wk∥∗ − ∥Wk∥2 ≤ ∥Wk∥∗ − ∥Wk∥2,
+(12)
+where ∥Wk∥2
+=
+∥W(τ1)
+k
+∥2 + Tr
+�
+e(τ1)
+k
+(e(τ1)
+k
+)H(Wk −
+W(τ1)
+k
+)
+�
+, and e(τ1)
+k
+is the eigenvector corresponding to the
+largest eigenvalue of W(τ1)
+k
+in the τ1-th iteration.
+By introducing (12) to the objective function (9a), we obtain
+the following problem:
+max
+Wk,Ak,Bk,Rk
+�K
+k=1Rk− 1
+η
+�
+k(∥Wk∥∗−∥Wk∥2)
+(13a)
+s.t.
+Rk ≥ Rk, Wk ⪰ 0, Rk ≥ Rmin
+k
+, ∀k, (13b)
+(9b) − (9e),
+(13c)
+where η > 0 is the penalty factor penalizing (13a) if Wk is
+not rank-one. It can be verified that, when η → 0, the solution
+{Wk} of problem (13) always satisfies equality (11).
+The reformulated problem (13) is a standard convex semi-
+definite programming (SDP), which can be efficiently solved
+via CVX. To obtain a high quality solution, we first initialize
+a large η to find a feasible starting point, and then gradually
+decrease η with η = µη, µ < 1 to a sufficiently small value to
+obtain an overall suboptimal solution. The process terminates
+when the penalty term satisfies the following criterion:
+max{∥Wk∥∗ − ∥Wk∥2, ∀k} ≤ ϵ1,
+(14)
+where ϵ1 denotes a predefined maximum violation of (11).
+B. MF-RIS Coefficient Design
+For the coefficient design at the MF-RIS, we define
+vk
+= [ur; 1] if user k is located at the space r; oth-
+erwise vk
+=
+[ut; 1]. Then, we define Vk
+=
+vkvH
+k ,
+with
+Vk
+⪰
+0
+and
+rank(Vk)
+=
+1.
+Let
+gk
+=
+[gk,1, gk,2, . . . , gk,M]H and Gk
+=
+Hwk, then we have
+Qk
+=
+diag
+��
+|gk,1|2, |gk,2|2, . . . , |gk,M|2��
+and
+�Gk
+=
+diag
+��
+|Gk,1|2, |Gk,2|2, . . . , |Gk,M|2��
++ σ2
+sIM. Given
+Qk =
+� Qk
+0
+0
+0
+�
+, Gk =
+� �Gk
+0
+0
+0
+�
+,
+(15)
+we can obtain
+∥ΘkHwk∥2 + ∥ΘkIM∥2
+F σ2
+s = Tr(VkGk),
+(16)
+∥gH
+k Θk∥2 = Tr(VkQk).
+(17)
+Thus, constraint (4c) can be replaced by (16).
+In order to handle the non-convex constraints (2) and (5), we
+define fk = diag(gH
+k )Gk, Rk = diag(gH
+k )H, ˜hk = ∥hH
+k ∥2,
+and dk = wH
+k hk, then we have
+Fk =
+� fkf H
+k
+fkd∗
+k
+dkf H
+k
+|dk|2
+�
+, Rk =
+� RkRH
+k
+Rkhk
+hH
+k RH
+k
+˜hk
+�
+.
+(18)
+
+4
+According to the above transformation, we can obtain
+|�hkwk|2 = |(hH
+k + gH
+k ΘkH)wk|2 = Tr(VkFk),(19)
+∥�hk∥2 = Tr(VkRk).
+(20)
+Based on (20), the decoding order in (2) is rewritten as
+Tr(V1R1) ≤ Tr(V2R2) ≤ · · · ≤ Tr(VKRK). (21)
+Then, given the active beamforming vector, the subproblem
+of MF-RIS coefficient design can be given by
+max
+Vk,Ak,Bk,Rk
+�K
+k=1 Rk
+(22a)
+s.t.
+Ak
+−1 ≤Tr(VkFk), ∀k,
+(22b)
+Bk≤
+K
+�
+i=k+1
+Tr(VkFi)+σ2
+sTr(VkQk)+σ2
+k, ∀k,(22c)
+�K
+k=1 Tr(VkGk) ≤ Po,
+(22d)
+Vk ⪰ 0, Rk ≥ Rmin
+k
+, ∀k,
+(22e)
+[Vk]m,m = βk
+m, [Vk]M+1,M+1 = 1, ∀k,
+(22f)
+rank(Vk) = 1, ∀k,
+(22g)
+θp
+m ∈ [0, 2π), (4d), (7b), (21), ∀m, ∀p,
+(22h)
+where Fi denotes Fk when wk is replaced by wi.
+Similar to (12), we replace the rank-one constraint in (22g)
+with the following form:
+∥Vk∥∗ − ∥Vk∥2 ≤ ∥Vk∥∗ − ∥Vk∥2,
+(23)
+where ∥Vk∥∗ and ∥Vk∥2 denote the nuclear norm and the
+spectral norm of matrix Vk, respectively. Besides, ∥Vk∥2 =
+∥V(τ2)
+k
+∥2 + Tr
+�
+z(τ2)
+k
+(z(τ2)
+k
+)H(Vk − V(τ2)
+k
+)
+�
+and z(τ2)
+k
+is the
+eigenvector corresponding to the largest eigenvalue of V(τ2)
+k
+in the τ2-th iteration.
+By introducing (10) into (7b), problem (22) can be refor-
+mulated as
+max
+Vk,Ak,Bk,Rk
+�K
+k=1Rk− 1
+ξ
+�
+k(∥Vk∥∗−∥Vk∥2)
+(24a)
+s.t.
+Rk ≥ Rk, θp
+m ∈ [0, 2π), ∀k, ∀m, ∀p, (24b)
+(4d), (21), (22b)−(22f),
+(24c)
+where ξ > 0 is the penalty factor to ensure Vk is rank-one.
+The problem (24) is a standard SDP problem. It can be
+solved by CVX. The termination criterion is given by
+max{∥Vk∥∗ − ∥Vk∥2, ∀k} ≤ ϵ2,
+(25)
+where ϵ2 denotes a predefined maximum violation.
+Based on the above derivation, we propose a penalty-
+based iterative algorithm to solve problem (4) efficiently. The
+details are given in Algorithm 1. Specifically, the initial points
+{W(0)
+k } and {V(0)
+k } are obtained by selecting the feasible
+ones from some random points. Since both the objectives of
+problems (13) and (24) are non-decreasing over iterations and
+the system throughout is upper-bounded by a finite value, the
+proposed Algorithm 1 is guaranteed to converge. Moreover,
+if the interior point method is employed, the complexity of
+Algorithm 1 is O(IoutIin(KN 3.5 + 2M 3.5)), where K, M
+and N are the numbers of users, BS antennas and MF-
+RIS elements, respectively. The terms Iin and Iout denote
+the number of the inner and outer iterations required for
+convergence, respectively.
+Algorithm 1 Penalty-Based Iterative Algorithm
+1: Initialize {W(0)
+k }, {V(0)
+k }, the error tolerance ∆, the
+maximum number of iteration T0,max, the penalty factors
+η and ξ, and the predefined threshold ϵ.
+2: repeat
+3:
+Set the iteration index τ0 = 0;
+4:
+repeat
+5:
+Given V(τ0)
+k
+, update W(τ0+1)
+k
+by solving (13);
+6:
+Given W(τ0+1)
+k
+, update V(τ0+1)
+k
+by solving (24);
+7:
+Update τ0 = τ0 + 1;
+8:
+until | R(τ0)
+sum −R(τ0−1)
+sum
+R(τ0−1)
+sum
+| < ∆ or τ0 > T0,max.
+9:
+Update {W(0)
+k , V(0)
+k } with {W(τ0)
+k
+, V(τ0)
+k
+};
+10:
+Update η = µη, ξ = µξ;
+11: until the constraints (14) and (25) satisfy ϵ;
+12: Output the converged solutions {W∗
+k} and {V∗
+k}.
+TABLE I: Simulation Parameters
+Parameter
+Value
+Path loss exponents of BS-MF-RIS,
+BS-users, MF-RIS-users links
+2.5, 3.5, 2.8
+Rician factors of all links
+3 dB
+Noise power at MF-RIS and users
+−80 dBm
+Minimum required QoS for users
+0.1 bit/s/Hz
+Maximum amplification power [13]
+Po = 10 dBm
+Maximum amplification factor
+βmax = 22 dB
+Convergence tolerance
+∆ = 10−6
+IV. SIMULATION RESULTS
+In this section, numerical results are provided to validate
+the performance of an MF-RIS-aided NOMA network. The
+BS and the MF-RIS are located at (0, 0, 0) and (0, 50, 20),
+respectively. Besides, the users are divided into two parts,
+distributed on circles centered on (0, 45, 0) and (0, 55, 0)
+with radius r = 3, respectively. We adopt Rician fading for
+all channels, and set K = 6, N = 16, M = 100, and
+Pmax = 20 dBm. Other parameters are listed in Table I. We
+compare the proposed MF-RIS with three existing RISs:
+• SF-RIS [7]: The SF-RIS only supports signal reflection
+or transmission, i.e., βmax = 1 and Θt or Θr = 0M×M.
+• Active RIS [13]: The active RIS simultaneously supports
+signal reflection and amplification, i.e., Θt = 0M×M.
+• STAR-RIS [9]: The STAR-RIS provides full space cov-
+erage by splitting signals to two sides, i.e., βmax = 1.
+Fig. 2(a) depicts the sum rate versus the maximum transmit
+power Pmax. It can be observed that the sum rates of all
+schemes increase with Pmax. Besides, the proposed MF-RIS
+always yields a better performance than other benchmarks.
+Specifically, when Pmax = 10 dBm, the MF-RIS enjoys
+a 59% higher sum rate than the SF-RIS. This is because
+the MF-RIS serves all users in full space through signal
+reflection, transmission and amplification functions. Besides,
+by providing additional energy to amplify the incident signal,
+the MF-RIS is able to efficiently mitigate the double-fading at-
+
+5
+5
+10
+15
+20
+25
+30
+2
+4
+6
+8
+10
+12
+14
+16
+Sum Rate (bps/Hz)
+Maximum transmit power,          (dBm)
+ MF-RIS
+ Active RIS
+ STAR-RIS
+ SF-RIS
+ Without RIS
+59%
+16%
+44%
+(a) Sum rate vs. the power budget.
+10
+20
+30
+40
+50
+60
+70
+80
+90
+100
+7
+8
+9
+10
+11
+12
+13
+Sum Rate (bps/Hz)
+Number of elements,     
+ MF-RIS
+ Active RIS
+ STAR-RIS
+ SF-RIS
+ Without RIS
+12%
+6%
+(b) Sum rate vs. the number of elements.
+0
+10
+20
+30
+40
+50
+7
+8
+9
+10
+11
+12
+13
+14
+Sum Rate (bps/Hz)
+Y-coordinate of RIS
+ MF-RIS 
+ Active RIS  
+ STAR-RIS   
+ SF-RIS
+ Without RIS 
+39%
+28%
+(c) Sum rate vs. the Y -coordinate of RIS.
+Fig. 2. Simulation results for the sum rate versus different transmit power, number of elements and RIS locations.
+tenuation, which helps to improve the channel gain of cascaded
+links. Furthermore, due to the limitations faced by the active
+RIS and STAR-RIS counterparts (i.e., half-space coverage
+and double-fading attenuation), the MF-RIS improves the rate
+performance by 16% and 44% when Pmax = 10 dBm,
+respectively. Additionally, it is evident that all RIS-aided
+schemes achieve significant gains than the scheme without
+RIS. This demonstrates the superiority of using RIS to improve
+the performance of wireless networks.
+Fig. 2(b) shows that the sum rate of all RIS-aided schemes
+increase with M. This is because a larger M enables a higher
+beamforming gain, thus improving the system performance. In
+addition, with more degree of freedoms to manipulate signal
+propagation, the STAR-RIS is capable of enjoying a 6% higher
+sum rate than the SF-RIS. Moreover, although only the users
+located in the reflection space are served by the active RIS,
+it outperforms the STAR-RIS with a 12% higher sum rate.
+This is because the performance gain obtained from the signal
+amplification of active RIS is greater than that from full-
+space coverage of STAR-RIS. This also implies that the signal
+amplification function plays an important role in improving the
+performance of RIS-aided networks.
+Fig. 2(c) illustrates the sum rate versus the Y -coordinate of
+RIS (from 0 to 50), where the RIS moves from the BS side
+to the user side. We can observe that the sum rates of the
+STAR-RIS and the SF-RIS first decrease and then increase.
+The reason behind this is that the channel gain decreases with
+the link distance. Specifically, when the STAR-RIS and the SF-
+RIS are located close to the middle point, the received signals
+at users are attenuated the most, resulting in the lowest sum
+rate. In contrast, owing to the signal amplification function,
+the MF-RIS and the active RIS are less affected by the
+double-fading attenuation, which achieve 39% and 28% gains
+in the middle point compared to the SF-RIS. Moreover, the
+corresponding sum rate maintains a continuous upward trend
+even when the MF-RIS and active RIS are far away from the
+BS. This is because as the RIS comes closer to the users, the
+power of the incident signal at the RIS is weaker. Thus, under
+a fixed amplification power budget, the MF-RIS can provide
+more amplification gain when deployed closer to users. This
+compensates for the attenuation caused by the double-fading
+issue. This observation also reveals that the MF-RIS should
+be deployed close to the users for better performance.
+V. CONCLUSION
+In this letter, we proposed a novel MF-RIS architecture to
+alleviate the double-fading attenuation via transmitting and
+reflecting the incident signal with power amplification. Then,
+we investigated the resource allocation problem in a downlink
+multiuser MF-RIS-aided NOMA network. Specifically, the
+active beamforming and MF-RIS coefficients were jointly
+optimized to maximize the achievable sum rate by leveraging
+SCA and penalty-based method. Numerical results validated
+the effectiveness of the proposed MF-RIS and the superiority
+of MF-RIS over traditional RISs. In the future, we are inter-
+ested in studying the coupled phase and hardware impairment
+problems of the MF-RIS. In addition, the robust beamforming
+under imperfect CSI cases deserves exploration as well.
+REFERENCES
+[1] Y. Liu, Z. Qin, Elkashlan et al., “Nonorthogonal multiple access for 5G
+and beyond,” Proc. IEEE, vol. 105, no. 12, pp. 2347–2381, Dec.2017.
+[2] M. Elhattab, M. A. Arfaoui, C. Assi et al., “RIS-assisted joint transmis-
+sion in a two-cell downlink NOMA cellular system,” IEEE J. Sel. Areas
+Commun., vol. 40, no. 4, pp. 1270–1286, Apr. 2022.
+[3] Y. Liu, X. Liu, X. Mu et al., “Reconfigurable intelligent surfaces:
+Principles and opportunities,” IEEE Commun. Surveys Tuts., vol. 23,
+no. 3, pp. 1546–1577, thirdquarter 2021.
+[4] A. S. d. Sena, D. Carrillo, F. Fang et al., “What role do intelligent re-
+flecting surfaces play in multi-antenna non-orthogonal multiple access?”
+IEEE Wireless Commun., vol. 27, no. 5, pp. 24–31, Oct. 2020.
+[5] Z. Ding and H. Vincent Poor, “A simple design of IRS-NOMA transmis-
+sion,” IEEE Commun. Lett., vol. 24, no. 5, pp. 1119–1123, Feb. 2020.
+[6] B. Zheng, Q. Wu, and R. Zhang, “Intelligent reflecting surface-assisted
+multiple access with user pairing: NOMA or OMA?” IEEE Commun.
+Lett., vol. 24, no. 4, pp. 753–757, Jan. 2020.
+[7] W. Ni, X. Liu, Y. Liu et al., “Resource allocation for multi-cell IRS-
+aided NOMA networks,” IEEE Trans. Wireless Commun., vol. 20, no. 7,
+pp. 4253–4268, Jul. 2021.
+[8] W. Wang, W. Ni, H. Tian et al., “Safeguarding NOMA networks via
+reconfigurable dual-functional surface under imperfect CSI,” IEEE J.
+Sel. Topics Signal Process., vol. 16, no. 5, pp. 950–966, Aug. 2022.
+[9] C. Wu, Y. Liu, X. Mu et al., “Coverage characterization of STAR-RIS
+networks: NOMA and OMA,” IEEE Commun. Lett., vol. 25, no. 9,
+pp.3036–3040, Sept. 2021.
+[10] T. Wang, M.-A. Badiu, G. Chen et al., “Performance analysis of IOS-
+assisted NOMA system with channel correlation and phase errors,” IEEE
+Trans. Veh. Technol., vol. 71, no. 11, pp. 11 861–11 875, Nov. 2022.
+[11] H. Liu, G. Li, X. Li et al., “Effective capacity analysis of STAR-RIS-
+assisted NOMA networks,” IEEE Wireless Commun. Lett., vol. 11, no. 9,
+pp. 1930–1934, Sept. 2022.
+[12] X. Li, Y. Zheng, M. Zeng et al., “Enhancing secrecy performance for
+STAR-RIS NOMA networks,” IEEE Trans. Veh. Technol., Oct. 2022.
+[13] R. Long, Y.-C. Liang, Y. Pei et al., “Active reconfigurable intelligent
+surface-aided wireless communications,” IEEE Trans. Wireless Com-
+mun., vol. 20, no. 8, pp. 4962–4975, Aug. 2021.
+

AtFRT4oBgHgl3EQfuDj6/content/tmp_files/load_file.txt

@@ -0,0 +1,428 @@
+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFRT4oBgHgl3EQfuDj6/content/2301.13630v1.pdf,len=427
+page_content='1  Enhancing NOMA Networks via Reconfigurable  Multi-Functional Surface  Ailing Zheng, Wanli Ni, Wen Wang, and Hui Tian  Abstract—By flexibly manipulating the radio propagation en-  vironment, reconfigurable intelligent surface (RIS) is a promising  technique for future wireless communications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFRT4oBgHgl3EQfuDj6/content/2301.13630v1.pdf'}
+page_content=' However, the  single-side coverage and double-fading attenuation faced by con-  ventional RISs largely restrict their applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFRT4oBgHgl3EQfuDj6/content/2301.13630v1.pdf'}
+page_content=' To address this  issue, we propose a novel concept of multi-functional RIS (MF-  RIS), which provides reflection, transmission, and amplification  simultaneously for the incident signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFRT4oBgHgl3EQfuDj6/content/2301.13630v1.pdf'}
+page_content=' With the aim of enhancing  the performance of a non-orthogonal multiple-access (NOMA)  downlink multiuser network, we deploy an MF-RIS to maximize  the sum rate by jointly optimizing the active beamforming and  MF-RIS coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFRT4oBgHgl3EQfuDj6/content/2301.13630v1.pdf'}
+page_content=' Then, an alternating optimization algorithm  is proposed to solve the formulated non-convex problem by  exploiting successive convex approximation and penalty-based  method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFRT4oBgHgl3EQfuDj6/content/2301.13630v1.pdf'}
+page_content=' Numerical results show that the proposed MF-RIS  outperforms conventional RISs under different settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFRT4oBgHgl3EQfuDj6/content/2301.13630v1.pdf'}
+page_content='  Index Terms—Multi-functional reconfigurable intelligent sur-  face, non-orthogonal multiple access, rate maximization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFRT4oBgHgl3EQfuDj6/content/2301.13630v1.pdf'}
+page_content='  I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFRT4oBgHgl3EQfuDj6/content/2301.13630v1.pdf'}
+page_content=' INTRODUCTION  Compared to orthogonal multiple access (OMA), non-  orthogonal multiple access (NOMA) is capable of achieving  high spectrum efficiency and massive connectivity [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFRT4oBgHgl3EQfuDj6/content/2301.13630v1.pdf'}
+page_content=' Prior  investigations have shown that the differences between users’  channel conditions can be exploited to enhance NOMA per-  formance [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFRT4oBgHgl3EQfuDj6/content/2301.13630v1.pdf'}
+page_content=' However, users in large-scale networks may  have poor or similar channel conditions, which hinders the  application of successive interference cancellation (SIC) and  the effective implementation of NOMA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFRT4oBgHgl3EQfuDj6/content/2301.13630v1.pdf'}
+page_content=' Therefore, adjusting  channel conditions and enhancing channel diversity are able  to release the potential of NOMA in practical networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFRT4oBgHgl3EQfuDj6/content/2301.13630v1.pdf'}
+page_content='  Recently, with the ability to reshape the wireless propaga-  tion environment, reconfigurable intelligent surface (RIS) has  emerged as a key technique to improve the performance of  NOMA networks [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFRT4oBgHgl3EQfuDj6/content/2301.13630v1.pdf'}
+page_content=' By properly designing the reflection  coefficients, RIS is able to smartly change the combined  channels to enhance the differences among users, thus boosting  the performance of NOMA in large-scale networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFRT4oBgHgl3EQfuDj6/content/2301.13630v1.pdf'}
+page_content=' Initial  investigations on RIS-aided NOMA networks in [3]–[7] had  verified the superiority of the integration of NOMA and RIS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFRT4oBgHgl3EQfuDj6/content/2301.13630v1.pdf'}
+page_content='  Specifically, the authors of [3] and [4] performed comprehen-  sive discussions of the main challenges and futuristic use cases  regarding RIS-aided NOMA networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFRT4oBgHgl3EQfuDj6/content/2301.13630v1.pdf'}
+page_content=' Moreover, the works in  [5]–[7] demonstrated the benefits brought by RISs to achieve  performance trade-off among multiple NOMA users through  This letter was supported by the Natural Science Foundation of Shandong  Province under Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFRT4oBgHgl3EQfuDj6/content/2301.13630v1.pdf'}
+page_content=' ZR2021LZH010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFRT4oBgHgl3EQfuDj6/content/2301.13630v1.pdf'}
+page_content=' The associate editor coordinating  the review of this letter and approving it for publication was Lina Bariah.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFRT4oBgHgl3EQfuDj6/content/2301.13630v1.pdf'}
+page_content='  (Corresponding author: Hui Tian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFRT4oBgHgl3EQfuDj6/content/2301.13630v1.pdf'}
+page_content=')  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFRT4oBgHgl3EQfuDj6/content/2301.13630v1.pdf'}
+page_content=' Zheng, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFRT4oBgHgl3EQfuDj6/content/2301.13630v1.pdf'}
+page_content=' Ni, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFRT4oBgHgl3EQfuDj6/content/2301.13630v1.pdf'}
+page_content=' Wang, and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFRT4oBgHgl3EQfuDj6/content/2301.13630v1.pdf'}
+page_content=' Tian are with the State Key Lab-  oratory of Networking and Switching Technology, Beijing University of  Posts and Telecommunications, Beijing 100876, China (e-mail: {ailing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFRT4oBgHgl3EQfuDj6/content/2301.13630v1.pdf'}
+page_content='zheng,  charleswall, wen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFRT4oBgHgl3EQfuDj6/content/2301.13630v1.pdf'}
+page_content='wang, tianhui}@bupt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFRT4oBgHgl3EQfuDj6/content/2301.13630v1.pdf'}
+page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFRT4oBgHgl3EQfuDj6/content/2301.13630v1.pdf'}
+page_content='cn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFRT4oBgHgl3EQfuDj6/content/2301.13630v1.pdf'}
+page_content='  smartly adjusting the decoding order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFRT4oBgHgl3EQfuDj6/content/2301.13630v1.pdf'}
+page_content=' However, the existing  literature on RIS-aided NOMA networks mostly uses single  functional RIS (SF-RIS) that only supports signal reflection  or transmission/refraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFRT4oBgHgl3EQfuDj6/content/2301.13630v1.pdf'}
+page_content=' This implies that only users located  in a single side can be served by the SF-RIS if no additional  operations are performed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFRT4oBgHgl3EQfuDj6/content/2301.13630v1.pdf'}
+page_content='  To overcome this limitation, the authors of [8] proposed the  concept of dual-functional RIS (DF-RIS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFRT4oBgHgl3EQfuDj6/content/2301.13630v1.pdf'}
+page_content=' Unlike SF-RIS, DF-  RIS refers to the reconfigurable dual-functional surface that  can conduct signal reflection and transmission simultaneously,  such as simultaneous transmitting and reflecting RIS (STAR-  RIS) [9] and intelligent omni-surface (IOS) [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFRT4oBgHgl3EQfuDj6/content/2301.13630v1.pdf'}
+page_content=' Specifically,  the coverage characterization of STAR-RIS-aided NOMA net-  works was investigated in [9] by studying a coverage range  maximization problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFRT4oBgHgl3EQfuDj6/content/2301.13630v1.pdf'}
+page_content=' The authors of [10] considered the  average rate maximization problem in an IOS-aided NOMA  networks with spatially correlated channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFRT4oBgHgl3EQfuDj6/content/2301.13630v1.pdf'}
+page_content=' Furthermore, the  effective capacity and secrecy outage probability of STAR-  RIS-aided NOMA networks were derived in [11] and [12],  respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFRT4oBgHgl3EQfuDj6/content/2301.13630v1.pdf'}
+page_content=' However, although the effective coverage can  be enhanced by the existing DF-RIS, the signals relayed by  the DF-RIS still suffer from channel fading twice due to  the features of cascaded channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFRT4oBgHgl3EQfuDj6/content/2301.13630v1.pdf'}
+page_content=' This double-fading effect  inevitably deteriorates the achievable performance of passive  RIS-assisted wireless networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFRT4oBgHgl3EQfuDj6/content/2301.13630v1.pdf'}
+page_content=' Therefore, it is necessary to  design new RIS architectures to mitigate the double-fading  attenuation problem faced by the existing RISs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFRT4oBgHgl3EQfuDj6/content/2301.13630v1.pdf'}
+page_content='  In this letter, a novel multi-functional RIS (MF-RIS) is  proposed to address the issues aforementioned.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFRT4oBgHgl3EQfuDj6/content/2301.13630v1.pdf'}
+page_content=' Specifically,  the proposed MF-RIS can not only divide the incident signal  into transmission and reflection two parts based on the field  equivalence principle, but also amplify the outgoing signal  with the help of active loads.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFRT4oBgHgl3EQfuDj6/content/2301.13630v1.pdf'}
+page_content=' Thus, the MF-RIS is able  to facilitate a full-space coverage and overcome the double-  fading issue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFRT4oBgHgl3EQfuDj6/content/2301.13630v1.pdf'}
+page_content=' Then, we investigate a sum rate maximization  problem in an MF-RIS-aided NOMA network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFRT4oBgHgl3EQfuDj6/content/2301.13630v1.pdf'}
+page_content=' Compared to  the existing problems formulated in [8] and [9], the newly  introduced MF-RIS constraints and highly coupled variables  make the performance optimization more complicated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFRT4oBgHgl3EQfuDj6/content/2301.13630v1.pdf'}
+page_content=' The  main contributions of this letter are summarized as follows:  1) We propose a new concept of MF-RIS by integrating the  surface electric and magnetic impedances, and power amplifier  into each element so that the incident signal can be reflected,  refracted, and amplified simultaneously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFRT4oBgHgl3EQfuDj6/content/2301.13630v1.pdf'}
+page_content=' 2) We formulate a  non-convex optimization problem to maximize the throughout  of an MF-RIS-aided NOMA network, where the MF-RIS is  deployed to constructively enhance the channel condition by  flexibly adjusting the radio propagation environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFRT4oBgHgl3EQfuDj6/content/2301.13630v1.pdf'}
+page_content=' 3) To  solve the formulated non-convex problem, we propose an  efficient iterative algorithm by alternatively optimizing the  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFRT4oBgHgl3EQfuDj6/content/2301.13630v1.pdf'}
+page_content='13630v1  [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFRT4oBgHgl3EQfuDj6/content/2301.13630v1.pdf'}
+page_content='IT]  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFRT4oBgHgl3EQfuDj6/content/2301.13630v1.pdf'}
+page_content='31 Jan 2023  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFRT4oBgHgl3EQfuDj6/content/2301.13630v1.pdf'}
+page_content='2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFRT4oBgHgl3EQfuDj6/content/2301.13630v1.pdf'}
+page_content='BS  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFRT4oBgHgl3EQfuDj6/content/2301.13630v1.pdf'}
+page_content='BS  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFRT4oBgHgl3EQfuDj6/content/2301.13630v1.pdf'}
+page_content='Transmission-only  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFRT4oBgHgl3EQfuDj6/content/2301.13630v1.pdf'}
+page_content='Reflection-only  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFRT4oBgHgl3EQfuDj6/content/2301.13630v1.pdf'}
+page_content='BS  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFRT4oBgHgl3EQfuDj6/content/2301.13630v1.pdf'}
+page_content='BS  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFRT4oBgHgl3EQfuDj6/content/2301.13630v1.pdf'}
+page_content='���  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFRT4oBgHgl3EQfuDj6/content/2301.13630v1.pdf'}
+page_content='���  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFRT4oBgHgl3EQfuDj6/content/2301.13630v1.pdf'}
+page_content='���  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFRT4oBgHgl3EQfuDj6/content/2301.13630v1.pdf'}
+page_content='���  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFRT4oBgHgl3EQfuDj6/content/2301.13630v1.pdf'}
+page_content='���  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFRT4oBgHgl3EQfuDj6/content/2301.13630v1.pdf'}
+page_content='���  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFRT4oBgHgl3EQfuDj6/content/2301.13630v1.pdf'}
+page_content='Reflected signal  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFRT4oBgHgl3EQfuDj6/content/2301.13630v1.pdf'}
+page_content='Transmitted signal  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFRT4oBgHgl3EQfuDj6/content/2301.13630v1.pdf'}
+page_content='Power supply  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFRT4oBgHgl3EQfuDj6/content/2301.13630v1.pdf'}
+page_content='Incident signal  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFRT4oBgHgl3EQfuDj6/content/2301.13630v1.pdf'}
+page_content='SF-RIS  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFRT4oBgHgl3EQfuDj6/content/2301.13630v1.pdf'}
+page_content='DF-RIS  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFRT4oBgHgl3EQfuDj6/content/2301.13630v1.pdf'}
+page_content='MF-RIS  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFRT4oBgHgl3EQfuDj6/content/2301.13630v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFRT4oBgHgl3EQfuDj6/content/2301.13630v1.pdf'}
+page_content='Users in the reflection space  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFRT4oBgHgl3EQfuDj6/content/2301.13630v1.pdf'}
+page_content='Users in the transmission space  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFRT4oBgHgl3EQfuDj6/content/2301.13630v1.pdf'}
+page_content='Incident signal  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFRT4oBgHgl3EQfuDj6/content/2301.13630v1.pdf'}
+page_content='coverage  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFRT4oBgHgl3EQfuDj6/content/2301.13630v1.pdf'}
+page_content='Signal amplification  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFRT4oBgHgl3EQfuDj6/content/2301.13630v1.pdf'}
+page_content='Passive relay  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFRT4oBgHgl3EQfuDj6/content/2301.13630v1.pdf'}
+page_content='Amplifier  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFRT4oBgHgl3EQfuDj6/content/2301.13630v1.pdf'}
+page_content='Phase shifter  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFRT4oBgHgl3EQfuDj6/content/2301.13630v1.pdf'}
+page_content='Power supply  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFRT4oBgHgl3EQfuDj6/content/2301.13630v1.pdf'}
+page_content='Reflected phase  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFRT4oBgHgl3EQfuDj6/content/2301.13630v1.pdf'}
+page_content='Transmitted phase  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFRT4oBgHgl3EQfuDj6/content/2301.13630v1.pdf'}
+page_content='Patch  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFRT4oBgHgl3EQfuDj6/content/2301.13630v1.pdf'}
+page_content='Patch  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFRT4oBgHgl3EQfuDj6/content/2301.13630v1.pdf'}
+page_content='max  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFRT4oBgHgl3EQfuDj6/content/2301.13630v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFRT4oBgHgl3EQfuDj6/content/2301.13630v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFRT4oBgHgl3EQfuDj6/content/2301.13630v1.pdf'}
+page_content='Incident signal  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFRT4oBgHgl3EQfuDj6/content/2301.13630v1.pdf'}
+page_content='Reflected signal  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFRT4oBgHgl3EQfuDj6/content/2301.13630v1.pdf'}
+page_content='Transmitted signal  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFRT4oBgHgl3EQfuDj6/content/2301.13630v1.pdf'}
+page_content='Power splitting  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFRT4oBgHgl3EQfuDj6/content/2301.13630v1.pdf'}
+page_content='Reflected signal Transmitted signal  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFRT4oBgHgl3EQfuDj6/content/2301.13630v1.pdf'}
+page_content='/  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFRT4oBgHgl3EQfuDj6/content/2301.13630v1.pdf'}
+page_content='max  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFRT4oBgHgl3EQfuDj6/content/2301.13630v1.pdf'}
+page_content='max  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFRT4oBgHgl3EQfuDj6/content/2301.13630v1.pdf'}
+page_content='[0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFRT4oBgHgl3EQfuDj6/content/2301.13630v1.pdf'}
+page_content='  ]  r t  m  r  t  m  m  �  �  �  �  �  �  �  �  360�  1  r  t  m  m  �  �  �  �  /  /  /  r t  m  j  r t  r t  m  m  m  y  e  s  �  �  �  /  /  /  r t  m  j  r t  r t  m  m  m  y  e  s  �  �  �  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFRT4oBgHgl3EQfuDj6/content/2301.13630v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFRT4oBgHgl3EQfuDj6/content/2301.13630v1.pdf'}
+page_content=' Conventional RIS vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFRT4oBgHgl3EQfuDj6/content/2301.13630v1.pdf'}
+page_content=' the proposed MF-RIS-aided NOMA networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFRT4oBgHgl3EQfuDj6/content/2301.13630v1.pdf'}
+page_content='  active beamforming and MF-RIS coefficients based on the  penalty-based method and successive convex approximation  (SCA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFRT4oBgHgl3EQfuDj6/content/2301.13630v1.pdf'}
+page_content=' 4) Simulation results show that the proposed MF-RIS-  aided NOMA network can provide up to about 59% sum rate  gain than the SF-RIS, and the MF-RIS prefers to be deployed  at the user side for better performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFRT4oBgHgl3EQfuDj6/content/2301.13630v1.pdf'}
+page_content='  II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFRT4oBgHgl3EQfuDj6/content/2301.13630v1.pdf'}
+page_content=' SYSTEM MODEL AND PROBLEM FORMULATION  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFRT4oBgHgl3EQfuDj6/content/2301.13630v1.pdf'}
+page_content=' System Model  We consider an MF-RIS-aided NOMA downlink network,  where an N-antenna BS communicates with K single-antenna  users with the aid of an MF-RIS comprising M elements, as  shown in the right of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFRT4oBgHgl3EQfuDj6/content/2301.13630v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFRT4oBgHgl3EQfuDj6/content/2301.13630v1.pdf'}
+page_content=' The sets of elements and users  are denoted by M = {1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFRT4oBgHgl3EQfuDj6/content/2301.13630v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFRT4oBgHgl3EQfuDj6/content/2301.13630v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFRT4oBgHgl3EQfuDj6/content/2301.13630v1.pdf'}
+page_content=' , M} and K = {1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFRT4oBgHgl3EQfuDj6/content/2301.13630v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFRT4oBgHgl3EQfuDj6/content/2301.13630v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFRT4oBgHgl3EQfuDj6/content/2301.13630v1.pdf'}
+page_content=' , K},  respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFRT4oBgHgl3EQfuDj6/content/2301.13630v1.pdf'}
+page_content=' The channels of BS-user, BS-RIS, and RIS-  user are denoted by hk  ∈  CN×1, H  ∈  CM×N, and  gk ∈ CM×1, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFRT4oBgHgl3EQfuDj6/content/2301.13630v1.pdf'}
+page_content=' Furthermore, we define up =  [  �  βp  1ejθp  1 ,  �  βp  2ejθp  2 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFRT4oBgHgl3EQfuDj6/content/2301.13630v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFRT4oBgHgl3EQfuDj6/content/2301.13630v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFRT4oBgHgl3EQfuDj6/content/2301.13630v1.pdf'}
+page_content=' ,  �  βp  Mejθp  M ]T  ∈ CM×1 as the  transmission (p = t) or reflection (p = r) beamforming vector,  where p ∈ {t, r} denotes the transmission and reflection  spaces, βp  m ∈ [0, βmax] and θp  m ∈ [0, 2π) represent the  amplitude and the phase shift response of the m-th element,  respectively, with the maximum amplification factor βmax ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFRT4oBgHgl3EQfuDj6/content/2301.13630v1.pdf'}
+page_content='  Due to the law of energy conservation, we have βr  m + βt  m ≤  βmax.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFRT4oBgHgl3EQfuDj6/content/2301.13630v1.pdf'}
+page_content=' If user k is located at the reflection space, the diagonal  matrix of the MF-RIS for user k is given by Θk = diag(ur);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFRT4oBgHgl3EQfuDj6/content/2301.13630v1.pdf'}
+page_content='  otherwise Θk = diag(ut).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFRT4oBgHgl3EQfuDj6/content/2301.13630v1.pdf'}
+page_content='  We assume that the perfect channel state information (CSI)  of all channels is available at the BS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFRT4oBgHgl3EQfuDj6/content/2301.13630v1.pdf'}
+page_content=' Then the signal received  at user k is expressed as  yk = (hH  k + gH  k ΘkH)x + gH  k Θkns + nk, ∀k,  (1)  where x = �  k wksk denotes the transmit signal, wk and sk ∈  CN(0, 1) represent the transmit precoder and the information  symbol for user k, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFRT4oBgHgl3EQfuDj6/content/2301.13630v1.pdf'}
+page_content=' ns ∈ CN(0, σ2  sIM) denotes  the dynamic noise at the MF-RIS with each element’s noise  power σ2  s, and nk ∈ CN(0, σ2  k) denotes the additive white  Gaussian noise at user k with power σ2  k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFRT4oBgHgl3EQfuDj6/content/2301.13630v1.pdf'}
+page_content='  By employing SIC, the strong user can mitigate the inter-  ference from weak users to improve the signal-to-interference-  plus-noise ratio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFRT4oBgHgl3EQfuDj6/content/2301.13630v1.pdf'}
+page_content=' Similar to [4]–[6], with the assistance of RIS  to flexibly adjust channel conditions of multiple users, we  assume that users’ indexes are ranked in an increasing order  with respect to their channel gains, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFRT4oBgHgl3EQfuDj6/content/2301.13630v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFRT4oBgHgl3EQfuDj6/content/2301.13630v1.pdf'}
+page_content=',  ∥�h1∥2 ≤ ∥�h2∥2 ≤ · · · ≤ ∥�hK∥2,  (2)  where �hk = hH  k +gH  k ΘkH is the equivalent combined channel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFRT4oBgHgl3EQfuDj6/content/2301.13630v1.pdf'}
+page_content='  For the fixed decoding order, the corresponding achievable  sum rate of user k is given by Rk = log2(1 + γk), where γk  can be obtained by  γk =  |�hkwk|2  �K  i=k+1(|�hkwi|2) + |gH  k Θkns|2 + σ2  k  , ∀k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFRT4oBgHgl3EQfuDj6/content/2301.13630v1.pdf'}
+page_content='  (3)  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFRT4oBgHgl3EQfuDj6/content/2301.13630v1.pdf'}
+page_content=' Problem Formulation  In this letter, we aim to maximize the achievable sum rate  of all users by jointly optimizing the active beamforming at  the BS and the coefficients at the MF-RIS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFRT4oBgHgl3EQfuDj6/content/2301.13630v1.pdf'}
+page_content=' Under the transmit  and amplification power constraints, and the quality-of-service  (QoS) requirement of users, the considered optimization prob-  lem can be formulated as  max  wk,Θk  �K  k=1 Rk  (4a)  s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFRT4oBgHgl3EQfuDj6/content/2301.13630v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFRT4oBgHgl3EQfuDj6/content/2301.13630v1.pdf'}
+page_content='  �K  k=1 ∥wk∥2 ≤ Pmax,  (4b)  �K  k=1(∥ΘkHwk∥2+∥ΘkIM∥2  F σ2  s)≤Po,  (4c)  βr  m + βt  m ≤ βmax, 0 ≤ βp  m ≤ βmax, ∀m, ∀p, (4d)  Rk ≥ Rmin  k  , θp  m ∈ [0, 2π), (2), ∀k, ∀m, ∀p, (4e)  where Pmax and Po denote the maximum transmit and am-  plification power at the BS and MF-RIS, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFRT4oBgHgl3EQfuDj6/content/2301.13630v1.pdf'}
+page_content=' Rmin  k  represents the minimum rate requirement of user k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFRT4oBgHgl3EQfuDj6/content/2301.13630v1.pdf'}
+page_content=' Specifi-  cally, the constraints for transmit power, amplification power,  the QoS requirements and the decoding order are given in  (4b)-(4e), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFRT4oBgHgl3EQfuDj6/content/2301.13630v1.pdf'}
+page_content=' It can be observed that the formulated  problem (4) is intractable due to the non-convex objective  function and constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFRT4oBgHgl3EQfuDj6/content/2301.13630v1.pdf'}
+page_content=' Besides, the active beamforming and  MF-RIS coefficients are highly coupled, making it difficult  to be solved directly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFRT4oBgHgl3EQfuDj6/content/2301.13630v1.pdf'}
+page_content=' Thus, we aim to transform problem  (4) into some tractable convex subproblems and solve them  3  separately and alternatively over iterations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFRT4oBgHgl3EQfuDj6/content/2301.13630v1.pdf'}
+page_content=' In the next section,  we adopt alternating optimization method to obtain the active  beamforming and the MF-RIS coefficients efficiently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFRT4oBgHgl3EQfuDj6/content/2301.13630v1.pdf'}
+page_content='  III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFRT4oBgHgl3EQfuDj6/content/2301.13630v1.pdf'}
+page_content=' PROPOSED SOLUTION  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFRT4oBgHgl3EQfuDj6/content/2301.13630v1.pdf'}
+page_content=' Active Beamforming Design  Given the MF-RIS coefficients, the active beamforming  optimization problem is still non-convex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFRT4oBgHgl3EQfuDj6/content/2301.13630v1.pdf'}
+page_content=' To solve it, we first  introduce an auxiliary variable set {Ak, Bk|k ∈ K}, where Ak  and Bk are defined as  Ak  −1 = |�hkwk|2,  (5)  Bk =  �K  i=k+1(|�hkwi|2) + |gH  k Θkns|2 + σ2  k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFRT4oBgHgl3EQfuDj6/content/2301.13630v1.pdf'}
+page_content='  (6)  Thus, the achievable data rate can be rewritten as Rk =  log2  �  1 + (AkBk)−1�  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFRT4oBgHgl3EQfuDj6/content/2301.13630v1.pdf'}
+page_content='  Then, the active beamforming optimization problem in (4)  can be equivalently expressed as  max  wk,Ak,Bk,Rk  �K  k=1 Rk  (7a)  s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFRT4oBgHgl3EQfuDj6/content/2301.13630v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFRT4oBgHgl3EQfuDj6/content/2301.13630v1.pdf'}
+page_content='  log2  �  1 + (AkBk)−1�  ≥ Rk, ∀k,  (7b)  Ak  −1 ≤ |�hkwk|2, ∀k,  (7c)  Bk ≥  K  �  i=k+1  (|�hkwi|2)+|gH  k Θkns|2+σ2  k, ∀k,(7d)  Rk ≥ Rmin  k  , (4b), (4c), ∀k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFRT4oBgHgl3EQfuDj6/content/2301.13630v1.pdf'}
+page_content='  (7e)  We further define �Hk = �hH  k �hk, Dk = (HHΘk)(HHΘk)H  and Wk = wkwH  k , where Wk ⪰ 0, and rank(Wk) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFRT4oBgHgl3EQfuDj6/content/2301.13630v1.pdf'}
+page_content='  Then, we have  |�hkwk|2 = Tr( �HkWk), ∥ΘkHwk∥2 = Tr(WkDk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFRT4oBgHgl3EQfuDj6/content/2301.13630v1.pdf'}
+page_content='  (8)  Therefore, problem (7) can be reformulated as  max  Wk,Ak,Bk,Rk  �K  k=1 Rk  (9a)  s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFRT4oBgHgl3EQfuDj6/content/2301.13630v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFRT4oBgHgl3EQfuDj6/content/2301.13630v1.pdf'}
+page_content='  Ak  −1 ≤ Tr( �HkWk), ∀k,  (9b)  Bk≥  K  �  i=k+1  Tr( �HkWi)+|gH  k Θkns|2+σ2  k, ∀k,(9c)  �K  k=1Tr(Wk) ≤ Pmax,  (9d)  �K  k=1  �  Tr(WkDk)+∥ΘkIM∥2σ2  s  �  ≤Po,  (9e)  rank(Wk) = 1, ∀k,  (9f)  Wk ⪰ 0, Rk ≥ Rmin  k  , (7b), ∀k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFRT4oBgHgl3EQfuDj6/content/2301.13630v1.pdf'}
+page_content='  (9g)  In order to deal with the non-convex constraint (7b), we  adopt the first-order Taylor expansion, and then we obtain the  lower bound as follows:  log2(1+  1  AkBk  )≥log2(1+  1  A(τ1)  k  B(τ1)  k  )− log2 e(Ak−A(τ1)  k  )  A(τ1)  k  (1+A(τ1)  k  B(τ1)  k  )  −  log2 e(Bk−B(τ1)  k  )  B(τ1)  k  (1+A(τ1)  k  B(τ1)  k  )  ∆= Rk,  (10)  where A(τ1)  k  and B(τ1)  k  are feasible points of Ak and Bk in  the τ1-th iteration, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFRT4oBgHgl3EQfuDj6/content/2301.13630v1.pdf'}
+page_content='  For the non-convex rank-one constraint in (9f), we assume  to transform it to a penalty term in the objective function,  which can be solved by SCA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFRT4oBgHgl3EQfuDj6/content/2301.13630v1.pdf'}
+page_content=' Thus, we firstly introduce an  equivalent equality:  ∥Wk∥∗ − ∥Wk∥2 = 0, ∀k,  (11)  where ∥Wk∥∗ = �  i εi(Wk) and ∥Wk∥2 = ε1(Wk) denote  the nuclear norm and the spectral norm of Wk, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFRT4oBgHgl3EQfuDj6/content/2301.13630v1.pdf'}
+page_content='  εi(Wk) is the i-th largest singular value of matrix Wk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFRT4oBgHgl3EQfuDj6/content/2301.13630v1.pdf'}
+page_content=' Thus,  when the matrix Wk is rank-one, equality (11) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFRT4oBgHgl3EQfuDj6/content/2301.13630v1.pdf'}
+page_content='  Next, we employ the penalty method to solve problem  (9) by adding (11) to the objective function (9a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFRT4oBgHgl3EQfuDj6/content/2301.13630v1.pdf'}
+page_content=' Since the  penalty term (11) makes the objective function not convex,  we apply the first-order Taylor expansion to obtain a convex  upper bound of (11) as follows:  ∥Wk∥∗ − ∥Wk∥2 ≤ ∥Wk∥∗ − ∥Wk∥2,  (12)  where ∥Wk∥2  =  ∥W(τ1)  k  ∥2 + Tr  �  e(τ1)  k  (e(τ1)  k  )H(Wk −  W(τ1)  k  )  �  , and e(τ1)  k  is the eigenvector corresponding to the  largest eigenvalue of W(τ1)  k  in the τ1-th iteration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFRT4oBgHgl3EQfuDj6/content/2301.13630v1.pdf'}
+page_content='  By introducing (12) to the objective function (9a), we obtain  the following problem:  max  Wk,Ak,Bk,Rk  �K  k=1Rk− 1  η  �  k(∥Wk∥∗−∥Wk∥2)  (13a)  s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFRT4oBgHgl3EQfuDj6/content/2301.13630v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFRT4oBgHgl3EQfuDj6/content/2301.13630v1.pdf'}
+page_content='  Rk ≥ Rk, Wk ⪰ 0, Rk ≥ Rmin  k  , ∀k, (13b)  (9b) − (9e),  (13c)  where η > 0 is the penalty factor penalizing (13a) if Wk is  not rank-one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFRT4oBgHgl3EQfuDj6/content/2301.13630v1.pdf'}
+page_content=' It can be verified that, when η → 0, the solution  {Wk} of problem (13) always satisfies equality (11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFRT4oBgHgl3EQfuDj6/content/2301.13630v1.pdf'}
+page_content='  The reformulated problem (13) is a standard convex semi-  definite programming (SDP), which can be efficiently solved  via CVX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFRT4oBgHgl3EQfuDj6/content/2301.13630v1.pdf'}
+page_content=' To obtain a high quality solution, we first initialize  a large η to find a feasible starting point, and then gradually  decrease η with η = µη, µ < 1 to a sufficiently small value to  obtain an overall suboptimal solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFRT4oBgHgl3EQfuDj6/content/2301.13630v1.pdf'}
+page_content=' The process terminates  when the penalty term satisfies the following criterion:  max{∥Wk∥∗ − ∥Wk∥2, ∀k} ≤ ϵ1,  (14)  where ϵ1 denotes a predefined maximum violation of (11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFRT4oBgHgl3EQfuDj6/content/2301.13630v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFRT4oBgHgl3EQfuDj6/content/2301.13630v1.pdf'}
+page_content=' MF-RIS Coefficient Design  For the coefficient design at the MF-RIS, we define  vk  = [ur;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFRT4oBgHgl3EQfuDj6/content/2301.13630v1.pdf'}
+page_content=' 1] if user k is located at the space r;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFRT4oBgHgl3EQfuDj6/content/2301.13630v1.pdf'}
+page_content=' oth-  erwise vk  =  [ut;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFRT4oBgHgl3EQfuDj6/content/2301.13630v1.pdf'}
+page_content=' 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFRT4oBgHgl3EQfuDj6/content/2301.13630v1.pdf'}
+page_content=' Then, we define Vk  =  vkvH  k ,  with  Vk  ⪰  0  and  rank(Vk)  =  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFRT4oBgHgl3EQfuDj6/content/2301.13630v1.pdf'}
+page_content='  Let  gk  =  [gk,1, gk,2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFRT4oBgHgl3EQfuDj6/content/2301.13630v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFRT4oBgHgl3EQfuDj6/content/2301.13630v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFRT4oBgHgl3EQfuDj6/content/2301.13630v1.pdf'}
+page_content=' , gk,M]H and Gk  =  Hwk, then we have  Qk  =  diag  ��  |gk,1|2, |gk,2|2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFRT4oBgHgl3EQfuDj6/content/2301.13630v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFRT4oBgHgl3EQfuDj6/content/2301.13630v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFRT4oBgHgl3EQfuDj6/content/2301.13630v1.pdf'}
+page_content=' , |gk,M|2��  and  �Gk  =  diag  ��  |Gk,1|2, |Gk,2|2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFRT4oBgHgl3EQfuDj6/content/2301.13630v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFRT4oBgHgl3EQfuDj6/content/2301.13630v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFRT4oBgHgl3EQfuDj6/content/2301.13630v1.pdf'}
+page_content=' , |Gk,M|2��  + σ2  sIM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFRT4oBgHgl3EQfuDj6/content/2301.13630v1.pdf'}
+page_content=' Given  Qk =  � Qk  0  0  0  �  , Gk =  � �Gk  0  0  0  �  ,  (15)  we can obtain  ∥ΘkHwk∥2 + ∥ΘkIM∥2  F σ2  s = Tr(VkGk),  (16)  ∥gH  k Θk∥2 = Tr(VkQk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFRT4oBgHgl3EQfuDj6/content/2301.13630v1.pdf'}
+page_content='  (17)  Thus, constraint (4c) can be replaced by (16).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFRT4oBgHgl3EQfuDj6/content/2301.13630v1.pdf'}
+page_content='  In order to handle the non-convex constraints (2) and (5), we  define fk = diag(gH  k )Gk, Rk = diag(gH  k )H, ˜hk = ∥hH  k ∥2,  and dk = wH  k hk, then we have  Fk =  � fkf H  k  fkd∗  k  dkf H  k  |dk|2  �  , Rk =  � RkRH  k  Rkhk  hH  k RH  k  ˜hk  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFRT4oBgHgl3EQfuDj6/content/2301.13630v1.pdf'}
+page_content='  (18)  4  According to the above transformation, we can obtain  |�hkwk|2 = |(hH  k + gH  k ΘkH)wk|2 = Tr(VkFk),(19)  ∥�hk∥2 = Tr(VkRk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFRT4oBgHgl3EQfuDj6/content/2301.13630v1.pdf'}
+page_content='  (20)  Based on (20), the decoding order in (2) is rewritten as  Tr(V1R1) ≤ Tr(V2R2) ≤ · · · ≤ Tr(VKRK).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFRT4oBgHgl3EQfuDj6/content/2301.13630v1.pdf'}
+page_content=' (21)  Then, given the active beamforming vector, the subproblem  of MF-RIS coefficient design can be given by  max  Vk,Ak,Bk,Rk  �K  k=1 Rk  (22a)  s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFRT4oBgHgl3EQfuDj6/content/2301.13630v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFRT4oBgHgl3EQfuDj6/content/2301.13630v1.pdf'}
+page_content='  Ak  −1 ≤Tr(VkFk), ∀k,  (22b)  Bk≤  K  �  i=k+1  Tr(VkFi)+σ2  sTr(VkQk)+σ2  k, ∀k,(22c)  �K  k=1 Tr(VkGk) ≤ Po,  (22d)  Vk ⪰ 0, Rk ≥ Rmin  k  , ∀k,  (22e)  [Vk]m,m = βk  m, [Vk]M+1,M+1 = 1, ∀k,  (22f)  rank(Vk) = 1, ∀k,  (22g)  θp  m ∈ [0, 2π), (4d), (7b), (21), ∀m, ∀p,  (22h)  where Fi denotes Fk when wk is replaced by wi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFRT4oBgHgl3EQfuDj6/content/2301.13630v1.pdf'}
+page_content='  Similar to (12), we replace the rank-one constraint in (22g)  with the following form:  ∥Vk∥∗ − ∥Vk∥2 ≤ ∥Vk∥∗ − ∥Vk∥2,  (23)  where ∥Vk∥∗ and ∥Vk∥2 denote the nuclear norm and the  spectral norm of matrix Vk, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFRT4oBgHgl3EQfuDj6/content/2301.13630v1.pdf'}
+page_content=' Besides, ∥Vk∥2 =  ∥V(τ2)  k  ∥2 + Tr  �  z(τ2)  k  (z(τ2)  k  )H(Vk − V(τ2)  k  )  �  and z(τ2)  k  is the  eigenvector corresponding to the largest eigenvalue of V(τ2)  k  in the τ2-th iteration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFRT4oBgHgl3EQfuDj6/content/2301.13630v1.pdf'}
+page_content='  By introducing (10) into (7b), problem (22) can be refor-  mulated as  max  Vk,Ak,Bk,Rk  �K  k=1Rk− 1  ξ  �  k(∥Vk∥∗−∥Vk∥2)  (24a)  s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFRT4oBgHgl3EQfuDj6/content/2301.13630v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFRT4oBgHgl3EQfuDj6/content/2301.13630v1.pdf'}
+page_content='  Rk ≥ Rk, θp  m ∈ [0, 2π), ∀k, ∀m, ∀p, (24b)  (4d), (21), (22b)−(22f),  (24c)  where ξ > 0 is the penalty factor to ensure Vk is rank-one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFRT4oBgHgl3EQfuDj6/content/2301.13630v1.pdf'}
+page_content='  The problem (24) is a standard SDP problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFRT4oBgHgl3EQfuDj6/content/2301.13630v1.pdf'}
+page_content=' It can be  solved by CVX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFRT4oBgHgl3EQfuDj6/content/2301.13630v1.pdf'}
+page_content=' The termination criterion is given by  max{∥Vk∥∗ − ∥Vk∥2, ∀k} ≤ ϵ2,  (25)  where ϵ2 denotes a predefined maximum violation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFRT4oBgHgl3EQfuDj6/content/2301.13630v1.pdf'}
+page_content='  Based on the above derivation, we propose a penalty-  based iterative algorithm to solve problem (4) efficiently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFRT4oBgHgl3EQfuDj6/content/2301.13630v1.pdf'}
+page_content=' The  details are given in Algorithm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFRT4oBgHgl3EQfuDj6/content/2301.13630v1.pdf'}
+page_content=' Specifically, the initial points  {W(0)  k } and {V(0)  k } are obtained by selecting the feasible  ones from some random points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFRT4oBgHgl3EQfuDj6/content/2301.13630v1.pdf'}
+page_content=' Since both the objectives of  problems (13) and (24) are non-decreasing over iterations and  the system throughout is upper-bounded by a finite value, the  proposed Algorithm 1 is guaranteed to converge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFRT4oBgHgl3EQfuDj6/content/2301.13630v1.pdf'}
+page_content=' Moreover,  if the interior point method is employed, the complexity of  Algorithm 1 is O(IoutIin(KN 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFRT4oBgHgl3EQfuDj6/content/2301.13630v1.pdf'}
+page_content='5 + 2M 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFRT4oBgHgl3EQfuDj6/content/2301.13630v1.pdf'}
+page_content='5)), where K, M  and N are the numbers of users, BS antennas and MF-  RIS elements, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFRT4oBgHgl3EQfuDj6/content/2301.13630v1.pdf'}
+page_content=' The terms Iin and Iout denote  the number of the inner and outer iterations required for  convergence, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFRT4oBgHgl3EQfuDj6/content/2301.13630v1.pdf'}
+page_content='  Algorithm 1 Penalty-Based Iterative Algorithm  1: Initialize {W(0)  k }, {V(0)  k }, the error tolerance ∆, the  maximum number of iteration T0,max, the penalty factors  η and ξ, and the predefined threshold ϵ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFRT4oBgHgl3EQfuDj6/content/2301.13630v1.pdf'}
+page_content='  2: repeat  3:  Set the iteration index τ0 = 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFRT4oBgHgl3EQfuDj6/content/2301.13630v1.pdf'}
+page_content='  4:  repeat  5:  Given V(τ0)  k  , update W(τ0+1)  k  by solving (13);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFRT4oBgHgl3EQfuDj6/content/2301.13630v1.pdf'}
+page_content='  6:  Given W(τ0+1)  k  , update V(τ0+1)  k  by solving (24);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFRT4oBgHgl3EQfuDj6/content/2301.13630v1.pdf'}
+page_content='  7:  Update τ0 = τ0 + 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFRT4oBgHgl3EQfuDj6/content/2301.13630v1.pdf'}
+page_content='  8:  until | R(τ0)  sum −R(τ0−1)  sum  R(τ0−1)  sum  | < ∆ or τ0 > T0,max.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFRT4oBgHgl3EQfuDj6/content/2301.13630v1.pdf'}
+page_content='  9:  Update {W(0)  k , V(0)  k } with {W(τ0)  k  , V(τ0)  k  };' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFRT4oBgHgl3EQfuDj6/content/2301.13630v1.pdf'}
+page_content='  10:  Update η = µη, ξ = µξ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFRT4oBgHgl3EQfuDj6/content/2301.13630v1.pdf'}
+page_content='  11: until the constraints (14) and (25) satisfy ϵ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFRT4oBgHgl3EQfuDj6/content/2301.13630v1.pdf'}
+page_content='  12: Output the converged solutions {W∗  k} and {V∗  k}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFRT4oBgHgl3EQfuDj6/content/2301.13630v1.pdf'}
+page_content='  TABLE I: Simulation Parameters  Parameter  Value  Path loss exponents of BS-MF-RIS,  BS-users, MF-RIS-users links  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFRT4oBgHgl3EQfuDj6/content/2301.13630v1.pdf'}
+page_content='5, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFRT4oBgHgl3EQfuDj6/content/2301.13630v1.pdf'}
+page_content='5, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFRT4oBgHgl3EQfuDj6/content/2301.13630v1.pdf'}
+page_content='8  Rician factors of all links  3 dB  Noise power at MF-RIS and users  −80 dBm  Minimum required QoS for users  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFRT4oBgHgl3EQfuDj6/content/2301.13630v1.pdf'}
+page_content='1 bit/s/Hz  Maximum amplification power [13]  Po = 10 dBm  Maximum amplification factor  βmax = 22 dB  Convergence tolerance  ∆ = 10−6  IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFRT4oBgHgl3EQfuDj6/content/2301.13630v1.pdf'}
+page_content=' SIMULATION RESULTS  In this section, numerical results are provided to validate  the performance of an MF-RIS-aided NOMA network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFRT4oBgHgl3EQfuDj6/content/2301.13630v1.pdf'}
+page_content=' The  BS and the MF-RIS are located at (0, 0, 0) and (0, 50, 20),  respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFRT4oBgHgl3EQfuDj6/content/2301.13630v1.pdf'}
+page_content=' Besides, the users are divided into two parts,  distributed on circles centered on (0, 45, 0) and (0, 55, 0)  with radius r = 3, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFRT4oBgHgl3EQfuDj6/content/2301.13630v1.pdf'}
+page_content=' We adopt Rician fading for  all channels, and set K = 6, N = 16, M = 100, and  Pmax = 20 dBm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFRT4oBgHgl3EQfuDj6/content/2301.13630v1.pdf'}
+page_content=' Other parameters are listed in Table I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFRT4oBgHgl3EQfuDj6/content/2301.13630v1.pdf'}
+page_content=' We  compare the proposed MF-RIS with three existing RISs:  SF-RIS [7]: The SF-RIS only supports signal reflection  or transmission, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFRT4oBgHgl3EQfuDj6/content/2301.13630v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFRT4oBgHgl3EQfuDj6/content/2301.13630v1.pdf'}
+page_content=', βmax = 1 and Θt or Θr = 0M×M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFRT4oBgHgl3EQfuDj6/content/2301.13630v1.pdf'}
+page_content='  Active RIS [13]: The active RIS simultaneously supports  signal reflection and amplification, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFRT4oBgHgl3EQfuDj6/content/2301.13630v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFRT4oBgHgl3EQfuDj6/content/2301.13630v1.pdf'}
+page_content=', Θt = 0M×M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFRT4oBgHgl3EQfuDj6/content/2301.13630v1.pdf'}
+page_content='  STAR-RIS [9]: The STAR-RIS provides full space cov-  erage by splitting signals to two sides, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFRT4oBgHgl3EQfuDj6/content/2301.13630v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFRT4oBgHgl3EQfuDj6/content/2301.13630v1.pdf'}
+page_content=', βmax = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFRT4oBgHgl3EQfuDj6/content/2301.13630v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFRT4oBgHgl3EQfuDj6/content/2301.13630v1.pdf'}
+page_content=' 2(a) depicts the sum rate versus the maximum transmit  power Pmax.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFRT4oBgHgl3EQfuDj6/content/2301.13630v1.pdf'}
+page_content=' It can be observed that the sum rates of all  schemes increase with Pmax.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFRT4oBgHgl3EQfuDj6/content/2301.13630v1.pdf'}
+page_content=' Besides, the proposed MF-RIS  always yields a better performance than other benchmarks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFRT4oBgHgl3EQfuDj6/content/2301.13630v1.pdf'}
+page_content='  Specifically, when Pmax = 10 dBm, the MF-RIS enjoys  a 59% higher sum rate than the SF-RIS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFRT4oBgHgl3EQfuDj6/content/2301.13630v1.pdf'}
+page_content=' This is because  the MF-RIS serves all users in full space through signal  reflection, transmission and amplification functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFRT4oBgHgl3EQfuDj6/content/2301.13630v1.pdf'}
+page_content=' Besides,  by providing additional energy to amplify the incident signal,  the MF-RIS is able to efficiently mitigate the double-fading at-  5  5  10  15  20  25  30  2  4  6  8  10  12  14  16  Sum Rate (bps/Hz)  Maximum transmit power,          (dBm)  MF-RIS  Active RIS  STAR-RIS  SF-RIS  Without RIS  59%  16%  44%  (a) Sum rate vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFRT4oBgHgl3EQfuDj6/content/2301.13630v1.pdf'}
+page_content=' the power budget.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFRT4oBgHgl3EQfuDj6/content/2301.13630v1.pdf'}
+page_content='  10  20  30  40  50  60  70  80  90  100  7  8  9  10  11  12  13  Sum Rate (bps/Hz)  Number of elements,  MF-RIS  Active RIS  STAR-RIS  SF-RIS  Without RIS  12%  6%  (b) Sum rate vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFRT4oBgHgl3EQfuDj6/content/2301.13630v1.pdf'}
+page_content=' the number of elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFRT4oBgHgl3EQfuDj6/content/2301.13630v1.pdf'}
+page_content='  0  10  20  30  40  50  7  8  9  10  11  12  13  14  Sum Rate (bps/Hz)  Y-coordinate of RIS  MF-RIS  Active RIS  STAR-RIS  SF-RIS  Without RIS  39%  28%  (c) Sum rate vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFRT4oBgHgl3EQfuDj6/content/2301.13630v1.pdf'}
+page_content=' the Y -coordinate of RIS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFRT4oBgHgl3EQfuDj6/content/2301.13630v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFRT4oBgHgl3EQfuDj6/content/2301.13630v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFRT4oBgHgl3EQfuDj6/content/2301.13630v1.pdf'}
+page_content=' Simulation results for the sum rate versus different transmit power, number of elements and RIS locations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFRT4oBgHgl3EQfuDj6/content/2301.13630v1.pdf'}
+page_content='  tenuation, which helps to improve the channel gain of cascaded  links.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFRT4oBgHgl3EQfuDj6/content/2301.13630v1.pdf'}
+page_content=' Furthermore, due to the limitations faced by the active  RIS and STAR-RIS counterparts (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFRT4oBgHgl3EQfuDj6/content/2301.13630v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFRT4oBgHgl3EQfuDj6/content/2301.13630v1.pdf'}
+page_content=', half-space coverage  and double-fading attenuation), the MF-RIS improves the rate  performance by 16% and 44% when Pmax = 10 dBm,  respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFRT4oBgHgl3EQfuDj6/content/2301.13630v1.pdf'}
+page_content=' Additionally, it is evident that all RIS-aided  schemes achieve significant gains than the scheme without  RIS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFRT4oBgHgl3EQfuDj6/content/2301.13630v1.pdf'}
+page_content=' This demonstrates the superiority of using RIS to improve  the performance of wireless networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFRT4oBgHgl3EQfuDj6/content/2301.13630v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFRT4oBgHgl3EQfuDj6/content/2301.13630v1.pdf'}
+page_content=' 2(b) shows that the sum rate of all RIS-aided schemes  increase with M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFRT4oBgHgl3EQfuDj6/content/2301.13630v1.pdf'}
+page_content=' This is because a larger M enables a higher  beamforming gain, thus improving the system performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFRT4oBgHgl3EQfuDj6/content/2301.13630v1.pdf'}
+page_content=' In  addition, with more degree of freedoms to manipulate signal  propagation, the STAR-RIS is capable of enjoying a 6% higher  sum rate than the SF-RIS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFRT4oBgHgl3EQfuDj6/content/2301.13630v1.pdf'}
+page_content=' Moreover, although only the users  located in the reflection space are served by the active RIS,  it outperforms the STAR-RIS with a 12% higher sum rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFRT4oBgHgl3EQfuDj6/content/2301.13630v1.pdf'}
+page_content='  This is because the performance gain obtained from the signal  amplification of active RIS is greater than that from full-  space coverage of STAR-RIS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFRT4oBgHgl3EQfuDj6/content/2301.13630v1.pdf'}
+page_content=' This also implies that the signal  amplification function plays an important role in improving the  performance of RIS-aided networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFRT4oBgHgl3EQfuDj6/content/2301.13630v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFRT4oBgHgl3EQfuDj6/content/2301.13630v1.pdf'}
+page_content=' 2(c) illustrates the sum rate versus the Y -coordinate of  RIS (from 0 to 50), where the RIS moves from the BS side  to the user side.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFRT4oBgHgl3EQfuDj6/content/2301.13630v1.pdf'}
+page_content=' We can observe that the sum rates of the  STAR-RIS and the SF-RIS first decrease and then increase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFRT4oBgHgl3EQfuDj6/content/2301.13630v1.pdf'}
+page_content='  The reason behind this is that the channel gain decreases with  the link distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFRT4oBgHgl3EQfuDj6/content/2301.13630v1.pdf'}
+page_content=' Specifically, when the STAR-RIS and the SF-  RIS are located close to the middle point, the received signals  at users are attenuated the most, resulting in the lowest sum  rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFRT4oBgHgl3EQfuDj6/content/2301.13630v1.pdf'}
+page_content=' In contrast, owing to the signal amplification function,  the MF-RIS and the active RIS are less affected by the  double-fading attenuation, which achieve 39% and 28% gains  in the middle point compared to the SF-RIS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFRT4oBgHgl3EQfuDj6/content/2301.13630v1.pdf'}
+page_content=' Moreover, the  corresponding sum rate maintains a continuous upward trend  even when the MF-RIS and active RIS are far away from the  BS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFRT4oBgHgl3EQfuDj6/content/2301.13630v1.pdf'}
+page_content=' This is because as the RIS comes closer to the users, the  power of the incident signal at the RIS is weaker.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFRT4oBgHgl3EQfuDj6/content/2301.13630v1.pdf'}
+page_content=' Thus, under  a fixed amplification power budget, the MF-RIS can provide  more amplification gain when deployed closer to users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFRT4oBgHgl3EQfuDj6/content/2301.13630v1.pdf'}
+page_content=' This  compensates for the attenuation caused by the double-fading  issue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFRT4oBgHgl3EQfuDj6/content/2301.13630v1.pdf'}
+page_content=' This observation also reveals that the MF-RIS should  be deployed close to the users for better performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFRT4oBgHgl3EQfuDj6/content/2301.13630v1.pdf'}
+page_content='  V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFRT4oBgHgl3EQfuDj6/content/2301.13630v1.pdf'}
+page_content=' CONCLUSION  In this letter, we proposed a novel MF-RIS architecture to  alleviate the double-fading attenuation via transmitting and  reflecting the incident signal with power amplification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFRT4oBgHgl3EQfuDj6/content/2301.13630v1.pdf'}
+page_content=' Then,  we investigated the resource allocation problem in a downlink  multiuser MF-RIS-aided NOMA network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFRT4oBgHgl3EQfuDj6/content/2301.13630v1.pdf'}
+page_content=' Specifically, the  active beamforming and MF-RIS coefficients were jointly  optimized to maximize the achievable sum rate by leveraging  SCA and penalty-based method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFRT4oBgHgl3EQfuDj6/content/2301.13630v1.pdf'}
+page_content=' Numerical results validated  the effectiveness of the proposed MF-RIS and the superiority  of MF-RIS over traditional RISs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFRT4oBgHgl3EQfuDj6/content/2301.13630v1.pdf'}
+page_content=' In the future, we are inter-  ested in studying the coupled phase and hardware impairment  problems of the MF-RIS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFRT4oBgHgl3EQfuDj6/content/2301.13630v1.pdf'}
+page_content=' In addition, the robust beamforming  under imperfect CSI cases deserves exploration as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFRT4oBgHgl3EQfuDj6/content/2301.13630v1.pdf'}
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+Probabilistic Neural Data Fusion for Learning from an Arbitrary
+Number of Multi-fidelity Data Sets
+Carlos Mora†1, Jonathan Tammer Eweis-Labolle†1, Tyler Johnson1, Likith Gadde2, and Ramin
+Bostanabad∗1
+1Department of Mechanical and Aerospace Engineering, University of California, Irvine
+2Northwood High School, Irvine
+Abstract
+In many applications in engineering and sciences analysts have simultaneous access to multiple data
+sources. In such cases, the overall cost of acquiring information can be reduced via data fusion or multi-
+fidelity (MF) modeling where one leverages inexpensive low-fidelity (LF) sources to reduce the reliance
+on expensive high-fidelity (HF) data. In this paper, we employ neural networks (NNs) for data fusion in
+scenarios where data is very scarce and obtained from an arbitrary number of sources with varying levels
+of fidelity and cost. We introduce a unique NN architecture that converts MF modeling into a nonlinear
+manifold learning problem. Our NN architecture inversely learns non-trivial (e.g., non-additive and non-
+hierarchical) biases of the LF sources in an interpretable and visualizable manifold where each data source is
+encoded via a low-dimensional distribution. This probabilistic manifold quantifies model form uncertainties
+such that LF sources with small bias are encoded close to the HF source. Additionally, we endow the output
+of our NN with a parametric distribution not only to quantify aleatoric uncertainties, but also to reformulate
+the network’s loss function based on strictly proper scoring rules which improve robustness and accuracy
+on unseen HF data. Through a set of analytic and engineering examples, we demonstrate that our approach
+provides a high predictive power while quantifying various sources uncertainties. Our codes and examples
+can be accessed via GitLab.
+Keywords: Multi-fidelity Modeling; Uncertainty Quantification; Bayesian Neural Networks; Inverse Prob-
+lems; Manifold Learning; Data Fusion.
+1
+Introduction
+In an increasing number of applications in engineering and sciences analysts have simultaneous access to
+multiple sources of information. For instance, materials’ properties can be estimated via multiple techniques
+such as (in decreasing order of cost and accuracy/fidelity) experiments, direct numerical simulations (DNS),
+a host of physics-based reduced order models (ROMs), or analytical methods [1–3]. In such applications,
+the overall cost of gathering information about the system of interest can be reduced via multi-fidelity (MF)
+modeling or data fusion where one leverages inexpensive low-fidelity (LF) sources to reduce the reliance
+on expensive high-fidelity (HF) data sources. In this paper, we employ neural networks (NNs) for MF
+modeling in scenarios where data is scarce and obtained from multiple sources with varying levels of fidelity
+and cost (i.e., data is unbalanced since more samples are available from cheaper sources). In particular,
+†Equal Contribution.
+∗Corresponding Author: [email protected]
+GitLab repository: https://gitlab.com/TammerUCI/pro-ndf
+1
+arXiv:2301.13271v1  [cs.LG]  30 Jan 2023
+
+our contributions are as follows (1) we introduce a unique NN architecture that not only facilitates data
+fusion, but also quantifies and visualizes the discrepancies/similarities between all data sources, and (2) we
+illustrate that a Bayesian treatment, besides alleviating overfitting and providing a probabilistic surrogate
+(i.e., an emulator), provides the means to develop a novel loss function (based on proper scoring rules) that
+improves the performance and robustness of the resulting MF NN emulator.
+Over the past few decades, many techniques have been developed for building MF surrogates which
+are used in outer-loop applications such as design optimization [4, 5], calibration of computer models [6],
+or Bayesian optimization [7]. The main motivation behind these techniques is to leverage the correlations
+between LF and HF data sources (and the fact that sampling from the former is typically cheaper) to improve
+the predictive performance of the surrogate while reducing the overall data acquisition costs. Early works
+in this field focused primarily on hierarchically linking bi-fidelity data. For instance, in space mapping [8–
+10] or multi-level [11–13] techniques the inputs of the LF data are mapped following formulations such as
+xl = F(xh) where xl and xh are the inputs of LF and HF sources, respectively. In this equation, F(·) is a
+transformation function whose predefined functional form is calibrated such that yl(F(xh)) approximates
+yh(xh) as closely as possible. These techniques are useful in applications where higher fidelity data are
+obtained by successively refining the discretization in simulations [11, 12], e.g., by refining the mesh when
+modeling the flow around an airfoil or estimating the fracture toughness of a microstructure. The main
+disadvantages of space mapping techniques are that (1) they rely on iterative and time-consuming analysis
+for choosing a near-optimal functional form for F(·), (2) they cannot jointly fuse more than two data sources
+at a time, (3) they quantify similarity/discrepancy between the sources based on pre-defined functions whose
+space may not include the true discrepancy, and (4) they do not quantify some uncertainty sources (such as
+lack of data) and are rarely formulated within a Bayesian setting that leverages prior information.
+A well-known hierarchical bi-fidelity modeling framework is that of Kennedy and O’Hagan (KOH) [14]
+who assume that the discrepancy between the LF and HF sources is additive (multiplicative terms have also
+been explored [15]) and that both sources as well as the discrepancy between them can be modeled via
+Gaussian processes (GPs). Upon this modeling assumption, KOH find the joint posterior of GPs’ hyperpa-
+rameters via either fully [16, 17] or modular Bayesian inference [18–21]. While KOH’s approach considers
+multiple uncertainties and has been successfully applied to a broad range of applications [22–24], it has three
+main limitations: (1) it only accommodates two data sources at a time, (2) it places a priori independence
+assumption between the GPs, and (3) it does not provide a low-dimensional, visualizable, and interpretable
+metric that quantifies the correlations between the data sources.
+Recent works have acknowledged the limitations of hierarchical methods and devised new methodologies
+to address them. For instance, MF modeling can be achieved via a recursive scheme [25] where a bi-fidelity
+method is repeatedly applied from the lowest to the highest fidelities. However, such recursive schemes
+inherit the limitations of bi-fidelity methods, cannot jointly fuse multi-source data sets, and are sensitive to
+the ordering (i.e., the relative accuracy of all sources must be known a priori).
+As another example, [26] presents MF networks (MFNets): an approach based on directed acyclic graphs
+that builds a MF surrogate using an arbitrary number of data sources. MFNets accommodate noisy data
+and are trained via gradient-based minimization of a nonlinear least squares objective. While MFNets can
+learn non-hierarchical relations between data sources, they: (1) rely on having prior knowledge on a set of
+latent variables that explain the relations between the sources, (2) assume each source can be surrogated via
+a linear subspace model, (3) are not probabilistic and also require regularization, (4) impose independence
+assumption among the data sources to derive the likelihood (i.e., the objective) function, and (5) rely on
+iterative approaches for finding the optimal graph structure.
+Other notable works that have studied the limitations of hierarchical techniques include [27–29] which are
+focused on identifying (and correcting) non-additive discrepancies between LF and HF sources. However,
+2
+
+the proposed solution in these works is intrusive and relies on some rather strong modeling assumptions that
+largely limit the applications. These limitations arise because the formulation of the discrepancy is learned
+via an embedded operator whose functional form and interaction with the LF source are constructed a priori.
+We have recently developed a GP-based approach [30] that addresses the above issues by converting MF
+modeling into a manifold learning problem where the relations between the sources are automatically quan-
+tified via an appropriately learnt distance measure. The conversion is achieved via latent map Gaussian
+processes [30] (LMGPs, see Section 2.1) which enable GPs to handle categorical variables and, correspond-
+ingly, data fusion: by augmenting the inputs via a categorical variable (which indicates the source of a data
+point) and then concatenating all the data sets, LMGPs can simultaneously learn from an arbitrary number of
+information sources. We have shown [30] that LMGP-based MF modeling consistently outperforms KOH’
+approach and can also handle calibration problems.
+Following the success of LMGPs in data fusion, in this work we examine the potentials of NNs in match-
+ing (and, hopefully, improving) LMGPs’ efficiency in MF modeling. Our current studies are motivated by
+the facts that (1) when viewed as (probabilistic or deterministic) graphical models [31], NNs provide unique
+opportunities to use MF data sets to uncover complex hidden relations between the corresponding sources,
+(2) the recent hardware and software advancements have dramatically accelerated architecture design and
+training of NNs, and (3) NNs scale to higher dimensions and big data significantly better than GPs.
+Over the past few years some NN-based approaches have been developed for MF modeling [26, 32–34].
+However, most of these works design the network architecture primarily based on hierarchical methods and
+consequently inherit their limitations. For instance, [32] builds two sequentially connected deterministic
+networks based on KOH’s method where the first and second NNs are tasked to emulate the LF and HF
+sources, respectively. In addition to sharing the limitations of KOH’s method, such a sequential bi-fidelity
+NN requires the LF and HF training data to be available at the same inputs (unless the two parts of the
+network are trained separately) and also relies on manual tuning of the architecture and loss function. It has
+been argued [33] that such sequentially trained NNs bridge MF modeling with transfer learning where the
+knowledge gained from the LF data is used in building the NN module that surrogates the HF source.
+Non-sequential NNs are rarely used for MF modeling (esp. with > 2 sources) due to the fact that searching
+for the optimum architecture (and effectively training it with small data) is a difficult task. We address this
+challenge by drawing inspiration from LMGPs where we design the architecture such that any number of
+MF data sets can be simultaneously fused and the overall discrepancies between sources are quantified with
+visualizable metrics. We also illustrate that making specific parts of the network probabilistic, in addition
+to being superior to both deterministic and all-probabilistic NNs, enables us to infuse a proper scoring rule
+[35] into the loss function and, in turn, improve the performance of the MF emulator. The particular rule
+that we adopt is interval score which is frequently used in testing the quality of probabilistic predictions but,
+to the best of our knowledge, has never been used in the training stage of a probabilistic NN. In summary,
+our major contributions are as follows:
+• We introduce a unique NN architecture for MF modeling that can fuse an arbitrary number of data
+sets and quantify both epistemic and aleatoric uncertainties.
+• We inversely learn the accuracy of the LF sources (with respect to the HF source) and visualize the
+learned relations in an interpretable manifold.
+• We show that a probabilistic setting allows us to develop a novel loss function (based on proper scoring
+rules) that improves the performance of the emulator.
+• We validate the performance of our approach on analytical and real-world examples and show that it
+3
+
+performs on par with the state of the art while providing improved scalability to high dimensions and
+big data.
+The rest of this paper is organized as follows. We review the relevant technical background in Section 2
+and then introduce our approach in Section 3. We test the performance of our approach on a host of analytical
+problems and real-world data sets in Section 4 and conclude the paper in Section 5.
+2
+Technical Preliminaries
+In this section we first review LMGPs which are extensions of GPs that handle categorical inputs and,
+thus, can readily fuse any number of data sets. Then, we provide some background on Bayesian neural
+networks (BNNs) which form the foundation of our neural data fusion framework.
+2.1
+Latent Map Gaussian Processes (LMGPs)
+Let us denote the output and inputs in the training data by y ∈ Y ≡ R and x = [x1, x2, . . . , xdx]T ∈ Rdx,
+respectively, with an individual training point i = 1, . . . , n denoted by the pair (y(i), x(i)). Assume the
+training data is a realization from a constant-mean1 GP and that the following relation holds:
+y(x) = m + ξ(x)
+(1)
+where m is the unknown constant mean and ξ(x) is a zero-mean GP whose covariance function or kernel
+is:
+cov(ξ(x), ξ(x
+′)) = c(x, x
+′) = s2r(x, x
+′)
+(2)
+where s2 is the variance of the process and r(·, ·) is a parametric correlation function such as the Gaussian:
+r(x, x
+′) = exp
+�
+−
+dx
+�
+i=1
+10ωi(xi − x
+′
+i)2
+�
+= exp
+�
+−
+�
+x − x
+′�T
+10Ω �
+x − x
+′��
+(3)
+where ω = [ω1, . . . , ωdx]T are the roughness or scale parameters and Ω = diag(ω).
+The training process and prediction formulas for a GP depend on the choice of the correlation function,
+which relies on a weighted Cartesian distance metric between any two inputs, see Equation (3). As we
+recently motivated in [36], to directly use GPs for mixed-variable modeling we reformulate r(·, ·) as detailed
+below such that it can handle categorical (qualitative) inputs.
+Let us denote the categorical inputs by t = [t1, . . . , tdt]T where the total number of distinct levels for
+qualitative variable ti is τi. To handle mixed inputs, LMGP learns a parametric function that maps cate-
+gorical variables to some points in a quantitative manifold or latent space2. These points (and hence the
+mapping function) can be incorporated into any standard correlation function, such as the Gaussian, which
+is reformulated as follows for mixed inputs:
+r
+�
+(x, t), (x
+′, t
+′)
+�
+= exp
+�
+−
+���z(t) − z(t
+′)
+���
+2
+2 −
+�
+x − x
+′�T
+10Ω �
+x − x
+′��
+(4)
+or, equivalently,
+r
+�
+(x, t), (x
+′, t
+′)
+�
+= exp
+�
+−
+dx
+�
+i=1
+10ωi(xi − x
+′
+i)2
+�
+× exp
+�
+−
+dz
+�
+i=1
+(zi(t) − zi(t
+′))2
+�
+(5)
+1GPs (and LMGPs) can also be formulated by using a linear combination of basis functions in place of the constant mean. This
+formulation relies on prior knowledge of the functional form of the output and can improve performance in extrapolation, see [30].
+2Multiple mapping functions can also be used to build multiple manifolds. We leverage this in Section 4 where we build two
+manifolds for data fusion problems with categorical or mixed inputs.
+4
+
+where ∥·∥2 denotes the Euclidean 2-norm and z(t) = [z1(t), . . . , zdz(t)]1×dz is the to-be-learned latent
+space point corresponding to the particular combination of categorical variables denoted by t. To find these
+points in the latent space, LMGP assigns a unique vector (i.e., a prior representation) to each combination of
+categorical variables. Then, it uses matrix multiplication3 to map each of these vectors to a point in a latent
+space of dimension dz:
+z(t) = ζ(t)A
+(6)
+where ζ(t) is the 1 × �dt
+i=1 τi unique prior vector representation of t and A is a �dt
+i=1 τi × dz matrix that
+maps ζ(t) to z(t). In this paper, we use dz = 2 since it simplifies visualization and has been shown to
+provide sufficient flexibility for learning the latent relations [36]. We construct ζ via a form of one-hot
+encoding where we first construct the 1 × τi vector vi =
+�
+vi
+1, vi
+2, . . . , vi
+τi
+�
+for each categorical variable ti
+such that vi
+j = 1 when ti is at level k = j and vi
+j = 0 when ti is at level k ̸= j for k ∈ 1, 2, . . . , τi. Then,
+we set ζ(t) = [v1, v2, . . . , vdt]. For example, for the two categorical variables t1 and t2 with 2 and 3 levels,
+ζ(t) = [0, 1, 0, 1, 0] encodes the combination where both variables are at level 2.
+To train an LMGP, we use maximum likelihood estimation (MLE) to jointly estimate all of its parameters:
+�
+ˆm, ˆs2, ˆω, ˆA
+�
+= argmax
+m,s2,ω,A
+��2πs2R
+��− 1
+2 × exp
+�
+−1
+2(y − 1m)T (s2R)−1(y − 1m)
+�
+(7)
+where |·| denotes the determinant operator, y = [y1, . . . , yn]T is the n × 1 vector of outputs in the training
+data, R is the n × n correlation matrix with the (i, j)th element Rij = r
+�
+(x(i), t(i)), (x(j), t(j))
+�
+for i, j =
+1, . . . , n, and 1 is a n × 1 vector of ones.
+After estimating the hyperparameters, we use the conditional distribution formulas to predict the response
+distribution at the arbitrary point p∗ = (x∗, t∗). The mean and variance of this normal distribution are:
+E [y (p∗)] = ˆm + rT (p∗) R−1 (y − 1 ˆm)
+(8)
+cov
+�
+y(p∗), y(p
+′)
+�
+= ˆs2r(p∗, p
+′) = ˆs2 �
+1 − rT (p∗)R−1r(p
+′) + g(p∗)(1T R−11)−1g(p
+′)
+�
+(9)
+where E denotes expectation, r (p∗) is an (n × 1) vector with the ith element r
+�
+p(i), p∗�
+, and g (p∗) =
+1 − 1T R−1r (p∗).
+To perform data fusion via LMGP, we re-frame multi-fidelity modeling as a manifold learning problem.
+Assume that we have ds data sources whose inputs and outputs are denoted by xsi, ysi, respectively, with
+i = 1, . . . , ds. We first pre-process the data by appending the inputs with a single categorical variable ts with
+ds levels (hereafter referred to as the source index variable) that distinguishes the data sources. Specifically,
+we add ts at level i for source si, i.e., xsi → [xsi, insi×1], where insi×1 is an nsi × 1 vector of i’s and nsi
+is the number of data points for source si. We then combine the data for all sources into one unified data set
+and fit an LMGP directly it, i.e., we fit LMGP to all of the data from all sources at once.
+The fitted LMGP can provide predictions for any desired data source based on the level used for ts and
+as such is an emulator for all of the data sources. Additionally, since the data sources are distinguished via
+a categorical variable, LMGP learns the correlations between them via a visualizable latent representation
+and uses these correlations to improve its predictions [30]. In the case that the raw inputs contain categorical
+variables tc, we use separate mappings for ts and tc, i.e., we assign unique priors ζ(ts) and ζ(tc) which
+LMGP uses to find mapping matrices As and Ac. The latent points corresponding to each mapping are then
+zs and zc, respectively.
+3More complex transformations based on, e.g., NNs, may also be used, although we do not do so in this paper.
+5
+
+Note that the correlation function in Equation (5) depends directly on the euclidean distance between a
+pair of latent points. This means that relative distances in the latent space directly correspond to correlations,
+e.g., if a pair of data sources ys1 and ys2 have corresponding latent points with a distance ∆ in the latent
+space then this directly implies by Equation (5) that LMGP has found those two sources to have a correlation
+of exp (−∆2).
+2.2
+Bayesian Neural Networks
+Feedforward neural networks (FFNNs) are one of the most common models used in deep learning and
+their main goal is to learn the underlying function f(x) that maps the inputs x to the target y [37]. To this
+end, an FFNN defines the mapping ˆf(x; θ) whose parameters θ are estimated such that ˆy = ˆf(x; θ) best
+approximates f(x). NN-based approaches for MF emulation can provide attractive advantages since they
+are universal function approximators [38] and can handle high-dimensional inputs and large data sets. In
+this subsection, we first describe the working principle of FFNNs and motivate the use of BNNs and Bayes
+by backprop [39].
+FFNNs propagate information from the inputs x to the output y through intermediate computations that
+define ˆf. They are traditionally built via a succession of L layers where L − 2 hidden layers are placed
+between the input and output layers. The output of layer k is denoted by zk and is obtained as follows:
+z1 = x,
+(10)
+zk = φk (W kzk−1 + bk)
+∀ k ∈ [2, L − 1],
+(11)
+ˆy = φL (W LzL−1 + bL)
+(12)
+where φ is the (typically non-linear) activation function. The parameters θk = (W k, bk), where W k and bk
+are the weight matrices and bias vectors, respectively, correspond to the connections between the (k − 1)th
+and kth layer. For brevity, we denote the parameters of the entire network by θ.
+From a statistical perspective, an FFNN aims to learn the conditional distribution P(y|x; θ) given the
+noisy data set D with independent and identically distributed samples:
+y(i) = f(x(i)) + ϵ ≈ ˆf(x(i); θ) + ϵ
+(13)
+where ϵ ∼ N(0, σ2) represents noise. Equation (13) indicates that P(y(i)|x(i)) ∼ N(f(x(i)), σ2) and
+hence the conditional probability P (D|θ) can be written as:
+P(D|θ) =
+n
+�
+i=1
+P(y(i)|x(i), θ)P(x(i)) =
+n
+�
+i=1
+N(y(i); ˆy(i), σ2)P(x(i))
+=
+n
+�
+i=1
+1
+σ
+√
+2π exp
+�
+− 1
+2σ2
+�
+y(i) − ˆy(i)�2�
+P(x(i))
+(14)
+Since the likelihood function L(θ) ≡ P(D|θ), the parameters θ can be estimated by maximizing L(θ) (the
+dependence on x is dropped for brevity):
+θMLE = arg max
+θ
+L(θ) ≡ arg min
+θ
+− log P(D|θ) = arg min
+θ
+1
+n
+n
+�
+i=1
+�
+y(i) − ˆf(x(i); θ)
+�2
+(15)
+which is equivalent to minimizing the mean squared error (MSE) of the predictions ˆy = ˆf(x; θ) with
+respect to the targets y. Equation (15) can be updated via Bayes rule to consider prior knowledge on θ in
+the optimization. These maximum a posteriori (MAP) estimates are obtained via:
+θMAP = arg max
+θ
+P (θ|D) = arg max
+θ
+log P (θ|D) = arg max
+θ
+log P (D|θ) + log P(θ)
+(16)
+6
+
+where the first term recovers MSE as in Equation (15) and the second term depends on the prior distribution
+assigned to the parameters. Equation (16) illustrates that Gaussian and Laplacian priors are equivalent to L2
+and L1 regularization, respectively [37, 39].
+FFNNs are likely to overfit in scenarios where data is scarce. Additionally, they cannot directly quantify
+prediction uncertainty and are often overconfident in extrapolation [40]. BNNs are developed to address
+these issues [41, 42]. In BNNs, the weights are endowed with probability distributions (rather than single
+point estimates) which naturally results in probabilistic predictions and can dramatically reduce overfitting
+via parameter regularization and model averaging.
+Predictions via a BNN requires sampling from the posterior distribution of the parameters, i.e., P (θ|D),
+which does not have a closed form and is highly complex. Over the past few years, various techniques
+have been developed to obtain samples from P (θ|D) (or an approximation thereof). The most popular
+techniques are based on either Markov Chain Monte Carlo (MCMC) [43] or variational inference (VI) [44]
+which, unlike MCMC, learns an approximation of the posterior distribution.
+Although MCMC methods are arguably the best techniques for sampling from the exact posterior, their
+lack of scalability makes them inefficient for BNNs of any practical size [45]. Hence, we employ Bayes by
+backprop [39] which is a variational method that approximates P (θ|D) with the parameterized distribution
+q(θ|ϕ) . The parameters ϕ are learned by minimizing the Kullback–Leibler (KL) divergence between the
+true and approximated posteriors:
+KL[q(θ|ϕ)||P(θ|D)] =
+�
+q(θ|ϕ) log
+� q(θ|ϕ)
+P(θ|D)
+�
+dθ =
+�
+q(θ|ϕ) log
+� q(θ|ϕ)P(D)
+P(D|θ)P(θ)
+�
+dθ
+=
+�
+q(θ|ϕ) log P(D)dθ +
+�
+q(θ|ϕ) log
+�q(θ|ϕ)
+P(θ)
+�
+dθ −
+�
+q(θ|ϕ) log P(D|θ)dθ
+= log P(D) + KL[q(θ||ϕ)|P(θ)] − Eq(θ|ϕ)[log P(D|θ)]
+(17)
+where Bayes rule is applied to P(θ|D) in the first line. Then, the parameters ϕ are estimated by minimizing
+Equation (17):
+ϕ∗ = argmin
+ϕ
+KL[q(θ|ϕ)||P(θ|D)] = argmin
+ϕ
+KL[q(θ|ϕ)||P(θ)] − Eq(θ|ϕ)[log P(D|θ)]
+(18)
+where the term log P(D) is excluded as it is constant. Equation (18) aims to minimize the sum of two terms.
+The second term corresponds to the expectation of the negative log-likelihood while the first term acts as a
+regularizer and corresponds to the KL divergence between the approximated posterior and the prior.
+3
+Probabilistic Neural Data Fusion
+Designing a multi-fidelity NN that leverages an ensemble of LF data sets to better learn an HF source is
+a very challenging task because of the following major reasons:
+1. The relations among the data sources can be unknown. For instance, in the Rational example (see
+Table 4 in Appendix A) there are three LF sources whose biases are not additive. Additionally, these
+LF sources are not hierarchically ordered in the sense that the second LF source is more accurate than
+the first one.
+2. There are typically (but not always) more LF data available since LF sources are generally cheaper
+compared to the HF source. Learning from such an unbalanced MF data is quite difficult especially in
+the presence of scarce HF data (as an example, see the sample sizes for the engineering applications
+described in Appendix B).
+7
+
+3. NNs can be built in many ways and, as shown in Section 4, their performance heavily depends on their
+architecture and training mechanism. Building an optimum4 NN with small, unbalanced, and MF data
+is even more difficult since the sensitivity to the architecture and training mechanism considerably
+increases.
+We propose to address the above challenges by converting MF modeling to a manifold5 learning problem
+which is then solved via an NN. We design the architecture, loss function, and training mechanism of this
+NN with a particular focus on uncertainty sources that include data scarcity (especially HF samples), noise
+with unknown variance (which can affect any of the data sources), non-trivial biases of LF sources, and data
+imbalances.
+As schematically demonstrated in Figure 1, we convert MF modeling to manifold learning by augmenting
+the input space with the categorical variable ts whose levels (e.g., {′1′,′ 2′, · · · } or {a, b, · · · }) indicate the
+source that generates a sample. We then map this source indicator variable to a low-dimensional manifold
+via a BNN (see Block 1 in Figure 1). If the original input space has the categorical variables tc, we similarly
+map them to a manifold (but this time we use a deterministic NN, see Block 2 in Figure 1). Afterwards,
+we combine the latent variables of these two manifolds with the quantitative inputs x via a deterministic
+NN, see Block 3 in Figure 1. As opposed to the other two blocks, we require Block 3 to produce a normal
+probability distribution in order to capture aleatoric uncertainties. Finally, we train the entire network on the
+entire6 data using our custom loss function that noticeably improves the prediction intervals.
+In the following subsections, we elaborate on our rationale for designing a multi-block architecture and
+a custom loss function in Section 3.1 and Section 3.2, respectively. Then, we provide some details on the
+training and inference stages in Section 3.3.
+3.1
+Multi-Block Architecture
+Each block of our network is designed to address particular challenges associated with MF modeling.
+Specifically, the BNN of Block 1 maps a quantitative prior representation ζ(ts) of the source indicator
+variable ts to a continuous manifold zs. We design ζ(ts) by one-hot encoding ts to merely inform the
+network about the source that generates a sample7. We build zs based on a categorical variable because
+it forces the manifold to uncover the relations between sources (i.e., the levels of ts). These relations are
+represented as distances in zs where sources that produce similar data are encoded with close-by points (see
+Section 4 for multiple examples). This distance learning is in sharp contrast to existing approaches since (1)
+it does not assume there is any hierarchy between the data sources, (2) it is scalable to an arbitrary number
+of data sets, (3) it enables training the entire network via all available samples, (4) it is visualizable and
+interpretable which helps in identifying anomalous data sources, and (5) it does not assume any specific
+form (e.g., additive, multiplicative, etc.) for the biases of LF sources.
+Block 1 is the only part of our network where the weights and biases are endowed with probability
+distributions. We make this choice to better learn model form errors and more accurately quantify the
+epistemic uncertainties due to lack of data and source-wise discrepancies. We note that, while the outputs
+of Block 1 do not parameterize a probability distribution, they are probabilistic by nature since they are
+obtained by propagating the deterministic vector ζ(ts) through some probabilistic hidden layers.
+Block 2 is an FFNN that maps the quantitative prior representation ζ(tc) of the categorical inputs tc to the
+4We measure optimality in terms of NN’s error in predicting unseen data from the HF source.
+5A manifold or a latent-space is a compact representation of a high-dimensional object such as an image.
+6By entire, we mean the combined data sets from all sources.
+7If there is some prior knowledge about the relation among the sources, ζ(ts) can be designed to reflect it. We do not pursue
+designing such informative priors in this work.
+8
+
+Targets
+Bayesian Neural Network
+Feedforward Neural Network
+Probabilistic Latent Mapping
+Probabilistic Output
+Source Indicator
+Feedforward Neural Network
+Deterministic Latent Mapping
+Numerical Inputs
+C
+Source
+Source
+Categorical Inputs
+Block 1
+Block 2
+Block 3
+Inputs
+Multi-fidelity data
+: Concatenation
+C
+Figure 1 Probabilistic neural data fusion (Pro-NDF): The proposed architecture allows to combine an arbitrary number of
+sources by appending a source indicator variable to the data sets and then concatenating them. Pro-NDF consists of three blocks
+that perform separate tasks related to MF modeling: (1) Block 1 is a BNN that maps a quantitative prior representation of the source
+indicator ζ(ts) to a continuous manifold, (2) Block 2 is an FFNN that maps a quantitative prior representation of the categorical
+inputs ζ(tc) to a continuous manifold, and (3) Block 3 is an FFNN with a probabilistic output that maps the numerical inputs and
+the latent variables to a parametric distribution.
+manifold zc (Block 2 is omitted if the original inputs are purely quantitative). Similar to Block 1, we design
+ζ(tc) via one-hot encoding and use deterministic outputs. However, unlike Block 1 we use a deterministic
+FFNN in Block 2 to map ζ(tc) into zc. We make this decision to reduce the number of parameters and also
+because the meaning (and hence effects) of categorical inputs across different sources is typically the same8.
+We set the manifold dimension to 2 for both Block 1 and Block 2, i.e., dzs = dzc = 2. While higher
+dimensions provide more learning capacities, our results in Section 4 and those reported elsewhere [46–
+51] indicate that low-dimensional manifolds are quite powerful in learning highly complex relations. For
+instance, [52] shows that a single latent variable can encode smiling in images of human faces which is a
+8Due to severe discrepancies such as large model form errors, the effects of a categorical variable on the response may be quite
+different across the sources.
+9
+
+high-dimensional and complex feature in the original data space. Additionally, our choice simplifies the
+visualization of the manifolds and reduces the chances of overfitting since we are primarily interested in
+scarce data applications.
+Block 3 is also an FFNN that maps the numerical inputs and the latent variables in both manifolds to
+a parametric distribution which represents the output. Block 3 has deterministic weights and biases since
+source-wise uncertainties are propagated to it via Block 1. However, we equip Block 3 with a probabilistic
+output because it: (1) quantifies aleatoric uncertainties that are inherent to the data sets9, and (2) enables
+designing a multi-task loss that considers the quality of the prediction intervals (detailed in Section 3.2).
+Additionally, Block 3 is responsible for learning the behavior for all data sources simultaneously, which
+allows it to leverage correlations between sources to augment predictions through a process akin to weight
+sharing.
+3.2
+Uncertainty-Focused Loss
+NNs typically provide overconfident predictions especially when they are trained on small and unbalanced
+data. As explained in Section 3.1, we aim to address this issue by making Block 1 and the network’s final
+output probabilistic. However, for these measures to work, we must develop an effective optimization10
+scheme where the loss function appropriately rewards prediction intervals (PIs) that are sufficiently wide
+(but not too wide) to cover unseen data (especially HF data). To design such a loss function, we draw
+inspiration from strictly proper scoring rules [35] and augment Equation (18) with the negatively oriented
+interval score. Our loss is defined as:
+L = LNLL + α1LKL + α2LIS + α3L2
+(19)
+where LNLL refers to the negative log-likelihood, LKL is the KL divergence between the prior and the vari-
+ational posterior distributions on the parameters (only applicable for the BNN from Block 1), LIS denotes
+the interval score term, and L2 is L2 regularization (only applicable for deterministic NNs, i.e., Block 2 and
+3). α1, α2 and α3 are hyperparameters that, respectively, determine the relative strengths of LKL, LIS and
+L2 compared to LNLL. The four terms in Equation (19) are calculated as:
+LNLL = − 1
+N
+N
+�
+i=1
+log N(y(i); ˆµ(i),
+�
+ˆσ(i)�2
+)
+(20)
+LKL = KL[q(θ|ϕ)||P(θ)]
+(21)
+LIS = 1
+N
+N
+�
+i=1
+[(ˆu(i) − ˆl(i)) + 2
+γ (ˆl(i) − y(i))1{y(i) < ˆl(i)} + 2
+γ (y(i) − ˆu(i))1{y(i) > ˆu(i)}]
+(22)
+L2 = |θ|2
+(23)
+where LKL is computed via a Monte Carlo approximation, N is the batch size, and 1{·} denotes the indica-
+tor function that returns 1 if the event in brackets is true and 0 otherwise. The three terms of Equation (19)
+compose a multi-task loss where: (1) the likelihood term LNLL penalizes the model if the predicted distri-
+bution does not match the target distribution, (2) the KL divergence term LKL favors variational posteriors
+that are similar to the assumed prior as per Equation (18), and (3) the interval score term LIS rewards nar-
+row PIs while penalizing the model for each observation y(i) that lies outside the (1 − γ) × 100% prediction
+interval that spans the range [ˆl(i), ˆu(i)] where ˆl(i) = ˆµ(i) − 1.96ˆσ(i) and ˆu(i) = ˆµ(i) + 1.96ˆσ(i). In this paper,
+we use γ = 5%, thus implying that LIS is minimized by a distribution whose 95% PI is as tight as possible
+while containing all the training data.
+9The predicted variance also includes epistemic uncertainties that are propagated from Block 1, see Section 3.3
+10Recall that we use Bayes by backprop which takes a variational approach towards finding the posteriors, see Section 2.2.
+10
+
+3.3
+Training and Prediction
+In BNNs, the variational posterior of θ is typically defined layer-wise as a multivariate Gaussian with
+mean µ ∈ Rck and covariance matrix Σ ∈ Rck×ck, i.e., N(µ, Σ), where ck is the total number of con-
+nections between two consecutive layers. Estimating the full covariance matrix requires learning O(c2
+k)
+parameters and is thus computationally prohibitive in most applications [45]. To reduce the costs, some
+simplifications have been adopted in the literature, such as learning diagonal or block diagonal [53] covari-
+ance matrices. However, our approach does not suffer from this computational issue since the only Bayesian
+part of our network is Block 1 (see Figure 1) whose size is typically very small (we use one hidden layer
+with 5 neurons for all the studies in Section 4). Hence, we estimate a dense covariance matrix between any
+two layers of Block 1 to improve its uncertainty quantification capacity. As for the prior, we use a zero
+mean Gaussian distribution with diagonal covariance matrix which makes the KL term equivalent to L2
+regularization with a rate defined by the standard deviation of the prior distribution [54]. Thus, the standard
+deviation is a hyperparameter that needs to be tuned specifically to each problem.
+BNNs represent their weights and biases by parameterized distributions which in our case are multivariate
+normal with dense covariance matrices. In a forward pass during either training or prediction, we take
+individual samples from these distributions and assign them to the weights and biases. In this way, instead
+of explicitly obtaining the true posterior distribution of the output of Block 1 (i.e., zs, see Figure 1), we
+obtain an empirical distribution in the zs manifold by taking a number of forward passes, see Figure 2. We
+refer to these forward passes as realizations and as explained below we use different number of passes in
+training versus prediction.
+To obtain the response (in training or testing) at the input u using Pro-NDF, which contains both a BNN
+component and a probabilistic output, we use ensemble prediction formulas [34]:
+ˆµ(u) = 1
+M
+M
+�
+j=1
+ˆµθj(u)
+(24)
+ˆσ(u) = 1
+M
+M
+�
+j=1
+�
+ˆσ2
+θj(u) + ˆµ2
+θj(u)
+�
+− ˆµ2(u)
+(25)
+where ˆµθj(u) and ˆσθj(u) are, respectively, the mean and standard deviation of the output distribution in
+the jth realization and θj are the associated network parameters. For predictions with a fitted NN, we use
+M = 1000 since it provides a higher accuracy in quantifying the uncertainty associated with learning the
+fidelity manifold (i.e., zs). While training the network, we use M = 200 to reduce the computational costs.
+The performance of an NN is highly sensitive to its architecture and hyperparameters if the training data
+is small, unbalanced, and multi-fidelity. To reduce this sensitivity and leverage the low costs of training a
+single NN on small data, we perform automated hyperparameter tuning 11. To this end, we use RayTune
+[55] and Hyperopt to find the optimum hyperparameters and architecture by minimizing the five-fold cross-
+validation errors on predicting the high-fidelity data.
+For our approach specifically, we apply the above tuning strategy to the architecture of Block 3, the
+learning rate of the Adam optimizer, α1, α2 and α3 in Equation (19), the prior standard deviation of weight
+matrices in Block 1, and the batch size. We fix the architectures of Block 1 and Block 2 to one hidden layer
+with 5 neurons and the dimension of both manifolds to 2. The activation function for all the neurons of
+Block 1 and 3 is hyperbolic tangent, whereas for Block 2 it is the sigmoid function. For more information
+and full details on implementation, please see our GitLab repository.
+11We use this approach for all NN-based data fusion approaches (including ours) in Section 4.
+11
+
+Block 1
+Block 2
+Block 3
+Probabilistic Fidelity Manifold
+Predictions with  
+Uncertainty Quantification
+Source Indicator
+Numerical Inputs
+Categorical Inputs
+realizations
+Get
+Categorical Inputs Manifold
+Figure 2 Outputs of Pro-NDF : We visualize the outputs of Pro-NDF after it is trained on the MF data of the HOIP data set, which
+does not have any numerical inputs (see Appendix B for more details). To provide probabilistic predictions that quantify both
+epistemic as well as aleatoric uncertainties, Pro-NDF learns a probabilistic fidelity manifold (where sources with similar behavior
+are encoded with close-by distributions) and a deterministic manifold for the categorical inputs.
+4
+Results and Discussions
+In this section, we validate our approach on three analytic and two real-world MF problems (detailed
+in Appendices A and B) and compare its performance against LMGP and two other existing NN-based
+approaches which are based on simple feedforward networks or sequential multi-fidelity (SMF) networks
+which are described in Appendix C. The hyperparameters of all the NN-based approaches are tuned as
+described in Section 3.3. We refer the reader to our GitLab repository for specific details on implementation,
+estimated hyperparameters, and training/test data. For LMGP, none of its architectural parameters (such as
+the kernel type, mean function, latent map, etc.) are tuned.
+We first conduct an ablation study in Section 4.1 to quantify the impacts of our designed architecture,
+loss function, and probabilistic elements. Then, we test the performance of the four MF approaches on the
+analytic and real-world problems in Section 4.2 and Section 4.3, respectively. In each problem, the goal
+is to model the HF source as accurately as possible, i.e., to obtain the lowest mean prediction error while
+maximizing the number of training/test samples that fall in the 95% PI. To this end, we use mean squared-
+error (MSE) and mean negatively oriented interval score (IS). Note that the FFNN and SMF approaches
+are not probabilistic, i.e., they provide point estimates rather than PIs and therefore they are only evaluated
+based on MSE.
+12
+
+0.8
+0.6
+0.4
+0.2
++×
+yli
+yl2
+0.2
+口
+yh
+1.5
+-0.5
+0.5
+21Level 1
+Level 2
+Level 3
+Level4
+Level 5
+Level 6
+Level 7
+Level 8
+Level 9
+Level 10
+Level 11
+Level 12
+Level 13
+Level 14
+Level15
+Level 160]
+.0
+8
+-10
+Predicted
+0
+20
+8
+00
+88
+30
+0
+OT
+Pro-NDF mean
+50
+Pro-NDF 95% PI
+35
+-30
+-25
+-20
+-15
+-10
+-5
+True4.1
+Ablation Study
+To evaluate the impact of the key components of Pro-NDF, we perform an ablation study on the Rational
+and DNS-ROM problems which are detailed in Appendices A and B, respectively. Namely, we analyze the
+impact of:
+1. Using a BNN rather than a deterministic FFNN in Block 1 for probabilistically learning the relations
+between the data sources.
+2. Considering LIS in the loss function of Equation (19).
+3. Fitting the model to the parameters of a distribution instead of a scalar, i.e., using a probabilistic
+output.
+4. Leveraging the fidelity map to detect the least accurate LF source and, in turn, assessing whether this
+source helps emulating the HF source.
+Regarding the third item above we note that we no longer use IS in the loss once the probabilistic output is
+removed. However, we still calculate the IS after training based on the empirical distribution of the fidelity
+manifold which is produced by the multiple realizations of the BNN component.
+We summarize the results of the ablation study on the two examples in Table 1. For both problems, we
+observe that using all components minimizes the test MSE and IS. Notably, both of our model’s probabilistic
+components significantly increase the performance: the probabilistic output enables Pro-NDF to not only
+capture aleatoric uncertainty, but also leverage IS in its loss function. Additionally, using a BNN improves
+Pro-NDF’s HF emulation capabilities by preventing overfitting in scarce data regions (since Block 1 is
+regularized) and by partially disentangling epistemic and aleatoric uncertainties which yields better PIs.
+We observe that without a probabilistic output, the IS (and hence the uncertainty quantification accuracy)
+drops quite significantly (compare V3 to V1 and the base in either of the problems) since the model can no
+longer account for aleatoric uncertainties. By comparing V1 to the base model in either of the problems in
+Table 1 Results of the ablation study: We evaluate the effect of removing individual components of Pro-NDF from it by reporting
+the MSE and IS on unseen HF data. All models are trained as discussed in Section 3 (e.g., all models benefit from automatic
+hyperparameter tuning). For both MSE and IS, lower numbers indicate better performance. The ticks indicate whether a component
+is used. The acronyms and symbols are defined as: HF: high-fidelity, LF1: low-fidelity 1, LF2: low-fidelity 2, LF3: low-fidelity
+3, LIS: negatively oriented interval score term in the loss function of Equation (19), PB1: probabilistic Block 1, PO: probabilistic
+output.
+Problem
+Model
+Version
+Input data
+Components
+MSE
+IS
+HF
+LF1
+LF2
+LF3
+LIS
+PB1
+PO
+Rational
+Base
+✓
+✓
+✓
+✓
+✓
+✓
+✓
+1.65 × 10−3
+0.22
+V1
+✓
+✓
+✓
+✓
+
+✓
+✓
+3.31 × 10−3
+0.26
+V2
+✓
+✓
+✓
+✓
+✓
+
+✓
+2.83 × 10−3
+0.24
+V3
+✓
+✓
+✓
+✓
+
+✓
+
+2.01 × 10−3
+0.40
+V4
+✓
+✓
+✓
+
+✓
+✓
+✓
+5.68 × 10−3
+0.98
+DNS-ROM
+Base
+✓
+✓
+✓
+✓
+✓
+✓
+✓
+8.99 × 109
+4.84 × 105
+V1
+✓
+✓
+✓
+✓
+
+✓
+✓
+1.23 × 1010
+5.89 × 105
+V2
+✓
+✓
+✓
+✓
+✓
+
+✓
+2.09 × 1010
+1.11 × 106
+V3
+✓
+✓
+✓
+✓
+
+✓
+
+1.70 × 1010
+4.07 × 106
+V4
+✓
+✓
+✓
+
+✓
+✓
+✓
+8.17 × 109
+5.06 × 105
+13
+
+Table 1 we see that for a model with a probabilistic output the optimal performance is obtained when LIS
+is used in the loss. That is, leveraging the IS in training improves both mean prediction and uncertainty
+quantification (measured via MSE and IS, respectively).
+In both problems, evaluating V1 through V3 against one another indicates that there is a trade-off between
+MSE and IS. That is, versions that perform well in terms of MSE, do not generally provide the smallest IS.
+However, when all of these components are included in Pro-NDF (see the base model in Table 1 for either
+of the problems), both MSE and IS are reduced. This improvement is due to the fact that the priors and
+LIS effectively regularize the model whose learning capacity is substantially increased by the probabilistic
+natures of Block 1 and the output.
+The probabilistic fidelity manifold (i.e., output of Block 1) provides an intuitive and visualizable tools
+to learn the similarity/discrepancy among the sources. Hence, once we fit the base model in each problem,
+we analyze the learned fidelity manifold to determine the LF source that has the least similarity to the HF
+source, see Figure 4(a) and Figure 5(a). Based on the distances in the fidelity manifold of each problem, we
+conclude that the third LF source is the least correlated one with the HF source in both cases. We exclude
+this source and its data from MF modeling and refit the base model to the rest of the data, see version V4
+for both problems.
+One of the major outputs of Pro-NDF is the learned fidelity manifold which indicates which LF source
+has the highest discrepancy compared to the HF source. Hence, after training a Pro-NDF and inversely
+identifying the least accurate LF source, we can build another Pro-NDF while excluding the data from this
+source. In the Rational problem, omitting the lowest-fidelity source results in much worse MSE and IS.
+We explain this observation by noting that this problem has an extremely small number of HF samples and
+therefore it is important to judiciously use all available data in training. However, in the DNS-ROM problem
+version V4 achieves the best MSE while Pro-NDF with all components achieves the best IS and second best
+MSE (compare base to V4 in Table 1). We explain this trend by noting that the size of the training data in the
+DNS-ROM problem is significantly higher than that in the Rational problem. Therefore, omitting a highly
+inaccurate data source improves mean prediction accuracy for the HF source in the DNS-ROM problem
+since the input-output relationships learned by Block 3 for the different sources are more similar. Omitting
+data from this source also increases the ratio of HF data available in the unified data set which helps in
+learning the HF behavior. However, using all data sources provides Pro-NDF with more information which
+improves the uncertainty quantification capability and hence a smaller error on IS.
+4.2
+Analytic Problems
+In this section, we validate our approach against LMGP and existing NN-based technologies for the
+Rational, Wing-weight, and Borehole examples detailed in Appendix A. These examples cover a wider range
+of input dimensionality, number of sources, and model form errors (e.g., additive and nonlinear biases).
+Similar to the previous section, we use MSE and IS on HF test data as the performance metrics. The input
+space of these three examples does not have categorical features and hence both Pro-NDF and LMGP learn
+a single manifold. We visualize the fidelity manifolds learned by Pro-NDF and LMGP to examine these
+models’ ability in inversely learning the relationships among the data sources (note that the LF sources are
+not ordered based on their accuracy). We highlight that, unlike Pro-NDF, the fidelity manifold of LMGP is
+not probabilistic and hence each data source is encoded with a single point in the manifold.
+The results for each approach on each problem are summarized in Table 2 and demonstrate that the
+probabilistic approaches, i.e., LMGP and Pro-NDF, significantly outperform the deterministic approaches
+in all problems. The FFNN approach performs significantly worse than LMGP and sometimes approaches
+the performance of Pro-NDF in MSE, while the SMF approach shows poor performance for all problems.
+14
+
+We explain SMF’s poor performance by noting that, as explained in Appendix C, hierarchical MF techniques
+such as SMF heavily rely on the knowledge of fidelity levels to process the data sources sequentially in the
+order of increasing accuracy. Since we assume in the problem setup that we only know which source has
+the highest fidelity and do not know the relative fidelity levels of the LF sources, the LF sources are ordered
+sub-optimally in the SMF approach which leads to a very poor prediction accuracy. The FFNN approach, by
+contrast, does not rely on the knowledge of fidelity levels and as such performs better than SMF. However,
+its performance lags behind that of LMGP and Pro-NDF because the architecture is not designed with MF
+problems in mind.
+LMGP, which is considered as our gold standard for MF problems with small data, outperforms Pro-NDF
+in both MSE and IS for the Wing-weight and Borehole problems, and in IS for the Rational problem. The
+Rational problem is simultaneously the most data deficient and least complex of the problems examined in
+this paper: as shown in Table 4, there are 4 data sources with only one being especially inaccurate, the input
+and output are both 1D, and there are only 5 training samples provided for the HF source. Pro-NDF and
+LMGP are well suited to tackle this problem as they both perform well for low-dimensional problems with
+simple underlying functional forms and well-correlated sources, and as such they have similar performance.
+Figure 3(a,b) reveals that LMGP captures all of the training points in a narrower 95% PI compared to Pro-
+NDFwhich explains LMGP’s lower IS in Table 2. However, Pro-NDF shows a better performance for this
+problem in terms of mean prediction accuracy and it also has a higher degree of agreement with the true
+function in extrapolation while LMGP reverts to its mean. We therefore conclude that both methods perform
+on par on the Rational problem.
+The learned fidelity manifold of Pro-NDF for the Rational problem is shown in Figure 4(a) which indicates
+that the network has inversely learned the true relationship between the data sources as yl1 and yl2 are
+encoded close to yh while yl3 is quite far from yh. These relative distances are proportional to the accuracy
+of the LF sources with respect to the HF source which are reported in Table 4. The fidelity manifold also
+shows a high spread in the distributions of the realizations for individual sources which indicates either a
+poor fit to the data or a lack of training samples. In this case, we attribute this spread to lack of data since
+the performance in IS and MSE is quite good.
+The Wing-weight and Borehole problems are both high-dimensional problems with relatively complex
+underlying functional forms and small amounts of data. LMGP is very well suited to tackle this type of
+problem[30] because the number of its hyperparameters scales much better than NN-based approaches such
+as Pro-NDF . Accordingly, we observe that LMGP achieves lower MSE and IS for both examples.
+Comparing the performance of Pro-NDF across the two high-dimensional problems, we observe that
+it performs much better on the Borehole problem. Examining the fidelity manifold learned by Pro-NDF
+and LMGP for the Wing-weight problem, see Figure 4(c) and Figure 4(d), respectively, we see that both ap-
+Table 2 Results on the analytic examples for different models: We test the performance of Pro-NDF against LMGP and existing
+NN-based technologies for the Rational, Wing-weight and Borehole examples detailed in Table 4. The training procedure for Pro-
+NDF, LMGP, FFNN and SMF is discussed in Section 3, Section 2.1, Appendix C.1 and, Appendix C.2 respectively. We report the
+MSE and IS on unseen HF data.
+Rational
+Wing-weight
+Borehole
+Model
+MSE
+IS
+MSE
+IS
+MSE
+IS
+Pro-NDF
+1.65 × 10−3
+0.22
+59.14
+37.09
+14.94
+19.24
+LMGP
+1.70 × 10−3
+0.20
+37.97
+29.70
+12.69
+17.57
+FFNN
+2.95 × 10−3
+-
+64.23
+-
+21.16
+-
+SMF
+8.08 × 10−3
+-
+542.74
+-
+172.87
+-
+15
+
+Figure 3 High-fidelity emulation on the Rational problem: Pro-NDF and LMGP approaches produce similar results in terms of
+mean prediction in interpolation. However, LMGP has a narrower 95% PI and reverts to its mean in extrapolation.
+proaches accurately determine the relationship between the sources as they agree with the RRMSEs reported
+in Table 4. Specifically, yl1 is closer to yh than yl2, which in turn is closer than yl3. Notably, both LMGP
+and Pro-NDF have the same relative ordering and positioning of the sources, i.e., (1) the mean position of all
+sources lies on an axis, and (2) yl1 is in the opposite direction relative to yh from yl2 and yl3. This reinforces
+our earlier assertion in Section 3: the positions of the sources in the fidelity manifold learned by Pro-NDF
+reflect correlations between the data sources. However, the relative distances between the LF sources in the
+latent space found by LMGP more accurately represents the true relationships between the sources because
+the position for yl3 is much more distant from yh than encoded positions of the other sources.
+In Figure 4(a) we observe a large spread in the realizations (i.e., the posterior distributions in the fidelity
+manifold are quite wide) which partially explains the poor12 performance in this problem. We attribute this
+performance level to the relative accuracy of the data sources since only one source, yl1, is at all accurate
+with respect to yh while the other LF sources are quite inaccurate. LMGP’s performance is not inherently
+hampered by including poorly correlated sources in the data fusion problem [30] since its performance,
+upon successful optimization, is at worse on par with fitting separate GPs to each source. By contrast, since
+Pro-NDF’s Block 3 is responsible for learning the relations between all sources and uses weight sharing,
+including especially inaccurate sources leads to relatively poor performance as shown in 4.1.
+The Borehole problem has five total sources where two LF sources (yl3 and yl4) are accurate while two
+LF sources (yl1 and yl2) are quite inaccurate with respect to yh. Since there are more total data available
+compared to the Wing-weight problem due to the additional data source, and since there are more high-
+accuracy LF sources, we observe that the spread of the realizations in the fidelity manifold of Pro-NDF is
+much smaller than in the Wing-weight, see Figure 4(e). This narrow spread indicates that a good fit has been
+achieved. We again observe that the relative directions and distances of the LF sources from yh are nearly
+identical in the manifolds of Pro-NDF and LMGP, see Figure 4(f), and that both methods have correctly
+identified the relationships between the sources, see Table 4. Based on these observations, it is no surprise
+that Pro-NDF achieves very good performance and nearly matches LMGP in terms of MSE and IS.
+12Poor with respect to LMGP. The performance of Pro-NDF is still much better than the other two NN-based approaches.
+16
+
+Figure 4 Pro-NDF and LMGP fidelity manifolds for the analytic problems: (a) Pro-NDF for Rational problem, (b) LMGP
+for Rational problem, (c) Pro-NDF for Wing-weight problem, (d) LMGP for Wing-weight problem, (e) Pro-NDF for Borehole
+problem, and (f) LMGP for Borehole problem.
+4.3
+Real-World Problems
+In this section, we validate our approach against LMGP and existing NN-based technologies on two
+engineering applications which are detailed in Appendix B. We again use MSE and IS on HF test data
+as our performance metrics and examine the manifolds learned by Pro-NDF and LMGP. In both of these
+applications, the input space has categorical features (so Pro-NDF and LMGP each build two manifolds)
+17
+
+Figure 5 Pro-NDF and LMGP fidelity manifolds for the real-world problems: (a) Pro-NDF for DNS-ROM data set, (b) LMGP
+for DNS-ROM data set, (c) Pro-NDF for HOIP data set, (d) LMGP for HOIP data set.
+and we do not know the underlying relationships between the data sources.
+The results for each approach on each problem are summarized in Table 3 and demonstrate that the prob-
+abilistic approaches again significantly outperform the deterministic ones. The FFNN approach performs
+nearly as well as LMGP and Pro-NDF in the DNS-ROM problem, but lags behing Pro-NDF and LMGP in
+the HOIP problem in terms of MSE. The SMF approach shows poor performance for both problems for the
+Table 3 Results on the real-world examples for different models: We test the performance of Pro-NDF against LMGP and
+existing NN-based technologies for the DNS-ROM and HOIP data sets detailed in Appendix B. We report the MSE and IS on
+unseen HF data.
+DNS-ROM
+HOIP
+Model
+MSE
+IS
+MSE
+IS
+Pro-NDF
+8.99 × 109
+4.84 × 105
+14.16
+14.84
+LMGP
+9.66 × 109
+6.60 × 105
+14.33
+20.08
+FFNN
+1.12 × 1010
+-
+21.55
+-
+SMF
+1.81 × 1010
+-
+28.09
+-
+18
+
+Figure 6 Pro-NDF categorical manifold for the HOIP problem: The combination of the categorical variables’ levels are color-
+coded based on: (a) the levels of tc
+1, (b) the levels of tc
+2, (c) the levels of tc
+3, (d) the average output value.
+same reasons provided in Section 4.2. Notably, Pro-NDF outperforms LMGP for both problems in terms
+of both metrics which we partially explain by noting that there are much more data are available in these
+real-world problems compared to the analytical examples of Appendix A. Being an NN-based approach,
+Pro-NDF scales very well with additional data while the performance of LMGP has diminishing returns and
+eventually plateaus (recall that the latent map and kernel of LMGP are not tuned which contribute to this
+plateauing performance).
+As shown in Figure 5(a-b), the fidelity manifolds learned by Pro-NDF and LMGP for the DNS-ROM
+problem are nearly analogous as the relative distances are quite similar. However, LMGP finds all sources to
+be on the diagonal axis while Pro-NDF learns a more nuanced relationship between the sources, which may
+contribute to its superior performance. We also observe that the spreads in the individual realizations for
+each point are fairly tight, which indicates that Pro-NDF is able to learn the relations between the sources
+reasonably well and, accordingly, provide good performance in terms of MSE and IS.
+The HOIP problem has three categorical inputs with 10, 3, and 16 levels and as such Pro-NDF uses two
+19
+
+Figure 7 LMGP categorical manifold for the HOIP problem: The combination of the categorical variables’ levels are color-
+coded based on: (a) the levels of tc
+1, (b) the levels of tc
+2, (c) the levels of tc
+3, (d) the average output value.
+separate latent transformations (one for the data source and the other for the three categorical variables) that
+correspond to Blocks 1 and 2 in Figure 1. The learned categorical manifolds for Pro-NDF and LMGP are
+shown in Figures 6 and 7 where the zc is visualized four times as the combinations of the categorical vari-
+ables are color-coded based on the levels of each of the three categorical variables and based on the average
+value of the output 13. Since there are no numerical features, the combined inputs are u = [ζ(ts), ζ(tc)] and
+ν = [zs, zc] are the inputs to Block 3 of Pro-NDF . Recall that we only use a BNN in Block 1 and as such
+we show only one realization for the manifold for Pro-NDF that encodes the categorical variables.
+Pro-NDF outperforms LMGP in terms of MSE by a small margin and IS by a significant margin for this
+problem which we attribute to the size of the data sets. Pro-NDF is able to leverage these additional data
+much more readily than LMGP which only uses simple mapping functions to handle categorical variables
+ts and tc. Pro-NDF also finds fairly tight spreads in the probabilistic fidelity manifold, see Figure 5(c);
+indicating that it has high certainty in its outputs and that we should expect good performance. We note
+13This average is obtained using the entire data set including both the training and test data
+20
+
+that all sources are found to be roughly on one axis and roughly spaced evenly from each other, which
+may indicate that Pro-NDF has failed to learn the more nuanced relationships between the sources. Equally
+likely, however, is that the relationships between the sources are simple enough to be represented in this
+way; since we do not know the underlying functional forms for this problem, we cannot give a definitive
+answer.
+We can also glean some information about the relationships between the categorical variable levels and
+their impact on the output by examining the corresponding manifolds in Figures 6 and 7. Figure 6(c) shows
+that Pro-NDF finds distinct clusters for all 16 levels of tc
+3 which indicates that distinguishing between the
+levels of tc
+3 is important to learning the output. Similarly, the levels of tc
+2 are distinguishable in Figure 6(b)
+as tc
+2 affects the response value. By contrast, Figure 6(a) shows no apparent trend between the 10 levels of tc
+1
+which implies that tc
+1 has little effect on the output as Pro-NDF does not learn to distinguish the levels from
+each other. By contrast, the manifold found by LMGP, shown in Figure 7 shows much less distinct clustering
+for each of the three categorical variables, which may help explain why it achieves a lower IS than Pro-NDF
+. Finally, we examine whether the latent positions for the categorical combinations are influenced by the
+average output value in Figure 6(d) and Figure 7(d). The manifold for Pro-NDF shows a clear trend of the
+average output value increasing as the latent points move from the bottom-left of the space to the top-right,
+while for LMGP there is no obvious trend. Based on these manifolds, Pro-NDF shows superior ability to
+discern relationships between the categorical combinations and between levels of categorical variables.
+5
+Conclusion
+In this paper, we introduce Pro-NDF for data fusion under uncertainty. Pro-NDF is based on a multi-block
+NN where each block is designed to take on specific tasks for MF modeling that arise in typical engineering
+applications. One of these blocks is probabilistic whose visualizable output can be used to detect LF sources
+with large model form errors. The final output of Pro-NDF is also probabilistic which enables to not only
+quantify aleatoric uncertainties, but also leverage strictly proper scoring rules during training.
+We validate each of the key components of Pro-NDF by performing an ablation study on an analytic and a
+real-world example. We also demonstrate that Pro-NDF outperforms other NN-based data fusion approaches
+by a large margin. Moreover, Pro-NDF performs on par to LMGP in low-dimensional cases with small data
+sets and slightly lags behind LMGP (a competing GP-based approach) in high-dimensional examples with
+very small data sets. However, as the size of the training data increases, Pro-NDF scales better than LMGP
+and provides smaller errors. In these studies, we test the performance on unseen HF data but note that
+Pro-NDF builds an MF emulator that probabilistically surrogates all the data sources simultaneously.
+A particularly useful output of Pro-NDF is its learnt fidelity manifold which encodes source-wise simi-
+larities/discrepancies. While the learnt distances in this manifold do not directly link correlation between
+the sources, we observe that the fidelity manifold of Pro-NDF and LMGP look quite similar in our stud-
+ies. Since the fidelity manifold of LMGP is embedded in its kernel and hence indicates the correlations,
+we believe the fidelity of Pro-NDF also estimates a scaled version of correlation. An added benefit of Pro-
+NDF’s fidelity manifold is that it is probabilistic where wide distributions can indicate if Pro-NDF is able
+to learn the relation between the data sources. Reducing this uncertainty via domain knowledge (especially
+qualitative information in engineering applications) is a future direction that we plan to investigate.
+The performance of any data fusion approach (including ours) can drop if there are one or more very
+inaccurate LF sources. With Pro-NDF , the learned fidelity manifold can be used to identify-discard such
+sources and then retrain Pro-NDF anew. This process can be repeated until all LF sources are encoded close
+to the HF source in the fidelity manifold. This iterative approach is, however, quite inefficient so we plan
+to develop an automated mechanism that perhaps leverages the fidelity manifold to adjust the loss function
+21
+
+and, in turn, prevent Pro-NDF from learning the highly inaccurate LF sources.
+6
+Acknowledgement
+We appreciate the support from National Science Foundation (award numbers OAC-2211908 and OAC-
+2103708) and the Early Career Faculty grant from NASA’s Space Technology Research Grants Program
+(award number 80NSSC21K1809).
+Appendices
+We provide the formulations of the analytic problems in Appendix A, the background and details of the
+real-world problems in Appendix B, and the methodology and details of the FFNN and SMF methods in
+Appendix C.
+A
+Table of Analytic Examples
+Table 4 details the analytic functions used for the examples covered in Section 4. For each multi-fidelity
+problem, we calculate the accuracy of each LF source with respect to the HF source via relative root mean
+squared error (RRMSE):
+RRMSE =
+�
+(yl − yh)T (yl − yh)
+10000 × var(yh)
+(A-1)
+where yl and yh are 10000×1 arrays of outputs sampled randomly via Sobol sequence from the LF and HF
+sources, respectively. We use the same sample locations and outputs as our test data when evaluating MSE
+and IS in Sections 4.1 and 4.2.
+B
+Background on Real-World Examples
+In the DNS-ROM problem, the goal is to predict the toughness of a multiscale metallic component with
+spatially varying porosity by combining four sources of data: (1) high-fidelity: direct numerical simula-
+tions (DNS) and (2, 3, 4) low-fidelity: a reduced-order model (ROM) with three different number of clusters
+(800, 1600, 3200) which balance accuracy against computational costs. The data sets have six numerical
+inputs that include pore volume fraction, number of pores, pore aspect ratio, average nearest neighbor dis-
+tance among the pores, evolutionary rate parameter, and critical effective plastic strain (the last two inputs
+govern the damage response of the material under load). The more clusters are used in the ROM, the more
+similar are its results compared to those of DNS at the expense of a higher computational burden. The data
+set contains nh = 70, nl1 = 110, nl2 = 170, nl3 = 250 samples. We use 80% of the available samples for
+each source for training and 20% for testing. For further details on this data set, we refer the reader to [2].
+In the HOIP problem, the goal is to predict the inter-molecular binding energy in hybrid organic-inorganic
+perovskite (HOIP) crystals. The data set has three categorical inputs with l1 = 10, l2 = 3, and l3 = 16
+levels which correspond to the elements present in each crystal. There are one HF and three LF data sets
+with unknown levels of fidelity and nh = 480, nl1 = 480, nl2 = 179, nl3 = 240. We use 90% of the
+available samples for each source for training and 10% for testing.
+C
+Other Multi-Fidelity NN-Based Approaches
+22
+
+Table 4 Table of analytic functions: The analytic examples have different input dimensionality, number of sources, and forms of
+model error. n denotes the number of samples, σ2 is the variance of the noise, and RRMSE is the relative root mean squared error
+of an LF source with respect to an HF source, see Equation (A-1).
+Name
+Source ID
+Formulation
+n
+σ2
+RRMSE
+Rational
+yh(x)
+1
+0.1x3+x2+x+1
+5
+0.001
+-
+yl1(x)
+1
+0.2x3+x2+x+1
+30
+0.001
+0.23
+yl2(x)
+1
+0×x3+x2+x+1
+30
+0.001
+0.15
+yl3(x)
+1
+0×x3+x2+0×x+1
+30
+0.001
+0.73
+Wing Weight
+yh(x)
+0.036S0.758
+ω
+W 0.0035
+fω
+�
+A
+cos2(Λ)
+�0.6
+q0.006×
+15
+25
+-
+λ0.04 �
+100tc
+cos(Λ)
+�−0.3
++ (NzWdg)0.49 + SωWp
+yl1(x)
+0.036S0.758
+ω
+W 0.0035
+fω
+�
+A
+cos2(Λ)
+�0.6
+q0.006×
+50
+25
+0.20
+λ0.04 �
+100tc
+cos(Λ)
+�−0.3
++ (NzWdg)0.49 + 1 × Wp
+yl2(x)
+0.036S0.8
+ω W 0.0035
+fω
+�
+A
+cos2(Λ)
+�0.6
+q0.006×
+50
+25
+1.14
+λ0.04 �
+100tc
+cos(Λ)
+�−0.3
++ (NzWdg)0.49 + 1 × Wp
+yl3(x)
+0.036S0.9
+ω W 0.0035
+fω
+�
+A
+cos2(Λ)
+�0.6
+q0.006×
+50
+25
+5.75
+λ0.04 �
+100tc
+cos(Λ)
+�−0.3
++ (NzWdg)0.49 + 0 × Wp
+Borehole
+yh(x)
+2πTu(Hu−Hl)
+ln
+�
+r
+rw
+��
+1+
+2LTu
+ln( r
+rw )r2wkw
++ Tu
+Tl
+�
+15
+6.25
+-
+yl1(x)
+2πTu(Hu−0.8Hl)
+ln
+�
+r
+rw
+��
+1+
+1LTu
+ln( r
+rw )r2wkw
++ Tu
+Tl
+�
+50
+6.25
+3.67
+yl2(x)
+2πTu(Hu−3Hl)
+ln
+�
+r
+rw
+��
+1+
+8LTu
+ln( r
+rw )r2wkw
++0.75 Tu
+Tl
+�
+50
+6.25
+3.73
+yl3(x)
+2πTu(1.1Hu−Hl)
+ln
+�
+4r
+rw
+��
+1+
+3LTu
+ln( r
+rw )r2wkw
++ Tu
+Tl
+�
+50
+6.25
+0.38
+yl4(x)
+2πTu(1.05Hu−Hl)
+ln
+�
+2r
+rw
+��
+1+
+2LTu
+ln( r
+rw )r2wkw
++ Tu
+Tl
+�
+50
+6.25
+0.19
+C.1
+Feedforward Neural Networks
+As depicted in Figure 8, for MF modeling via an FFNN we simply feed the numerical inputs x, the prior
+representation of the source indicator ζ(ts) and the prior representation of the categorical inputs ζ(tc) into
+the FFNN to produce the output. This approach has two clear disadvantages with respect to Pro-NDF: (1)
+it does not provide a tool such as the fidelity manifold of Pro-NDF that provides a direct visualization of
+the correlation between the data sources, and (2) it has a fully deterministic setting which does not enable
+uncertainty quantification and thus using a loss function based on proper scoring rules. In particular, we use
+23
+
+Targets
+Source Indicator
+Numerical Inputs
+C
+Categorical Inputs
+Inputs
+Multi-fidelity data
+Feedforward Neural Network
+Source
+Source
+: Concatenation
+C
+Figure 8 FFNN for multi-fidelity modeling: As Pro-NDF , this approach allows to use an arbitrary number of sources by ap-
+pending a source indicator variable to each data set and concatenating them. The FFNN maps the numerical inputs x, a priori
+representation of the source indicator ζ(ts), and categorical inputs ζ(tc) to the output.
+the following loss function for training the FFNN:
+L = LMSE + βL2
+(C-2)
+where LMSE is the mean squared error of the predictions and L2 is L2 regularization:
+LMSE = 1
+N
+N
+�
+i=1
+(y(i) − ˆy(i))2
+(C-3)
+L2 = |θ|2
+(C-4)
+We employ Adam as the optimizer and use RayTune [55] and Hyperopt with five-fold cross-validation to
+find the optimum architecture and hyperparameters which include the learning rate, regularization parameter
+β, and batch size N. For further details on implementation, please see our GitLab repository.
+C.2
+Sequential Multi-Fidelity Networks
+Unlike the other methods presented in this paper, multi-fidelity modeling via SMF requires training a
+separate sorrgate for each data source. As depicted in Figure 9, individual FFNNs are trained for each
+source in the sequence that ends with the HF source. After a sorrugate is trained for a data source, its
+outputs are used to augment the inputs of the next model in the sequence and hence the resulting input-
+output relationships are:
+24
+
+Multi-fidelity data
+Targets
+LF Source
+Inputs
+Targets
+LF Source
+Inputs
+Targets
+HF Source
+Inputs
+Numerical Inputs
+Categorical Inputs
+Feedforward Neural Network
+Numerical Inputs
+Categorical Inputs
+Feedforward Neural Network
+Numerical Inputs
+Categorical Inputs
+Feedforward Neural Network
+Figure 9 Sequential Multi-fidelity (SMF) Networks: SMF is a hierarchical approach that relies on sequentially training a model
+(e.g., an FFNN) for each data source in an ascending order based on the fidelities. The inputs of a model are augmented with the
+outputs of the previous one until reaching the model of the HF source.
+ˆys1 = ˆfs1 (us1)
+ˆys2 = ˆfs2 (us2, ˆys1(us2))
+· · ·
+ˆysds = ˆfsds (usds, ˆysds−1(usds))
+(C-5)
+where ˆysi is the output of the FFNN , ˆfsi is the mapping defined by the FFNN, usi is the combined numeric
+and categorical input u = [x, ζ(tc)], and i denotes the data source with i = ds being the HF source.
+Each individual FFNN employs the same loss function and optimizer as in the FFNN method presented in
+Appendix C.1.
+Unlike the other three MF methods we study in this paper, the SMF approach is highly sensitive to the
+ordering of the data sources in the sequence. In the case that the fidelity levels are known, they are assigned
+in the order of increasing fidelity, i.e., source 1 is the least accurate LF source while source ds−1 is the most
+25
+
+accurate. With this ordering, the SMF approach leverages the entire data set to achieve good HF prediction
+accuracy by minimizing the complexity of the mapping learned by each successive FFNN. However, in
+the case that the fidelities are not known, the order of the LF sources is assigned randomly. In this case,
+the mappings of the successive FFNNs no longer monotonically approaches that of the HF function, and
+the SMF approach is unable to properly leverage the additional LF data. In this paper, we assume that the
+fidelity levels are unknown and therefore assign the data source ordering randomly when using SMF.
+Similar to the FFNN approach, the SMF approach does not provide a latent mapping and is entirely deter-
+ministic. Like all hierarchical approaches, it also requires knowledge of fidelity levels for good performance.
+These factors lead to a marked disadvantage in the context of the problems examined in this paper, and we
+therefore expect the SMF method to perform poorly.
+We use RayTune and Hyperopt with five-fold cross-validation to find the optimum architecture and hyper-
+parameters for each FFNN in the SMF method. Namely, we tune the learning rate, regularization parameter
+β, and batch size N. We also tune an additional parameter that determines whether to use the numeric and
+categorical inputs u in the final FFNN, since the mapping may be simple enough to learn from just the
+previous FFNN outputs in the case that the last LF source is highly accurate. For further details on imple-
+mentation, please see our GitLab repository.
+26
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@@ -0,0 +1,3322 @@
+CLIP-Driven Universal Model for Organ Segmentation and Tumor Detection
+Jie Liu1, Yixiao Zhang2, Jie-Neng Chen2, Junfei Xiao2, Yongyi Lu2, Bennett A. Landman3,
+Yixuan Yuan4,5, Alan Yuille2, Yucheng Tang3,6,∗, and Zongwei Zhou2,*
+1City University of Hong Kong
+2Johns Hopkins University
+3Vanderbilt University
+4Chinese University of Hong Kong
+5CUHK Shenzhen Research Institute
+6NVIDIA
+Project: https://github.com/ljwztc/CLIP-Driven-Universal-Model
+Abstract
+An increasing number of public datasets have shown
+a marked clinical impact on assessing anatomical struc-
+tures. However, each of the datasets is small, partially la-
+beled, and rarely investigates severe tumor subjects. More-
+over, current models are limited to segmenting specific or-
+gans/tumors, which can not be extended to novel domains
+and classes. To tackle these limitations, we introduce em-
+bedding learned from Contrastive Language–Image Pre-
+training (CLIP) to segmentation models, dubbed the CLIP-
+Driven Universal Model. The Universal Model can better
+segment 25 organs and 6 types of tumors by exploiting the
+semantic relationship between abdominal structures. The
+model is developed from an assembly of 14 datasets with
+3,410 CT scans and evaluated on 6,162 external CT scans
+from 3 datasets. We rank first on the public leaderboard of
+the Medical Segmentation Decathlon (MSD) and achieve
+the state-of-the-art results on Beyond The Cranial Vault
+(BTCV). Compared with dataset-specific models, the Uni-
+versal Model is computationally more efficient (6× faster),
+generalizes better to CT scans from varying sites, and shows
+stronger transfer learning performance on novel tasks. The
+design of CLIP embedding enables the Universal Model to
+be easily extended to new classes without catastrophically
+forgetting the previously learned classes.
+1. Introduction
+Enormous advances in medical imaging benefit from the
+ever-growing amount of annotated datasets [1,30,38,39,68].
+Although a total of around 5,000 annotated abdominal CT
+scans are publicly available, an ingrained impression re-
+mains: medical imaging datasets are too small to develop
+robust AI models [13,48,58,64,82,83]. One of the reasons
+for this impression is that these public datasets are associ-
+*Corresponding authors: Yucheng Tang ([email protected]) and
+Zongwei Zhou ([email protected])
+Figure 1. Cosine similarity between CLIP embeddings. The
+CLIP embedding reveals the intrinsic semantics of the anatomical
+structures by mapping similar concepts close to each other in the
+embedding space. For example, “Liver” has a large similarity with
+“Liver Tumor” and “Hepatic Vessel” (the hepatic vessel returns
+low-oxygen blood from your liver back to the heart, which has a
+high anatomical relationship with the liver); “Left Kidney” has a
+large similarity with “Right Kidney”.
+ated with imaging competitions. As per the fairness con-
+cern, these datasets must be used in isolation (no external
+data are allowed). Since each institute has limited time,
+budget, and particular clinical purposes, the number of CT
+scans in each dataset is limited, and the types of annotated
+organs vary significantly from institute to institute. What is
+more, only a small proportion (100’s) of public CT scans
+contain annotated tumors performed by experts [1,3,25].
+The potential of AI models, trained on a combination of
+existing public datasets, for multi-organ segmentation and
+tumor detection is unknown. This has motivated us to re-
+lax the requirement that no external data is allowed, exploit
+the public datasets with partial labels, and demonstrate the
+clinical impact of AI performance, including model gen-
+eralizability (i.e., robust to CT scans from various hospi-
+tals) [39], transferability (i.e., generic image representation
+1
+arXiv:2301.00785v1  [eess.IV]  2 Jan 2023
+
+Left Kidney
+Liver
+Tumor
+Kidneythat is transferable to multiple downstream tasks) [86], and
+extensibility (i.e., adaptable to novel classes without for-
+getting previously learned classes) [36]. Specifically, we
+have assembled 14 publicly available datasets, including
+3,672 CT scans with 25 partially annotated organs and 6
+tumors. Furthermore, we anticipate that most of the medi-
+cal datasets, released in the near future, will also focus on a
+small set of organs/tumors and that some current unlabeled
+organs/tumors, such as the vermiform appendix, would be
+annotated. This requires us to develop new strategies that
+can continually deal with more partially labeled datasets
+with novel classes from a variety of institutes.
+Formidable challenges exist in assembling partially an-
+notated datasets. First, label inconsistency in three aspects.
+(i) Index inconsistency. The same organ can be labeled as
+different indexes. For example, the stomach is labeled ‘7’
+in BTCV, but ‘5’ in WORD. (ii) Name inconsistency. Nam-
+ing can be confusing if multiple labels refer to the same
+anatomical structure. For example, “postcava” in AMOS22
+and “inferior vena cava�� in BTCV. (iii) Background incon-
+sistency. For example, when combining Pancreas-CT and
+MSD-Spleen, the pancreas is marked as the background
+in MSD-Spleen whereas it should have been marked as
+the foreground. Second, label orthogonality. Most seg-
+mentation methods, trained with one-hot labels [77], dis-
+miss the semantic relationship between classes.
+Given
+liver [1,0,0], liver tumor [0,1,0], and pancreas [0,0,1], there
+is no semantic difference between liver↔liver tumor and
+liver↔pancreas. A possible solution is few-hot labels [54],
+with which, the liver, liver tumor, and pancreas can be en-
+coded as [1,0,0], [1,1,0], and [0,0,1]. Although few-hot la-
+bels could indicate that liver tumors are part of the liver, the
+relationship between organs remains orthogonal and, more
+importantly, neither one-hot nor few-hot labels are easy to
+extend to more classes [6, 23]. Adding novel classes re-
+quires increasing the dimensionality of one- or few-hot la-
+bels and retraining the previously trained model.
+To address the label inconsistency, we maintain a revised
+label taxonomy from a collection of public datasets, gener-
+ate a binary segmentation mask for each class, and compute
+loss only for the classes with available labels. To address
+the label orthogonality, inspired by Guo et al. [19], one- or
+few-hot labels are replaced by the text embedding gener-
+ated by the pre-trained text encoder from CLIP1. Figure 1
+illustrates that CLIP embedding presents the relationship
+between organs and tumors. More importantly, the fixed-
+length CLIP embedding allows us to adapt the pre-trained
+model to open-vocabulary segmentation and extend to novel
+classes without forgetting previously learned classes.
+In this work, we propose a CLIP-driven Universal Model
+1CLIP (Contrastive Language–Image Pre-training) was pre-trained on
+400 million image-text pairs (some are medical images and text [5]), ex-
+ploiting the semantic relationship between images and language.
+that can segment 25 organs and detect 6 tumors with state-
+of-the-art performance, generalize to CT scans from dif-
+ferent institutes, and can be extended to more classes, in
+contrast with existing dataset-specific models [59]. Specif-
+ically, experimental results have demonstrated six advan-
+tages of the CLIP-driven Universal Model.
+1. High abdominal organ segmentation performance. We
+rank first in the MSD and BTCV challenges, leading
+to substantial performance improvement over others.
+2. A higher specificity of tumor detection than existing
+models while maintaining compelling sensitivity.
+3. Computationally more efficient than dataset-specific
+models, accelerating the testing speed by order of six.
+4. The performance of organ segmentation and tumor de-
+tection is generalized to CT scans from a variety of
+hospitals without additional tuning and adaptation.
+5. An effective Foundation Model for numerous down-
+stream tasks, showing a strong transferability on tasks
+across multiple diseases, organs, and datasets.
+6. The extensibility to novel classes shows the capability
+of quickly adapting to novel classes without forgetting
+previously learned classes.
+With the Universal Model, we have also created a large
+dataset of 3,672 CT scans with 6 organs annotated by either
+experts or the model. Refinement of model prediction for
+some cases is performed. This dataset, comprising multi-
+center, multi-vendor, multi-phase, and multi-disease cases,
+provides a diverse test bed to develop high-performance AI
+models for organ segmentation and tumor detection.
+2. Related Work
+Partial label problem. Publicly available datasets for ab-
+dominal imaging focus on different organs and tumors [30,
+33, 38, 39], e.g., AbdomenCT-1K dataset for 4 organ seg-
+mentation [39], WORD dataset for 16 organ segmenta-
+tion [38] and TotalSegmentor dataset for 104 anatomical
+structure segmentation [68]. The partial label problem oc-
+curs when training AI models on a combination of these
+datasets due to their inconsistent label taxonomy. To ex-
+ploit the partial labels, several approaches have been in-
+vestigated [18, 77, 78, 81], aiming for a single model that
+can perform organ segmentation [12, 35] and tumor detec-
+tion [2,37,41,43,71,73,89]. These studies have the follow-
+ing limitations. (1) Due to the small scale of the dataset as-
+sembly2, the potential of assembling datasets was not con-
+vincing. Their performance was similar to dataset-specific
+2Zhou et al. [81] assembled 150 CT scans from 4 datasets; Fang et
+al. [18] assembled 548 CT scans from 4 datasets; Zhang et al. [77] assem-
+bled 1,155 CT scans from 7 datasets.
+2
+
+Figure 2. Overview. We have developed a Universal Model from an assembly of 14 public datasets of 3,410 CT scans. In total, 25 organs
+and 6 types of tumors are partially labeled (detailed in Appendix Table 7). To deal with partial labels, Universal Model consists of a text
+branch (purple) and a vision branch (blue) (§3.2). The official test set of MSD and BTCV are used to benchmark the performance of
+organ segmentation (§4.1) and tumor detection (§4.2). 3D-IRCADb and TotalSegmentator are used for independent, external validation of
+model generalizability (§5.2), and transferability (§5.3). In addition to public datasets, the Universal Model has also been evaluated on a
+large-scale private dataset, consisting of 5,038 CT scans with 21 annotated organs, to investigate the extensibility to new classes (§5.4).
+models and was not evaluated on the official benchmark.
+(2) Due to the one-hot labels, the semantic relationship be-
+tween organs and tumors was discarded. Table 1 reveals
+that the introduction of CLIP embedding is a salient factor
+to our proposed framework.
+CLIP in medical imaging. With the widespread success of
+large models in the field of language processing and under-
+standing [4,15,56], large-scale pre-trained vision-language
+models (VLM), e.g., CLIP [14], have recently been applied
+to multiple vision tasks [5,47,50,66], but rarely to the med-
+ical domain [16, 67]. Qin et al. [49] suggested that VLM
+could be used for zero-shot learning in the medical domain
+and recognize novel classes with well-designed prompts.
+Grounded by these two findings, we are among the first to
+introduce CLIP embedding to medical segmentation tasks
+using partial labels, in which we underline the importance
+of the semantic relationship between anatomical structures
+in segmentation and incremental learning.
+3. Methodology
+3.1. Background
+Problem definition.
+Let M and N be the total number
+of datasets to combine and data points in the combina-
+tion of the datasets, respectively.
+Given a dataset D =
+{(X1, Y1), (X2, Y2), ..., (XN, YN)}, there are a total of
+K unique classes.
+For ∀n ∈ [1, N], if the presence of
+∀k ∈ [1, K] classes in Xi is annotated in Yi, D is a fully
+labeled dataset; otherwise, D is a partially labeled dataset.
+Previous solutions. Two groups of solutions were proposed
+to address the partial label problem. Given a data point
+Xn, n ∈ [1, N], the objective is to train a model F(·) us-
+ing the assembly dataset DA = {D1, D2, ..., DM}, and the
+model can predict all K classes, if presented in Xn.
+• Solution #1 [11,18,27,54,54,61,72,81] aims to solve
+Fθ(Xn) = P k
+n , n ∈ [1, N], k ∈ [1, K], where the
+prediction Pn is one-hot encoding with length k.
+• Solution #2 [31, 77, 88] aims to solve Fθ(Xn, wk) =
+Pn, n ∈ [1, N], k ∈ [1, K], where wk is an one-hot
+vector to indicate which class to be predicted.
+According to Zhang et al. [77], both solutions have sim-
+ilar segmentation performance, whereas #2 is computation-
+ally more efficient. However, both solutions rely on one-hot
+labels, sharing two limitations. First, they dismiss the se-
+mantic and anatomical relationship between organs and tu-
+mors. Second, they are inappropriate in segmenting various
+subtypes of tumors (and novel classes). To address these
+limitations, we modify wk in Solution #2 to CLIP embed-
+ding and introduce in-depth in the following sections.
+3.2. CLIP-Driven Universal Model
+The overall framework of CLIP-Driven Universal Model
+(see Figure 2) has a text branch and a vision branch. The
+text branch first generates the CLIP embedding for each or-
+gan and tumor using an appropriate medical prompting (Ta-
+ble 1), and then the vision branch takes both CT scans and
+CLIP embedding to predict the segmentation mask.
+Text branch. Let wk be the CLIP embedding of the k-th
+class, produced by the pre-trained text encoder in CLIP and
+3
+
+ACT of
+text
+a [CLS].
+encoder
+100 3D-IRCADb (13;0)
+CLIP embedding
+classes
+prompt temp
+Total=6162
+1024 TotalSegmentator (104;0)
+5038 JHH (21;0)
+text-based controller
+ global pooling
+for testing
+dataset 
+processing
+Param. 0 
+000
+ 82 Pancreas-CT (1;0)
+ 201 LiTS (1;1)
+网 300 KiTS (1;1)
+dataset 2
+text-driven Segmentor
+ 1000 AbdomenCT-1K (4;0)
+standardized I
+vision
+140 CT-ORG (4;0)
+encoder
+Cony
+Conv
+Cony
+Total=3410
+ 40 CHAOS (4;0)
+ 947 MSD (7;4)
+50 BTCV (13;0)
+e.g., Swin UNETR
+网 500 AMOS (15;0)
+150 WORD (16;0)
+dataset 14
+for training
+public datasets
+partial labelsEmbedding
+prompt
+DSC
+One-hot [77]
+-
+67.18
+CLIP V1
+A photo of a [CLS].
+69.70
+CLIP V2
+There is [CLS] in this computerized tomography.
+73.49
+CLIP V3
+A computerized tomography of a [CLS].
+73.86
+Table 1. Medical prompt templates. The DSC is the average
+of 25 organs and 6 tumors in the 5-fold cross-validation of MSD.
+All three prompts can elicit knowledge from CLIP, achieving sig-
+nificant improvement over the conventional one-hot labels (DoD-
+Net [77]) on the MSD dataset.
+a medical prompt (e.g., “a computerized tomography of a
+[CLS]”, where [CLS] is a concrete class names). We first
+concatenate the CLIP embedding (wk) and the global im-
+age feature (f) and then input it to a multi-layer perceptron
+(MLP), namely text-based controller [62], to generate pa-
+rameters (θk), i.e.,
+θk = MLP(wk ⊕ f),
+(1)
+where ⊕ is the concatenation. Although CLIP embedding
+significantly outperforms one-hot labels [77], we mark that
+the choice of medical prompt template is critical. Table 1
+presents the effectiveness of three prompt templates. More-
+over, the introduction of CLIP embedding addresses the
+label orthogonality problem by encoding hierarchical rela-
+tionship among organs and tumors (illustrated in Figure 1).
+The CLIP embedding also allows us to extend the Univer-
+sal Model to novel classes (e.g., organs, tumors, bones, and
+other anatomical structures) because the length of CLIP em-
+bedding is fixed and the incremental learning of new classes
+will not affect other classes (elaborated in §5.4).
+Vision branch. We pre-process CT scans using isotropic
+spacing and uniformed intensity scale to reduce the domain
+gap among various datasets3. The standardized and normal-
+ized CT scans are then processed by the vision encoder. Let
+F be the image features extracted by the vision encoder.
+To process F , we use three sequential convolutional layers
+with 1 × 1 × 1 kernels, namely text-driven segmentor. The
+first two layers have 8 channels, and the last one has 1 chan-
+nel, corresponding to the class of [CLS]k. The prediction
+for the class [CLS]k is computed as
+Pk = Sigmoid (((F ∗ θk1) ∗ θk2) ∗ θk3) ,
+(2)
+where θk1, θk2, θk3 are computed by Equation 1, and * rep-
+resents the convolution. For each class [CLS]k, we gener-
+ate the prediction using one vs. all manner (i.e., Sigmoid
+instead of Softmax).
+3A standardized and normalized CT pre-processing is important when
+combining multiple datasets. Substantial differences in CT scans can occur
+in image quality and technical display, originating from different acquisi-
+tion parameters, reconstruction kernels, contrast enhancements, intensity
+variation, and so on [20,46,74].
+Masked back-propagation. To address the label inconsis-
+tency problem, we proposed the masked back-propagation
+technique. The BCE loss function is utilized for supervi-
+sion. We masked the loss terms of these classes that are not
+contained in Y and only back-propagate the accurate super-
+vision to update the whole framework. The masked back-
+propagation addresses the label inconsistency in the partial
+label problem. Specifically, partially labeled datasets anno-
+tate some other organs as background, leading to the dis-
+ability of existing training schemes (Solution #1).
+3.3. Assembling Public Datasets
+To fully explore the potential of the proposed Universal
+Model in multi-organ segmentation and tumor detectio, we
+have thus far assembled a total of 3,410 CT scans from 14
+public datasets (many more can be included when datasets
+are available). However, it is non-trivial to assemble par-
+tially labeled datasets due to several challenges. Apart from
+the label inconsistency and label orthogonality that we have
+addressed, two challenges remain.
+Inconsistent label protocols. In AMOS, “Aorta” refers to
+the entire region of Aorta, but in AbdomenCT-1K, a part of
+the upper regions annotation is missing. Aorta is consid-
+ered and annotated (see Appendix Figure 10). It is because
+of the inconsistent definitions in different datasets and this
+requires considerable manual corrections when assembling
+these datasets together.
+Long-tail problem. The assembly of public datasets leads
+to severe class imbalance problems, especially for small tu-
+mors. Appendix Figure 11 entails the proportion of each
+class in the datasets. In this paper, we utilize data augmen-
+tation to alleviate the long-tail problem, but more research
+is encouraged to explore the solution to these two problems.
+4. Experiments & Results
+Datasets and evaluation metrics.
+A total of 14 public
+datasets consisting of 3,410 CT scans are assembled for
+training.
+Other 2 public and 1 private datasets are used
+for testing.
+Due to page limits, dataset details and pre-
+processing are described in Appendix §B. Dice Similarity
+Coefficient (DSC) and Normalized Surface Distance (NSD)
+are evaluated for organ/tumor segmentation; Sensitivity and
+Specificity are evaluated for tumor detection.
+Implementation details. The Universal Model is trained us-
+ing the AdamW optimizer with a warm-up cosine scheduler
+of 50 epochs. The segmentation experiments use batch-size
+of 6 per GPU with a patch size of 96 × 96 × 96. Default
+initial learning rate of 4e−4, momentum of 0.9 and decay
+of 1e−5 on multi-GPU (4) with DDP. The framework is im-
+plemented in MONAI 0.9.04. The five-fold cross validation
+4https://monai.io/
+4
+
+Task03 Liver
+Task07 Pancreas
+Method
+Dice1
+Dice2
+Avg.
+NSD1
+NSD2
+Avg.
+Dice1
+Dice2
+Avg.
+NSD1
+NSD2
+Avg.
+Kim et al. [32]
+94.25
+72.96
+83.61
+96.76
+88.58
+92.67
+80.61
+51.75
+66.18
+95.83
+73.09
+84.46
+Trans VW [21]
+95.18
+76.90
+86.04
+97.86
+92.03
+94.95
+81.42
+51.08
+66.25
+96.07
+70.13
+83.10
+C2FNAS [75]
+94.98
+72.89
+83.94
+98.38
+89.15
+93.77
+80.76
+54.41
+67.59
+96.16
+75.58
+85.87
+Models Gen. [86]
+95.72
+77.50
+86.61
+98.48
+91.92
+95.20
+81.36
+50.36
+65.86
+96.16
+70.02
+83.09
+nnUNet [28]
+95.75
+75.97
+85.86
+98.55
+90.65
+94.60
+81.64
+52.78
+67.21
+96.14
+71.47
+83.81
+DiNTS [24]
+95.35
+74.62
+84.99
+98.69
+91.02
+94.86
+81.02
+55.35
+68.19
+96.26
+75.90
+86.08
+Swin UNETR [61]
+95.35
+75.68
+85.52
+98.34
+91.59
+94.97
+81.85
+58.21
+70.71
+96.57
+79.10
+87.84
+Universal Model
+95.44
+77.27
+86.36
+98.44
+92.60
+95.52
+82.20
+62.21
+72.21
+96.69
+84.91
+90.80
+Task08 Hepatic Vessel
+Task06 Lung
+Task09 Spleen
+Task10 Colon
+Method
+Dice1
+Dice2
+Avg.
+NSD1
+NSD2
+Avg.
+Dice1
+NSD1
+Dice1
+NSD1
+Dice1
+NSD1
+Kim et al. [32]
+62.34
+68.63
+65.49
+83.22
+78.43
+80.83
+63.10
+62.51
+91.92
+94.83
+49.32
+62.21
+Trans VW [21]
+65.80
+71.44
+68.62
+84.01
+80.15
+82.08
+74.54
+76.22
+97.35
+99.87
+51.47
+60.53
+C2FNAS [75]
+64.30
+71.00
+67.65
+83.78
+80.66
+82.22
+70.44
+72.22
+96.28
+97.66
+58.90
+72.56
+Models Gen. [86]
+65.80
+71.44
+68.62
+84.01
+80.15
+82.08
+74.54
+76.22
+97.35
+99.87
+51.47
+60.53
+nnUNet [28]
+66.46
+71.78
+69.12
+84.43
+80.72
+82.58
+73.97
+76.02
+97.43
+99.89
+58.33
+68.43
+DiNTS [24]
+64.50
+71.76
+68.13
+83.98
+81.03
+82.51
+74.75
+77.02
+96.98
+99.83
+59.21
+70.34
+Swin UNETR [61]
+65.69
+72.20
+68.95
+84.83
+81.62
+83.23
+76.60
+77.40
+96.99
+99.84
+59.45
+70.89
+Universal Model
+66.34
+75.19
+70.77
+85.27
+85.16
+85.22
+78.47
+78.67
+96.92
+99.69
+61.17
+72.86
+Table 2. Leaderboard performance on MSD. The results are evaluated in the server on the MSD competition test dataset. All Dice and
+NSD metrics are obtained from the MSD public leaderboard.
+Figure 3. Benchmark on MSD validation dataset. We compare Universal Model with Swin UNETR [61] (previously ranked first on the
+MSD leaderboard) on 5-hold cross-validation of the MSD dataset. Universal Model achieves overall better segmentation performance and
+offers substantial improvement in the tasks of segmenting liver tumors (+14%), pancreatic tumors (+8%), and colon tumors (+11%).
+Figure 4. Intra-observer variability. We obtain similar perfor-
+mance between pseudo labels generated by the Universal Model
+(AI) and annotations performed by two human experts (Dr1,2) on
+6 organs. Spleen (Spl), liver (Liv), kidneys (Kid), stomach (Sto),
+gallbladder (Gall), and pancreas (Pan) can be annotated by AI with
+a similar intra-observer variability to humans. Examples of pseudo
+labels and human annotations are provided in Appendix Figure 7.
+strategy is performed. We select the best model in each
+fold by evaluating the validation best metrics. Models are
+trained on NVIDIA RTX A5000 cards.
+4.1. Organ Segmentation on MSD and BTCV
+We offer the top #1 solution in both Medical Segmen-
+tation Decathlon (MSD)5 and Beyond The Cranial Vault
+5decathlon-10.grand-challenge.org/evaluation/challenge/leaderboard/
+(BTCV), surpassing the runners-up by a considerable mar-
+gin. Table 2 and Figure 3 present detailed comparison on
+the official test set and 5-hold cross validation on MSD, re-
+spectively. Table 3 compares Universal Model with other
+methods in the validation set of BTCV, offering at least
+3.5% improvements over the second best.
+Manual annotations have inter-rater and intra-rater vari-
+ance [29], particularly in segmentation tasks, because some
+of the organs’ boundaries are blurry and ambiguous. We
+assess the quality of pseudo labels predicted by Universal
+Model and manual annotation performed by human experts.
+17 CT scans in BTCV have been annotated by two indepen-
+dent groups of radiologists from different institutes (not test
+server labels). As a result, each CT scan is associated with
+AI prediction, and two human annotations (Dr1 and Dr2).
+Figure 4 presents their mutual DSC scores, i.e., AI↔Dr1,
+AI↔Dr2, and Dr1↔Dr2. We find the DSC between AI
+and humans is slightly larger than the DSC between humans
+in segmenting 6 types of organs (i.e., spleen, liver, kidney,
+stomach, and pancreas). With this high-quality AI predic-
+tion, we assemble a large dataset of 3,410 CT scans from
+a diverse set of hospitals (Figure 2 and generate pseudo la-
+5
+
+96.5 94.1
+96.7 95.8
+100 -
+82.7 80.1
+62.1
+DSC (%)
+80
+71.9
+67.1 68.9
+60.8
+69.4 68.6
+H
+62.6 62.3
+ Universal Mode
+57.9
+50.5
+60
+工
+52.5
+ Swin UNETR (SOTA)
+工
+40 
+Liv
+Liv Tumor
+Pan
+Pan Tumor
+HepaticVes
+Hepatic Tumor
+Spl
+Lung Tumor100
+堂古
+丰丰
+90
+(Al, Dr1)
+C
+DS
+(Al, Dr2)
+80
+(Dr1, Dr2)
+70
+Spl
+Liv
+Kid
+Sto
+Gall
+PanMethods
+Spl
+RKid
+LKid
+Gall
+Eso
+Liv
+Sto
+Aor
+IVC
+Veins
+Pan
+AG
+Avg.
+ASPP [8]
+94.19
+91.24
+88.02
+63.58
+72.64
+93.61
+80.05
+86.20
+80.11
+71.49
+74.29
+64.58
+78.81
+PaNN [81]
+94.04
+90.21
+88.58
+64.96
+73.20
+93.50
+80.87
+87.26
+80.32
+70.59
+75.28
+64.92
+79.13
+TransUNet [7]
+94.10
+90.22
+88.84
+65.49
+73.19
+93.24
+80.85
+87.47
+80.48
+71.47
+74.26
+64.76
+79.16
+CoTr* [69]
+95.60
+89.22
+88.58
+67.54
+74.95
+95.97
+81.95
+88.58
+81.20
+72.83
+76.69
+65.81
+80.36
+CoTr [69]
+95.51
+88.03
+89.19
+68.49
+75.83
+95.93
+81.84
+89.01
+82.32
+73.39
+75.12
+65.78
+80.48
+RandPatch [60]
+95.82
+88.52
+90.14
+68.31
+75.01
+96.48
+82.93
+88.96
+82.49
+73.54
+75.48
+66.09
+80.76
+TransBTS [28]
+94.59
+89.23
+90.47
+68.50
+75.59
+96.14
+83.72
+88.85
+82.28
+74.25
+75.12
+66.74
+80.94
+nnFormer [28]
+94.51
+88.49
+93.39
+65.51
+74.49
+96.10
+83.83
+88.91
+80.58
+75.94
+77.71
+68.19
+81.22
+UNETR [22]
+94.91
+92.10
+93.12
+76.98
+74.01
+96.17
+79.98
+89.74
+81.20
+75.05
+80.12
+62.60
+81.43
+nnU-Net [28]
+95.92
+88.28
+92.62
+66.58
+75.71
+96.49
+86.05
+88.33
+82.72
+78.31
+79.17
+67.99
+82.01
+Swin UNETR [61]
+95.44
+93.38
+93.40
+77.12
+74.14
+96.39
+80.12
+90.02
+82.93
+75.08
+81.02
+64.98
+82.06
+Universal Model
+95.82
+94.28
+94.11
+79.52
+76.55
+97.05
+92.59
+91.63
+86.00
+77.54
+83.17
+70.52
+86.13
+Table 3. Benchmark on BTCV validation dataset. For a fair comparison, we did not use model ensemble during the evaluation. All
+experiments are under the same data splits, computing resources, and testing conditions. Evaluated on the BTCV validation set, Universal
+Model achieves the overall best performance, yielding at least +3.5% DSC improvement over the state-of-the-art method.
+Methods
+Liver Tumor
+Kidney Tumor
+Pancreatic Tumor
+Sen. (LiTS)
+Spec. (CHAOS)
+Sen. (KiTS)
+Spec. (CHAOS)
+Sen. (MSD-Pancreas)
+Spec. (Pancreas-CT)
+nnU-Net [28]
+94.44
+75.00
+96.88
+85.00
+95.18
+88.75
+UNet++ [85]
+94.44
+80.00
+N/A
+N/A
+N/A
+N/A
+UNETR [22]
+86.11
+95.00
+93.75
+95.00
+90.36
+81.25
+Swin UNETR [61]
+91.67
+85.00
+97.91
+70.00
+97.59
+87.50
+Universal Model
+88.89
+95.00
+91.67
+95.00
+93.98
+91.25
+Table 4. Tumor detection performance. The CT scans in LiTS [3], KiTS [26], and MSD Pancreas [1] contain tumors in liver, kidney and
+pancreas, respectively. These scans are used to compute the sensitivity of tumor detection. To perform an alternative check of specificity,
+we use CHAOS [63] and Pancreas-CT [52]. It has been confirmed that CHAOS has no liver or kidney tumor, and Pancreas-CT has no
+pancreatic tumor in the CT scans. Universal Model achieves a higher specificity than most baseline models, while maintaining a compelling
+sensitivity. High specificity is clinically important because it reveals that Universal Model does not predict many false positives.
+bels for 25 organs and 6 tumors6. Pseudo-label refinement
+has been performed for a few CT scans where AI’s predic-
+tion is uncertain. This fully annotated dataset will be re-
+leased. Now that these 6 organs can be segmented by AI
+with a similar variance to human experts, we encourage the
+research community to concentrate on creating annotations
+for harder organs and tumors.
+4.2. Tumor Detection on Five Datasets
+Figure 3 demonstrates that Universal Model surpasses
+Swin UNETR by a large margin in segmenting liver, pan-
+creatic, and colon tumors, leading to 14%, 8%, and 12%
+improvement in DSC scores, respectively. However, DSC
+scores cannot faithfully reveal the tumor detection perfor-
+mance because, by default, they are only calculated on ab-
+normal CT scans (with tumors) [28]. The AI might gener-
+ate numerous false positives when encountering normal CT
+scans (that have no tumor) [53]. Therefore, we also eval-
+uate patient-level Sensitivity and Specificity for detecting
+the three types of tumors. To obtain normal CT scans, we
+adopt the CHAOS and Pancreas-CT datasets because these
+two datasets provide pathological verification that no tu-
+6The quality of 19 other organs and 6 tumors has not been compared
+with human annotations because there is no publicly available CT scans
+that have been annotated by multiple independent groups on these objects.
+mors are present [52, 63]. Table 4 indicates that Universal
+Model achieves the highest Specificity of 95%, 95%, and
+91% on normal CT scans, while maintaining a compelling
+Sensitivity of 89%, 92%, and 94% on abnormal CT scans.
+Moreover, Rows 1–3 in Figure 5 depict the prediction of
+small/medium/large pancreatic tumors; Row 4 shows that
+Universal Model can precisely segment the pancreas and
+reduce the number of false positives on normal CT scans.
+Compared with dataset-specific models, the smaller number
+of false positives predicted by our Universal Model under-
+lines the necessity of assembling diverse datasets, benefit-
+ing from not only sufficient positive examples for training
+but also a larger number of negative examples as a control.
+5. Intriguing Properties
+5.1. Efficiency: FLOPs vs. DSC Scores
+It is clinically important to make AI models faster [9,17].
+The floating-point operations per second (FLOPS) are used
+to indicate the inference speed. Figure 6 presents a speed-
+performance plot, showing that Universal Model is com-
+putationally more efficient compared with dataset-specific
+models (>6× faster), while maintaining the highest DSC
+score of 74% on average.
+6
+
+CT scan
+ground
+truth
+Universal
+Model
+Swin
+UNETR
+zoom-in
+UNETR
+nnU-Net
+UNesT
+nnFormer
+Length: 60.4mm 
+Length: 16.2mm 
+Length: 9.6mm 
+Figure 5. Pancreatic tumor detection. Qualitative visualizations of the proposed Universal Model and five competitive baseline methods.
+We review the detection results of tumors from smaller to larger sizes (Rows 1–3). When it comes a CT scan without tumor from other
+hospitals, the Universal Model generalize well in organ segmentation and does not generate many false positives of tumors (Row 4; §4.2;
+§5.2). The visualization of tumor detection in other organs (e.g., liver tumors and kidney tumors) can be found in Appendix Figures 8–9.
+3D-IRCADb
+spleen
+kidneyR
+kidneyL
+gallbladder liver
+stomach
+pancreas
+lungR
+lungL
+mDSC*
+mDSC
+SegResNet [55]
+94.08
+80.01
+91.60
+69.59
+95.62
+89.53
+79.19
+N/A
+N/A
+85.66
+N/A
+nnFormer [79]
+93.75
+88.20
+90.11
+62.22
+94.93
+87.93
+78.90
+N/A
+N/A
+85.14
+N/A
+UNesT [76]
+94.02
+84.90
+94.95
+68.58
+95.10
+89.28
+79.94
+N/A
+N/A
+86.68
+N/A
+TransBTS [65]
+91.33
+76.22
+88.87
+62.50
+94.42
+85.87
+63.90
+N/A
+N/A
+80.44
+N/A
+TransUNet [7]
+94.09
+82.07
+89.92
+63.07
+95.55
+89.12
+79.53
+N/A
+N/A
+84.76
+N/A
+UNETR [22]
+92.23
+91.28
+94.19
+56.20
+94.25
+86.73
+72.56
+91.56
+93.31
+83.92
+85.81
+Swin UNETR [61]
+93.51
+66.34
+90.63
+61.05
+94.73
+87.37
+73.77
+93.72
+92.17
+81.05
+83.69
+Universal Model
+95.76
+94.99
+94.42
+88.79
+97.03
+89.36
+80.99
+97.71
+96.72
+91.62
+92.86
+JHH
+spleen
+kidneyR
+kidneyL
+gallbladder liver
+stomach
+pancreas
+arota
+postcava
+vein
+mDSC
+SegResNet [55]
+93.11
+89.92
+87.84
+74.62
+95.37
+87.90
+76.33
+84.05
+79.36
+57.13
+82.56
+nnFormer [79]
+86.71
+87.03
+84.28
+63.37
+91.64
+73.18
+71.88
+84.73
+78.61
+55.31
+77.67
+UNesT [76]
+93.82
+90.42
+89.04
+76.40
+95.30
+89.65
+78.97
+84.36
+79.61
+59.70
+83.73
+TransBTS [65]
+85.47
+81.58
+82.00
+60.58
+92.50
+72.29
+63.25
+83.47
+75.07
+55.38
+75.16
+TransUNet [7]
+94.63
+89.86
+89.61
+77.28
+95.85
+88.95
+79.98
+85.06
+81.02
+59.76
+84.20
+UNETR [22]
+91.89
+89.07
+87.60
+66.97
+91.48
+83.18
+70.56
+82.92
+75.20
+57.53
+79.64
+Swin UNETR [61]
+92.23
+84.34
+82.95
+74.06
+94.91
+82.28
+71.17
+85.50
+79.18
+55.11
+80.17
+Universal Model
+93.94
+91.53
+90.21
+84.15
+96.25
+92.51
+82.72
+77.35
+79.64
+57.10
+84.54
+Table 5. Generalizability: Results on external datasets. We evaluate Universal Model and eight models on data from two external
+sources without additional fine-tuning or domain adaptation. mDSC* is the average dice score of the first seven organs. Compared with
+dataset-specific models, our Universal Model performs more robustly to CT scans taken from a variety of scanners, protocols, and institutes.
+5.2. Generalizability: External Datasets Results
+A key expectation of reliable medical AI models is their
+generalizability, i.e., performance on new data across many
+hospitals, rather than the performance tailored to a sin-
+gle dataset [42, 45]. Compared with dataset-specific mod-
+els, Universal Model was trained on the order of magni-
+tude more diverse CT scans, therefore demonstrating sig-
+nificantly better generalizability (i.e., directly testing the
+model on external data without adaptation or fine-tuning).
+We conduct the evaluation on a public dataset 3D-IRCADb
+and a private dataset JHH, which are absolutely not seen
+in the training and can be regarded as external validation.
+As shown in Table 5, Universal Model substantially outper-
+forms the previous methods on 3D-IRCADb and JHH with
+7
+
+pancreas_414_0000
+10cmpancreas_382_0000
+10cm382-00000000382
+0000S13B200000000
+cm682
+000082
+0000
+cm382
+0000pancreas_402_0000
+10cm402_00000000000000004020000DOOC0000PANCREAS 00030000
+A
+R
+10 cm
+PPANCREAS00030000
+A
+R
+1cmPANCREAS00030000
+A
+R
+1cmPANCREAS 00030000
+A
+R
+1cmPANCREAS 00030000
+A
+R
+1cmPANCREAS 00030000
+A
+R
+1cmPANCREAS OO030O00
+A
+R
+1cmPANCREAS OO030000
+A
+R
+1cmPANCREAS OO030000
+A
+R
+1cmMethod
+TotalSeg vertebrae
+TotalSeg cardiac
+TotalSeg muscles
+TotalSeg organs
+JHH cardiac
+JHH organs
+Scratch
+81.06
+84.47
+88.83
+86.42
+71.63
+89.08
+MedicalNet [10]
+82.28
+87.40
+91.36
+86.90
+58.07
+77.68
+Models Gen. [87]
+85.12
+86.51
+89.96
+85.78
+74.25
+88.64
+Swin UNETR [61]
+86.23
+87.91
+92.39
+88.56
+67.85
+87.21
+UniMiSS [70]
+85.12
+88.96
+92.86
+88.51
+69.33
+82.53
+Universal model
+86.49
+89.57
+94.43
+88.95
+72.06
+89.37
+Table 6. Transferability: Fine-tuning performance. Fine-tuning Universal Model significantly outperforms learning from scratch on
+two downstream datasets (i.e., TotalSegmentator and JHH). Moreover, Universal Model, trained by image segmentation as proxy task, can
+extract better visual representation—more related to segmentation tasks—than other pre-trained models developed in the medical domain.
+Due to the space, the per-class evaluation of TotalSegmentator and JHH can be found in Appendix Tables 9–12 and Table 13, respectively.
+Figure 6. Efficiency: FLOPs vs. DSC. We plot the average DSC
+score on the 6 MSD tasks against the FLOPs (Floating-point op-
+erations per second). The FLOPs is computed based on input with
+spatial size 96 × 96 × 96. The size of each circle indicates the
+number of parameters (‘#P’). In the inference, Universal Model is
+faster than nnU-Net (2nd best in performance) and Swin UNETR
+(3rd best) by 19× and 6× measured by FLOPs, respectively.
+a DSC improvement of 5% and 4%, respectively.
+5.3. Transferability: Fine-tuning Results
+Another property of Universal Model is serving as a
+powerful pre-training model for segmentation.
+Through
+pre-training by assembly dataset directly and fine-tuning
+to other datasets, the Universal Model achieves the high-
+est DSC compared with other pre-training methods with
+86.49%, 89.57%, 94.43% and 88.95% for four tasks in To-
+talSegmentator dataset, as shown in Table 6.
+Since the
+other unrelated pre-training tasks (reconstruction, coloriza-
+tion, jigsaw) can not capture the fine-grained information
+for segmentation.
+5.4. Extensibility: Incremental Learning Results
+Unlike existing models (e.g., [28, 84]), introducing ad-
+ditional classes (CLIP embeddings) would not affect the
+segmentation model architecture, allowing the Universal
+Model to perform incremental learning. As a result, Uni-
+versal Model is easily extended to upcoming datasets in
+the future with novel anatomical structures annotated. As
+a demonstration, our model can be easily adapted to seg-
+ment the renal veins in the JHH dataset (which are absent in
+the current public dataset). To segment the left and right re-
+nal veins, two new CLIP embeddings for the left and right
+renal veins are concatenated to the original CLIP embed-
+dings. Another text-driven segmentor for the new classes is
+also added to the model, and the segmentation output for the
+new classes is generated by this new segmentor. The other
+parameters of the model remain the same. In this way, the
+performance of the previously learned organs and tumors
+will not be affected. We perform continual fine-tuning to
+the Universal Model (pre-trained on 14 datasets) using the
+AdamW optimizer with a learning rate 1e−4 for 100 epochs.
+Our model achieved DSC 28.6% and 21.5% on the left and
+right renal veins, respectively.
+6. Conclusion
+In this work, we present a CLIP-Driven Universal Model
+for abdominal organ segmentation and tumor detection. We
+represent the first attempt to explore the potential of med-
+ical AI models in an assembly dataset, which consists of
+3,672 CT scans with 25 partially annotated organs and 6
+tumors. To solve the label inconsistency and orthogonality
+problem and build up an extensible framework, this work
+proposes a Universal Model with CLIP embedding to en-
+able a flexible and powerful segmentor, allowing the model
+to learn from the assembly of partially labeled datasets for
+high-performing organ segmentation and tumor detection
+(ranking first in both MSD and BTCV). More importantly,
+we have demonstrated that CLIP embedding can establish
+the anatomical relationship and enables the model to be
+extended to novel organs and tumors. Finally, the experi-
+ment results validate several clinically important merits of
+the CLIP-Driven Universal Model, such as compelling effi-
+ciency, generalizability, transferability, and extensibility.
+Acknowledgments. This work was supported by the Lust-
+garten Foundation for Pancreatic Cancer Research and Na-
+tional Natural Science Foundation of China (62001410).
+We thank Tong Li, Wenxuan Li for their feedback and con-
+structive suggestions at several stages of the project.
+8
+
+80
+75
+Swin UNETR
+Universal Model
+(#P 371.94 M)
+(#P 62.25 M)
+70
+DSC (%)
+nnU-Net
+(#P 370.74 M)
+UNETR
+65
+nnFormer
+(#P 555.71 M)
+(#P 894.48 M)
+60
+Worse
+DoDNet
+(#P 17.535M)
+55
+200
+400
+800
+1600
+3200
+6400
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+Appendix: CLIP-Driven Universal Model
+Abstract. In this supplementary material, we provide addi-
+tional information about the CLIP-Driven Universal Model
+and the assembly of 14 public datasets, as well as more
+detailed experimental results than those in the main paper.
+Appendix A discusses the influence of the medical prompt
+template. Appendix B provides the specifications for the
+assembly of datasets. Appendix C elaborates on the imple-
+mentation details, including the data augmentations, model
+network structures and evaluation metrics used in the main
+paper. Appendix D supplements the qualitative and quan-
+titative analysis in the main paper, including the visualiza-
+tion of kidney tumors and liver tumors, public results on the
+MSD leaderboard, and complete evaluation results of the
+transfer learning experiment. Finally, Appendix E visual-
+izes several open challenges discussed in §3.3 of the main
+paper.
+A. Medical Prompt Template
+To fully explore the effect of templates on CLIP embed-
+ding, an experiment is performed in the whole assembly of
+datasets as shown in Table 1. Four text templates are em-
+ployed to show the context, i.e., “V1: A computerized to-
+mography of a [CLS].”, “V2: There is [CLS] in this comput-
+erized tomography.”, “V3: This computerized tomography
+has a [CLS].”, “V4: A photo of a [CLS].”. The effective-
+ness of the prompt template is slightly different from the
+toy experiment. With increasing organ numbers, templates
+V1 and V2 still show better performance in encoding the
+relationship, whereas template V3 would deteriorate the re-
+sults. In addition, a widely used template V4 could also
+promote the segmentation performance.
+As known, the prompt template is a crucial factor for text
+model [40, 80]. How select an appropriate template is still
+an open problem for the medical image text-vision models.
+We encourage more future work to explore this area.
+B. Assembly of Datasets
+The assembly of datasets consists of 14 publicly avail-
+able datasets for training and 2 public datasets and 1 large-
+scale private dataset for testing (summarized in Table 7). It
+is non-trivial to assemble datasets annotated from various
+institutions since the annotation protocols are inconsistent.
+As mentioned in the main paper, we unify the label index for
+all datasets. The corresponding relationship is as follows.
+(Spleen, 1); (Right Kidney, 2); (Left Kidney, 3); (Gall Blad-
+der, 4); (Esophagus, 5); (Liver, 6); (Stomach, 7); (Aorta, 8);
+(Postcava, 9); (Portal Vein and Splenic Vein, 10); (Pancreas,
+11); (Right Adrenal Gland, 12); (Left Adrenal Gland, 13);
+(Duodenum, 14); (Hepatic Vessel, 15); (Right Lung, 16);
+(Left Lung, 17); (Colon, 18); (Intestine, 19); (Rectum, 20);
+(Bladder, 21); (Prostate/Uterus, 22); (Head of Femur Left,
+23); (Head of Femur Right, 24); (Celiac Truck, 25); (Kid-
+ney Tumor, 26); (Liver Tumor, 27); (Pancreas Tumor, 28);
+(Hepatic Vessel Tumor, 29); (Lung Tumor, 30); (Colon Tu-
+mor, 31); (Kidney Cyst, 32). Firstly, we map all the datasets
+into the standard index template. Then, for these datasets
+(KiTS, WORD, AbdomenCT-1K, and CT-ORG), which do
+not distinguish between the left and right organs, we split
+the organ (Kidney, Adrenal Gland, and Lung) into left part
+and right part through the script. In addition, we have taken
+the inclusion relation into consideration, e.g., the organ tu-
+mor is part of the organ, and the hepatic vessel is inside
+the liver. Fortunately, we formulate each organ segmenta-
+tion result as a binary mask. Thus, we can organize the
+segmentation ground truth for these overlapped organs in-
+dependently in a binary mask manner.
+Pancreas-CT [52] consists of 82 contrast-enhanced ab-
+dominal CT volumes. This dataset only provides the pan-
+creas label annotated by an experienced radiologist, and all
+CT scans have no pancreatic tumor.
+LiTS [3] contains 131 and 70 contrast-enhanced 3-D ab-
+dominal CT scans for training and testing, respectively. The
+data set was acquired by different scanners and protocols at
+six different clinical sites, with a largely varying in-plane
+resolution from 0.55 to 1.0 mm and slice spacing from 0.45
+to 6.0 mm.
+KiTS [25] includes 210 training cases and 90 testing cases
+with annotations provided by the University of Minnesota
+Medical Center. Each CT scan has one or more kidney tu-
+mors.
+AbdomenCT-1K [39] consists of 1112 CT scans from five
+datasets with liver, kidney, spleen, and pancreas annota-
+tions.
+CT-ORG [51] is composed of 140 CT images containing 6
+organ classes, which are from 8 different medical centers.
+Most of the images exhibit liver lesions, both benign and
+malignant.
+CHAOS [63] provides 20 patients for multi-organ segmen-
+tation. All CT scans have no liver tumor.
+MSD CT Tasks [1] includes liver, lung, pancreas, colon,
+hepatic vessel, and spleen tasks for a total of 947 CT scans
+with 4 organs and 5 tumors.
+BTCV [34] consists of 50 abdominal CT scans from
+metastatic liver cancer patients or post-operative ventral
+hernia patients. They are collected from the Vanderbilt Uni-
+versity Medical Center.
+AMOS22 [30] is the abbreviation of the multi-modality ab-
+dominal multi-organ segmentation challenge of 2022. The
+AMOS dataset contains 500 CT with voxel-level annota-
+tions of 15 abdominal organs.
+WORD [38] collects 150 CT scans from 150 patients be-
+fore the radiation therapy in a single center. All of them are
+13
+
+Datasets
+# Targets
+# Scans
+Annotated Organs or Tumors
+1. Pancreas-CT [52]
+1
+82
+Pancreas
+2. LiTS [3]
+2
+201
+Liver, Liver Tumor∗
+3. KiTS [25]
+2
+300
+Kidney, Kidney Tumor∗
+4. AbdomenCT-1K [39]
+4
+1000
+Spleen, Kidney, Liver, Pancreas
+5. CT-ORG [51]
+4
+140
+Lung, Liver, Kidneys and Bladder
+6. CHAOS [63]
+4
+40
+Liver, Left Kidney, Right Kidney, Spl
+7-11. MSD CT Tasks [1]
+9
+947
+Spl, Liver and Tumor∗, Lung Tumor∗, Colon Tumor∗, Pan and Tumor∗, Hepatic Vessel and
+Tumor∗
+12. BTCV [34]
+13
+50
+Spl, RKid, LKid, Gall, Eso, Liv, Sto, Aor, IVC, R&SVeins, Pan, RAG, LAG
+13. AMOS22 [30]
+15
+500
+Spl, RKid, LKid, Gall, Eso, Liv, Sto, Aor, IVC, Pan, RAG, LAG, Duo, Bla, Pro/UTE
+14. WORD [38]
+16
+150
+Spl, RKid, LKid, Gall, Eso, Liv, Sto, Pan, RAG, Duo, Col, Int, Rec, Bla, LFH, RFH
+15. 3D-IRCADb [57]
+13
+20
+Liv, Liv Cyst, RLung, LLung, Venous, PVein, Aor, Spl, RKid, LKid, Gall, IVC
+16. TotalSegmentator [68]
+104
+1,024
+Clavicula, Humerus, Scapula, Rib 1-12, Vertebrae C1-7, Vertebrae T1-9, Vertebrae L1-5, Hip,
+Sacrum, Femur, Aorta, Pulmonary Artery, Right Ventricle, Right Atrium, Left Atrium, Left Ven-
+tricle, Myocardium, PVein, SVein, IVC, Iliac Artery, Iliac Vena, Brain, Trachea, Lung Upper
+Lobe, Lung Middle Lobe, Lung Lower Lobe, AG, Spl, Liv, Gall, Pan, Kid, Eso, Sto, Duo, Small
+Bowel, Colon, Bla, Autochthon, Iliopsoas, Gluteus Minimus, Gluteus Medius, Gluteus Maximus
+17. JHH (private)
+21
+5,038
+Aor, AG, CBD, Celiac AA, Colon, duo, Gall, IVC, Lkid, RKid, Liv, Pan, Pan Duct, SMA, Small
+bowel, Spl, Sto, Veins, Kid LtRV, Kid RtRV, CBD Stent, PDAC∗, PanNET∗, Pancreatic Cyst∗
+Table 7. The information for an assembly of datasets. We have developed a Universal Model from an assembly of 1–14 public
+datasets. The official test and validation sets of Medical Segmentation Decathlon (MSD) and Beyond the Cranial Vault (BTCV) are used
+to benchmark the performance of organ segmentation (§4.1) and tumor detection (§4.2). 3D-IRCADb (15) and TotalSegmentator (16) are
+used for independent evaluation of model generalizability (§5.2), transferability (§5.3). In addition to public datasets, the Universal Model
+has also been evaluated on a large-scale private dataset (17), consisting of 5,038 CT scans with 21 annotated organs, to investigate the
+extensibility to new classes (§5.4). This list will continue to grow when more annotated datasets become available.
+scanned by a SIEMENS CT scanner without appearance en-
+hancement. Each CT volume consists of 159 to 330 slices
+of 512 × 512 pixels.
+3D-IRCADb [57] contains 20 venous phase enhanced CT
+scans. Each CT scan has various annotations, and only an-
+notated organs are tested to validate the model’s generaliz-
+ability.
+TotalSegmentator [68] collects 1024 CT scans randomly
+sampled from PACS over the timespan of the last 10 years.
+The dataset contains CT images with different sequences
+(native, arterial, portal venous, late phase, dual-energy),
+with and without contrast agent, with different bulb volt-
+ages, with different slice thicknesses and resolution and
+with different kernels (soft tissue kernel, bone kernel).
+JHH (private) contains 5038 CT scans with 21 annotated
+organs, where each case was scanned by contrast-enhanced
+CT in both venous and arterial phases, acquired on Siemens
+MDCT scanners. The JHH dataset is used to investigate the
+extensibility of new classes.
+C. Implementation Details
+C.1. Data Augmentation
+Our data augmentation is implemented in python with
+MONAI7. The orientation of CT scans is changed into spec-
+ified axcodes. Isotropic spacing is adopted to re-slice each
+7https://monai.io/
+Task
+SwinUNETR [61]
+Ours
+Task 03
+Liver
+94.12±2.34
+96.49±0.23
+Liver Tumor
+57.86±4.72
+71.94±3.74
+Task 06
+Lung Tmuor
+68.90±5.44
+67.15±5.81
+Task 07
+Pancreas
+80.06±0.83
+82.70±1.96
+Panc. Tumor
+52.53±3.76
+60.82±10.2
+Task 08
+Hepat. Ves.
+62.33±2.44
+62.55±3.64
+Ves. Tumor
+68.56±3.82
+69.39±2.29
+Task 09
+Spleen
+95.80±0.56
+96.71±0.21
+Task 10
+Col. Tumor
+50.45±10.1
+62.14±17.8
+Table 8.
+The 5-fold cross-validation performance on MSD.
+These are the tabular comparison between Universal Model and
+Swin UNETR [61] (previously ranked first on the MSD leader-
+board). The performance is evaluated by DSC scores.
+scan to the same voxel size of 1.5 × 1.5 × 1.5mm3. We
+truncate the intensity in each scan to the range [−175, 250]
+and linearly normalize them to [0, 1]. Considering the valid
+part is part of the whole medical image, we crop only the
+foreground object based on the images. During training, we
+crop random fixed-sized 96×96×96 regions with the center
+being a foreground or background voxel based on the pre-
+defined ratio. Also, we randomly rotate the input patch by
+90 degrees and shift intensity with 0.1 offset with 0.1 and
+0.2 probability. To avoid confusion between the organ in the
+right and left parts, we do not use mirroring augmentation.
+14
+
+C.2. Network Structures
+Text branch. We apply the pre-trained text encoder “ViT-
+B/32” of the CLIP as the text branch8. We can extract and
+store the text features to reduce overhead brought by the text
+encoder in the training and inference stage since the CLIP
+embedding only depends on the dictionary, which is fixed.
+Vision branch. We adopt Swin UNETR as a vision en-
+coder.
+The Swin UNETR consists of 4 attention stages
+comprising 2 transformer blocks and 5 convolution stages
+comprising of CNN-based structure. In the attention stage,
+a patch merging layer is used to reduce the resolution by
+a factor of 2. Stage 1 consists of a linear embedding layer
+and transformer blocks that maintain the number of tokens
+as H
+2 × W
+2 × D
+2 . a patch merging layer groups patches with
+resolution 2 × 2 × 2 and concatenates them, resulting in a
+4C-dimensional feature embedding. A linear layer is then
+used to down-sample the resolution by reducing the dimen-
+sion to 2C. The same procedure continues in stages 2, 3,
+and 4 [61]. The text-based controller is a single convolu-
+tional layer, which takes the CLIP embedding and global
+pooling feature from the last convolution stages in the vi-
+sion encoder as input.
+C.3. Evaluation Metrics
+The Dice similarity coefficient (DSC) and Normalized
+Surface Distance (NSD) are used as measurements for 3D
+segmentation results. The DSC metric is defined as:
+DSC =
+2 �I
+i=1 Yi ˆYi
+�I
+i=1 Yi + �I
+i=1 ˆYi
+,
+(3)
+where Y and ˆY denote the ground truth and prediction of
+voxel values. The details of Normalized Surface Distance
+(NSD) could refer to Sec. 4.6 in [44].
+D. Additional Evaluations
+Table 8 shows the detailed numerical result between
+Universal Model and Swin UNETR. Tables 9–12 and Ta-
+ble 13 show the per-class evaluation of TotalSegmentator
+and JHH, which validates the transferability of the proposed
+Universal Model. Figure 7 exhibits the contour line compar-
+ison among Universal Model and two human experts. We
+can see the model predictions are roughly similar to human
+annotation, which validates the effectiveness of the pseudo
+label generated by our Universal Model. Figure 9 and Fig-
+ure 8 shows several kidney and liver tumor cases compar-
+ison among the proposed Universal Model and four com-
+petitive baseline methods. Our method can not only detect
+small and big tumors in various organs but also not generate
+false positives of tumors.
+8https://github.com/openai/CLIP
+E. Discussion of Open Challenges
+The first open challenge is the inconsistent annotation
+protocol. The annotation standard is different from institu-
+tion to institution. For example, the aorta annotation is dif-
+ferent in AbdomenCT-12organ and other datasets, as shown
+in Figure 10. A part of the upper aorta region is missing
+in AbdomenCT-12organ, while the annotation is complete
+in BTCV and AMOS. This open challenge requires several
+experienced radiology experts to re-annotate the CT scans.
+The second challenge is the long tail problem. We count
+the proportion of each organ and tumor in Figure 11. The
+assembly of datasets has a severe long-tail distribution,
+which would lead to unsatisfactory performance of tumor
+classes.
+Mitigating the long-tail distribution would con-
+tribute to more robust detection of the tumor.
+15
+
+Figure 7. Contour line comparison among pseudo labels and two human experts. The red line represents the annotation from Doctor
+1; green line indicates the annotation from Doctor 2; blue line shows the results generated by Universal Model. Examples of CT scans
+annotated by our pseudo labels and two human experts with contour line comparison. The prediction results of these organs generated by
+the medical model are comparable with human experts.
+16
+
+Case
+2
+Case
+3
+Case
+4
+Case
+5
+Case
+Spleen
+Liver
+Stomach
+Gall Bladder
+PancreasCT scan
+ground
+truth
+ours
+Swin
+UNETR
+zoom-in
+Length: 8.5mm 
+Length: 11.8mm 
+Length: 10.5mm 
+UNETR
+nnU-Net
+nnFormer
+Length: 22.1mm 
+UNesT
+Length: 21.3mm 
+Figure 8. Liver tumor detection. Qualitative visualizations of the proposed Universal Model and four competitive baseline methods. We
+review the detection results of tumors from smaller to larger sizes (Rows 1–4). The Universal Model succeeds in detecting small tumors
+ignored by other methods and in detecting multiple tumors in one CT. In addition, it avoids the false positive prediction, which validates
+the good practicability of Universal model.
+CT scan
+ground
+truth
+ours
+Swin
+UNETR
+zoom-in
+Length: 27.5mm 
+Length: 17.8mm 
+Length: 137.2mm 
+UNETR
+nnU-Net
+nnFormer
+Length: 31.5mm 
+UNesT
+Figure 9. Kidney tumor detection. Qualitative visualizations of the proposed Universal Model and four competitive baseline methods.
+We review the detection results of tumors from smaller to larger sizes (Rows 1–4). The Universal Model can detect well not only on the
+kidneys (red region), but also kidney tumors (green region) and cysts (blue region).
+17
+
+img_liver_40
+R
+10 cmimg_liver_40
+cmimg_liver 92
+A
+R
+cmimgliver31
+A
+R
+cmimg_liver_40
+R
+CImg_liver 92
+R
+cmimg_liver_92
+A
+R
+1
+10 cmimgver31
+P
+R
+cmimg_liver_40
+:
+cmimg_liver_31
+R
+10cmimg_liver 92
+A
+R
+cmimgliver31
+A
+R
+cmimg_liver_40
+R
+cmimg_liver 92
+A
+R
+cmimgliver31
+R
+cmimg_liver_40
+:
+cmimg_liver 92
+A
+R
+cmimgliver31
+A
+R
+cmimg_liver_73
+R
+L
+10 cm
+Pimg_liver_73
+A
+R
+工
+P
+1cmimg_liver_40
+cmimg_liver_73
+A
+R
+工
+P
+1cmimg_liver_73
+A
+R
+工
+P
+1cmimg_liver_73
+A
+R
+cmimg_liver_73
+A
+R
+工
+P
+1cmimg_liver_73
+A
+R
+工
+P
+1cmimg_liver_73
+A
+R
+工
+P
+1cmimg_liver 92
+A
+R
+cmimg_liver_40
+:
+cmimg_liver 92
+A
+R
+cmimg_liver_73
+A
+R
+工
+P
+1cmimgliver31
+A
+R
+cmimg_liver_40
+Cimg_liver 92
+A
+R
+cmimgliver31
+R
+cmimg_img0026
+A
+R
+10 cm
+Pimg_img0026
+R
+1cmimg_img0182
+R
+cmimg_img0042
+10cmimg_img0026
+R
+1cmimg_img0182
+R
+cmimg_img0182
+A
+R
+10cmimg_img0042
+10cmimg_img0026
+R
+Fcmimg_img0042
+R
+10 cmimg_img0182
+R
+cmimg_img0042
+10cmimg_img0026
+R
+1cmimg_img0182
+R
+cmimg_img0042
+10cmimg_img0026
+R
+1cmimg_img0182
+R
+cmimg_img0042
+10 cmimg_img0155
+R
+10 cmimg.imgo155
+R
+cmimg_img0026
+R
+Fcmimg.imgo155
+R
+cmimg.imgo155
+R
+1cmimg.imgo155
+R
+cmimg.imgo155
+R
+cmimg.imgo155
+R
+cmimg.imgo155
+R
+cmimg_img0182
+R
+cmimg_img0026
+R
+1cmimg_img0182
+R
+cmimg_img0042
+10 cmimg.imgo155
+R
+1cmimg_img0042
+10 cmimg_img0026
+R
+Fcmimg_img0182
+A
+R
+cmimg_img0042
+10 cmMethod
+L5
+L4
+L3
+L2
+L1
+T12
+T11
+T10
+T9
+T8
+T7
+T6
+Scratch
+86.68
+88.37
+89.83
+84.28
+91.98
+87.45
+88.29
+86.78
+83.50
+75.70
+77.73
+75.84
+MedicalNet [10]
+91.72
+91.01
+86.03
+84.73
+91.52
+89.98
+89.06
+89.35
+85.71
+82.99
+81.54
+79.74
+Models Gen. [87]
+89.64
+89.24
+89.38
+82.85
+90.79
+88.62
+90.11
+90.43
+89.22
+85.21
+80.83
+77.40
+Swin UNETR [61]
+89.56
+90.80
+93.08
+86.38
+94.35
+89.65
+92.02
+91.99
+89.65
+82.20
+85.01
+81.06
+UniMiSS [70]
+89.20
+91.21
+94.16
+86.61
+91.57
+87.29
+90.18
+90.56
+88.09
+83.47
+80.73
+76.40
+Universal Model
+88.95
+91.38
+93.82
+87.04
+93.53
+88.96
+90.50
+91.40
+89.18
+84.25
+83.63
+79.95
+Method
+T5
+T4
+T3
+T2
+T1
+C7
+C6
+C5
+C4
+C3
+C2
+C1
+Average
+Scratch
+73.14
+72.26
+77.12
+80.36
+85.76
+83.39
+69.80
+70.23
+69.82
+85.74
+83.35
+78.18
+81.06
+MedicalNet [10]
+77.28
+76.60
+76.57
+80.94
+85.54
+83.05
+76.05
+73.04
+80.55
+74.35
+74.67
+72.91
+82.28
+Models Gen. [87]
+79.59
+78.73
+82.01
+84.63
+90.02
+88.20
+81.09
+78.90
+78.21
+89.69
+88.06
+80.23
+85.12
+Swin UNETR [61]
+82.33
+77.74
+81.78
+83.53
+88.22
+87.81
+78.38
+80.36
+83.00
+92.68
+87.97
+80.16
+86.23
+UniMiSS [70]
+78.97
+76.60
+82.33
+85.14
+90.04
+88.68
+79.18
+79.17
+79.00
+88.19
+86.38
+79.80
+85.12
+Universal Model
+83.07
+78.67
+82.97
+86.06
+90.67
+88.75
+77.03
+80.87
+83.05
+92.94
+88.20
+80.87
+86.49
+Table 9. The complete evaluation of TotalSeg vertebrae. The results are evaluated by DSC. Our Universal Model represents the best
+transferability.
+Method
+esophagus
+trachea
+HM
+HA left
+HV left
+HA right
+HV right
+PA
+brain
+Scratch
+84.73
+90.72
+85.53
+91.78
+91.15
+90.10
+88.25
+87.20
+93.79
+MedicalNet [10]
+89.43
+94.08
+88.71
+93.50
+92.17
+90.90
+90.83
+89.51
+95.11
+Models Gen. [87]
+87.96
+93.47
+87.40
+93.61
+92.23
+92.02
+89.74
+89.34
+94.99
+Swin UNETR [61]
+89.77
+94.37
+88.85
+94.42
+92.99
+92.61
+90.40
+88.91
+95.14
+UniMiSS [70]
+90.45
+94.51
+90.29
+94.34
+93.70
+93.10
+91.46
+89.67
+94.99
+Universal Model
+90.97
+94.71
+90.88
+94.64
+93.72
+93.30
+91.66
+90.80
+95.34
+Method
+IA left
+IA right
+IV left
+IV right
+small bow.
+duodenum
+colon
+UB
+face
+Average
+Scratch
+80.32
+79.78
+79.80
+81.69
+81.97
+72.21
+82.51
+89.59
+69.40
+84.47
+MedicalNet [10]
+87.06
+84.90
+86.93
+86.46
+83.14
+72.01
+84.22
+90.43
+73.85
+87.40
+Models Gen. [87]
+85.71
+83.09
+85.77
+85.79
+81.75
+69.37
+85.25
+90.31
+69.42
+86.51
+Swin UNETR [61]
+88.26
+86.44
+87.13
+87.59
+83.29
+70.71
+87.50
+89.93
+74.08
+87.91
+UniMiSS [70]
+89.18
+87.81
+89.04
+88.55
+84.83
+74.74
+88.16
+91.83
+74.76
+88.96
+Universal Model
+89.89
+88.54
+89.58
+89.27
+84.85
+76.23
+89.06
+92.07
+76.81
+89.57
+Table 10. The complete evaluation of TotalSeg cardiac. The results are evaluated by DSC. Our Universal Model represents the best
+transferability. The abbreviation in the table is listed as follows. HM (heart myocardium), HA (heart atrium), HV (heart ventricle), PA
+(pulmonary artery), IA (iliac artery), IV (iliac vena), UB (urinary bladder).
+Method
+Humerus L Humerus R Scapula L
+Scapula R
+Clav. L
+Clav. R
+Femur L Femur R
+Hip L
+Hip R
+Sacrum
+Scratch
+84.27
+84.44
+91.71
+89.78
+80.38
+75.81
+93.41
+93.02
+92.90
+88.66
+83.63
+MedicalNet [10]
+87.25
+85.67
+88.68
+92.62
+94.35
+93.96
+84.85
+96.59
+96.98
+96.31
+95.19
+Models Gen. [87]
+90.61
+79.73
+88.56
+92.06
+91.19
+92.57
+86.08
+93.57
+85.35
+82.40
+87.91
+Swin UNETR [61]
+88.32
+86.35
+90.82
+93.88
+94.90
+94.52
+85.92
+97.71
+97.42
+97.49
+95.73
+UniMiSS [70]
+89.73
+92.30
+91.72
+94.77
+94.57
+93.66
+84.92
+97.67
+97.35
+97.11
+96.18
+Universal Model
+91.32
+93.87
+93.11
+95.59
+95.00
+95.88
+86.79
+98.48
+98.04
+98.32
+96.94
+Method
+GMa L
+GMa R
+GMe L
+GMe R
+GMi L
+GMi R
+Aotu. L
+Aotu. R
+Iliopsoas L Iliopsoas R Average
+Scratch
+95.53
+91.78
+85.27
+94.80
+86.54
+93.01
+95.17
+93.44
+87.99
+83.95
+88.83
+MedicalNet [10]
+94.69
+95.72
+92.17
+89.15
+89.76
+90.77
+94.45
+94.24
+80.29
+84.94
+91.36
+Models Gen. [87]
+96.19
+92.06
+90.07
+94.99
+92.12
+92.60
+95.86
+95.93
+85.64
+83.82
+89.96
+Swin UNETR [61]
+95.32
+96.34
+93.57
+89.87
+90.75
+91.74
+95.16
+94.86
+83.53
+86.00
+92.39
+UniMiSS [70]
+95.53
+96.37
+93.80
+90.28
+90.87
+93.02
+95.17
+95.48
+85.71
+84.02
+92.86
+Universal Model
+96.68
+96.99
+95.55
+91.36
+93.19
+94.52
+96.31
+96.34
+86.92
+88.89
+94.29
+Table 11. The complete evaluation of TotalSeg muscles. The results are evaluated by DSC. Our Universal Model represents the best
+transferability. The abbreviation in the table is listed as follows. Clav. (Clavicula), GMa (gluteus maximus), GMe (gluteus medius), GMi
+(gluteus minimus), Aotu. (Autochthon)
+18
+
+Method
+spleen
+Kidney R
+Kidney L
+gallbladder
+liver
+stomach
+aorta
+IVC
+PSV
+Scratch
+93.58
+94.09
+87.73
+73.86
+96.79
+89.17
+90.68
+82.10
+71.35
+MedicalNet [10]
+95.54
+92.43
+90.86
+79.36
+97.10
+91.53
+90.12
+86.18
+73.34
+Models Gen. [87]
+95.60
+94.37
+88.51
+78.39
+97.39
+91.68
+93.18
+85.94
+74.58
+Swin UNETR [61]
+89.77
+94.37
+88.85
+74.42
+92.99
+92.61
+90.40
+88.91
+75.14
+UniMiSS [70]
+95.78
+94.75
+89.35
+79.14
+97.39
+91.87
+93.50
+86.19
+75.26
+Universal Model
+96.24
+94.67
+91.43
+81.48
+97.63
+92.76
+92.22
+87.87
+76.10
+Method
+pancreas
+AG R
+AG L
+LUL L
+LLL L
+LUL R
+LML R
+LLL R
+Average
+Scratch
+80.80
+78.94
+72.83
+95.88
+91.66
+87.17
+88.91
+93.71
+86.42
+MedicalNet [10]
+83.11
+79.15
+69.22
+93.64
+89.88
+86.38
+87.08
+92.40
+86.90
+Models Gen. [87]
+82.97
+83.05
+75.49
+95.79
+92.90
+90.10
+91.06
+94.65
+85.78
+Swin UNETR [61]
+85.24
+81.86
+74.33
+95.06
+92.16
+88.37
+89.45
+94.04
+88.56
+UniMiSS [70]
+82.11
+79.37
+73.12
+96.08
+93.18
+90.31
+91.99
+95.43
+88.51
+Universal Model
+85.21
+82.25
+75.01
+95.04
+92.28
+88.21
+89.69
+94.06
+88.95
+Table 12. The complete evaluation of TotalSeg organs. The results are evaluated by DSC. Our Universal Model represents the best
+transferability. The abbreviation in the table is listed as follows. IVC (inferior vena cava), PSV (portal vein and splenic vein), AG (adrenal
+gland), LUL (lung upper lobe), LLL (lung lower lobe), LML (lung middle lobe)
+Method
+spleen
+Kidney R
+Kidney L
+gallbladder
+liver
+stomach
+Scratch
+95.66
+94.43
+93.69
+86.14
+96.74
+94.30
+MedicalNet [10]
+91.08
+88.63
+86.60
+61.23
+93.29
+88.22
+Models Gen. [87]
+95.02
+93.44
+93.07
+84.73
+94.12
+94.05
+Swin UNETR [61]
+94.71
+93.95
+92.27
+81.75
+96.00
+92.79
+UniMiSS [70]
+88.35
+91.49
+90.41
+82.91
+93.80
+89.57
+Universal Model
+95.98
+94.71
+94.00
+87.18
+96.87
+94.50
+Method
+aorta
+IVC
+pancreas
+PSV
+AG
+CAA
+Average
+Scratch
+87.68
+79.73
+85.03
+68.48
+66.61
+50.61
+81.98
+MedicalNet [10]
+83.27
+75.32
+70.67
+46.82
+41.69
+26.87
+68.88
+Models Gen. [87]
+89.46
+81.50
+84.23
+71.79
+70.46
+54.23
+82.81
+Swin UNETR [61]
+87.43
+80.89
+81.19
+66.71
+65.04
+36.38
+79.55
+UniMiSS [70]
+88.50
+77.98
+71.86
+61.68
+51.82
+49.16
+76.10
+Universal Model
+88.36
+79.98
+85.82
+69.38
+65.88
+50.53
+82.24
+Table 13. The complete evaluation of JHH. The results are evaluated by DSC. IVC (inferior vena cava), PSV (portal vein and splenic
+vein), AG (adrenal gland), CAA (celiac abdominal aorta)
+19
+
+AbdomenCT-12organ
+BTCV
+AMOS
+Ground Truth
+Zoom-in
+CT Scan
+Figure 10. Inconsistent Label Protocol. The aorta annotation standard is inconsistent in AbdomenCT-12organ and other datasets. A part
+of the upper aorta region is missing in AbdomenCT-12organ, while the aorta annotation is complete in BTCV and AMOS.
+20
+
+Organ12_0003_0000
+R
+10 cmimg0002
+S
+R
+10cmimg0002
+R
+I
+cmimg0002
+R
+I
+cmOrgan12_0003_0000
+R
+cmOrgan12_0003_0000
+R
+cmOrgan12_0034_0000
+S
+R
+10 cmOrgan12_0034L0000
+S
+ROrgan12_0034L0000
+Samos_0036
+R
+10.cmamos_0036
+S
+R
+cmamos_0036
+S
+R
+cmFigure 11. The proportion of 32 classes. We observe that the assembly of datasets presents severe long-tail distribution.
+21
+
+Liver
+Right Lung
+Left Lung
+Intestine
+Colon
+Kidney Tumor
+Hepatic Vessel Tumor
+Spleen
+Pancreas
+Right Head of Femur
+Left Head of Femur
+Stomach
+Bladder
+Right Kidney
+Liver Tumor
+Left Kidney
+Prostate
+Hepatic Vessel
+Duodenum
+Rectum
+Postcava
+Arota
+Pancreas Tumor
+Gall Bladder
+Lung Tumor
+Kidney Cyst
+Colon Tumor
+Esophagus
+Portal Vein and Splenic Vein
+Left Adrenal Gland
+Right Adrenal Gland
+Celiac Truck
+0
+00
+8T0
+80'0
+0.12
+Percentage

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+arXiv:2301.01966v1  [math.PR]  5 Jan 2023
+Noname manuscript No.
+(will be inserted by the editor)
+Ruin Probabilities for a Sparre Andersen Model with
+Investments: the Case of Annuity Payments
+Yuri Kabanov · Platon Promyslov
+January 6, 2023
+Dedicated to the memory of Tomas Bj¨ork.
+Abstract This note is a complement to the paper by Eberlein, Kabanov, and Schmidt
+on the asymptotic of the ruin probability in a Sparre Andersen non-life insurance
+model with investments a risky asset whose price follows a geometric L´evy process.
+Using the techniques of semi-Markov processes we extend the result of the mentioned
+paper to the case of annuities and models with two-sided jumps.
+Keywords Ruin probabilities · Sparre Andersen model · Actuarial models with
+investments · Renewal processes · Annuities · Distributional equations
+Mathematics Subject Classification (2010) 60G44
+JEL Classification G22 · G23
+1 Introduction
+In the classical Sparre Andersen model of insurance company the counts of claims
+form a renewal process. In recent studies, see [1], [2] and references therein, this
+model was enriched by the assumption that the capital reserve of the insurance com-
+pany is fully invested in a risky asset whose price evolves as a geometric L´evy pro-
+cess. In the paper [2] by Eberlein, Kabanov, and Schmidt it was considered the non-
+life insurance version of such a model. It was shown that under rather mild hypothe-
+Lomonosov Moscow State University, Federal Research Center “Computer Science and Control” of
+the Russian Academy of Sciences Moscow, Russia, and Universit´e de Franche-Comt´e, Laboratoire de
+Math´ematiques, UMR CNRS 6623, 16 Route de Gray, 25030 Besanc¸on, France
+E-mail: [email protected].
+Lomonosov Moscow State University, Moscow, Russia
+E-mail: [email protected]
+
+2
+Yuri Kabanov, Platon Promyslov
+ses on the business process the asymptotic behavior of the (ultimate) ruin probability
+is essentially the same as in the Cram´er–Lundberg model with risky investments.
+Namely, the ruin probability decays, up to a multiplicative constant, as the function
+u−β where u, the initial capital, tends to infinity. The decay rate β depends only of
+characteristics of the price process. The method of analysis in [2] is based heavily on
+the assumption that the risk process has only downward jumps and, therefore, crosses
+the zero level only by a jump. This specific feature allows a straightforward reduction
+to a discrete-time Markovian framework.
+The approach of [2] left an open question whether the results hold also in the case
+of upward jumps. This is a feature of the annuity model when the risk process crosses
+the zero level in a continuous way. In a less popular mixed model with two-sided
+jumps the crossing may happen in both way. Of course, a positive answer is expected
+here: this was already established for the Cram´er–Lundberg models with investments
+analyzed by Kabanov, Pergamenshchikov,and Pukhlyakov, [5], [7], as well as in very
+general L´evy Ornstein–Uhlenbeck models introduced and studied by Paulsen, see [8],
+[9], [10], [11], and a more recent paper [6] by Kabanov and Pergamenshchikov.
+Our note, based on the study [2], gives a positive answer for a Sparre Andersen
+model with investments in its annuity version, with the upward jumps. We discuss
+briefly the needed changes leading to a result for a model with upward and downward
+jumps used serving to describe the evolution of the capital reserve of a company with
+two types of the business activity. Our techniques is based on the imbedding of a
+semi-Markov process into a Markov one by increasing the dimensionality.
+In the paper we use standard notations of stochastic calculus and concepts dis-
+cussed in details in [6], [2].
+2 The model
+The Sparre Andersen model with risky investments considered contains two ingredi-
+ents:
+1. The price process of a risky financial asset S = (St)t≥0. It is of the form
+S = E(R) where E is the stochastic exponential, R is a L´evy process with the L´evy
+triplet (a, σ2, Π) and such that Π((−∞, −1]) = 0. The latter condition ensures that
+the jumps ∆R > −1, hence, the price S > 0. In such a case, S = eV where V = ln S
+is again a L´evy process which can be given by the formula
+Vt = at − 1
+2σ2t + σWt + h ∗ (µ − ν)t + (ln(1 + x) − h) ∗ µt,
+(2.1)
+where h(x) := xI{|x|≤1}. The L´evy triplet of V is (aV , σ2, ΠV ) with
+aV = a − σ2
+2 + Π(h(ln(1 + x)) − h)
+and ΠV = Πϕ−1, ϕ : x �→ ln(1 + x).
+It is assume that R is non-deterministic, that is, at least one of the parameters σ2
+or Π is not zero.
+
+Ruin Probabilities for a Sparre Andersen Model with Investments
+3
+2. The ”business process“. It is an independent of S compound renewal process
+P = (Pt). Classically, it can be written in the form
+Pt = ct +
+Nt
+�
+i=1
+ξi,
+(2.2)
+where N = (Nt) is a counting renewal process with the interarrival times (lengths
+of the inter jump intervals) Ui := Ti − Ti−1, i ≥ 2, forming an i.i.d. sequence
+independent of the i.i.d. sequence of random variables ξi = ∆PTi, i ≥ 1, with the
+common law Fξ, Fξ({0}) = 0. In the sequel, a “generic“ r.v. with such a law is
+denoted by ξ. As usual, T0 := 0. The common law of Ui we denoted by F and use
+the same character for its distribution function.
+The risk process X = Xu, u > 0, is defined as the solution of the non-homogene-
+ous linear stochastic equation
+Xt = u +
+� t
+0
+Xs−dRs + Pt.
+(2.3)
+The ruin probability is the function of the initial capital Ψ(u) := P[τ u < ∞]
+where τ u := inf{t : Xu
+t ≤ 0}.
+The cases of major interest are: c > 0 and ξi < 0 (a non-life insurance model,
+considered in [2]) and c < 0 and ξi > 0 (annuities payments model). The latter case
+studied here, is often interpreted as a model of venture company paying salary and
+selling innovations. The case where Fξ charges both half-axes can be viewed as a
+model of company combined two types of activity, see, e.g., [1]. and we study also
+the case where ξi may take positive and negative values.
+If c ≥ 0 and ξ > 0 the ruin never happens and this case is excluded from consid-
+erations.
+Standing assumption. The cumulant generating function H : q → ln E e−qVT1
+of the random variable VT1 has a root β > 0 not laying on the boundary of the
+effective domain of H. That is, if the int dom H = (q, ¯q), there is a unique root
+β ∈ (0, ¯q).
+We are looking for conditions under which
+0 < lim inf
+u→∞ uβΨ(u) ≤ lim sup
+u→∞ uβΨ(u) < ∞.
+(2.4)
+The paper [2] treats the case of the non-life insurance. We formulate its main
+result in a more transparent form.
+Theorem 2.1 ([2]) Suppose that the drift c ≥ 0, the law Fξ is concentrated on
+(−∞, 0), E[|ξ|β] < ∞, and E[eεT1] < ∞ for some ε > 0. Then (2.4) holds if at
+least one of the following conditions are fulfilled:
+1. σ ̸= 0 or ξ is unbounded from below.
+2a. Π((−1, 0)) > 0 and Π((0, ∞)) > 0.
+2b. Π((−1, 0)) = 0 and Π(h) = ∞.
+2c. Π((0, ∞)) = 0 and Π(|h|) = ∞.
+2d. Π((−∞, 0)) = 0, 0 < Π(h) < ∞, F((0, t)) > 0 for every t > 0.
+2e. Π((0, ∞)) = 0, 0 < Π(|h|) < ∞, F((0, t)) > 0 for every t > 0.
+
+4
+Yuri Kabanov, Platon Promyslov
+The proof in [2] used heavily the assumption that the business process has a posi-
+tive drift and negative claims corresponding to the non-life insurance setting. In such
+a case the ruin may happen only at an instant of jump and, therefore, one needs to
+monitor the risk process only at T1, T2, and so on. Such a reduction to a discrete-time
+ruin model does not work if ξi > 0.
+In our paper we consider the annuity model of the Sparre Andersen type where
+the ruin occurs because exhausting resources and the risk process reaches zero in a
+continuous way. The main result can be formulated as follows.
+Theorem 2.2 Suppose that the drift c < 0, the law Fξ is concentrated on (0, ∞),
+E[ξβ] < ∞, and E[eεT1] < ∞ for some ε > 0. Then (2.4) holds if at least one of the
+following conditions are fulfilled:
+1. σ ̸= 0.
+2a. Π((−1, 0)) > 0 and Π((0, ∞)) > 0.
+2b. Π((−1, 0)) = 0 and Π(h) = ∞.
+2c. Π((0, ∞)) = 0 and Π(|h|) = ∞.
+2d. Π((−1, 0)) = 0, 0 < Π(h) < ∞, F((t, ∞)) > 0 for every t > 0.
+2e. Π((0, ∞)) = 0, 0 < Π(|h|) < ∞, F((t, ∞)) > 0 for every t > 0.
+For the mixed case we have the following result.
+Theorem 2.3 Suppose that the drift c ∈ R, the law Fξ charges both half-lines
+(−∞, 0) and (0, ∞), E[|ξ|β] < ∞, and E[eεT1] < ∞ for some ε > 0. Then (2.4)
+holds if at least one of the following conditions are fulfilled:
+1. σ ̸= 0 or |ξ| is unbounded.
+2a. Π((−1, 0)) > 0 and Π((0, ∞)) > 0.
+2b. Π((−1, 0)) = 0 and Π(h) = ∞.
+2c. Π((0, ∞)) = 0 and Π(|h|) = ∞.
+2d. Π((−1, 0)) = 0, 0 < Π(h) < ∞, F((t, ∞)) > 0 for every t > 0 in the case
+c < 0 and Fξ((0, ε)) > 0 for every ε > 0 in the case c ≥ 0.
+2e. Π((0, ∞)) = 0, 0 < Π(|h|) < ∞, F((t, ∞)) > 0 for every t > 0 in the case
+c < 0 and Fξ((0, ε)) > 0 for every ε > 0 in the case c ≥ 0.
+The annuity and the mixed setting require a different approach inspired by the the-
+ory of semi-Markov processes. Namely, we consider the business process P as the
+component of the two-dimensional Markov process (P, D) where the second compo-
+nent D = Dr is a “clock”, i.e. a process measuring the elapsed time after the instant
+of last claim. We assume that the law of U = T1 may be different from the common
+law of the further interarrival times: at the instant zero a portion r of the interarrival
+time is already elapsed. This feature admits obvious justifications: e.g., the venture
+company may change the governance when a project was still in progress.
+Here and throughout the paper we use the superscript r to emphasize that the law
+of a random variable or a process depends on r, skipping usually r = 0.
+Formally, the “clock”, Dr = (Dr
+t ), is a process with the initial value Dr
+0 = r,
+Dr
+t = r+t on the interval [0, T1), and Dr
+t := t−T r
+n on all other interarrival intervals
+[T r
+n, T r
+n+1), n ≥ 1. That is, the “clock” restarts from zero at each instant T r
+n. We
+
+Ruin Probabilities for a Sparre Andersen Model with Investments
+5
+denote by F r the law of the first interarrival time T r
+1 = T r
+1 − T0. In accordance with
+our convention F 0 = F.
+Alternatively, Dr can be representing as the solution of the linear equation
+Dr
+t = r + t −
+�
+[0,t]
+Dr
+s−dNs.
+Typically, P[T r
+1 > t] = P[Ti > t + r]/P[Ti > r], i > 1. In the case of
+exponential distribution F r = F for all r ≥ 0 (“absence of memory”).
+We assume that F r ≥ F.
+Recall that the assumed independence of P r and R implies that the joint quadratic
+characteristic [P r, R] is zero and the ruin process Xu,r can be written in the form
+resembling the Cauchy formula for solutions of liner differential equations:
+Xu,r
+t
+= eVt(u − Y r
+t ),
+(2.5)
+where
+Y r
+t := −
+�
+(0,t]
+E−1
+s− (R)dP r
+s = −
+�
+(0,t]
+e−Vs−dP r
+s .
+(2.6)
+The strict positivity of the process E(R) = eV implies that the ruin time
+τ u,r := inf{t ≥ 0 : Xu,r
+t
+≤ 0} = inf{t ≥ 0 : Y r
+t ≥ u}.
+The crucial element of our study is the following
+Lemma 2.4 Suppose that Y r
+t → Y r
+∞ almost surely as t → ∞ where Y r
+∞ is a finite
+random variable such that ¯G(u, r) := P[Y r
+∞ > u] > 0 for every u > 0 and r ≥ 0. If
+¯G∗ := infq ¯G(0, q) > 0, then
+¯G(u, r) ≤ Ψ(u, r) =
+¯G(u, r)
+E
+� ¯G(Xu,r
+τ u,r, Dr
+τ u,r) | τ u,r < ∞
+� ≤ 1
+¯G∗
+¯G(u, r).
+(2.7)
+Proof. Let τ be an arbitrary stopping time with respect to the filtration (FP,D,R
+t
+). As
+we assume that the finite limit Y r
+∞ exists, the random variable
+Y r
+τ,∞ :=
+�
+− limN→∞
+�
+(τ,τ+N] e−(Vs−−Vτ )dP r
+s ,
+τ < ∞,
+0,
+τ = ∞,
+is well defined. On the set {τ < ∞}
+Y r
+τ,∞ = eVτ (Y r
+∞ − Y r
+τ ) = Xu,r
+τ
++ eVτ (Y r
+∞ − u).
+(2.8)
+Let ζ be a FP,D,R
+τ
+-measurable random variable.
+Using the strong Markov property, we get that
+P
+�
+Y r
+τ,∞ > ζ, τ < ∞
+�
+= E
+� ¯G(ζ, Dr
+τ)I{τ<∞}
+�
+(2.9)
+
+6
+Yuri Kabanov, Platon Promyslov
+Noting that Ψ(u, r) := P [τu,r < ∞] ≥ P [Y r
+∞ > u] > 0, we deduce from here
+using (2.8) that
+¯G(u, r) = P [Y r
+∞ > u, τ u,r < ∞] = P
+�
+Y r
+τ u,r,∞ > Xu,r
+τ u,r, τ u,r < ∞
+�
+= P[τ u,r < ∞]E
+� ¯G(Xu,r
+τ u,r, Dr
+τ u,r) | τ u,r < ∞
+�
+≥ P[τ u,r < ∞]E
+� ¯G(0, Dr
+τ u,r) | τ u,r < ∞
+�
+≥ P[τ u,r < ∞] inf
+q
+¯G(0, q)
+and get the result. ✷
+In view of the above lemma the proof of the main theorem is reduced to estab-
+lishing the existence of finite limits Y r
+∞ and finding the asymptotic of the tail of their
+distributions.
+Let us introduce the notations
+Qr
+k := −
+�
+(T r
+k−1,T r
+k ]
+e
+−(Vs−−VT r
+k−1 )dP r
+s ,
+M r
+k := e
+−(VT r
+k −VT r
+k−1).
+(2.10)
+Lemma 2.5 The random variables Y r
+∞ admit the representations
+Y r
+∞ = Qr
+1 + M r
+1 ˜Y r
+∞,
+where
+Qr
+1 := −
+�
+[0,T r
+1 ]
+e−Vs−dP r
+s ,
+M r
+1 := e−VT r
+1 ,
+(2.11)
+(Qr
+1, M r
+1) and ˜Y r
+∞ are independent, and the laws of ˜Y r
+∞ and Y 0
+∞ coincide.
+Proof. Note that
+Y r
+T r
+n = −
+�
+[0,T r
+1 ]
+e−Vs−dP r
+s −
+n
+�
+k=2
+e
+VT r
+k−1
+�
+(T r
+k−1,T r
+k ]
+e
+−(Vs−−VT r
+k−1)dP r
+s
+= Qr
+1 + M r
+1
+�
+Qr
+2 +
+n
+�
+k=3
+M r
+2 ...M r
+k−1Qr
+k
+�
+,
+where the random variable in the parentheses is independent of (Qr
+1, M r
+1) and has the
+same distribution as Y 0
+n−1. ✷
+Lemma 2.6 Suppose that Y∞ is unbounded from above. If c < 0, then
+inf
+q
+¯G(0, q) ≥ E[ ¯G(ξ, 0)] > 0.
+If c ∈ R and the distribution function F r ≤ F, then
+inf
+q
+¯G(0, q) > 0.
+
+Ruin Probabilities for a Sparre Andersen Model with Investments
+7
+Proof. Using Lemma 2.5 we have:
+¯G(0, r) = P[Y r
+∞ > 0] = P[Qr
+1/M r
+1 + ˜Y r
+∞ > 0]
+=
+�
+P
+�
+|c|eVt
+�
+[0,t]
+e−Vsds − ξ1 + ˜Y r
+∞ > 0
+�
+FT r
+1 (dt)
+≥ P[ ˜Y r
+∞ > ξ1] =
+�
+P[ ˜Y r
+∞ > x]Fξ(dx) =
+�
+¯G(x, 0)Fξ(dx) > 0
+since Y∞ is unbounded.
+The inspection of the proof reveals that the majority of arguments does work with
+minor changes also for the case where c is of an arbitrary sign and the law Fξ charges
+(−∞, 0) and (0, ∞). In particular, the proof that the finite limit Y∞ exists remains
+the same.
+Put ft := |ξ1| + |c|te2V ∗
+t , where V ∗
+t := sups≤t |Vs|. Then
+|Qr
+1|/M r
+1 ≤ |ξ1| + |c|eVT r
+1
+�
+[0,T r
+1 ]
+e−Vsds ≤ fT r
+1 .
+It follows that
+¯G(0, r) = P[ ˜Y r
+∞ > −Qr
+1/M r
+1 ] = E ¯G(−Qr
+1/M r
+1, 0) ≥ E ¯G(|Qr
+1|/M r
+1, 0)
+≥ E
+�
+¯G(ft, 0)F r(dt) = −E
+�
+F r(t)d ¯G(ft, 0)
+≥ −E
+�
+F(t)d ¯G(ft, 0) ≥ E
+�
+¯G(ft, 0)F(dt) > 0,
+where we use the property F r ≥ F. Thus, infr ¯G(0, r) > 0. ✷
+3 Tails of solutions of distributional equations
+As a number of results on the ruin with investments, the proof is based on the implicit
+renewal theory. As in [2] we shall use the following formulation combining several
+useful facts:
+Theorem 3.1 Suppose that for some β > 0,
+E[M β] = 1,
+E[M β (ln M)+] < ∞,
+E[|Q|β] < ∞.
+(3.1)
+Let Y∞ be the solution of the distributional equation Y∞
+d= Q + MY∞ and let
+¯G(u) := P[Y∞ > u]. Then lim sup uβ ¯G(u) < ∞. If the random variable Y∞ is
+unbounded from above, then lim inf uβ ¯G(u) > 0.
+In the previous section we introduce a process Y = (Yt). Assuming that it has
+at infinity a limit Y∞, which is a finite unbounded from above random variable, we
+have proved that it solves the required distributional equation and its tail function
+gives lower and upper bounds for the ruin probability. It remains to check that the
+hypotheses of Theorem 2.2 ensure the assumed properties and get the result applying
+the above theorem. We do this in the next sections.
+
+8
+Yuri Kabanov, Platon Promyslov
+4 The existence of the limit Y r
+∞
+First, we recall several results from [2].
+Lemma 4.1 ( [2], Lemma 2.1) Let T > 0 be a random variable independent of R.
+Suppose that E[eεT ] < ∞ for some ε > 0. Let β ∈ (0, ¯q) be the root of the equation
+H(q) = 0. If q ∈ [β, ¯q) is such that H(q) ≤ ε/2, then
+E
+�
+sup
+s≤T
+e−qVs
+�
+< ∞.
+(4.1)
+Corollary 4.2 Suppose that E[eεT1] < ∞ for some ε > 0. Let
+�Q1 := sup
+t≤T1
+|e−V− · Pt|.
+If E[|ξ1|β] < ∞, then E[| �Qβ
+1] < ∞.
+Though the above assertion is a bit more general than Corollary 2.2 in [2], the
+proof is exactly the same. Note also that it does not depend on the sign of c or ξ1 and
+needs only the integrability of |ξ1|β. It implies, in particular, that E[|Qβ
+1|] < ∞
+Lemma 4.3 Suppose that E[eεT1] < ∞ and E[|ξ1|β∧ε∧1] < ∞ for some ε > 0.
+Then Yt → Y∞ almost surely as t → ∞ where Y∞ is a finite random variable.
+Proof. The convergence a.s. of the sequence YTn, n ≥ 1, to a finite r.v. Y∞ has
+been proven in Lemma 4.1 of [2] as well as the fact that ρ := E[M p
+1 ] < 1 for any
+p ∈ (0, β ∧ ε ∧ 1).
+Put In := (Tn−1, Tn] and
+∆n := sup
+v∈In
+�����
+�
+(Tn−1,v]
+e−Vs−dPs
+����� =
+n−1
+�
+i=1
+Mi sup
+v∈In
+�����
+�
+(Tn−1,v]
+e−(Vs−−VTn−1)dPs
+����� .
+By virtue of the Borel–Cantelli lemma, to get the announced result it is sufficient
+to show that for every δ > 0
+∞
+�
+n=1
+P[∆n ≥ δ] < ∞.
+But this is true because the Chebyshev inequality and the Corollary 4.2 imply that
+P[∆n ≥ δ] ≤ δ−pρpE[ �Q1|p]. ✷
+By Lemma 2.5 the sequence Y r
+T r
+n converges a.s. to Y r
+∞ and the same arguments
+as above allows us to conclude that Y r
+t also converges.
+
+Ruin Probabilities for a Sparre Andersen Model with Investments
+9
+5 When the distribution of Y∞ is unbounded from above?
+The question in the title of the section is studied in [2] for the non-life insurance
+case, i.e. when c < 0 and Fξ((0, ∞)) = 1. In the present paper we provide sufficient
+conditions for the unboundedness from above for all new cases using the techniques
+developed in the mentioned paper. It is based on the following elementary observa-
+tion: if f : X × Y → R is a measurable function, r.v. η and ζ are independent and
+have the laws Fη and Fζ, then the r.v. f(η, ζ) is unbounded from above provided that
+there exists a measurable set X0 ⊆ X with Fη(X0) > 0 such that the r.v. f(x, ζ) is
+unbounded from above for every x ∈ X0.
+Let An := M1...Mn, n ≥ 1, A0 := 0.
+A tractable sufficient condition is give by the following
+Lemma 5.1 ([2], Lemma 5.1) If there exists n ≥ 1 such that the random variables
+Q1 and (Q1+· · ·+An−1Qn)/An are unbounded from above, then Y∞ is unbounded
+from above.
+It usually works already with n = 1 but sometimes we need it with n = 2. A
+short look at the expressions
+Q1 = −c
+� T1
+0
+e−Vrdr − e−VT1ξ1
+(5.1)
+Q1/A1 = −ceVT1
+� T1
+0
+e−Vrdr − ξ1,
+(5.2)
+Q1/A2 + Q2/M2 = −ceVT2
+� T2
+0
+e−Vrdr − ξ1eVT2−VT1 − ξ2
+(5.3)
+shows that Y∞ is unbounded from above when ξ is unbounded from below (of course,
+the latter property is not fulfilled for the annuity model).
+Using the above sufficient condition of the unboundedness, we examine various
+cases.
+1. Let σ ̸= 0. In this case the following lemma is helpful:
+Lemma 5.2 Let K > 0, σ ̸= 0 and 0 ≤ s < t. Then the random variables
+ζ := KeσWt −
+� t
+0
+eσWrdr,
+˜ζ := Keσ(Wt−Ws) − eσWt
+� t
+0
+eσWrdr,
+(5.4)
+are unbounded from below and from above.
+The property that ζ and ˜ζ are unbounded from above has been proven in [2],
+Lemma 5.2. The unboundedness from below can be established by similar arguments.
+It is also clear, that if K = 0, then ζ and ˜ζ are unbounded from below.
+The process ¯V := V − σW is independent of the Wiener process W. If c < 0,
+then
+Q1 ≥ |c| inf
+r≤T1 e− ¯Vr
+� T1
+0
+e−σWrdr − ξ1e− ¯VT1e−σWT1 .
+
+10
+Yuri Kabanov, Platon Promyslov
+Using the conditioning with respect to ¯V , ξ1, T1 and the previous lemma, we get that
+Q1 is unbounded from above. Since
+Q1/A1 ≥ |c|e
+¯VT1 inf
+r≤T1 e− ¯VreσWT1
+� T1
+0
+e−σWrdr − ξ1,
+we conclude in the same way that Q1/A1 is unbounded from above. If c ≥ 0, then
+necessary Fξ(−∞, 0)) > 0 (recall that we exclude the case c ≥ 0, ξ > 0 when the
+ruin is impossible). Lemma 5.2 implies that the random variables Q1 and Q1/A2 +
+Q2/M2 are unbounded from above.
+2. Let σ = 0 and let ξ be bounded from below. We treat separately several sub-
+cases.
+2a. Π((−1, 0)) > 0 and Π((0, ∞)) > 0. Fix ε > 0 such that Π((−1, −ε)) > 0
+and Π((ε, ∞)) > 0 and put
+V (1) := I{−1<x<−ε} ln(1 + x) ∗ µ + I{x>ε} ln(1 + x) ∗ µ.
+Then the processes V (1) and V (2) := V − V (1) are independent.
+Note that V (1) is the sum of two independent compound Poisson processes with
+negative and positive jumps, respectively, and the absolute values of jumps are larger
+than some constant cε > 0.
+Lemma 5.3 Let K > 0, t > 0. Then the random variable
+ζ := Ke−V (1)
+t
+−
+� t
+0
+e−V (1)
+r
+dr
+(5.5)
+is unbounded from above and from below, the random variable
+�ζ := e−V (1)
+t
+� t
+0
+e−V (1)
+r
+dr.
+is unbounded from above.
+Proof. The arguments are simple and we explain only the idea. One can consider
+trajectories where V (1) has a lot of negative jumps in a neighborhood of zero while all
+positive jumps are concentrate in a neighborhood of t. Choosing suitable parameters
+and using the independence of processes with positive and negative jumps we obtain
+that, with a strictly positive probability, the first term in the definition of ζ is arbitrary
+close to zero while the integral is arbitrary large. Thus, ζ is unbounded from below.
+Symmetric arguments lead to the conclusion that ζ is unbounded from above. ✷
+Let c < 0. Then the following bounds are obvious:
+Q1 ≥ |c| inf
+r≤T1 e−V (2)
+r
+� T1
+0
+e−V (1)
+r
+dr − ξ1e− ¯V (2)
+T1 e−V (1)
+T1 ,
+Q1/A1 ≥ |c|eV (2)
+T1
+inf
+r≤T1 e−V (2)
+r
+eV (1)
+T1
+� T1
+0
+e−V (1)
+r
+dr − ξ1.
+
+Ruin Probabilities for a Sparre Andersen Model with Investments
+11
+By conditioning with respect to the random variables V (2), T1, ξ1, which are inde-
+pendent of V (1), and using Lemma 5.3 we easily obtain that the random variables Q1
+and Q1/A1 are unbounded from above and, by Lemma 5.1, so is Y∞.
+Let c ≥ 0. The same arguments as in [2] show that Q1 and Q1/A2 + Q2/M2 are
+unbounded from above on the non-null set {ξ1 < 0, ξ2 < 0}.
+2b. Π((−1, 0)) = 0, Π(h) = ∞.
+We use the decomposition of V depending on the choice of ε ∈ (0, 1). Namely,
+put
+V ε := I{x≤ε}h ∗ (µ − ν) + I{x≤ε}(ln(1 + x) − h) ∗ µ,
+(5.6)
+˜V ε := I{x>ε}h ∗ (µ − ν) + I{x>ε}(ln(1 + x) − h) ∗ µ.
+(5.7)
+Note that Vt = at + V ε
+t + ˜V ε
+t and
+˜V ε = I{x>ε} ln(1 + x) ∗ µ − I{x>ε}h ∗ ν.
+Lemma 5.4 Let K > 0, t > 0. Then the random variables
+η :=
+� t
+0
+e−Vrdr − Ke−Vt,
+η′ := eVt
+� t
+0
+e−Vrdr
+(5.8)
+are unbounded from above.
+Proof. Without loss of generality we assume that a = 0. Fix N > 0 and choose ε > 0
+small enough to ensure that Π(I{x>ε}h) ≥ N. Let Γε := {sups≤t |V ε
+s | ≤ 1}. Let
+denoting by Jε and ¯Jε the processes in the lhs of (5.6). Using the Doob inequality
+and the elementary bound x − ln(1 + x) ≤ x2/2 for x > 0, we get that
+P
+�
+sup
+s≤t
+|V ε
+s | > 1
+�
+≤ P
+�
+sup
+s≤t
+|Jε
+s | > 1/2
+�
++ P
+�
+| ¯Jε
+t | > 1/2
+�
+≤ 2E
+�
+sup
+s≤t
+|Jε
+s |
+�
++ 2E
+�
+| ¯Jε
+t |
+�
+≤ 2(I{x≤ε}h2 ∗ νt)1/2 + I{x≤ε}h2 ∗ νt → 0,
+ε → 0.
+Thus, the set Γε is non-null, at least, for sufficiently small ε, and on this set
+η ≥ 1
+e
+� t
+0
+e− ˜V ε
+r dr − Ke− ˜V ε
+t +1.
+(5.9)
+On the intersection Γε∩{I{x>ε}h∗µt/2 = 0, ln(1+ε)µ((t/2, t]×(ε, 1]) ≥ Nt+1}
+we have
+η ≥ 1
+e
+� t/2
+0
+eNrdr − Ke =
+1
+eN (eNt/2 − 1) − Ke.
+Due to the independence of V ε and ˜V ε this intersection is a non-null set. Since N is
+arbitrary large, the required property of η holds.
+The analysis of η′ follows the same line. At the first stage we replace V by ˜V
+and compensate the linear decrease of V by a large number of positive jumps on the
+second half of the interval [0, t]. ✷
+
+12
+Yuri Kabanov, Platon Promyslov
+Let c < 0. Then the random variables
+Q1 = |c|
+� T1
+0
+e−Vrdr − e−VT1ξ1,
+Q1/A1 = |c|eVT1
+� T1
+0
+e−Vrdr − ξ1
+are unbounded from above by virtue of the lemma.
+Let c ≥ 0. The same arguments as in [2] lead to the conclusion that that Q1 and
+Q1/A2 + Q2/M2 are unbounded from above on the non-null set {ξ1 < 0, ξ2 < 0}.
+2c. Π((0, ∞)) = 0, Π(|h|) = ∞.
+Let c < 0. Again we use the decomposition of V depending on the choice of
+ε ∈ (0, 1). Put
+V ε := I{x≥−ε}h ∗ (µ − ν) + I{x≥−ε}(ln(1 + x) − h) ∗ µ,
+(5.10)
+˜V ε := I{x<−ε}h ∗ (µ − ν) + I{x<−ε}(ln(1 + x) − h) ∗ µ,
+(5.11)
+L := I{x<−ε} ln(1 + x) ∗ µ and Πε := Π(I{x<−ε}|h|) ↑ ∞ as ε → 0. Then
+˜V ε
+t = I{x<−ε} ln(1 + x) ∗ µt − I{x<−ε}h ∗ ν = Lt + Πεt.
+To prove that Q1 is inbounded from above, we argue as follows. As in the previous
+subcase 2b. we reduce the problem to checking that the random variable
+˜η =
+� t
+0
+e− ˜V ε
+r dr − Ke− ˜V ε
+r
+is unbounded from above. Let t1 := t2/2 where t2 := 1/Πε. Note that t2 ≤ t when
+ε > 0 is sufficiently small. On the set {Lt = Lt1} we have that
+˜η ≥ (t2 − t1)e|¯Lt1|e−Πεt2 − Ke
+¯Lt1e−Πεt
+= e|¯Lt1|(1/(2eΠε) − Kee−Πεt) ≥ e|¯Lt1|/(4eΠε)
+for sufficiently small ε. Since the r.v. |¯Lt1| is unbounded from above, so is Q1.
+On the set {Lt = 0, ξ1 ≤ K} non-null for any ε > 0 and sufficiently large K,
+we have the bound
+Q1/A1 ≥ (|c|/Πε)(eΠεt − 1) − K.
+It follows that Q1/A1 is unbounded from above.
+The case c ≥ 0 is treated as in [2].
+2d. Π((−∞, 0)) = 0, 0 < Π(h) < ∞, and F((t, ∞)) > 0 for every t > 0.
+In this subcase Vt = Lt − bt, where Lt := ln(1 + x) ∗ µt is an increasing process
+and b := Π(h) − a. Note that b > 0, otherwise we get a contradiction with the
+existence of β > 0 such that ln Ee−βV1 = 0.
+Let c < 0. Take arbitrary N > 0. Let s > 0 be the solution of the equation
+(|c|/b)(ebs − 1) = N. Take K large enough to ensure that P[ξ ≤ K] > 0. Chose
+t > s such that F((s, t)) > 0. On the non-null set
+{Ls = 0, T1 ∈ (s, t), eLt−Ls ≥ Kebt, ξ < K}
+
+Ruin Probabilities for a Sparre Andersen Model with Investments
+13
+we have that
+Q1 ≥ (|c|/b)(ebs − 1) − ebt−Ltξ ≥ N − 1.
+Thus, Q1 is unbounded from above.
+To prove that Q1/A1 is unbounded from above we take ε > 0 and K > 0 such
+that F((t, ∞)) > 0 and Fξ((0, K)) > 0. Setting cε := (|c|/b)(e−bε − e−2bε) we
+have that on the non-null set
+{T1 > ε, LT1−ε = 0, LT1 ≥ ln((N + K)/cε), ξ < K}
+we have
+Q1/A1 ≥ (|c|/b)eLT1e−bT1(eb(T1−ε) − 1) − ξ ≥ cεeLt − K ≥ N.
+So, by Lemma 5.1 Y∞ is unbounded.
+If c ≥ 0, we proceed as in [2], using the assumption that F charges any neighbor-
+hood of zero.
+2e. Π((0, ∞)) = 0, 0 < Π(|h|) < ∞, and F((t, ∞)) > 0 for every t > 0.
+We have again V = Lt − bt but now the jump process L is decreasing and the
+constant b < 0.
+Let c < 0. Fix N > 0. Let s, t > 0 be such that F((s, t)) > 0. On the non-null
+set
+{T1 ∈ (s, t), |Ls/2| ≥ N, Ls/2 = Lt, ξ < e|Ls/2|}
+we have
+Q1 ≥ |c|(T1/2)e|L(1/2)T1|−|b|T1 − e|L(1/2)T1|−|b|T1ξ ≥ |c|(s/2)eN−|b|t − 1.
+Since N is arbitrary, Q1 is unbounded from above.
+For any t > 0 and K > 0 on the non-null set {T1 ≥ t, Lt = 0, ξ ≤ K}
+Q1/A1 ≤ |c/b|(e|b|t − 1) − K.
+This implies that Q1/A1 is unbounded from above.
+If c ≥ 0, we again proceed as in [2].
+Acknowledgements
+The research is funded by the grant of RSF n◦ 20-68-47030 ”Econometric and prob-
+abilistic methods for the analysis of financial markets with complex structure“.
+
+14
+Yuri Kabanov, Platon Promyslov
+References
+1. Albrecher, H., Constantinescu, C., Thomann, E.: Asymptotic results for renewal risk models with
+risky investments. Stoch. Proc. Appl. 122, 3767–3789 (2012)
+2. Eberlein, E., Kabanov, Yu., Schmidt, T.: Ruin probabilities for a Sparre Andersen model with invest-
+ments. Stoch. Proc. Appl. 144, 72–84 (2022)
+3. Goldie, C.M.: Implicit renewal theory and tails of solutions of random equations. Ann. Appl. Probab.
+1, 126–166 (1991)
+4. Grandell, I.: Aspects of Risk Theory. Springer, Berlin (1990)
+5. Kabanov, Yu., Pergamenshchikov, S.: In the insurance business risky investments are dangerous: the
+case of negative risk sums. Finance Stoch. 20, 355–379 (2016)
+6. Kabanov, Yu., Pergamenshchikov, S.: Ruin probabilities for a L´evy-driven generalised Ornstein–
+Uhlenbeck process. Finance Stoch. 24, 39–69 (2020)
+7. Kabanov, Yu., Pukhlyakov, N.: Ruin probabilities with investments: smoothness, IDE and ODE,
+asymptotic behavior. J. Appl. Probab. 59, 556–570 (2020)
+8. Paulsen, J.: Risk theory in stochastic economic environment. Stoch. Proc. Appl. 46, 327–361 (1993)
+9. Paulsen, J., Stochastic Calculus with Applications to Risk Theory. Lecture Notes, Univ. of Bergen
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+11. Paulsen, J., Gjessing, H.K.: Ruin theory with stochastic return on investments. Adv. Appl. Probab.
+29, 965–985 (1997)
+

DdA0T4oBgHgl3EQfAv9Y/content/tmp_files/load_file.txt

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+page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdA0T4oBgHgl3EQfAv9Y/content/2301.01966v1.pdf'}
+page_content='01966v1  [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdA0T4oBgHgl3EQfAv9Y/content/2301.01966v1.pdf'}
+page_content='PR]  5 Jan 2023  Noname manuscript No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdA0T4oBgHgl3EQfAv9Y/content/2301.01966v1.pdf'}
+page_content='  (will be inserted by the editor)  Ruin Probabilities for a Sparre Andersen Model with  Investments: the Case of Annuity Payments  Yuri Kabanov · Platon Promyslov  January 6, 2023  Dedicated to the memory of Tomas Bj¨ork.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdA0T4oBgHgl3EQfAv9Y/content/2301.01966v1.pdf'}
+page_content='  Abstract This note is a complement to the paper by Eberlein, Kabanov, and Schmidt  on the asymptotic of the ruin probability in a Sparre Andersen non-life insurance  model with investments a risky asset whose price follows a geometric L´evy process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdA0T4oBgHgl3EQfAv9Y/content/2301.01966v1.pdf'}
+page_content='  Using the techniques of semi-Markov processes we extend the result of the mentioned  paper to the case of annuities and models with two-sided jumps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdA0T4oBgHgl3EQfAv9Y/content/2301.01966v1.pdf'}
+page_content='  Keywords Ruin probabilities · Sparre Andersen model · Actuarial models with  investments · Renewal processes · Annuities · Distributional equations  Mathematics Subject Classification (2010) 60G44  JEL Classification G22 · G23  1 Introduction  In the classical Sparre Andersen model of insurance company the counts of claims  form a renewal process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdA0T4oBgHgl3EQfAv9Y/content/2301.01966v1.pdf'}
+page_content=' In recent studies, see [1], [2] and references therein, this  model was enriched by the assumption that the capital reserve of the insurance com-  pany is fully invested in a risky asset whose price evolves as a geometric L´evy pro-  cess.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdA0T4oBgHgl3EQfAv9Y/content/2301.01966v1.pdf'}
+page_content=' In the paper [2] by Eberlein, Kabanov, and Schmidt it was considered the non-  life insurance version of such a model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdA0T4oBgHgl3EQfAv9Y/content/2301.01966v1.pdf'}
+page_content=' It was shown that under rather mild hypothe-  Lomonosov Moscow State University, Federal Research Center “Computer Science and Control” of  the Russian Academy of Sciences Moscow, Russia, and Universit´e de Franche-Comt´e, Laboratoire de  Math´ematiques, UMR CNRS 6623, 16 Route de Gray, 25030 Besanc¸on, France  E-mail: ykabanov@univ-fcomte.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdA0T4oBgHgl3EQfAv9Y/content/2301.01966v1.pdf'}
+page_content='fr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdA0T4oBgHgl3EQfAv9Y/content/2301.01966v1.pdf'}
+page_content='  Lomonosov Moscow State University, Moscow, Russia  E-mail: platon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdA0T4oBgHgl3EQfAv9Y/content/2301.01966v1.pdf'}
+page_content='promyslov@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdA0T4oBgHgl3EQfAv9Y/content/2301.01966v1.pdf'}
+page_content='com  2  Yuri Kabanov, Platon Promyslov  ses on the business process the asymptotic behavior of the (ultimate) ruin probability  is essentially the same as in the Cram´er–Lundberg model with risky investments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdA0T4oBgHgl3EQfAv9Y/content/2301.01966v1.pdf'}
+page_content='  Namely, the ruin probability decays, up to a multiplicative constant, as the function  u−β where u, the initial capital, tends to infinity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdA0T4oBgHgl3EQfAv9Y/content/2301.01966v1.pdf'}
+page_content=' The decay rate β depends only of  characteristics of the price process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdA0T4oBgHgl3EQfAv9Y/content/2301.01966v1.pdf'}
+page_content=' The method of analysis in [2] is based heavily on  the assumption that the risk process has only downward jumps and, therefore, crosses  the zero level only by a jump.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdA0T4oBgHgl3EQfAv9Y/content/2301.01966v1.pdf'}
+page_content=' This specific feature allows a straightforward reduction  to a discrete-time Markovian framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdA0T4oBgHgl3EQfAv9Y/content/2301.01966v1.pdf'}
+page_content='  The approach of [2] left an open question whether the results hold also in the case  of upward jumps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdA0T4oBgHgl3EQfAv9Y/content/2301.01966v1.pdf'}
+page_content=' This is a feature of the annuity model when the risk process crosses  the zero level in a continuous way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdA0T4oBgHgl3EQfAv9Y/content/2301.01966v1.pdf'}
+page_content=' In a less popular mixed model with two-sided  jumps the crossing may happen in both way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdA0T4oBgHgl3EQfAv9Y/content/2301.01966v1.pdf'}
+page_content=' Of course, a positive answer is expected  here: this was already established for the Cram´er–Lundberg models with investments  analyzed by Kabanov, Pergamenshchikov,and Pukhlyakov, [5], [7], as well as in very  general L´evy Ornstein–Uhlenbeck models introduced and studied by Paulsen, see [8],  [9], [10], [11], and a more recent paper [6] by Kabanov and Pergamenshchikov.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdA0T4oBgHgl3EQfAv9Y/content/2301.01966v1.pdf'}
+page_content='  Our note, based on the study [2], gives a positive answer for a Sparre Andersen  model with investments in its annuity version, with the upward jumps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdA0T4oBgHgl3EQfAv9Y/content/2301.01966v1.pdf'}
+page_content=' We discuss  briefly the needed changes leading to a result for a model with upward and downward  jumps used serving to describe the evolution of the capital reserve of a company with  two types of the business activity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdA0T4oBgHgl3EQfAv9Y/content/2301.01966v1.pdf'}
+page_content=' Our techniques is based on the imbedding of a  semi-Markov process into a Markov one by increasing the dimensionality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdA0T4oBgHgl3EQfAv9Y/content/2301.01966v1.pdf'}
+page_content='  In the paper we use standard notations of stochastic calculus and concepts dis-  cussed in details in [6], [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdA0T4oBgHgl3EQfAv9Y/content/2301.01966v1.pdf'}
+page_content='  2 The model  The Sparre Andersen model with risky investments considered contains two ingredi-  ents:  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdA0T4oBgHgl3EQfAv9Y/content/2301.01966v1.pdf'}
+page_content=' The price process of a risky financial asset S = (St)t≥0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdA0T4oBgHgl3EQfAv9Y/content/2301.01966v1.pdf'}
+page_content=' It is of the form  S = E(R) where E is the stochastic exponential, R is a L´evy process with the L´evy  triplet (a, σ2, Π) and such that Π((−∞, −1]) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdA0T4oBgHgl3EQfAv9Y/content/2301.01966v1.pdf'}
+page_content=' The latter condition ensures that  the jumps ∆R > −1, hence, the price S > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdA0T4oBgHgl3EQfAv9Y/content/2301.01966v1.pdf'}
+page_content=' In such a case, S = eV where V = ln S  is again a L´evy process which can be given by the formula  Vt = at − 1  2σ2t + σWt + h ∗ (µ − ν)t + (ln(1 + x) − h) ∗ µt,  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdA0T4oBgHgl3EQfAv9Y/content/2301.01966v1.pdf'}
+page_content='1)  where h(x) := xI{|x|≤1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdA0T4oBgHgl3EQfAv9Y/content/2301.01966v1.pdf'}
+page_content=' The L´evy triplet of V is (aV , σ2, ΠV ) with  aV = a − σ2  2 + Π(h(ln(1 + x)) − h)  and ΠV = Πϕ−1, ϕ : x �→ ln(1 + x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdA0T4oBgHgl3EQfAv9Y/content/2301.01966v1.pdf'}
+page_content='  It is assume that R is non-deterministic, that is, at least one of the parameters σ2  or Π is not zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdA0T4oBgHgl3EQfAv9Y/content/2301.01966v1.pdf'}
+page_content='  Ruin Probabilities for a Sparre Andersen Model with Investments  3  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdA0T4oBgHgl3EQfAv9Y/content/2301.01966v1.pdf'}
+page_content=' The ”business process“.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdA0T4oBgHgl3EQfAv9Y/content/2301.01966v1.pdf'}
+page_content=' It is an independent of S compound renewal process  P = (Pt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdA0T4oBgHgl3EQfAv9Y/content/2301.01966v1.pdf'}
+page_content=' Classically, it can be written in the form  Pt = ct +  Nt  �  i=1  ξi,  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdA0T4oBgHgl3EQfAv9Y/content/2301.01966v1.pdf'}
+page_content='2)  where N = (Nt) is a counting renewal process with the interarrival times (lengths  of the inter jump intervals) Ui := Ti − Ti−1, i ≥ 2, forming an i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdA0T4oBgHgl3EQfAv9Y/content/2301.01966v1.pdf'}
+page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdA0T4oBgHgl3EQfAv9Y/content/2301.01966v1.pdf'}
+page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdA0T4oBgHgl3EQfAv9Y/content/2301.01966v1.pdf'}
+page_content=' sequence  independent of the i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdA0T4oBgHgl3EQfAv9Y/content/2301.01966v1.pdf'}
+page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdA0T4oBgHgl3EQfAv9Y/content/2301.01966v1.pdf'}
+page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdA0T4oBgHgl3EQfAv9Y/content/2301.01966v1.pdf'}
+page_content=' sequence of random variables ξi = ∆PTi, i ≥ 1, with the  common law Fξ, Fξ({0}) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdA0T4oBgHgl3EQfAv9Y/content/2301.01966v1.pdf'}
+page_content=' In the sequel, a “generic“ r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdA0T4oBgHgl3EQfAv9Y/content/2301.01966v1.pdf'}
+page_content='v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdA0T4oBgHgl3EQfAv9Y/content/2301.01966v1.pdf'}
+page_content=' with such a law is  denoted by ξ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdA0T4oBgHgl3EQfAv9Y/content/2301.01966v1.pdf'}
+page_content=' As usual, T0 := 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdA0T4oBgHgl3EQfAv9Y/content/2301.01966v1.pdf'}
+page_content=' The common law of Ui we denoted by F and use  the same character for its distribution function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdA0T4oBgHgl3EQfAv9Y/content/2301.01966v1.pdf'}
+page_content='  The risk process X = Xu, u > 0, is defined as the solution of the non-homogene-  ous linear stochastic equation  Xt = u +  � t  0  Xs−dRs + Pt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdA0T4oBgHgl3EQfAv9Y/content/2301.01966v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdA0T4oBgHgl3EQfAv9Y/content/2301.01966v1.pdf'}
+page_content='3)  The ruin probability is the function of the initial capital Ψ(u) := P[τ u < ∞]  where τ u := inf{t : Xu  t ≤ 0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdA0T4oBgHgl3EQfAv9Y/content/2301.01966v1.pdf'}
+page_content='  The cases of major interest are: c > 0 and ξi < 0 (a non-life insurance model,  considered in [2]) and c < 0 and ξi > 0 (annuities payments model).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdA0T4oBgHgl3EQfAv9Y/content/2301.01966v1.pdf'}
+page_content=' The latter case  studied here, is often interpreted as a model of venture company paying salary and  selling innovations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdA0T4oBgHgl3EQfAv9Y/content/2301.01966v1.pdf'}
+page_content=' The case where Fξ charges both half-axes can be viewed as a  model of company combined two types of activity, see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdA0T4oBgHgl3EQfAv9Y/content/2301.01966v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdA0T4oBgHgl3EQfAv9Y/content/2301.01966v1.pdf'}
+page_content=', [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdA0T4oBgHgl3EQfAv9Y/content/2301.01966v1.pdf'}
+page_content=' and we study also  the case where ξi may take positive and negative values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdA0T4oBgHgl3EQfAv9Y/content/2301.01966v1.pdf'}
+page_content='  If c ≥ 0 and ξ > 0 the ruin never happens and this case is excluded from consid-  erations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdA0T4oBgHgl3EQfAv9Y/content/2301.01966v1.pdf'}
+page_content='  Standing assumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdA0T4oBgHgl3EQfAv9Y/content/2301.01966v1.pdf'}
+page_content=' The cumulant generating function H : q → ln E e−qVT1  of the random variable VT1 has a root β > 0 not laying on the boundary of the  effective domain of H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdA0T4oBgHgl3EQfAv9Y/content/2301.01966v1.pdf'}
+page_content=' That is, if the int dom H = (q, ¯q), there is a unique root  β ∈ (0, ¯q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdA0T4oBgHgl3EQfAv9Y/content/2301.01966v1.pdf'}
+page_content='  We are looking for conditions under which  0 < lim inf  u→∞ uβΨ(u) ≤ lim sup  u→∞ uβΨ(u) < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdA0T4oBgHgl3EQfAv9Y/content/2301.01966v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdA0T4oBgHgl3EQfAv9Y/content/2301.01966v1.pdf'}
+page_content='4)  The paper [2] treats the case of the non-life insurance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdA0T4oBgHgl3EQfAv9Y/content/2301.01966v1.pdf'}
+page_content=' We formulate its main  result in a more transparent form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdA0T4oBgHgl3EQfAv9Y/content/2301.01966v1.pdf'}
+page_content='  Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdA0T4oBgHgl3EQfAv9Y/content/2301.01966v1.pdf'}
+page_content='1 ([2]) Suppose that the drift c ≥ 0, the law Fξ is concentrated on  (−∞, 0), E[|ξ|β] < ∞, and E[eεT1] < ∞ for some ε > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdA0T4oBgHgl3EQfAv9Y/content/2301.01966v1.pdf'}
+page_content=' Then (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdA0T4oBgHgl3EQfAv9Y/content/2301.01966v1.pdf'}
+page_content='4) holds if at  least one of the following conditions are fulfilled:  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdA0T4oBgHgl3EQfAv9Y/content/2301.01966v1.pdf'}
+page_content=' σ ̸= 0 or ξ is unbounded from below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdA0T4oBgHgl3EQfAv9Y/content/2301.01966v1.pdf'}
+page_content='  2a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdA0T4oBgHgl3EQfAv9Y/content/2301.01966v1.pdf'}
+page_content=' Π((−1, 0)) > 0 and Π((0, ∞)) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdA0T4oBgHgl3EQfAv9Y/content/2301.01966v1.pdf'}
+page_content='  2b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdA0T4oBgHgl3EQfAv9Y/content/2301.01966v1.pdf'}
+page_content=' Π((−1, 0)) = 0 and Π(h) = ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdA0T4oBgHgl3EQfAv9Y/content/2301.01966v1.pdf'}
+page_content='  2c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdA0T4oBgHgl3EQfAv9Y/content/2301.01966v1.pdf'}
+page_content=' Π((0, ∞)) = 0 and Π(|h|) = ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdA0T4oBgHgl3EQfAv9Y/content/2301.01966v1.pdf'}
+page_content='  2d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdA0T4oBgHgl3EQfAv9Y/content/2301.01966v1.pdf'}
+page_content=' Π((−∞, 0)) = 0, 0 < Π(h) < ∞, F((0, t)) > 0 for every t > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdA0T4oBgHgl3EQfAv9Y/content/2301.01966v1.pdf'}
+page_content='  2e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdA0T4oBgHgl3EQfAv9Y/content/2301.01966v1.pdf'}
+page_content=' Π((0, ∞)) = 0, 0 < Π(|h|) < ∞, F((0, t)) > 0 for every t > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdA0T4oBgHgl3EQfAv9Y/content/2301.01966v1.pdf'}
+page_content='  4  Yuri Kabanov, Platon Promyslov  The proof in [2] used heavily the assumption that the business process has a posi-  tive drift and negative claims corresponding to the non-life insurance setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdA0T4oBgHgl3EQfAv9Y/content/2301.01966v1.pdf'}
+page_content=' In such  a case the ruin may happen only at an instant of jump and, therefore, one needs to  monitor the risk process only at T1, T2, and so on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdA0T4oBgHgl3EQfAv9Y/content/2301.01966v1.pdf'}
+page_content=' Such a reduction to a discrete-time  ruin model does not work if ξi > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdA0T4oBgHgl3EQfAv9Y/content/2301.01966v1.pdf'}
+page_content='  In our paper we consider the annuity model of the Sparre Andersen type where  the ruin occurs because exhausting resources and the risk process reaches zero in a  continuous way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdA0T4oBgHgl3EQfAv9Y/content/2301.01966v1.pdf'}
+page_content=' The main result can be formulated as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdA0T4oBgHgl3EQfAv9Y/content/2301.01966v1.pdf'}
+page_content='  Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdA0T4oBgHgl3EQfAv9Y/content/2301.01966v1.pdf'}
+page_content='2 Suppose that the drift c < 0, the law Fξ is concentrated on (0, ∞),  E[ξβ] < ∞, and E[eεT1] < ∞ for some ε > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdA0T4oBgHgl3EQfAv9Y/content/2301.01966v1.pdf'}
+page_content=' Then (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdA0T4oBgHgl3EQfAv9Y/content/2301.01966v1.pdf'}
+page_content='4) holds if at least one of the  following conditions are fulfilled:  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdA0T4oBgHgl3EQfAv9Y/content/2301.01966v1.pdf'}
+page_content=' σ ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdA0T4oBgHgl3EQfAv9Y/content/2301.01966v1.pdf'}
+page_content='  2a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdA0T4oBgHgl3EQfAv9Y/content/2301.01966v1.pdf'}
+page_content=' Π((−1, 0)) > 0 and Π((0, ∞)) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdA0T4oBgHgl3EQfAv9Y/content/2301.01966v1.pdf'}
+page_content='  2b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdA0T4oBgHgl3EQfAv9Y/content/2301.01966v1.pdf'}
+page_content=' Π((−1, 0)) = 0 and Π(h) = ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdA0T4oBgHgl3EQfAv9Y/content/2301.01966v1.pdf'}
+page_content='  2c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdA0T4oBgHgl3EQfAv9Y/content/2301.01966v1.pdf'}
+page_content=' Π((0, ∞)) = 0 and Π(|h|) = ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdA0T4oBgHgl3EQfAv9Y/content/2301.01966v1.pdf'}
+page_content='  2d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdA0T4oBgHgl3EQfAv9Y/content/2301.01966v1.pdf'}
+page_content=' Π((−1, 0)) = 0, 0 < Π(h) < ∞, F((t, ∞)) > 0 for every t > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdA0T4oBgHgl3EQfAv9Y/content/2301.01966v1.pdf'}
+page_content='  2e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdA0T4oBgHgl3EQfAv9Y/content/2301.01966v1.pdf'}
+page_content=' Π((0, ∞)) = 0, 0 < Π(|h|) < ∞, F((t, ∞)) > 0 for every t > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdA0T4oBgHgl3EQfAv9Y/content/2301.01966v1.pdf'}
+page_content='  For the mixed case we have the following result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdA0T4oBgHgl3EQfAv9Y/content/2301.01966v1.pdf'}
+page_content='  Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdA0T4oBgHgl3EQfAv9Y/content/2301.01966v1.pdf'}
+page_content='3 Suppose that the drift c ∈ R, the law Fξ charges both half-lines  (−∞, 0) and (0, ∞), E[|ξ|β] < ∞, and E[eεT1] < ∞ for some ε > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdA0T4oBgHgl3EQfAv9Y/content/2301.01966v1.pdf'}
+page_content=' Then (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdA0T4oBgHgl3EQfAv9Y/content/2301.01966v1.pdf'}
+page_content='4)  holds if at least one of the following conditions are fulfilled:  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdA0T4oBgHgl3EQfAv9Y/content/2301.01966v1.pdf'}
+page_content=' σ ̸= 0 or |ξ| is unbounded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdA0T4oBgHgl3EQfAv9Y/content/2301.01966v1.pdf'}
+page_content='  2a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdA0T4oBgHgl3EQfAv9Y/content/2301.01966v1.pdf'}
+page_content=' Π((−1, 0)) > 0 and Π((0, ∞)) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdA0T4oBgHgl3EQfAv9Y/content/2301.01966v1.pdf'}
+page_content='  2b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdA0T4oBgHgl3EQfAv9Y/content/2301.01966v1.pdf'}
+page_content=' Π((−1, 0)) = 0 and Π(h) = ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdA0T4oBgHgl3EQfAv9Y/content/2301.01966v1.pdf'}
+page_content='  2c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdA0T4oBgHgl3EQfAv9Y/content/2301.01966v1.pdf'}
+page_content=' Π((0, ∞)) = 0 and Π(|h|) = ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdA0T4oBgHgl3EQfAv9Y/content/2301.01966v1.pdf'}
+page_content='  2d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdA0T4oBgHgl3EQfAv9Y/content/2301.01966v1.pdf'}
+page_content=' Π((−1, 0)) = 0, 0 < Π(h) < ∞, F((t, ∞)) > 0 for every t > 0 in the case  c < 0 and Fξ((0, ε)) > 0 for every ε > 0 in the case c ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdA0T4oBgHgl3EQfAv9Y/content/2301.01966v1.pdf'}
+page_content='  2e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdA0T4oBgHgl3EQfAv9Y/content/2301.01966v1.pdf'}
+page_content=' Π((0, ∞)) = 0, 0 < Π(|h|) < ∞, F((t, ∞)) > 0 for every t > 0 in the case  c < 0 and Fξ((0, ε)) > 0 for every ε > 0 in the case c ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdA0T4oBgHgl3EQfAv9Y/content/2301.01966v1.pdf'}
+page_content='  The annuity and the mixed setting require a different approach inspired by the the-  ory of semi-Markov processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdA0T4oBgHgl3EQfAv9Y/content/2301.01966v1.pdf'}
+page_content=' Namely, we consider the business process P as the  component of the two-dimensional Markov process (P, D) where the second compo-  nent D = Dr is a “clock”, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdA0T4oBgHgl3EQfAv9Y/content/2301.01966v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdA0T4oBgHgl3EQfAv9Y/content/2301.01966v1.pdf'}
+page_content=' a process measuring the elapsed time after the instant  of last claim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdA0T4oBgHgl3EQfAv9Y/content/2301.01966v1.pdf'}
+page_content=' We assume that the law of U = T1 may be different from the common  law of the further interarrival times: at the instant zero a portion r of the interarrival  time is already elapsed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdA0T4oBgHgl3EQfAv9Y/content/2301.01966v1.pdf'}
+page_content=' This feature admits obvious justifications: e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdA0T4oBgHgl3EQfAv9Y/content/2301.01966v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdA0T4oBgHgl3EQfAv9Y/content/2301.01966v1.pdf'}
+page_content=', the venture  company may change the governance when a project was still in progress.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdA0T4oBgHgl3EQfAv9Y/content/2301.01966v1.pdf'}
+page_content='  Here and throughout the paper we use the superscript r to emphasize that the law  of a random variable or a process depends on r, skipping usually r = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdA0T4oBgHgl3EQfAv9Y/content/2301.01966v1.pdf'}
+page_content='  Formally, the “clock”, Dr = (Dr  t ), is a process with the initial value Dr  0 = r,  Dr  t = r+t on the interval [0, T1), and Dr  t := t−T r  n on all other interarrival intervals  [T r  n, T r  n+1), n ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdA0T4oBgHgl3EQfAv9Y/content/2301.01966v1.pdf'}
+page_content=' That is, the “clock” restarts from zero at each instant T r  n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdA0T4oBgHgl3EQfAv9Y/content/2301.01966v1.pdf'}
+page_content=' We  Ruin Probabilities for a Sparre Andersen Model with Investments  5  denote by F r the law of the first interarrival time T r  1 = T r  1 − T0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdA0T4oBgHgl3EQfAv9Y/content/2301.01966v1.pdf'}
+page_content=' In accordance with  our convention F 0 = F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdA0T4oBgHgl3EQfAv9Y/content/2301.01966v1.pdf'}
+page_content='  Alternatively, Dr can be representing as the solution of the linear equation  Dr  t = r + t −  �  [0,t]  Dr  s−dNs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdA0T4oBgHgl3EQfAv9Y/content/2301.01966v1.pdf'}
+page_content='  Typically, P[T r  1 > t] = P[Ti > t + r]/P[Ti > r], i > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdA0T4oBgHgl3EQfAv9Y/content/2301.01966v1.pdf'}
+page_content=' In the case of  exponential distribution F r = F for all r ≥ 0 (“absence of memory”).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdA0T4oBgHgl3EQfAv9Y/content/2301.01966v1.pdf'}
+page_content='  We assume that F r ≥ F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdA0T4oBgHgl3EQfAv9Y/content/2301.01966v1.pdf'}
+page_content='  Recall that the assumed independence of P r and R implies that the joint quadratic  characteristic [P r, R] is zero and the ruin process Xu,r can be written in the form  resembling the Cauchy formula for solutions of liner differential equations:  Xu,r  t  = eVt(u − Y r  t ),  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdA0T4oBgHgl3EQfAv9Y/content/2301.01966v1.pdf'}
+page_content='5)  where  Y r  t := −  �  (0,t]  E−1  s− (R)dP r  s = −  �  (0,t]  e−Vs−dP r  s .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdA0T4oBgHgl3EQfAv9Y/content/2301.01966v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdA0T4oBgHgl3EQfAv9Y/content/2301.01966v1.pdf'}
+page_content='6)  The strict positivity of the process E(R) = eV implies that the ruin time  τ u,r := inf{t ≥ 0 : Xu,r  t  ≤ 0} = inf{t ≥ 0 : Y r  t ≥ u}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdA0T4oBgHgl3EQfAv9Y/content/2301.01966v1.pdf'}
+page_content='  The crucial element of our study is the following  Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdA0T4oBgHgl3EQfAv9Y/content/2301.01966v1.pdf'}
+page_content='4 Suppose that Y r  t → Y r  ∞ almost surely as t → ∞ where Y r  ∞ is a finite  random variable such that ¯G(u, r) := P[Y r  ∞ > u] > 0 for every u > 0 and r ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdA0T4oBgHgl3EQfAv9Y/content/2301.01966v1.pdf'}
+page_content=' If  ¯G∗ := infq ¯G(0, q) > 0, then  ¯G(u, r) ≤ Ψ(u, r) =  ¯G(u, r)  E  � ¯G(Xu,r  τ u,r, Dr  τ u,r) | τ u,r < ∞  � ≤ 1  ¯G∗  ¯G(u, r).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdA0T4oBgHgl3EQfAv9Y/content/2301.01966v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdA0T4oBgHgl3EQfAv9Y/content/2301.01966v1.pdf'}
+page_content='7)  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdA0T4oBgHgl3EQfAv9Y/content/2301.01966v1.pdf'}
+page_content=' Let τ be an arbitrary stopping time with respect to the filtration (FP,D,R  t  ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdA0T4oBgHgl3EQfAv9Y/content/2301.01966v1.pdf'}
+page_content=' As  we assume that the finite limit Y r  ∞ exists, the random variable  Y r  τ,∞ :=  �  − limN→∞  �  (τ,τ+N] e−(Vs−−Vτ )dP r  s ,  τ < ∞,  0,  τ = ∞,  is well defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdA0T4oBgHgl3EQfAv9Y/content/2301.01966v1.pdf'}
+page_content=' On the set {τ < ∞}  Y r  τ,∞ = eVτ (Y r  ∞ − Y r  τ ) = Xu,r  τ  + eVτ (Y r  ∞ − u).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdA0T4oBgHgl3EQfAv9Y/content/2301.01966v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdA0T4oBgHgl3EQfAv9Y/content/2301.01966v1.pdf'}
+page_content='8)  Let ζ be a FP,D,R  τ  measurable random variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdA0T4oBgHgl3EQfAv9Y/content/2301.01966v1.pdf'}
+page_content='  Using the strong Markov property, we get that  P  �  Y r  τ,∞ > ζ, τ < ∞  �  = E  � ¯G(ζ, Dr  τ)I{τ<∞}  �  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdA0T4oBgHgl3EQfAv9Y/content/2301.01966v1.pdf'}
+page_content='9)  6  Yuri Kabanov, Platon Promyslov  Noting that Ψ(u, r) := P [τu,r < ∞] ≥ P [Y r  ∞ > u] > 0, we deduce from here  using (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdA0T4oBgHgl3EQfAv9Y/content/2301.01966v1.pdf'}
+page_content='8) that  ¯G(u, r) = P [Y r  ∞ > u, τ u,r < ∞] = P  �  Y r  τ u,r,∞ > Xu,r  τ u,r, τ u,r < ∞  �  = P[τ u,r < ∞]E  � ¯G(Xu,r  τ u,r, Dr  τ u,r) | τ u,r < ∞  �  ≥ P[τ u,r < ∞]E  � ¯G(0, Dr  τ u,r) | τ u,r < ∞  �  ≥ P[τ u,r < ∞] inf  q  ¯G(0, q)  and get the result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdA0T4oBgHgl3EQfAv9Y/content/2301.01966v1.pdf'}
+page_content=' ✷  In view of the above lemma the proof of the main theorem is reduced to estab-  lishing the existence of finite limits Y r  ∞ and finding the asymptotic of the tail of their  distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdA0T4oBgHgl3EQfAv9Y/content/2301.01966v1.pdf'}
+page_content='  Let us introduce the notations  Qr  k := −  �  (T r  k−1,T r  k ]  e  −(Vs−−VT r  k−1 )dP r  s ,  M r  k := e  −(VT r  k −VT r  k−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdA0T4oBgHgl3EQfAv9Y/content/2301.01966v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdA0T4oBgHgl3EQfAv9Y/content/2301.01966v1.pdf'}
+page_content='10)  Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdA0T4oBgHgl3EQfAv9Y/content/2301.01966v1.pdf'}
+page_content='5 The random variables Y r  ∞ admit the representations  Y r  ∞ = Qr  1 + M r  1 ˜Y r  ∞,  where  Qr  1 := −  �  [0,T r  1 ]  e−Vs−dP r  s ,  M r  1 := e−VT r  1 ,  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdA0T4oBgHgl3EQfAv9Y/content/2301.01966v1.pdf'}
+page_content='11)  (Qr  1, M r  1) and ˜Y r  ∞ are independent, and the laws of ˜Y r  ∞ and Y 0  ∞ coincide.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdA0T4oBgHgl3EQfAv9Y/content/2301.01966v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdA0T4oBgHgl3EQfAv9Y/content/2301.01966v1.pdf'}
+page_content=' Note that  Y r  T r  n = −  �  [0,T r  1 ]  e−Vs−dP r  s −  n  �  k=2  e  VT r  k−1  �  (T r  k−1,T r  k ]  e  −(Vs−−VT r  k−1)dP r  s  = Qr  1 + M r  1  �  Qr  2 +  n  �  k=3  M r  2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdA0T4oBgHgl3EQfAv9Y/content/2301.01966v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdA0T4oBgHgl3EQfAv9Y/content/2301.01966v1.pdf'}
+page_content='M r  k−1Qr  k  �  ,  where the random variable in the parentheses is independent of (Qr  1, M r  1) and has the  same distribution as Y 0  n−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdA0T4oBgHgl3EQfAv9Y/content/2301.01966v1.pdf'}
+page_content=' ✷  Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdA0T4oBgHgl3EQfAv9Y/content/2301.01966v1.pdf'}
+page_content='6 Suppose that Y∞ is unbounded from above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdA0T4oBgHgl3EQfAv9Y/content/2301.01966v1.pdf'}
+page_content=' If c < 0, then  inf  q  ¯G(0, q) ≥ E[ ¯G(ξ, 0)] > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdA0T4oBgHgl3EQfAv9Y/content/2301.01966v1.pdf'}
+page_content='  If c ∈ R and the distribution function F r ≤ F, then  inf  q  ¯G(0, q) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdA0T4oBgHgl3EQfAv9Y/content/2301.01966v1.pdf'}
+page_content='  Ruin Probabilities for a Sparre Andersen Model with Investments  7  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdA0T4oBgHgl3EQfAv9Y/content/2301.01966v1.pdf'}
+page_content=' Using Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdA0T4oBgHgl3EQfAv9Y/content/2301.01966v1.pdf'}
+page_content='5 we have:  ¯G(0, r) = P[Y r  ∞ > 0] = P[Qr  1/M r  1 + ˜Y r  ∞ > 0]  =  �  P  �  |c|eVt  �  [0,t]  e−Vsds − ξ1 + ˜Y r  ∞ > 0  �  FT r  1 (dt)  ≥ P[ ˜Y r  ∞ > ξ1] =  �  P[ ˜Y r  ∞ > x]Fξ(dx) =  �  ¯G(x, 0)Fξ(dx) > 0  since Y∞ is unbounded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdA0T4oBgHgl3EQfAv9Y/content/2301.01966v1.pdf'}
+page_content='  The inspection of the proof reveals that the majority of arguments does work with  minor changes also for the case where c is of an arbitrary sign and the law Fξ charges  (−∞, 0) and (0, ∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdA0T4oBgHgl3EQfAv9Y/content/2301.01966v1.pdf'}
+page_content=' In particular, the proof that the finite limit Y∞ exists remains  the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdA0T4oBgHgl3EQfAv9Y/content/2301.01966v1.pdf'}
+page_content='  Put ft := |ξ1| + |c|te2V ∗  t , where V ∗  t := sups≤t |Vs|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdA0T4oBgHgl3EQfAv9Y/content/2301.01966v1.pdf'}
+page_content=' Then  |Qr  1|/M r  1 ≤ |ξ1| + |c|eVT r  1  �  [0,T r  1 ]  e−Vsds ≤ fT r  1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdA0T4oBgHgl3EQfAv9Y/content/2301.01966v1.pdf'}
+page_content='  It follows that  ¯G(0, r) = P[ ˜Y r  ∞ > −Qr  1/M r  1 ] = E ¯G(−Qr  1/M r  1, 0) ≥ E ¯G(|Qr  1|/M r  1, 0)  ≥ E  �  ¯G(ft, 0)F r(dt) = −E  �  F r(t)d ¯G(ft, 0)  ≥ −E  �  F(t)d ¯G(ft, 0) ≥ E  �  ¯G(ft, 0)F(dt) > 0,  where we use the property F r ≥ F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdA0T4oBgHgl3EQfAv9Y/content/2301.01966v1.pdf'}
+page_content=' Thus, infr ¯G(0, r) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdA0T4oBgHgl3EQfAv9Y/content/2301.01966v1.pdf'}
+page_content=' ✷  3 Tails of solutions of distributional equations  As a number of results on the ruin with investments, the proof is based on the implicit  renewal theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdA0T4oBgHgl3EQfAv9Y/content/2301.01966v1.pdf'}
+page_content=' As in [2] we shall use the following formulation combining several  useful facts:  Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdA0T4oBgHgl3EQfAv9Y/content/2301.01966v1.pdf'}
+page_content='1 Suppose that for some β > 0,  E[M β] = 1,  E[M β (ln M)+] < ∞,  E[|Q|β] < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdA0T4oBgHgl3EQfAv9Y/content/2301.01966v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdA0T4oBgHgl3EQfAv9Y/content/2301.01966v1.pdf'}
+page_content='1)  Let Y∞ be the solution of the distributional equation Y∞  d= Q + MY∞ and let  ¯G(u) := P[Y∞ > u].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdA0T4oBgHgl3EQfAv9Y/content/2301.01966v1.pdf'}
+page_content=' Then lim sup uβ ¯G(u) < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdA0T4oBgHgl3EQfAv9Y/content/2301.01966v1.pdf'}
+page_content=' If the random variable Y∞ is  unbounded from above, then lim inf uβ ¯G(u) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdA0T4oBgHgl3EQfAv9Y/content/2301.01966v1.pdf'}
+page_content='  In the previous section we introduce a process Y = (Yt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdA0T4oBgHgl3EQfAv9Y/content/2301.01966v1.pdf'}
+page_content=' Assuming that it has  at infinity a limit Y∞, which is a finite unbounded from above random variable, we  have proved that it solves the required distributional equation and its tail function  gives lower and upper bounds for the ruin probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdA0T4oBgHgl3EQfAv9Y/content/2301.01966v1.pdf'}
+page_content=' It remains to check that the  hypotheses of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdA0T4oBgHgl3EQfAv9Y/content/2301.01966v1.pdf'}
+page_content='2 ensure the assumed properties and get the result applying  the above theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdA0T4oBgHgl3EQfAv9Y/content/2301.01966v1.pdf'}
+page_content=' We do this in the next sections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdA0T4oBgHgl3EQfAv9Y/content/2301.01966v1.pdf'}
+page_content='  8  Yuri Kabanov, Platon Promyslov  4 The existence of the limit Y r  ∞  First, we recall several results from [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdA0T4oBgHgl3EQfAv9Y/content/2301.01966v1.pdf'}
+page_content='  Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdA0T4oBgHgl3EQfAv9Y/content/2301.01966v1.pdf'}
+page_content='1 ( [2], Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdA0T4oBgHgl3EQfAv9Y/content/2301.01966v1.pdf'}
+page_content='1) Let T > 0 be a random variable independent of R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdA0T4oBgHgl3EQfAv9Y/content/2301.01966v1.pdf'}
+page_content='  Suppose that E[eεT ] < ∞ for some ε > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdA0T4oBgHgl3EQfAv9Y/content/2301.01966v1.pdf'}
+page_content=' Let β ∈ (0, ¯q) be the root of the equation  H(q) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdA0T4oBgHgl3EQfAv9Y/content/2301.01966v1.pdf'}
+page_content=' If q ∈ [β, ¯q) is such that H(q) ≤ ε/2, then  E  �  sup  s≤T  e−qVs  �  < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdA0T4oBgHgl3EQfAv9Y/content/2301.01966v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdA0T4oBgHgl3EQfAv9Y/content/2301.01966v1.pdf'}
+page_content='1)  Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdA0T4oBgHgl3EQfAv9Y/content/2301.01966v1.pdf'}
+page_content='2 Suppose that E[eεT1] < ∞ for some ε > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdA0T4oBgHgl3EQfAv9Y/content/2301.01966v1.pdf'}
+page_content=' Let  �Q1 := sup  t≤T1  |e−V− · Pt|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdA0T4oBgHgl3EQfAv9Y/content/2301.01966v1.pdf'}
+page_content='  If E[|ξ1|β] < ∞, then E[| �Qβ  1] < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdA0T4oBgHgl3EQfAv9Y/content/2301.01966v1.pdf'}
+page_content='  Though the above assertion is a bit more general than Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdA0T4oBgHgl3EQfAv9Y/content/2301.01966v1.pdf'}
+page_content='2 in [2], the  proof is exactly the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdA0T4oBgHgl3EQfAv9Y/content/2301.01966v1.pdf'}
+page_content=' Note also that it does not depend on the sign of c or ξ1 and  needs only the integrability of |ξ1|β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdA0T4oBgHgl3EQfAv9Y/content/2301.01966v1.pdf'}
+page_content=' It implies, in particular, that E[|Qβ  1|] < ∞  Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdA0T4oBgHgl3EQfAv9Y/content/2301.01966v1.pdf'}
+page_content='3 Suppose that E[eεT1] < ∞ and E[|ξ1|β∧ε∧1] < ∞ for some ε > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdA0T4oBgHgl3EQfAv9Y/content/2301.01966v1.pdf'}
+page_content='  Then Yt → Y∞ almost surely as t → ∞ where Y∞ is a finite random variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdA0T4oBgHgl3EQfAv9Y/content/2301.01966v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdA0T4oBgHgl3EQfAv9Y/content/2301.01966v1.pdf'}
+page_content=' The convergence a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdA0T4oBgHgl3EQfAv9Y/content/2301.01966v1.pdf'}
+page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdA0T4oBgHgl3EQfAv9Y/content/2301.01966v1.pdf'}
+page_content=' of the sequence YTn, n ≥ 1, to a finite r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdA0T4oBgHgl3EQfAv9Y/content/2301.01966v1.pdf'}
+page_content='v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdA0T4oBgHgl3EQfAv9Y/content/2301.01966v1.pdf'}
+page_content=' Y∞ has  been proven in Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdA0T4oBgHgl3EQfAv9Y/content/2301.01966v1.pdf'}
+page_content='1 of [2] as well as the fact that ρ := E[M p  1 ] < 1 for any  p ∈ (0, β ∧ ε ∧ 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdA0T4oBgHgl3EQfAv9Y/content/2301.01966v1.pdf'}
+page_content='  Put In := (Tn−1, Tn] and  ∆n := sup  v∈In  �����  �  (Tn−1,v]  e−Vs−dPs  ����� =  n−1  �  i=1  Mi sup  v∈In  �����  �  (Tn−1,v]  e−(Vs−−VTn−1)dPs  ����� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdA0T4oBgHgl3EQfAv9Y/content/2301.01966v1.pdf'}
+page_content='  By virtue of the Borel–Cantelli lemma, to get the announced result it is sufficient  to show that for every δ > 0  ∞  �  n=1  P[∆n ≥ δ] < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdA0T4oBgHgl3EQfAv9Y/content/2301.01966v1.pdf'}
+page_content='  But this is true because the Chebyshev inequality and the Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdA0T4oBgHgl3EQfAv9Y/content/2301.01966v1.pdf'}
+page_content='2 imply that  P[∆n ≥ δ] ≤ δ−pρpE[ �Q1|p].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdA0T4oBgHgl3EQfAv9Y/content/2301.01966v1.pdf'}
+page_content=' ✷  By Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdA0T4oBgHgl3EQfAv9Y/content/2301.01966v1.pdf'}
+page_content='5 the sequence Y r  T r  n converges a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdA0T4oBgHgl3EQfAv9Y/content/2301.01966v1.pdf'}
+page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdA0T4oBgHgl3EQfAv9Y/content/2301.01966v1.pdf'}
+page_content=' to Y r  ∞ and the same arguments  as above allows us to conclude that Y r  t also converges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdA0T4oBgHgl3EQfAv9Y/content/2301.01966v1.pdf'}
+page_content='  Ruin Probabilities for a Sparre Andersen Model with Investments  9  5 When the distribution of Y∞ is unbounded from above?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdA0T4oBgHgl3EQfAv9Y/content/2301.01966v1.pdf'}
+page_content='  The question in the title of the section is studied in [2] for the non-life insurance  case, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdA0T4oBgHgl3EQfAv9Y/content/2301.01966v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdA0T4oBgHgl3EQfAv9Y/content/2301.01966v1.pdf'}
+page_content=' when c < 0 and Fξ((0, ∞)) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdA0T4oBgHgl3EQfAv9Y/content/2301.01966v1.pdf'}
+page_content=' In the present paper we provide sufficient  conditions for the unboundedness from above for all new cases using the techniques  developed in the mentioned paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdA0T4oBgHgl3EQfAv9Y/content/2301.01966v1.pdf'}
+page_content=' It is based on the following elementary observa-  tion: if f : X × Y → R is a measurable function, r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdA0T4oBgHgl3EQfAv9Y/content/2301.01966v1.pdf'}
+page_content='v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdA0T4oBgHgl3EQfAv9Y/content/2301.01966v1.pdf'}
+page_content=' η and ζ are independent and  have the laws Fη and Fζ, then the r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdA0T4oBgHgl3EQfAv9Y/content/2301.01966v1.pdf'}
+page_content='v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdA0T4oBgHgl3EQfAv9Y/content/2301.01966v1.pdf'}
+page_content=' f(η, ζ) is unbounded from above provided that  there exists a measurable set X0 ⊆ X with Fη(X0) > 0 such that the r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdA0T4oBgHgl3EQfAv9Y/content/2301.01966v1.pdf'}
+page_content='v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdA0T4oBgHgl3EQfAv9Y/content/2301.01966v1.pdf'}
+page_content=' f(x, ζ) is  unbounded from above for every x ∈ X0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdA0T4oBgHgl3EQfAv9Y/content/2301.01966v1.pdf'}
+page_content='  Let An := M1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdA0T4oBgHgl3EQfAv9Y/content/2301.01966v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdA0T4oBgHgl3EQfAv9Y/content/2301.01966v1.pdf'}
+page_content='Mn, n ≥ 1, A0 := 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdA0T4oBgHgl3EQfAv9Y/content/2301.01966v1.pdf'}
+page_content='  A tractable sufficient condition is give by the following  Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdA0T4oBgHgl3EQfAv9Y/content/2301.01966v1.pdf'}
+page_content='1 ([2], Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdA0T4oBgHgl3EQfAv9Y/content/2301.01966v1.pdf'}
+page_content='1) If there exists n ≥ 1 such that the random variables  Q1 and (Q1+· · ·+An−1Qn)/An are unbounded from above, then Y∞ is unbounded  from above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdA0T4oBgHgl3EQfAv9Y/content/2301.01966v1.pdf'}
+page_content='  It usually works already with n = 1 but sometimes we need it with n = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdA0T4oBgHgl3EQfAv9Y/content/2301.01966v1.pdf'}
+page_content=' A  short look at the expressions  Q1 = −c  � T1  0  e−Vrdr − e−VT1ξ1  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdA0T4oBgHgl3EQfAv9Y/content/2301.01966v1.pdf'}
+page_content='1)  Q1/A1 = −ceVT1  � T1  0  e−Vrdr − ξ1,  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdA0T4oBgHgl3EQfAv9Y/content/2301.01966v1.pdf'}
+page_content='2)  Q1/A2 + Q2/M2 = −ceVT2  � T2  0  e−Vrdr − ξ1eVT2−VT1 − ξ2  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdA0T4oBgHgl3EQfAv9Y/content/2301.01966v1.pdf'}
+page_content='3)  shows that Y∞ is unbounded from above when ξ is unbounded from below (of course,  the latter property is not fulfilled for the annuity model).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdA0T4oBgHgl3EQfAv9Y/content/2301.01966v1.pdf'}
+page_content='  Using the above sufficient condition of the unboundedness, we examine various  cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdA0T4oBgHgl3EQfAv9Y/content/2301.01966v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdA0T4oBgHgl3EQfAv9Y/content/2301.01966v1.pdf'}
+page_content=' Let σ ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdA0T4oBgHgl3EQfAv9Y/content/2301.01966v1.pdf'}
+page_content=' In this case the following lemma is helpful:  Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdA0T4oBgHgl3EQfAv9Y/content/2301.01966v1.pdf'}
+page_content='2 Let K > 0, σ ̸= 0 and 0 ≤ s < t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdA0T4oBgHgl3EQfAv9Y/content/2301.01966v1.pdf'}
+page_content=' Then the random variables  ζ := KeσWt −  � t  0  eσWrdr,  ˜ζ := Keσ(Wt−Ws) − eσWt  � t  0  eσWrdr,  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdA0T4oBgHgl3EQfAv9Y/content/2301.01966v1.pdf'}
+page_content='4)  are unbounded from below and from above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdA0T4oBgHgl3EQfAv9Y/content/2301.01966v1.pdf'}
+page_content='  The property that ζ and ˜ζ are unbounded from above has been proven in [2],  Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdA0T4oBgHgl3EQfAv9Y/content/2301.01966v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdA0T4oBgHgl3EQfAv9Y/content/2301.01966v1.pdf'}
+page_content=' The unboundedness from below can be established by similar arguments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdA0T4oBgHgl3EQfAv9Y/content/2301.01966v1.pdf'}
+page_content='  It is also clear, that if K = 0, then ζ and ˜ζ are unbounded from below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdA0T4oBgHgl3EQfAv9Y/content/2301.01966v1.pdf'}
+page_content='  The process ¯V := V − σW is independent of the Wiener process W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdA0T4oBgHgl3EQfAv9Y/content/2301.01966v1.pdf'}
+page_content=' If c < 0,  then  Q1 ≥ |c| inf  r≤T1 e− ¯Vr  � T1  0  e−σWrdr − ξ1e− ¯VT1e−σWT1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdA0T4oBgHgl3EQfAv9Y/content/2301.01966v1.pdf'}
+page_content='  10  Yuri Kabanov, Platon Promyslov  Using the conditioning with respect to ¯V , ξ1, T1 and the previous lemma, we get that  Q1 is unbounded from above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdA0T4oBgHgl3EQfAv9Y/content/2301.01966v1.pdf'}
+page_content=' Since  Q1/A1 ≥ |c|e  ¯VT1 inf  r≤T1 e− ¯VreσWT1  � T1  0  e−σWrdr − ξ1,  we conclude in the same way that Q1/A1 is unbounded from above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdA0T4oBgHgl3EQfAv9Y/content/2301.01966v1.pdf'}
+page_content=' If c ≥ 0, then  necessary Fξ(−∞, 0)) > 0 (recall that we exclude the case c ≥ 0, ξ > 0 when the  ruin is impossible).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdA0T4oBgHgl3EQfAv9Y/content/2301.01966v1.pdf'}
+page_content=' Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdA0T4oBgHgl3EQfAv9Y/content/2301.01966v1.pdf'}
+page_content='2 implies that the random variables Q1 and Q1/A2 +  Q2/M2 are unbounded from above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdA0T4oBgHgl3EQfAv9Y/content/2301.01966v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdA0T4oBgHgl3EQfAv9Y/content/2301.01966v1.pdf'}
+page_content=' Let σ = 0 and let ξ be bounded from below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdA0T4oBgHgl3EQfAv9Y/content/2301.01966v1.pdf'}
+page_content=' We treat separately several sub-  cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdA0T4oBgHgl3EQfAv9Y/content/2301.01966v1.pdf'}
+page_content='  2a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdA0T4oBgHgl3EQfAv9Y/content/2301.01966v1.pdf'}
+page_content=' Π((−1, 0)) > 0 and Π((0, ∞)) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdA0T4oBgHgl3EQfAv9Y/content/2301.01966v1.pdf'}
+page_content=' Fix ε > 0 such that Π((−1, −ε)) > 0  and Π((ε, ∞)) > 0 and put  V (1) := I{−1<x<−ε} ln(1 + x) ∗ µ + I{x>ε} ln(1 + x) ∗ µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdA0T4oBgHgl3EQfAv9Y/content/2301.01966v1.pdf'}
+page_content='  Then the processes V (1) and V (2) := V − V (1) are independent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdA0T4oBgHgl3EQfAv9Y/content/2301.01966v1.pdf'}
+page_content='  Note that V (1) is the sum of two independent compound Poisson processes with  negative and positive jumps, respectively, and the absolute values of jumps are larger  than some constant cε > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdA0T4oBgHgl3EQfAv9Y/content/2301.01966v1.pdf'}
+page_content='  Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdA0T4oBgHgl3EQfAv9Y/content/2301.01966v1.pdf'}
+page_content='3 Let K > 0, t > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdA0T4oBgHgl3EQfAv9Y/content/2301.01966v1.pdf'}
+page_content=' Then the random variable  ζ := Ke−V (1)  t  −  � t  0  e−V (1)  r  dr  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdA0T4oBgHgl3EQfAv9Y/content/2301.01966v1.pdf'}
+page_content='5)  is unbounded from above and from below, the random variable  �ζ := e−V (1)  t  � t  0  e−V (1)  r  dr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdA0T4oBgHgl3EQfAv9Y/content/2301.01966v1.pdf'}
+page_content='  is unbounded from above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdA0T4oBgHgl3EQfAv9Y/content/2301.01966v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdA0T4oBgHgl3EQfAv9Y/content/2301.01966v1.pdf'}
+page_content=' The arguments are simple and we explain only the idea.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdA0T4oBgHgl3EQfAv9Y/content/2301.01966v1.pdf'}
+page_content=' One can consider  trajectories where V (1) has a lot of negative jumps in a neighborhood of zero while all  positive jumps are concentrate in a neighborhood of t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdA0T4oBgHgl3EQfAv9Y/content/2301.01966v1.pdf'}
+page_content=' Choosing suitable parameters  and using the independence of processes with positive and negative jumps we obtain  that, with a strictly positive probability, the first term in the definition of ζ is arbitrary  close to zero while the integral is arbitrary large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdA0T4oBgHgl3EQfAv9Y/content/2301.01966v1.pdf'}
+page_content=' Thus, ζ is unbounded from below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdA0T4oBgHgl3EQfAv9Y/content/2301.01966v1.pdf'}
+page_content='  Symmetric arguments lead to the conclusion that ζ is unbounded from above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdA0T4oBgHgl3EQfAv9Y/content/2301.01966v1.pdf'}
+page_content=' ✷  Let c < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdA0T4oBgHgl3EQfAv9Y/content/2301.01966v1.pdf'}
+page_content=' Then the following bounds are obvious:  Q1 ≥ |c| inf  r≤T1 e−V (2)  r  � T1  0  e−V (1)  r  dr − ξ1e− ¯V (2)  T1 e−V (1)  T1 ,  Q1/A1 ≥ |c|eV (2)  T1  inf  r≤T1 e−V (2)  r  eV (1)  T1  � T1  0  e−V (1)  r  dr − ξ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdA0T4oBgHgl3EQfAv9Y/content/2301.01966v1.pdf'}
+page_content='  Ruin Probabilities for a Sparre Andersen Model with Investments  11  By conditioning with respect to the random variables V (2), T1, ξ1, which are inde-  pendent of V (1), and using Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdA0T4oBgHgl3EQfAv9Y/content/2301.01966v1.pdf'}
+page_content='3 we easily obtain that the random variables Q1  and Q1/A1 are unbounded from above and, by Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdA0T4oBgHgl3EQfAv9Y/content/2301.01966v1.pdf'}
+page_content='1, so is Y∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdA0T4oBgHgl3EQfAv9Y/content/2301.01966v1.pdf'}
+page_content='  Let c ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdA0T4oBgHgl3EQfAv9Y/content/2301.01966v1.pdf'}
+page_content=' The same arguments as in [2] show that Q1 and Q1/A2 + Q2/M2 are  unbounded from above on the non-null set {ξ1 < 0, ξ2 < 0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdA0T4oBgHgl3EQfAv9Y/content/2301.01966v1.pdf'}
+page_content='  2b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdA0T4oBgHgl3EQfAv9Y/content/2301.01966v1.pdf'}
+page_content=' Π((−1, 0)) = 0, Π(h) = ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdA0T4oBgHgl3EQfAv9Y/content/2301.01966v1.pdf'}
+page_content='  We use the decomposition of V depending on the choice of ε ∈ (0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdA0T4oBgHgl3EQfAv9Y/content/2301.01966v1.pdf'}
+page_content=' Namely,  put  V ε := I{x≤ε}h ∗ (µ − ν) + I{x≤ε}(ln(1 + x) − h) ∗ µ,  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdA0T4oBgHgl3EQfAv9Y/content/2301.01966v1.pdf'}
+page_content='6)  ˜V ε := I{x>ε}h ∗ (µ − ν) + I{x>ε}(ln(1 + x) − h) ∗ µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdA0T4oBgHgl3EQfAv9Y/content/2301.01966v1.pdf'}
+page_content='  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdA0T4oBgHgl3EQfAv9Y/content/2301.01966v1.pdf'}
+page_content='7)  Note that Vt = at + V ε  t + ˜V ε  t and  ˜V ε = I{x>ε} ln(1 + x) ∗ µ − I{x>ε}h ∗ ν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdA0T4oBgHgl3EQfAv9Y/content/2301.01966v1.pdf'}
+page_content='  Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdA0T4oBgHgl3EQfAv9Y/content/2301.01966v1.pdf'}
+page_content='4 Let K > 0, t > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdA0T4oBgHgl3EQfAv9Y/content/2301.01966v1.pdf'}
+page_content=' Then the random variables  η :=  � t  0  e−Vrdr − Ke−Vt,  η′ := eVt  � t  0  e−Vrdr  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdA0T4oBgHgl3EQfAv9Y/content/2301.01966v1.pdf'}
+page_content='8)  are unbounded from above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdA0T4oBgHgl3EQfAv9Y/content/2301.01966v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdA0T4oBgHgl3EQfAv9Y/content/2301.01966v1.pdf'}
+page_content=' Without loss of generality we assume that a = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdA0T4oBgHgl3EQfAv9Y/content/2301.01966v1.pdf'}
+page_content=' Fix N > 0 and choose ε > 0  small enough to ensure that Π(I{x>ε}h) ≥ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdA0T4oBgHgl3EQfAv9Y/content/2301.01966v1.pdf'}
+page_content=' Let Γε := {sups≤t |V ε  s | ≤ 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdA0T4oBgHgl3EQfAv9Y/content/2301.01966v1.pdf'}
+page_content=' Let  denoting by Jε and ¯Jε the processes in the lhs of (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdA0T4oBgHgl3EQfAv9Y/content/2301.01966v1.pdf'}
+page_content='6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdA0T4oBgHgl3EQfAv9Y/content/2301.01966v1.pdf'}
+page_content=' Using the Doob inequality  and the elementary bound x − ln(1 + x) ≤ x2/2 for x > 0, we get that  P  �  sup  s≤t  |V ε  s | > 1  �  ≤ P  �  sup  s≤t  |Jε  s | > 1/2  �  + P  �  | ¯Jε  t | > 1/2  �  ≤ 2E  �  sup  s≤t  |Jε  s |  �  + 2E  �  | ¯Jε  t |  �  ≤ 2(I{x≤ε}h2 ∗ νt)1/2 + I{x≤ε}h2 ∗ νt → 0,  ε → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdA0T4oBgHgl3EQfAv9Y/content/2301.01966v1.pdf'}
+page_content='  Thus, the set Γε is non-null, at least, for sufficiently small ε, and on this set  η ≥ 1  e  � t  0  e− ˜V ε  r dr − Ke− ˜V ε  t +1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdA0T4oBgHgl3EQfAv9Y/content/2301.01966v1.pdf'}
+page_content='  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdA0T4oBgHgl3EQfAv9Y/content/2301.01966v1.pdf'}
+page_content='9)  On the intersection Γε∩{I{x>ε}h∗µt/2 = 0, ln(1+ε)µ((t/2, t]×(ε, 1]) ≥ Nt+1}  we have  η ≥ 1  e  � t/2  0  eNrdr − Ke =  1  eN (eNt/2 − 1) − Ke.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdA0T4oBgHgl3EQfAv9Y/content/2301.01966v1.pdf'}
+page_content='  Due to the independence of V ε and ˜V ε this intersection is a non-null set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdA0T4oBgHgl3EQfAv9Y/content/2301.01966v1.pdf'}
+page_content=' Since N is  arbitrary large, the required property of η holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdA0T4oBgHgl3EQfAv9Y/content/2301.01966v1.pdf'}
+page_content='  The analysis of η′ follows the same line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdA0T4oBgHgl3EQfAv9Y/content/2301.01966v1.pdf'}
+page_content=' At the first stage we replace V by ˜V  and compensate the linear decrease of V by a large number of positive jumps on the  second half of the interval [0, t].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdA0T4oBgHgl3EQfAv9Y/content/2301.01966v1.pdf'}
+page_content=' ✷  12  Yuri Kabanov, Platon Promyslov  Let c < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdA0T4oBgHgl3EQfAv9Y/content/2301.01966v1.pdf'}
+page_content=' Then the random variables  Q1 = |c|  � T1  0  e−Vrdr − e−VT1ξ1,  Q1/A1 = |c|eVT1  � T1  0  e−Vrdr − ξ1  are unbounded from above by virtue of the lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdA0T4oBgHgl3EQfAv9Y/content/2301.01966v1.pdf'}
+page_content='  Let c ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdA0T4oBgHgl3EQfAv9Y/content/2301.01966v1.pdf'}
+page_content=' The same arguments as in [2] lead to the conclusion that that Q1 and  Q1/A2 + Q2/M2 are unbounded from above on the non-null set {ξ1 < 0, ξ2 < 0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdA0T4oBgHgl3EQfAv9Y/content/2301.01966v1.pdf'}
+page_content='  2c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdA0T4oBgHgl3EQfAv9Y/content/2301.01966v1.pdf'}
+page_content=' Π((0, ∞)) = 0, Π(|h|) = ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdA0T4oBgHgl3EQfAv9Y/content/2301.01966v1.pdf'}
+page_content='  Let c < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdA0T4oBgHgl3EQfAv9Y/content/2301.01966v1.pdf'}
+page_content=' Again we use the decomposition of V depending on the choice of  ε ∈ (0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdA0T4oBgHgl3EQfAv9Y/content/2301.01966v1.pdf'}
+page_content=' Put  V ε := I{x≥−ε}h ∗ (µ − ν) + I{x≥−ε}(ln(1 + x) − h) ∗ µ,  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdA0T4oBgHgl3EQfAv9Y/content/2301.01966v1.pdf'}
+page_content='10)  ˜V ε := I{x<−ε}h ∗ (µ − ν) + I{x<−ε}(ln(1 + x) − h) ∗ µ,  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdA0T4oBgHgl3EQfAv9Y/content/2301.01966v1.pdf'}
+page_content='11)  L := I{x<−ε} ln(1 + x) ∗ µ and Πε := Π(I{x<−ε}|h|) ↑ ∞ as ε → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdA0T4oBgHgl3EQfAv9Y/content/2301.01966v1.pdf'}
+page_content=' Then  ˜V ε  t = I{x<−ε} ln(1 + x) ∗ µt − I{x<−ε}h ∗ ν = Lt + Πεt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdA0T4oBgHgl3EQfAv9Y/content/2301.01966v1.pdf'}
+page_content='  To prove that Q1 is inbounded from above, we argue as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdA0T4oBgHgl3EQfAv9Y/content/2301.01966v1.pdf'}
+page_content=' As in the previous  subcase 2b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdA0T4oBgHgl3EQfAv9Y/content/2301.01966v1.pdf'}
+page_content=' we reduce the problem to checking that the random variable  ˜η =  � t  0  e− ˜V ε  r dr − Ke− ˜V ε  r  is unbounded from above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdA0T4oBgHgl3EQfAv9Y/content/2301.01966v1.pdf'}
+page_content=' Let t1 := t2/2 where t2 := 1/Πε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdA0T4oBgHgl3EQfAv9Y/content/2301.01966v1.pdf'}
+page_content=' Note that t2 ≤ t when  ε > 0 is sufficiently small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdA0T4oBgHgl3EQfAv9Y/content/2301.01966v1.pdf'}
+page_content=' On the set {Lt = Lt1} we have that  ˜η ≥ (t2 − t1)e|¯Lt1|e−Πεt2 − Ke  ¯Lt1e−Πεt  = e|¯Lt1|(1/(2eΠε) − Kee−Πεt) ≥ e|¯Lt1|/(4eΠε)  for sufficiently small ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdA0T4oBgHgl3EQfAv9Y/content/2301.01966v1.pdf'}
+page_content=' Since the r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdA0T4oBgHgl3EQfAv9Y/content/2301.01966v1.pdf'}
+page_content='v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdA0T4oBgHgl3EQfAv9Y/content/2301.01966v1.pdf'}
+page_content=' |¯Lt1| is unbounded from above, so is Q1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdA0T4oBgHgl3EQfAv9Y/content/2301.01966v1.pdf'}
+page_content='  On the set {Lt = 0, ξ1 ≤ K} non-null for any ε > 0 and sufficiently large K,  we have the bound  Q1/A1 ≥ (|c|/Πε)(eΠεt − 1) − K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdA0T4oBgHgl3EQfAv9Y/content/2301.01966v1.pdf'}
+page_content='  It follows that Q1/A1 is unbounded from above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdA0T4oBgHgl3EQfAv9Y/content/2301.01966v1.pdf'}
+page_content='  The case c ≥ 0 is treated as in [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdA0T4oBgHgl3EQfAv9Y/content/2301.01966v1.pdf'}
+page_content='  2d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdA0T4oBgHgl3EQfAv9Y/content/2301.01966v1.pdf'}
+page_content=' Π((−∞, 0)) = 0, 0 < Π(h) < ∞, and F((t, ∞)) > 0 for every t > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdA0T4oBgHgl3EQfAv9Y/content/2301.01966v1.pdf'}
+page_content='  In this subcase Vt = Lt − bt, where Lt := ln(1 + x) ∗ µt is an increasing process  and b := Π(h) − a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdA0T4oBgHgl3EQfAv9Y/content/2301.01966v1.pdf'}
+page_content=' Note that b > 0, otherwise we get a contradiction with the  existence of β > 0 such that ln Ee−βV1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdA0T4oBgHgl3EQfAv9Y/content/2301.01966v1.pdf'}
+page_content='  Let c < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdA0T4oBgHgl3EQfAv9Y/content/2301.01966v1.pdf'}
+page_content=' Take arbitrary N > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdA0T4oBgHgl3EQfAv9Y/content/2301.01966v1.pdf'}
+page_content=' Let s > 0 be the solution of the equation  (|c|/b)(ebs − 1) = N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdA0T4oBgHgl3EQfAv9Y/content/2301.01966v1.pdf'}
+page_content=' Take K large enough to ensure that P[ξ ≤ K] > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdA0T4oBgHgl3EQfAv9Y/content/2301.01966v1.pdf'}
+page_content=' Chose  t > s such that F((s, t)) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdA0T4oBgHgl3EQfAv9Y/content/2301.01966v1.pdf'}
+page_content=' On the non-null set  {Ls = 0, T1 ∈ (s, t), eLt−Ls ≥ Kebt, ξ < K}  Ruin Probabilities for a Sparre Andersen Model with Investments  13  we have that  Q1 ≥ (|c|/b)(ebs − 1) − ebt−Ltξ ≥ N − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdA0T4oBgHgl3EQfAv9Y/content/2301.01966v1.pdf'}
+page_content='  Thus, Q1 is unbounded from above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdA0T4oBgHgl3EQfAv9Y/content/2301.01966v1.pdf'}
+page_content='  To prove that Q1/A1 is unbounded from above we take ε > 0 and K > 0 such  that F((t, ∞)) > 0 and Fξ((0, K)) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdA0T4oBgHgl3EQfAv9Y/content/2301.01966v1.pdf'}
+page_content=' Setting cε := (|c|/b)(e−bε − e−2bε) we  have that on the non-null set  {T1 > ε, LT1−ε = 0, LT1 ≥ ln((N + K)/cε), ξ < K}  we have  Q1/A1 ≥ (|c|/b)eLT1e−bT1(eb(T1−ε) − 1) − ξ ≥ cεeLt − K ≥ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdA0T4oBgHgl3EQfAv9Y/content/2301.01966v1.pdf'}
+page_content='  So, by Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdA0T4oBgHgl3EQfAv9Y/content/2301.01966v1.pdf'}
+page_content='1 Y∞ is unbounded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdA0T4oBgHgl3EQfAv9Y/content/2301.01966v1.pdf'}
+page_content='  If c ≥ 0, we proceed as in [2], using the assumption that F charges any neighbor-  hood of zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdA0T4oBgHgl3EQfAv9Y/content/2301.01966v1.pdf'}
+page_content='  2e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdA0T4oBgHgl3EQfAv9Y/content/2301.01966v1.pdf'}
+page_content=' Π((0, ∞)) = 0, 0 < Π(|h|) < ∞, and F((t, ∞)) > 0 for every t > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdA0T4oBgHgl3EQfAv9Y/content/2301.01966v1.pdf'}
+page_content='  We have again V = Lt − bt but now the jump process L is decreasing and the  constant b < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdA0T4oBgHgl3EQfAv9Y/content/2301.01966v1.pdf'}
+page_content='  Let c < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdA0T4oBgHgl3EQfAv9Y/content/2301.01966v1.pdf'}
+page_content=' Fix N > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdA0T4oBgHgl3EQfAv9Y/content/2301.01966v1.pdf'}
+page_content=' Let s, t > 0 be such that F((s, t)) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdA0T4oBgHgl3EQfAv9Y/content/2301.01966v1.pdf'}
+page_content=' On the non-null  set  {T1 ∈ (s, t), |Ls/2| ≥ N, Ls/2 = Lt, ξ < e|Ls/2|}  we have  Q1 ≥ |c|(T1/2)e|L(1/2)T1|−|b|T1 − e|L(1/2)T1|−|b|T1ξ ≥ |c|(s/2)eN−|b|t − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdA0T4oBgHgl3EQfAv9Y/content/2301.01966v1.pdf'}
+page_content='  Since N is arbitrary, Q1 is unbounded from above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdA0T4oBgHgl3EQfAv9Y/content/2301.01966v1.pdf'}
+page_content='  For any t > 0 and K > 0 on the non-null set {T1 ≥ t, Lt = 0, ξ ≤ K}  Q1/A1 ≤ |c/b|(e|b|t − 1) − K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdA0T4oBgHgl3EQfAv9Y/content/2301.01966v1.pdf'}
+page_content='  This implies that Q1/A1 is unbounded from above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdA0T4oBgHgl3EQfAv9Y/content/2301.01966v1.pdf'}
+page_content='  If c ≥ 0, we again proceed as in [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdA0T4oBgHgl3EQfAv9Y/content/2301.01966v1.pdf'}
+page_content='  Acknowledgements  The research is funded by the grant of RSF n◦ 20-68-47030 ”Econometric and prob-  abilistic methods for the analysis of financial markets with complex structure“.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdA0T4oBgHgl3EQfAv9Y/content/2301.01966v1.pdf'}
+page_content='  14  Yuri Kabanov, Platon Promyslov  References  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdA0T4oBgHgl3EQfAv9Y/content/2301.01966v1.pdf'}
+page_content=' Albrecher, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdA0T4oBgHgl3EQfAv9Y/content/2301.01966v1.pdf'}
+page_content=', Constantinescu, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdA0T4oBgHgl3EQfAv9Y/content/2301.01966v1.pdf'}
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@@ -0,0 +1,1116 @@
+GUAP: Graph Universal Attack Through Adversarial Patching
+Xiao Zang1 , Jie Chen2∗ and Bo Yuan1
+1Department of Electrical and Computer Engineering, Rutgers University
+2MIT-IBM Watson AI Lab, IBM Research
+[email protected], [email protected], [email protected]
+Abstract
+Graph neural networks (GNNs) are a class of ef-
+fective deep learning models for node classifi-
+cation tasks; yet their predictive capability may
+be severely compromised under adversarially de-
+signed unnoticeable perturbations to the graph
+structure and/or node data.
+Most of the current
+work on graph adversarial attacks aims at low-
+ering the overall prediction accuracy, but we ar-
+gue that the resulting abnormal model performance
+may catch attention easily and invite quick coun-
+terattack. Moreover, attacks through modification
+of existing graph data may be hard to conduct if
+good security protocols are implemented. In this
+work, we consider an easier attack harder to be no-
+ticed, through adversarially patching the graph with
+new nodes and edges. The attack is universal: it
+targets a single node each time and flips its con-
+nection to the same set of patch nodes. The attack
+is unnoticeable: it does not modify the predictions
+of nodes other than the target. We develop an al-
+gorithm, named GUAP, that achieves high attack
+success rate but meanwhile preserves the predic-
+tion accuracy. GUAP is fast to train by employ-
+ing a sampling strategy. We demonstrate that a 5%
+sampling in each epoch yields 20x speedup in train-
+ing, with only a slight degradation in attack perfor-
+mance. Additionally, we show that the adversarial
+patch trained with the graph convolutional network
+transfers well to other GNNs, such as the graph at-
+tention network.
+1
+Introduction
+Graph structured data are ubiquitous, with examples ranging
+from molecules, social networks, power systems, to knowl-
+edge graphs. Graph representation learning is one of the key
+areas of machine learning, with several extensively explored
+downstream tasks including node classification, graph clas-
+sification, and community detection. Over the past decade,
+a plethora of learning methods were proposed, ranging from
+∗Contact Author
+unsupervised embedding approaches (e.g., DeepWalk [Per-
+ozzi et al., 2014] and node2vec [Grover and Leskovec,
+2016]) to supervised/semi-supervised graph neural network
+(GNN) models (e.g., GCN [Kipf and Welling, 2017] and
+GAT [Veliˇckovi´c et al., 2017]). GNN models steadily im-
+prove the performance of downstream tasks and achieve state-
+of-the-art results.
+The seminal work of [Szegedy et al., 2013] and [Goodfel-
+low et al., 2014] point out that despite achieving high predic-
+tion accuracy, deep models are fragile to adversarially manip-
+ulated inputs, stirring a proliferation of research on designing
+adversarial attacks and defense schemes against them. GNNs
+as an emerging class of deep models tailored for graph struc-
+tured data also urge scrutiny. A major development in this
+context focuses on node classifications, in part because of
+their economic and societal importance. For example, bad
+actors in a financial network may hide themselves through
+manipulating contacts and transactions to benign actors, dev-
+astating the predictive power of GNNs in identifying illicit
+activities.
+Much prior work [Dai et al., 2018; Wu et al., 2019;
+Z¨ugner and G¨unnemann, 2019] studying adversarial attacks
+on GNNs aims at lowering the classification accuracy toward
+all nodes in the graph, through either poisoning the training
+data to weaken training, or modifying the test data to mislead
+trained models. However, in many practical scenarios, tak-
+ing control of existing data proves to be challenging and thus
+modifying the graph data is less realistic.
+In this work, we consider attacking a trained model through
+adversarially patching the graph data with new nodes and
+edges. These new edges must involve the new nodes; they
+should not change the connections between existing ones. For
+example, in a social network setting, the patching amounts to
+creating new accounts and setting their friendship. The key is
+that such new nodes are adversarial and their effect is secre-
+tive: When a target is being attacked, its connections to the
+patch nodes are all flipped so that its prediction is changed,
+while the predictions to other nodes are not. See Figure 1 for
+illustration.
+The idea of adversarial patching exists in prior graph-attack
+work. Greedy-GAN [Wang et al., 2018] adopts a greedy ap-
+proach and NIPA [Sun et al., 2019] uses reinforcement learn-
+ing to compute the new edges. Our work differs from them in
+several aspects. First, these methods aim at lowering the pre-
+arXiv:2301.01731v1  [cs.LG]  4 Jan 2023
+
+Figure 1: Illustration of GUAP. A set of patch nodes {13, 14, 15} and edges are inserted. One attacks node 7 through flipping its connections
+with the patch. Predictions of other nodes remain unchanged.
+diction accuracy whereas ours barely. Hence, they often need
+a large patch (e.g., 20% of the original node set in [Wang et
+al., 2018] and 10% in [Sun et al., 2019], for the Cora data set)
+but our patch is rather small (e.g., 1%). Second, these meth-
+ods attack the entire node set all at once whereas ours targets a
+single node each time. Third, even though the work of [Wang
+et al., 2018] additionally considers attacking single targets,
+such an attack needs a new optimization whenever the target
+changes, incurring expensive computation. On the contrary,
+our work computes the patch once for all and uses the flip-
+ping mechanism to perform attack, which is computationally
+economic. Our attack is of the universal type.
+We propose an algorithm, named GUAP, to compute the
+adversarial patch. It consists of two parts: node generation
+and edge training. Features of the new nodes are random and
+they are generated based on the statistics of those of the orig-
+inal graph. We find that the random generation is robust in
+the sense that once the patch is computed, regenerating the
+node features using the same mechanism barely affects the
+attack performance. For edge training, we treat the connec-
+tions involving the new nodes as parameters to optimize. The
+optimization achieves two goals: (i) it alters the prediction of
+the attack target and (ii) it maintains the prediction of other
+nodes. The latter goal distinguishes our work from most of
+the prior work.
+We summarize the contributions as follows:
+1. We present a novel attack scenario, which patches given
+graph data without modifying its original content and at-
+tacks one target at a time through flipping its connections
+to the patch nodes.
+2. We propose a universal attack algorithm, which achieves
+high attack success rate (ASR) while maintaining the orig-
+inal prediction accuracy.
+3. We show that attack training can be speeded up through
+sampling the training set in each epoch, without sacrificing
+the attack performance a lot.
+4. We demonstrate that our method admits good transferabil-
+ity of attack performance to a model different from the one
+used for training.
+2
+Related Work
+Universal
+attack.
+Universal
+attacks
+compute
+input-
+independent perturbations to fool the classifier. They more
+often appear in the computer vision literature.
+The work
+of [Moosavi-Dezfooli et al., 2017] perturbs every pixel
+of an image whereas the work of [Brown et al., 2017]
+computes an adversarial patch that is attached to an image
+at random locations. For graphs, research is sporadic. The
+work of [Zang et al., 2020] selects a set of anchor nodes so
+that attacking a target amounts to flipping the connections
+between the target and the anchors.
+Graph adversarial attack.
+Much work devotes to the
+modification of the graph structure, resulting in poor qual-
+ity of node embeddings [Chen et al., 2018b; Xu et al., 2019a;
+Wang and Gong, 2019; Bojchevski and G¨unnemann, 2018;
+Dai et al., 2018; Xu et al., 2019b]. Nettack [Z¨ugner et al.,
+2018] modifies not only the graph structure but also the node
+features. Specifically, targeting each node, Nettack modifies
+the node feature entry or graph structure step by step, to max-
+imize the prediction loss of the attacked node in a greedy
+manner. Modifying the graph is generally a discrete opti-
+mization problem, which invites greedy algorithms, but the
+work of [Liu et al., 2019; Bose et al., 2019] proposes a prob-
+abilistic framework under which continuous optimization is
+performed.
+The work of [Z¨ugner and G¨unnemann, 2019]
+poisons the graph by treating it as a hyperparameter and op-
+timizing through hypergradient updates. Closest to our work
+are [Wang et al., 2018] and [Sun et al., 2019], both of which
+inject adversarial nodes to the graph. The former greedily in-
+serts nodes and uses a discriminator to compute node features
+so that one cannot distinguish new nodes from the original
+ones. The latter computes adversarial nodes by using rein-
+forcement learning. Neither approach attacks a single node
+through connection flipping as we do.
+
+Patch Nodes
+Original Graph
+Patch Nodes: 13, 14, 15.
+13 14 15
+Node colors represents different classes
+8
+Graph Learning
+8
+-14 15
+Node
+13
+14
+Target
+15
+Target
+generation
+10
+12
+GCN
+Edge
+8
+8
+DeepWalk
+Targeted
+3
+Input
+Predict
+training
+3 
+New Graph
+attack
+node2vec
+1314
+15
+13--14. 15
+12
+10
+GAT
+8
+8
+Attack by flipping edges
+Only prediction of 7 changes
+3
+9
+10
+Generate adversarially patched graph3
+Preliminaries
+We use GCN [Kipf and Welling, 2017] as the attack model.
+Denote by G = (A, X) the given graph, where A is the n×n
+adjacency matrix and X is the n × d node feature matrix.
+Throughout, we assume that the graph is unweighted; thus, A
+is binary. Denote by f(A, X) the GCN model, which reads
+Z := f(A, X) = softmax( �A·ReLU( �AXW (0))·W (1)), (1)
+where �A = �D− 1
+2 �A �D− 1
+2 is the normalized adjacency matrix
+with �A = A + I and �D = diag(�
+j �Aij). The matrices W (0)
+and W (1) are trainable parameters whose sizes are respec-
+tively d × d′ and d′ × K, where K is the number of classes.
+Consequently, Z is the n×K output matrix, whose ith row is
+the probability vector for the ith node. Let VL be the training
+set and Y be the labels. The training of GCN minimizes the
+cross-entropy loss
+L = −
+�
+i∈VL
+K
+�
+k=1
+1{Yi = k} ln Zik.
+(2)
+4
+Graph Universal Attack Through
+Adversarial Patching
+Denote by Gnew = (Anew, Xnew) the new graph with m
+patch nodes. For convenience we order them after the original
+nodes, so that Anew =
+� A
+C
+CT
+B
+�
+and Xnew =
+�
+X
+Xpatch
+�
+.
+Here, C is n×m, denoting the connections between the orig-
+inal nodes and the patch ones; B is m × m, denoting the
+connections between the patch nodes themselves; and Xpatch
+is m × d, denoting the feature matrix of the patch nodes. We
+will discuss the generation of Xpatch and the computation of
+C and B in the following subsections, respectively.
+Additionally, we only consider undirected graphs in this
+paper. This is because even for directed ones, a majority of
+graph neural networks (e.g., GCN [Kipf and Welling, 2017])
+remove the edge directions and take the symmetric adjacency
+matrix as input. Our method is evaluated on three undirected
+graph benchmark data sets.
+4.1
+Node Generation
+Realistic node features may be generated by using a genera-
+tive model (e.g., GAN) [Wang et al., 2018], but the learning
+from existing nodes suffers many challenges, including high
+dimensionality and small training set. We opt for a simple
+mechanism that is sufficiently robust.
+We treat each feature dimension independently. In general,
+for numeric features we fit a normal distribution for each fea-
+ture dimension and sample from it. Without a priori knowl-
+edge, a normal distribution appears to be the most straightfor-
+ward parameterization. Depending on data sets, more accu-
+rate distributions may be fit or even learned.
+For some of the data sets we experiment with, the node fea-
+tures are binary. Hence, we perform binarization and make
+the feature value 0 if the Gaussian sample is smaller than 0.5,
+or 1 otherwise. If the training set contains 1 with probability
+p and 0 with probability 1 − p, then the fitted normal distri-
+bution has mean p and variance p(1−p). Thus, the new sam-
+ples take 1 with probability 1
+2
+�
+1 − erf
+�
+1/2−p
+√
+2p(1−p)
+��
+, which
+is approximately p. In other words, the general approach of
+fitting a normal distribution covers well the special case of
+Bernoulli distribution.
+4.2
+Edge Training
+Denote by ˆl(A, X, i) the predicted label of node i given
+graph adjacency matrix A and node feature matrix X. It is
+where the largest entry of f(A, X)i resides; i.e., ˆl(A, X, i) =
+arg max f(A, X)i. Our attack aims at two goals: changing
+the prediction of i while preserving those of other nodes.
+Both goals involve the graph adjacency matrix A′
+new when
+node i is being attacked. They may be mathematically sum-
+marized as:
+for each i in the training set VL,
+�
+ˆl(A′
+new, Xnew, i) ̸= ˆl(A, X, i),
+ˆl(A′
+new, Xnew, j) = ˆl(A, X, j),
+∀j ̸= i.
+(3)
+Note that A′
+new is i-dependent but we suppress the depen-
+dency in notation to avoid cluttering.
+We elaborate how A′
+new is computed from Anew. Let p =
+[0, . . . , 0, 1, . . . , 1] be an (n + m)-vector, where the first n
+entries are 0 and the rest are 1. We call p the attack vector,
+since the 1 entries will be used to flip the connections with
+the patch nodes. We extend p to the attack matrix P, whose
+ith row and ith column are the same as p and zero otherwise.
+Thus, Pij denotes whether the connection between node i and
+j is flipped. One easily derives that
+A′
+new := attack(Anew, i) = (1−P)◦Anew+P◦(10−Anew),
+(4)
+where ◦ stands for element-wise product, 1 is a matrix of all
+ones, and 10 is analogous except that the diagonal is set as
+zero.
+Throughout training, we will also need to revert the at-
+tacked graph back to the patched graph. Such an “unattack”
+operation is simple to conduct by using the attack matrix to
+flip back:
+Anew := unattack(A′
+new, i) = attack(A′
+new, i).
+(5)
+Outer Loop: GUAP
+Recall that Anew =
+� A
+C
+CT
+B
+�
+. The overall algorithm is to
+start with an initial Anew (specifically, B = 0 and C = 0) and
+iteratively update it by using certain perturbation ∆Anew =
+�
+0
+∆C
+∆CT
+∆B
+�
+that reflects the two goals summarized in (3).
+See Algorithm 1.
+Concretely, the training is conducted in several epochs,
+each of which iterates over the training set VL. At node i,
+we compute the attacked adjacency matrix A′
+new and check
+if i’s prediction changes. If not, we use an inner procedure
+IGP (to be elaborated subsequently) to generate a perturba-
+tion ∆Anew. Then we update A′
+new with this perturbation
+
+Algorithm 1 Graph Universal Attack Through Adversarial
+Patching (GUAP)
+Input: A, X
+Output: Adjacency matrix Anew of the patched graph and
+node features Xnew
+Initialize Anew and generate Xnew
+epoch ← 0
+while epoch < max epoch do
+for node i in training set do
+A′
+new ← attack(Anew, i)
+if ˆl(A′
+new, Xnew, i) = ˆl(A, X, i) then
+∆Anew ← IGP(A′
+new, Xnew, i)
+A′
+new ← A′
+new + ∆Anew
+Anew ← unattack(A′
+new, i)
+Anew ← L2-projection(Anew, radius)
+Anew ← Anew.clip(0, 1)
+Anew.diagonal ← 0
+end if
+end for
+Anew ← (Anew > 0.5) ? 1 : 0
+Compute ASR using Equation (6) and record the highest
+value
+epoch ← epoch + 1
+end while
+return Anew at the epoch of highest ASR and Xnew
+and revert it to the unattacked matrix Anew. Because the
+perturbation may gradually modify Anew to an incompara-
+ble magnitude, we apply L2 projection as well as clipping to
+prevent Anew from exploding. The L2 projection is applied
+to each patch node indivually so that the vector of edges to
+such a node has an L2 norm radius. We also set the diagonal
+of B to be zero to prevent self loops.
+After the entire training set is iterated, the B and C blocks
+contain real values within (0, 1). We binarize them to main-
+tain unweightedness. Then, the attack success rate
+ASR(VL) :=
+1
+|VL|
+|VL|
+�
+i=1
+1{ˆl(A′
+new, Xnew, i) ̸= ˆl(A, X, i)}
+(6)
+is computed as the metric of attack performance.
+Inner Loop: IGP
+The inner procedure iterative graph perturbation, IGP, com-
+putes a perturbation ∆Anew to the current attacked matrix
+A′
+new to gear it toward the two goals summarized in (3). For
+the first goal, the strategy is to push the prediction toward the
+decision boundary of another class; whereas for the second
+goal, the strategy is to progress toward a smaller loss for all
+nodes except i:
+L′
+new := −
+�
+j∈VL\i
+K
+�
+k=1
+1{Yj = k} ln f(A′
+new, Xnew)jk.
+(7)
+The procedure is summarized in Algorithm 2.
+Algorithm 2 is a while-loop that iteratively computes the
+perturbation till the prediction of node i changes (or the iter-
+Algorithm 2 Iterative Graph Perturbation (IGP)
+Input: Attacked adjacency matrix A′
+new, feature matrix
+Xnew, node i
+Output: Perturbation ∆Anew
+Initialize empty ∆Anew
+E′
+new ← A′
+new
+iter ← 0
+pred ← ˆl(A, X, i)
+while ˆl(A′
+new, Xnew, i) = pred and iter < max iter do
+v ←
+|∆fk|
+||∆wk||2
+2 ∆wk according to Equation (8)
+v[0 : n] ← 0
+∆Anew[i, :] ← ∆Anew[i, :] + (1 + overshoot) · v and
+analogously for ∆Anew[:, i]
+E′
+new ← A′
+new + ∆Anew
+E′
+new ← E′
+new.clip(0, 1)
+grad ← ∇L′
+new(E′
+new)
+# see loss function (7)
+grad[0 : n, 0 : n] ← 0
+grad[i, :] ← 0 and analogously for grad[:, i]
+grad ← (grad + gradT )/2
+∆Anew ← ∆Anew − step · grad
+iter ← iter + 1
+end while
+return ∆Anew
+ation count reaches maximum). The loop contains two part,
+corresponding to the two goals respectively. The first part in-
+tends to attack i. Denote the prediction pred; i.e., pred =
+ˆl(A, X, i).
+According to [Moosavi-Dezfooli et al., 2016;
+Zang et al., 2020], the minimum perturbation v on the ith
+row (and column) of A′
+new that sends node i to the decision
+boundary of the closest class k can be calculated as
+k = arg min
+c̸=pred
+|∆fc|
+∥∆wc∥2
+, v =
+|∆fk|
+||∆wk||2
+2
+∆wk,
+(8)
+where ∆fc = f(Anew, Xnew)i,c − f(Anew, Xnew)i,pred
+and ∆wc = ∇f(Anew, Xnew)i,c − ∇f(Anew, Xnew)i,pred.
+Here, gradient is taken with respect to the ith row (and sym-
+metrically column) of Anew. We set the first n entries of v to
+be zero because the original graph should not change. We also
+use a small overshoot constant to send node i to the other
+side of the decision boundary. We introduce a temporary no-
+tation E′
+new to denote the updated A′
+new for subsequent use.
+We also apply clipping, similar to what is done in the outer
+loop.
+The second part intends to lower the loss (7). We calcu-
+late its gradient grad at E′
+new and set the first n × n block
+to be zero, We also set the ith row and column zero because
+prediction of node i is not to be preserved. After numeri-
+cal symmetrization, we update the perturbation ∆Anew along
+the gradient descent direction, completing one iteration of the
+while-loop.
+5
+Experiments
+In this section,
+we perform a set of comprehensive
+experiments to demonstrate the effectiveness of GUAP.
+
+We investigate the patch size, show speedup of train-
+ing
+under
+the
+sampling
+strategy,
+compare
+with
+sev-
+eral related methods, and present transferability results.
+Code is available at https://anonymous.4open.science/r/
+ffd4fad9-367f-4a2a-bc65-1a7fe23d9d7f/.
+5.1
+Data Sets and Details
+We use three commonly used benchmark data sets: Cora,
+Citeseer, and Pol.Blogs. Their information is summarized in
+Table 1. Training is based on the standard GCN model and
+hence we also list its test accuracy in the table.
+Table 1: Node Classification Datasets. Only the largest connected
+component (LCC) is considered.
+Statistics
+Cora
+Citeseer
+Pol.Blogs
+# Nodes(LCC)
+2708
+3327
+1222
+# Edges(LCC)
+5278
+4676
+16714
+# Classes
+7
+6
+2
+Train/teset Set
+140/1000
+120/1000
+121/1101
+Accuracy(GCN)
+81.4%
+70.4%
+94.3%
+The hyperparameters for Algorithms 1 and 2 are:
+max epoch
+=
+50, max iter
+=
+30, radius
+=
+10,
+overshoot = 0.02, and step = 10. All experiments are
+repeated ten times under the same hyperparameters.
+5.2
+Compared Methods
+Universal attacks on graphs are rarely studied; hence, for
+comparison are a combination of existing universal attack
+method, non-universal method, and variants of the proposed
+method.
+1. Graph Universal Attack (GUA) [Zang et al., 2020]. As op-
+posed to adversarial patching, GUA seeks a set of anchor
+nodes from the graph and attacks a target through flipping
+its connections to these anchors. For a fair comparison, the
+number of anchors is the same as the patch size in GUAP.
+2. Fast Gradient Attack (FGA) [Chen et al., 2018b]. FGA
+is not a universal attack method. Targeting each node, it
+iteratively modifies the patch connection with the largest
+absolute gradient value. For a fair comparison, FGA can
+modify up to the same number of patch edges as GUAP.
+3. GUAP without patching edges. This variant of GUAP in-
+troduces only patch nodes but no edges. In other words,
+when a node is attacked, it will be connected through
+edges to all patch nodes.
+4. GUAP with randomly patched edges. Rather than per-
+forming the sophisticated edge training, this variant in-
+troduces random edges to the patch nodes. In this case,
+the existence of an edge follows a Bernoulli distribution
+with certain success probability. We experiment with two
+cases: one such that the number of new edges is approxi-
+mately the same as the that in GUAP; and the other merely
+setting probability = 0.5, introducing many more edges.
+5. GUAP with regenerated node features. This variant first
+computes the patched graph as does GUAP; then, it regen-
+erates the patch node features. Note that the training of
+Table 2: Average ASR of GUAP with and without clipping. The
+percentages of the patch nodes are 1%, 1%, 5% for Cora, Citeseer,
+and Pol.Blogs, respectively. The projection radius ξ = 10.
+Method
+Cora
+Citeseer
+Pol.Blogs
+GUAP w/o clipping
+82.24%
+83.41%
+45.76%
+GUAP w/ clipping
+91.37%
+85.05%
+53.24%
+the patched graph relies on the initial features. Hence, it is
+interesting to see how change of features affects attack.
+5.3
+Results
+We use two metrics for evaluation: ASR and ∆Acc (change
+of prediction accuracy).
+Patch size.
+First we determine a reasonable patch size. Fig-
+ure 2 reveals a common pattern across data sets: the ASR in-
+creases as more and more nodes are patched into the graph,
+before peak, whereas the accuracy is fairly stable. The ASR
+curve climbs quickly, indicating that a small patch size suf-
+fices to achieve good ASR. We thus set the patch size to be
+1%, 1%, and 5% of the original node set for Cora, Citeseer,
+and Pol.Blogs, respectively.
+The L2 Projection Radius.
+We then keep the plateaued
+percentages of patch nodes found in Figure 2 and investigate
+the influence of one important hyper-parameter: L2 projec-
+tion radius ξ. In Figure 3, we report the average ASR, predic-
+tion accuracy, and the number of patch edges by increasing
+ξ. It shows that when ξ = 10, the ASRs achieve the highest
+value on the three benchmarks, while preserving the overall
+prediction accuracy. Afterward, the ASR will not increase
+further.
+Moreover, the number of patch edges quickly climbs up
+with larger ξ. This is because the number of patch edges is
+implicitly controlled by the projection radius. Increasing ξ
+will densify the patch. In real situations, we can adjust the
+projection radius to make a balance in the trade-off between
+ASR and number of patch edges, so that the edge density for
+the added patches is indistinguishable from real ones. Never-
+theless, in subsequent experiments, we adopt ξ = 10 for the
+highest ASR, regardless of the density of patch edges.
+Necessity of clipping.
+Based on [Zang et al., 2020], we also
+adopt clipping to encourage the stability of results. In Table 2,
+we list the average ASR of two variants of GUAP using clip-
+ping versus not. It shows that clipping significantly increases
+the ASR of GUAP. Therefore, clipping is a necessary ingre-
+dient of the method for achieving high attack performance.
+Training cost and acceleration.
+Next we investigate the
+computational cost. An estimate is O(max epoch · |VL| ·
+m(m + 2n)), where the factor |VL| (denoting the training set
+size) comes from the for-loop in Algorithm 1, whereas the
+factor m(m + 2n) (denoting the difference in matrix size be-
+tween the original graph and the patched graph) comes from
+the inner procedure Algorithm 2. Because the node set size n
+is given and the patch size m is implicitly controlled by the
+desired ASR (see the preceding experiment), one factor we
+may adjust to scale the cost better is the length of the for-loop.
+
+(a) Cora.
+(b) Citeseer.
+(c) Pol.Blogs.
+Figure 2: ASR and accuracy as number of patch nodes increases (in percentage of the node set size).
+(a) Cora.
+(b) Citeseer.
+(c) Pol.Blogs.
+Figure 3: Average number of patch edges, ASR, and accuracy as the L2 projection radius ξ increases.
+Inside each epoch, rather than iterating over the entire training
+set, we deal with a random subset only. Table 3 shows that
+as one uses a smaller subset, the training time reduces pro-
+portionally, whereas ASR suffers only slightly and accuracy
+barely changes. Hence, for a large graph with large training
+set, the sampling scheme effectively accelerates training.
+Comparison with related methods.
+Now we compare
+GUAP with several of its variants, as well as GUA and FGA.
+See Table 4. Same as GUAP, GUA preserves the prediction
+accuracy but achieves a lower ASR. The preservation of ac-
+curacy indicates that most nodes have a robust neighborhood
+and the compromise of only the target as one of the neigh-
+bors affects little. The observation similarly applies to GUAP
+without patching any edges, although in this case the ASR is
+significantly dropped. The observation also applies to GUAP
+with randomly patched edges, because the number of such
+edges is quite small. In this case, the ASR also suffers, al-
+though to a less extent than the case of not patching any edges.
+However, when more and more random edges are patched,
+these edges play an increasingly significant role to the neigh-
+borhood, leading to substantial compromise in the prediction
+accuracy. Next, GUAP with regenerated node features barely
+changes the ASR and the accuracy. This observation, together
+with earlier ones, indicates that node features are much less
+important than edges, the effort of training which pays. The
+non-universal attack method FGA also barely changes the
+accuracy, but in some cases it achieves a higher ASR than
+GUAP while in others not.
+Nettack [Z¨ugner et al., 2018] is a popular non-universal at-
+tack method. Due to its high computational cost for a compa-
+rable perturbation, it is infeasible to conduct experiments with
+Nettack in a fair setting. Here, we highlight the computational
+costs. GUAP takes time O(max epoch · |VL| · m(m + 2n)).
+|VL| is typically a small fraction of n and grows much slower
+than n, which can be further reduced by sampling. On the
+other hand, to attack all nodes, Nettack costs O(n2(E · T +
+F))), where E and F represent the number of edge and fea-
+ture perturbations, respectively, and T is the average size of
+a 2-hop neighborhood. In practice, the n2 factor renders Net-
+tack a rather slower method to run, if the aim is to attack all
+nodes.
+Transferability to other models.
+We apply the patch
+trained with GCN on other GNN models: GAT [Veliˇckovi´c et
+al., 2017], FastGCN [Chen et al., 2018a], AS-GCN [Huang
+et al., 2018], and two embedding models node2vec [Grover
+and Leskovec, 2016] and DeepWalk [Perozzi et al., 2014].
+Table 5 summarizes the results.
+GAT is developed based
+on GCN through incorporating the attention mechanism.
+Node2vec and DeepWalk update node embeddings by ex-
+ploring the local structure via random walk. Different from
+the other models, instead of using the whole graph, FastGCN
+and AS-GCN use importance sampling to sample layer-wise
+nodes to reduce training cost.
+One sees that the attack performance is well maintained
+on GAT, except the ASR of Pol.Blogs. For this exception
+and all cases of node2vec and DeepWalk, the ASR still is
+
+-O- ASR
+.A.. NewAccO- ASR
+..A.. NewAccO- ASR
+A.. NewAcc口
+# Patch Edges
+O ASR
+NewAcc# Patch Edges
+O ASR
+NewAcc# Patch Edges
+O ASR
+-NewAccTable 3: ASR and change of accuracy under different sampling rate of the training set. Values in () next to the data set name denote the patch
+set size as a percentage of the node set size.
+Cora (1%)
+Citeseer (1%)
+Pol.Blogs (5%)
+Sample Rate
+ASR
+∆Acc
+ASR
+∆Acc
+ASR
+∆Acc
+100%
+91.37%
+−0.12%
+85.05%
+−0.14%
+53.24%
++0.26%
+40% (3x speedup)
+89.57%
+−0.17%
+86.93%
+−0.14%
+52.92%
++0.35%
+20% (5x speedup)
+87.25%
+−0.17%
+87.53%
+−0.19%
+53.01%
++0.26%
+10% (10x speedup)
+82.42%
++0.01%
+83.41%
+−0.15%
+52.78%
++0.36%
+5% (20x speedup)
+80.01%
+−0.16%
+79.43%
+−0.09%
+52.77%
++0.36%
+Table 4: Comparison of with related methods. Values in () next to the data set name denote the patch set size.
+Cora (29)
+Citeseer (33)
+Pol.Blogs (45)
+Baselines
+ASR
+∆Acc
+ASR
+∆Acc
+ASR
+∆Acc
+GUA
+86.48%
+−0.07%
+82.23%
+−0.07%
+48.36%
++0.38%
+GUAP w/o patch edges
+28.34%
+−0.01%
+25.02%
+−0.01%
+14.62%
++0.39%
+GUAP w/ random edges
+58.27%
+−0.79%
+62.73%
+−0.88%
+19.99%
++0.31%
+GUAP w/ more rand. edges
+68.81%
+−46.03%
+77.74%
+−48.47%
+18.69%
+−17.26%
+GUAP
+91.42%
+−0.11%
+85.03%
+−0.15%
+51.10%
++0.36%
+GUAP + regen. features
+91.41%
+−0.02%
+85.00%
+−0.02%
+51.08%
++0.36%
+FGA (Not universal)
+94.90%
+−0.66%
+92.91%
+−0.20%
+42.74%
+−0.14%
+Table 5: Attack performance when using the patched graph trained
+with GCN on other models. The patch set percentage is 1%, 1%,
+5% for Cora, Citeseer and Pol.Blogs, respectively.
+Methods
+Cora
+Citeseer
+Pol.Blogs
+GCN(ASR)
+91.37%
+85.05%
+53.24%
+GCN(∆Acc)
+−0.12%
+−0.14%
++0.26%
+GAT(ASR)
+90.91%
+85.04%
+40.02%
+GAT(∆Acc)
+−0.36%
+−0.19%
+−0.04%
+node2vec(ASR)
+74.89%
+84.24%
+43.07%
+node2vec(∆Acc)
++2.58%
++3.66%
++2.83%
+DeepWalk(ASR)
+81.02%
+82.41%
+41.32%
+DeepWalk(∆Acc)
+−41.42%
+−21.51%
+−3.50%
+FastGCN(ASR)
+41.39%
+34.74%
+36.59%
+FastGCN(∆Acc)
+−2.43%
+−0.40%
+−2.08%
+AS-GCN(ASR)
+36.68%
+31.09%
+39.24%
+AS-GCN(∆Acc)
+−2.42%
+−1.46%
+−2.24%
+reasonably similar to the GCN case. However, the node2vec
+accuracy surprisingly increases and the DeepWalk accuracy
+significantly drops.
+Additionally, both FastGCN and AS-GCN reveal robust-
+ness against our attack, which is by and large owing to the
+use of sampling. Such an observation is not surprising. The
+patch is quite small, constituting only 1% of the nodes in Cora
+and Citeseer. Consequently, the patch nodes are likely to be
+ignored in sampling and thus voiding attacks. Furthermore,
+the patches do not negatively impact the overall accuracy sig-
+nificantly.
+Based on the above findings, we see that the patch opti-
+mized for GCN is not guaranteed to work similarly on all
+other models, although it does perform equally well on GAT
+and also reasonably close on DeepWalk and node2vec in
+terms of ASR. Such a result is expected, since GAT has a
+similar architecture to GCN whereas the other models oper-
+ate quite differently. Overall, we conclude that our approach
+transfers well to neural architectures similar to the one trained
+on.
+6
+Conclusion
+In this paper, we consider a novel type of graph universal at-
+tacks that do not modify the existing nodes and edges and
+that do not change the prediction of nodes other than the tar-
+get. The attack adversarially patches a small number of new
+nodes and edges to the original graph. It compromises any
+target through flipping its connections to the patch. We de-
+velop an algorithm, GUAP, to find such a patch and demon-
+strate high attack success rate. We show that the algorithm
+can be accelerated through sampling the training set in each
+epoch without sacrificing attack performance, hinting feasi-
+bility for large graphs. For example, a 5% sampling leads
+to a 20x speedup in training. GUAP achieves a higher ASR
+than the recently proposed universal attack GUA. Moreover,
+the patch trained with GCN can be used to effectively attack
+other models, such as GAT, as well.
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