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|---|---|---|---|---|---|---|---|---|
**Elliptic Curve**
|
Curves
|
Mar 27, 2022
|
Alan Jo
|
Alan Jo
|
May 30, 2022
|
9c8eae521cd54185b67c7fda669b8638
|
||
Spline
|
Curves
|
Nov 14, 2022
|
Alan Jo
|
Alan Jo
|
Nov 14, 2022
|
### Splines
|Title|
|:-:|
|[Cubic B-Splines](https://texonom.com/cubic-b-splines-cd868a25b69c4112b1aaede96f33ca2c)|
|[Cubic Hermite Spline](https://texonom.com/cubic-hermite-spline-a09ded1c44ca44b5920f2354c594bc82)|
> [The Beauty of Bรฉzier Curves](https://www.youtube.com/watch?v=aVwxzDHniEw)
|
46228d53c9ab4ebc97a53d27ca07e245
|
|
Cubic Bezier Curve
|
Bรฉzier curves
|
Nov 14, 2022
|
Alan Jo
|
Alan Jo
|
Nov 14, 2022
|
3f18b9925ac8437b85412f8806109a8a
|
||
Cubic B-Splines
|
Splines
|
Nov 14, 2022
|
Alan Jo
|
Alan Jo
|
Nov 14, 2022
|
cd868a25b69c4112b1aaede96f33ca2c
|
||
Cubic Hermite Spline
|
Splines
|
Nov 14, 2022
|
Alan Jo
|
Alan Jo
|
Nov 14, 2022
|
a09ded1c44ca44b5920f2354c594bc82
|
||
Linear Interpolation
|
Interpolations
|
May 27, 2022
|
Alan Jo
|
Alan Jo
|
Apr 19, 2023
|
### lerp
[Bilinear Interpolation](https://texonom.com/bilinear-interpolation-49ec30618b704bfcb2f513a427092f96)
|
d329ad3559dc490eb7592506de0d435e
|
|
Bilinear Interpolation
|
Linear Interpolation
| null | null | null | null | null |
49ec30618b704bfcb2f513a427092f96
|
|
Bicubic Patch
|
Surface Notion
|
Nov 14, 2022
|
Alan Jo
|
Alan Jo
|
Nov 14, 2022
|
> [Bicubic Patches](https://www.inf.ed.ac.uk/teaching/courses/cg/d3/bezierPatch.html)
|
6c488ea8e7fe4d369fc84d763773a174
|
|
Implicit Surface
|
Surface Notion
|
Nov 14, 2022
|
Alan Jo
|
Alan Jo
|
Nov 14, 2022
|
b093380ece4b490cafc67fadee3dbd5b
|
||
Differentiable Manifold
|
Complex Geometry Notion
|
Feb 22, 2022
|
Alan Jo
|
Alan Jo
|
Feb 22, 2022
|
## smooth manifold
### Differentiable Manifolds
|Title|
|:-:|
|[Complex Manifold](https://texonom.com/complex-manifold-f01eac66b38e434db8eacd87da44cfd1)|
|[symplectic manifold](https://texonom.com/symplectic-manifold-f973a0bc77494e6ead89d1758e4e3236)|
|[generalized complex manifold](https://texonom.com/generalized-complex-manifold-1ca2b87c2d844f4cb67071f7d4319e91)|
|
13ccf7d24d92437d8524988d7f863f0e
|
|
Complex Manifold
|
Differentiable Manifolds
|
Feb 22, 2022
|
Alan Jo
|
Alan Jo
|
Feb 22, 2022
|
### ๋ณต์๋ค์์ฒด
๊ตญ์์ ์ผ๋ก ๋ณต์ ๊ณต๊ฐย {\displaystyle \mathbb {C} ^{n}}์ผ๋ก ๊ฐ์ฃผํ ์ ์๋ย [๋งค๋๋ฌ์ด ๋ค์์ฒด](https://ko.wikipedia.org/wiki/%EB%A7%A4%EB%81%84%EB%9F%AC%EC%9A%B4_%EB%8B%A4%EC%96%91%EC%B2%B4)
### Complex Manifolds
|Title|
|:-:|
|[Hermitian manifold](https://texonom.com/hermitian-manifold-4d106bc7abe4467eb63d09e2189ca08b)|
> [๋ณต์๋ค์์ฒด - ์ํค๋ฐฑ๊ณผ, ์ฐ๋ฆฌ ๋ชจ๋์ ๋ฐฑ๊ณผ์ฌ์ ](https://ko.wikipedia.org/wiki/%EB%B3%B5%EC%86%8C%EB%8B%A4%EC%96%91%EC%B2%B4)
|
f01eac66b38e434db8eacd87da44cfd1
|
|
generalized complex manifold
|
Differentiable Manifolds
|
Feb 22, 2022
|
Alan Jo
|
Alan Jo
|
Feb 22, 2022
|
1ca2b87c2d844f4cb67071f7d4319e91
|
||
symplectic manifold
|
Differentiable Manifolds
|
Feb 22, 2022
|
Alan Jo
|
Alan Jo
|
Feb 22, 2022
|
f973a0bc77494e6ead89d1758e4e3236
|
||
Hermitian manifold
|
Complex Manifolds
|
Feb 22, 2022
|
Alan Jo
|
Alan Jo
|
Jun 25, 2023
|
๋ณต์ ๊ธฐํํ์์ย [Riemannian Manifold](https://texonom.com/riemannian-manifold-2772641c3daf4ef98d7b897769cda5e1) ์ ๋์๋๋ ๊ฐ๋
|
4d106bc7abe4467eb63d09e2189ca08b
|
|
Computational Geometry Algorithm
|
Computational Geometry Notion
|
Sep 16, 2021
|
Alan Jo
|
Alan Jo
|
Apr 7, 2023
|
### Computational Geometry Algorithms
|Title|
|:-:|
|
689f89080f5f42eaa95f98961d9b5217
|
|
Kรคhler manifold
|
Differential Geometry Fields
|
Jul 27, 2022
|
Alan Jo
|
Alan Jo
|
Jul 27, 2022
|
### Kรคhler manifold Notion
|Title|
|:-:|
|[Hodge Theory](https://texonom.com/hodge-theory-63342a4cc3fd43c489e9e1df190c83f8)|
|
a3ca08b54edb4e69be8acdc3f9b3b6be
|
|
Hodge Theory
|
Kรคhler manifold Notion
|
Jul 27, 2022
|
Alan Jo
|
Alan Jo
|
Jul 27, 2022
|
[June Huh](https://texonom.com/june-huh-1b5e90c4634b473aab142f7b2423cf06)
|
### Hodge Theory Notion
|Title|
|:-:|
|[Hodge Structure](https://texonom.com/hodge-structure-c71b95d4f8cf4a66a9432219ef84cdc6)|
|
63342a4cc3fd43c489e9e1df190c83f8
|
Hodge Structure
|
Hodge Theory Notion
|
Jul 27, 2022
|
Alan Jo
|
Alan Jo
|
Jul 27, 2022
|
๋ก๊ทธ ์ค๋ชฉ์ฑ
### Template Gallary
|Title|
|:-:|
|[Template Page](https://texonom.com/template-page-f4881e476a8045b6b510570b4cd328c8)|
|
c71b95d4f8cf4a66a9432219ef84cdc6
|
|
Template Page
|
Template Gallary
|
Jul 27, 2022
|
Alan Jo
|
Alan Jo
|
Jul 27, 2022
|
f4881e476a8045b6b510570b4cd328c8
|
||
Concave Set
|
Euclid Geometry Notion
|
May 18, 2023
|
Alan Jo
|
Alan Jo
|
May 18, 2023
|
[Convex Set](https://texonom.com/convex-set-b732e3e2d0fa492cb51471bbd45403b7)
|
### Concave Set Notion
|Title|
|:-:|
|
01174d4fbb66477f87ca8b7cddddcc45
|
Convex Set
|
Euclid Geometry Notion
|
Mar 9, 2023
|
Alan Jo
|
Alan Jo
|
May 18, 2023
|
vector space that contains all line segments connecting any two points within the set
$$A \; set \; C \subset R^d \; is \; convex \; if$$
$$\lambda x + (1 - \lambda)y \in C, \forall \lambda \in [0,1]$$
### Convex Set Notion
|Title|
|:-:|
|[Convex Function](https://texonom.com/convex-function-b61779c9cb814c0e9515c9dc71ca9596)|
|
b732e3e2d0fa492cb51471bbd45403b7
|
|
Euler Angle
|
Euclid Geometry Notion
|
Aug 20, 2021
|
Alan Jo
|
Alan Jo
|
Mar 9, 2023
|
5daeab8525864f5dad9a7d054cd8f360
|
||
Rodrigues formula
|
Euclid Geometry Notion
|
Aug 20, 2021
|
Alan Jo
|
Alan Jo
|
Mar 9, 2023
|
[EulerโRodrigues formula](https://texonom.com/eulerrodrigues-formula-748cb8cd3bc94900874c8053e3e568a4)
|
0708f6eb71a240379a067234503c64ed
|
|
Convex Function
|
Convex Set Notion
|
May 18, 2023
|
Alan Jo
|
Alan Jo
|
May 18, 2023
|
For a convex set $C \subset R^d$, a function $f: C \rightarrow R$ is convex if
$$f(\lambda x+ (1 - \lambda)y) \le \lambda f(x) + (1 - \lambda)f(y), \forall x, y \in C and \forall \lambda \in [0, 1]$$
|
b61779c9cb814c0e9515c9dc71ca9596
|
|
EulerโRodrigues formula
|
Rodrigues formula
| null | null | null | null | null |
748cb8cd3bc94900874c8053e3e568a4
|
|
Affinity
|
Euclidean Geometry Notion
|
Oct 31, 2022
|
Alan Jo
|
Alan Jo
|
Apr 18, 2023
|
[Matrix](https://texonom.com/matrix-cbbb0614f0214e9ab9fd092a3b818132) [Convexity](https://texonom.com/convexity-f4f40f0e40664c40ad939e0588332c98)
|
## AffineTransformation
### linear transformation + Translation
์ํ ๊ธฐํํ์ ์ฑ์ง๋ค์ ๋ณด์กดํ๋ ๋ ์ํ ๊ณต๊ฐ ์ฌ์ด์ ํจ์
$$Ma + \vec{b}$$
### Affine Function
ํจ์๊ฐ affineํ๋ค๋ ๊ฒ์ ํจ์๊ฐ ์๋์ ๊ฐ์ ํจ์๋ก ํํ๋ ์ ์๋ค๋ ๊ฒ์ ์๋ฏธ
$$f(x) = Ax + b$$
> [Affine transformation](https://en.wikipedia.org/wiki/Affine_transformation)
> [What are affine transformations?](https://www.youtube.com/watch?v=E3Phj6J287o)
|
6bba36af687f4be5adc4276ebc5a3d9a
|
Euclidean Space
|
Euclidean Geometry Notion
|
Mar 10, 2023
|
Alan Jo
|
Alan Jo
|
Mar 10, 2023
|
b03a1df4f1d34009a3eeca6016cf1c93
|
||
**Hilbert space**
|
Euclidean Geometry Notion
|
Apr 17, 2023
|
Alan Jo
|
Alan Jo
|
Apr 17, 2023
|
generalizing the methods of [Euclidean Space](https://texonom.com/euclidean-space-b03a1df4f1d34009a3eeca6016cf1c93) to infinite-dimension
> [Hilbert space](https://en.wikipedia.org/wiki/Hilbert_space)
|
ef68e0eabd7045609d94265ec7a4902f
|
|
H**omogeneous coordinate**
|
Projective Geometry Notion
|
Oct 26, 2022
|
Alan Jo
|
Alan Jo
|
Apr 19, 2023
|
We canโt do compute perspective projection
์ฌ์๊ธฐํํ์์ n์ฐจ์ ์ฌ์ ๊ณต๊ฐ์ n+1๊ฐ์ ์ขํ๋ก ๋ํ๋ด๋ ์ขํ๋ฅผ ๋์ฐจ์ขํ๋ผ ํ๋ค
$$\begin{pmatrix}x \\y \\z \\w \\\end{pmatrix}=\begin{pmatrix}x /w \\y /w \\z /w \\\end{pmatrix}$$
w๋ก 1 ๋ง์ด ์ฌ์ฉ
normal coordinate is heterogeneous
> [Homogeneous Coordinates | Yasen Hu](https://yasenh.github.io/post/homogeneous-coordinates/)
|
1c52d6e0a29d4c30a298306eddc6bdaf
|
|
Projection
|
Projective Geometry Notion
|
Oct 26, 2022
|
Alan Jo
|
Alan Jo
|
Oct 26, 2022
|
72a75777396c4eba9c36f596364b09eb
|
||
Projective Plane
|
Projective Geometry Notion
|
Oct 26, 2022
|
Alan Jo
|
Alan Jo
|
Oct 26, 2022
|
์์ ์ ์ง๋๋ ์ง์ ๋ค๋ก ๋ง๋ค์ด์ง ๊ณต๊ฐ
|
37c3457e2da345839771100f3d9689e2
|
|
**Ricci curvature**
| null | null | null | null | null | null |
๋ฆฌ๋ง ๋ค์์ฒด์ ๊ณก๋ฅ ์ ๋ํ๋ด๋ 2์ฐจ ํ
์์ฅ
๋ฆฌ๋ง ๊ณก๋ฅ ํ
์์ ๋๊ฐํฉ์ด๋ค. ๋ถํผ์ ์๊ณก์ ๋ํ๋ด๋ ๊ฒ์ผ๋ก ํด์
|
0e5be06f9dd34ef1b75b4f3530e411dc
|
Riemann curvature tensor
|
Riemannian Geometry Notion
|
Mar 28, 2023
|
Alan Jo
|
Alan Jo
|
Mar 28, 2023
|
[Bernhard Riemann](https://texonom.com/bernhard-riemann-166d61214be5439ca5f7ba1090063a3e)
|
์ด๋ ๋๋ฌธ์ 4์ฐจ์ ์ด์์ฐจ์๋ ์ค๋ฅ์๋ ์ํ์ ์์ ๊ฐ๋ฅ
|
5cfe4cbcd3764dfabab0a7a0e8daafb8
|
Riemannian Manifold
| null | null | null | null | null | null |
2772641c3daf4ef98d7b897769cda5e1
|
|
Angle
|
Geometry Notion
|
Jun 4, 2021
|
Alan Jo
|
Alan Jo
|
Feb 22, 2022
|
### Angles
|Title|
|:-:|
|[Euler Angle](https://texonom.com/euler-angle-336ea99239024f3fae587e1180c0798f)|
|
fe43202d6a8f481d99db2552657191bb
|
|
Coordinate
|
Geometry Notion
|
Mar 21, 2023
|
Alan Jo
|
Alan Jo
|
Mar 21, 2023
|
### Coordinates
|Title|
|:-:|
|[Polar Coordinate](https://texonom.com/polar-coordinate-affa11aeb26241cdabf2a592e73daf91)|
|[Rectangular Coordinate](https://texonom.com/rectangular-coordinate-46c47322fa3540a8b8469746fba555c0)|
|
85b71f77113e49338924246b458a3e06
|
|
Displacement
|
Geometry Notion
|
Mar 24, 2021
|
Alan Jo
|
Alan Jo
|
Feb 22, 2022
|
๋ณ์
|
6959089ae02445bebbc88af90e33928f
|
|
Distance
|
Geometry Notion
|
May 30, 2023
|
Alan Jo
|
Alan Jo
|
Jun 11, 2023
|
### Distance Types
|Title|
|:-:|
|[Euclidean Distance](https://texonom.com/euclidean-distance-0d3a8b5dc5e04ee595dc85365b2d3fd9)|
|[Geodesic Distance](https://texonom.com/geodesic-distance-b3fce1b60ef848e481da3e9d0a65e188)|
|[Statistical Distance](https://texonom.com/statistical-distance-5c0ad7397f3e4d008e8d6479784a19f0)|
|
3e9ed62771dd42f4893c6209c36fed81
|
|
Fractal
| null | null | null | null | null | null |
c4ae0e4ab1cf4e8db2c9d4f3e7412114
|
|
Paraboloid
|
Polyhedrons
|
Apr 19, 2023
|
Alan Jo
|
Alan Jo
|
Sep 3, 2023
|
$$f(x,y) = x^2 + y^2$$
if we slice at some z then โ [Circle](https://texonom.com/circle-ee04d780f37b4376ae197ea28a8e9a1e)
[Eigenvector](https://texonom.com/eigenvector-d2424f7eaea4443d81105606ebf4e975)
|
ea4b361db9d342909a517fdabbf50ef6
|
|
Plane
|
Geometry Notion
|
Sep 26, 2022
|
Alan Jo
|
Alan Jo
|
Apr 6, 2023
|
[Normal Vector](https://texonom.com/normal-vector-84412bdb0b8c43e3a206cd54bc9901f7)
|
p is normal vector
$$(x-p) \cdot n = 0$$
### Planes
|Title|
|:-:|
|[Hyperplane](https://texonom.com/hyperplane-cce08b97fc4a461d806aee1ce68a1911)|
### Plane Notion
|Title|
|:-:|
|[Normal Vector](https://texonom.com/normal-vector-84412bdb0b8c43e3a206cd54bc9901f7)|
|[Coplanar](https://texonom.com/coplanar-e32a7186c1e44fdfabe9af64705f4a1f)|
|
2ec9eca0e67a403b9c63ed26b44e1881
|
P**olyhedron**
|
Polytopes
|
Jun 20, 2022
|
Alan Jo
|
Alan Jo
|
Sep 3, 2023
|
### ๋ค๋ฉด์ฒด
### P**olyhedrons**
|Title|
|:-:|
|[Regular Polyhedron](https://texonom.com/regular-polyhedron-90f91a9623f645c8906d4a904d754ff9)|
|[Sphere](https://texonom.com/sphere-7ca05d25d2cc4dfa99654d64103dc667)|
|[Paraboloid](https://texonom.com/paraboloid-ea4b361db9d342909a517fdabbf50ef6)|
### P**olyhedron Notion**
|Title|
|:-:|
|[Crystallography](https://texonom.com/crystallography-3c5c6dab988546959b1a085ba81ed676)|
> [Polyhedron](https://en.wikipedia.org/wiki/Polyhedron)
|
9aff2db4adca4dc1acbaf371eb4b7b73
|
|
Pythagorean Theorem
|
Geometry Notion
|
Mar 10, 2023
|
Alan Jo
|
Alan Jo
|
Mar 10, 2023
|
### ํผํ๊ณ ๋ผ์ค ์ ๋ฆฌ
can be generalized even if there is not a [Euclidean Space](https://texonom.com/euclidean-space-b03a1df4f1d34009a3eeca6016cf1c93)
|
12e01244f5db459886d08bbc809ea964
|
|
Quadratics
|
Geometry Notion
|
Mar 23, 2023
|
Alan Jo
|
Alan Jo
|
Mar 23, 2023
|
2cb9782695f74ece92a1b1fa49c7953d
|
||
Quaternion
|
Geometry Notion
|
Jul 29, 2020
|
Alan Jo
|
Alan Jo
|
Sep 3, 2023
|
### Quaternions, Hamilton number
๋ณต์์๋ฅผ ํ์ฅํด ๋ง๋ ์ ์ฒด๊ณ์ด๋ค. ๋ค ๊ฐ์ ์ค์ ์ฑ๋ถ์ ๊ฐ์ง๋ฉฐ, ๋ง์
๊ณผ ๊ณฑ์
์ ๊ฒฐํฉ๋ฒ์น ๋ฐ ๋ง์
์ ๊ตํ๋ฒ์น์ ๋ง์กฑ์ํค์ง๋ง ๊ณฑ์
์ ๊ตํ๋ฒ์น์ ์ฑ๋ฆฝํ์ง ์๋๋ค
- Non commutative Algebra
- ํ์ ์ ํํํ๊ธฐ ์ํด ์ฌ์ฉ
introduce multiple complex number (canโt make from 3, we need 4)
$$i^2 = j^2 = k^2 = ijk = -1$$
> [์ฌ์์](https://ko.wikipedia.org/wiki/%EC%82%AC%EC%9B%90%EC%88%98)
|
d3fb67e838e34633ad70cc2f0f8454c1
|
|
Sinusoid
|
Geometry Notion
|
Mar 21, 2023
|
Alan Jo
|
Alan Jo
|
Mar 21, 2023
|
### ์ ํํ
### Sinusoids
|Title|
|:-:|
|[Cosine](https://texonom.com/cosine-be8bdb6b9a714ee1bb748930e7cfccfe)|
|[Sine](https://texonom.com/sine-4c32ef7f18c84089a02d7c29f9d14e98)|
### Sinusoid Notion
|Title|
|:-:|
|[Euler's Formula](https://texonom.com/eulers-formula-71f5337221904ba5b26e1398af5de0ab)|
|[Sinc Function](https://texonom.com/sinc-function-ecbf029e361a476184147df258961a3a)|
|
262738d8c4cc47b8bf7e87524a396311
|
|
Sphere
|
Polyhedrons
|
Sep 26, 2022
|
Alan Jo
|
Alan Jo
|
Sep 3, 2023
|
$$||x - c^2|| = r^2$$
### Spheres
|Title|
|:-:|
|[Hypersphere](https://texonom.com/hypersphere-02397359500f4cd885654cbc1505c23b)|
|[Circle](https://texonom.com/circle-ee04d780f37b4376ae197ea28a8e9a1e)|
|
7ca05d25d2cc4dfa99654d64103dc667
|
|
Euler Angle
|
Angles
|
Jun 4, 2021
| null | null | null |
๊ฐ์ฒด๊ฐ ๋์ธ ๋ฐฉํฅ์ 3์ฐจ์ ๊ณต๊ฐ์ ํ์ํ๊ธฐ ์ํด ๋ ์จํ๋ฅดํธ ์ค์ผ๋ฌ๊ฐ ๋์
ํ ์ธ ๊ฐ์ ๊ฐ๋
3์ฐจ์ ํ์ ๊ตฐ SO์ ํ ์ขํ๊ณ
[gimbal lock](https://texonom.com/gimbal-lock-73f2be32809d48f8b54058c46b36f9db)
### Euler Angles
|Title|
|:-:|
|[Tait Bryan Angle](https://texonom.com/tait-bryan-angle-1942f607253d423da17011e5fe6b786a)|
|
336ea99239024f3fae587e1180c0798f
|
|
gimbal lock
|
Euler Angle
| null | null | null | null | null |
euler angle ์ ๋ฐ์ํ๋ ๋ฌธ์
[Quaternion](https://texonom.com/quaternion-d3fb67e838e34633ad70cc2f0f8454c1) ์ ์ฌ์ฉํ๋ฉด ํด๊ฒฐ
|
73f2be32809d48f8b54058c46b36f9db
|
Tait Bryan Angle
|
Euler Angles
|
Jun 4, 2021
| null | null | null | null |
1942f607253d423da17011e5fe6b786a
|
|
Polar Coordinate
|
Coordinates
|
Mar 21, 2023
|
Alan Jo
|
Alan Jo
|
Mar 21, 2023
|
[Euler's Formula](https://texonom.com/eulers-formula-71f5337221904ba5b26e1398af5de0ab)
|
### Polar transformation
|
affa11aeb26241cdabf2a592e73daf91
|
Rectangular Coordinate
|
Coordinates
|
Mar 21, 2023
|
Alan Jo
|
Alan Jo
|
Mar 21, 2023
|
46c47322fa3540a8b8469746fba555c0
|
||
Euclidean Distance
|
Distance Types
|
Mar 10, 2023
|
Alan Jo
|
Alan Jo
|
May 30, 2023
|
### **Pythagorean distance**
> [Euclidean distance](https://en.wikipedia.org/wiki/Euclidean_distance)
|
0d3a8b5dc5e04ee595dc85365b2d3fd9
|
|
Geodesic Distance
|
Distance Types
|
Aug 26, 2022
|
Alan Jo
|
Alan Jo
|
May 30, 2023
|
### ์ง์ ์ ๊ฐ๋
์ ๊ตฝ์ ๊ณต๊ฐ์ผ๋ก ์ผ๋ฐํํ ๊ฒ
generalized shortest path for curved space
> [Why Gravity is not a force | Uncovering the Law of Gravity | Explained with General Relativity](https://www.youtube.com/watch?v=cbLRF-XY-b0)
|
b3fce1b60ef848e481da3e9d0a65e188
|
|
Statistical Distance
|
Distance Types
|
Jun 11, 2023
|
Alan Jo
|
Alan Jo
|
Jun 11, 2023
|
### Statistical Distances
|Title|
|:-:|
|[Divergence Distance](https://texonom.com/divergence-distance-300140936f34407799d9d105dd99ad9c)|
|
5c0ad7397f3e4d008e8d6479784a19f0
|
|
Divergence Distance
|
Statistical Distances
|
Jun 11, 2023
|
Alan Jo
|
Alan Jo
|
Jun 11, 2023
|
### divergences can be viewed as generalizations of SED
### Divergence Distances
|Title|
|:-:|
|[KL Divergence](https://texonom.com/kl-divergence-c7964872f5184a7baaf312605405aef6)|
|[Jensen-Shannon divergence](https://texonom.com/jensen-shannon-divergence-4749b6791d9b4fdd9dd72ded54e9ac99)|
|[Chi-square Divergence](https://texonom.com/chi-square-divergence-c95e678a6d9b4d538690a8aec066636a)|
|[SED](https://texonom.com/sed-3cf0805eb3814622a82f09f63ab130e4)|
|[Bregman divergence](https://texonom.com/bregman-divergence-fb4ede67a13e4399b59444923c604d89)|
|[f-divergence](https://texonom.com/f-divergence-c05d2237e72f4b518a426ceb101c2828)|
|[Bhattacharyya distance](https://texonom.com/bhattacharyya-distance-9ee4f443e3744555a167bf9b7c2f885b)|
|[Hamming distance](https://texonom.com/hamming-distance-7b2c99ae58644103900c25798460bf4e)|
> [Divergence (statistics)](https://en.wikipedia.org/wiki/Divergence_(statistics))
|
300140936f34407799d9d105dd99ad9c
|
|
**Bhattacharyya distance**
|
Divergence Distances
|
Jun 11, 2023
|
Alan Jo
|
Alan Jo
|
Jun 11, 2023
| null |
9ee4f443e3744555a167bf9b7c2f885b
|
|
**Bregman divergence**
|
Divergence Distances
|
Jun 11, 2023
|
Alan Jo
|
Alan Jo
|
Jun 11, 2023
| null |
### **Bregman divergences**
|Title|
|:-:|
|[Mahalanobis distance](https://texonom.com/mahalanobis-distance-091ab31287fe4d53b88568c081d16b21)|
> [Bregman divergence](https://en.wikipedia.org/wiki/Bregman_divergence)
|
fb4ede67a13e4399b59444923c604d89
|
Chi-square Divergence
|
Divergence Distances
|
Jun 6, 2023
|
Alan Jo
|
Alan Jo
|
Jun 11, 2023
| null |
c95e678a6d9b4d538690a8aec066636a
|
|
***f*****-divergence**
|
Divergence Distances
|
Jun 11, 2023
|
Alan Jo
|
Alan Jo
|
Jun 11, 2023
| null |
c05d2237e72f4b518a426ceb101c2828
|
|
**Hamming distance**
|
Divergence Distances
|
Jun 11, 2023
|
Alan Jo
|
Alan Jo
|
Jun 11, 2023
| null |
7b2c99ae58644103900c25798460bf4e
|
|
**Jensen-Shannon divergence**
|
Divergence Distances
|
Jun 1, 2023
|
Alan Jo
|
Alan Jo
|
Jun 11, 2023
| null |
KL-divergence๋ฅผ 2๊ฐ์ง๋ฅผ ๊ตฌํ๊ณ ๋ ํ๊ท
Distance like
|
4749b6791d9b4fdd9dd72ded54e9ac99
|
KL Divergence
| null | null | null | null | null | null |
## Relative Entropy, Kullback Leibler Divergence, **I-divergence**
a metric to compare two distributions
๋ ๋ถํฌ๊ฐ ๊ฐ๋ค๋ฉด 0, ๊ทธ ์ด์ธ์ ๊ฒฝ์ฐ์๋ 0๋ณด๋ค ํฌ๋ค (์ฆ ๊ธฐ๋ํ ์ ๋ณด์ ๊ฒฐ๊ณผ์ ์ฐจ์ด prior, posterior)
divergence means just difference
bigger than 0 due to the [Gibbs' inequality](https://texonom.com/gibbs-inequality-924d4f81cbb345e4a3c4436d64e29abe)
KL-divergence๋ฅผ minimizeํ๋ ๊ฒ ๋ํ ๊ฒฐ๊ตญ log likelihood๋ฅผ maximizeํ๋ ๊ฒ๊ณผ ๊ฐ๋ค
### Popular Distance
Cross-entropy์์ entropy๋ฅผ ๋บ ๊ฐ. ๊ฑฐ๋ฆฌ ๊ฐ๋
์ด ์๋๋ค
In case of KL divergence, we have the correspondence between MLE and KL matching
has an analytic solution if both p and q follows the normal distribution
> [KL divergence - ๊ณต๋์ด์ ์ํ์ ๋ฆฌ๋
ธํธ (Angelo's Math Notes)](https://angeloyeo.github.io/2020/10/27/KL_divergence.html)
|
c7964872f5184a7baaf312605405aef6
|
SED
|
Divergence Distances
|
Jun 11, 2023
|
Alan Jo
|
Alan Jo
|
Jun 11, 2023
| null |
**Squared Euclidean distance**
|
3cf0805eb3814622a82f09f63ab130e4
|
**Mahalanobis distance**
|
Bregman divergences
|
Jun 5, 2023
|
Alan Jo
|
Alan Jo
|
Jun 11, 2023
| null |
> [๋งํ ๋ผ๋
ธ๋น์ค ๊ฑฐ๋ฆฌ - ๊ณต๋์ด์ ์ํ์ ๋ฆฌ๋
ธํธ (Angelo's Math Notes)](https://angeloyeo.github.io/2022/09/28/Mahalanobis_distance.html)
|
091ab31287fe4d53b88568c081d16b21
|
Coplanar
|
Plane Notion
|
Apr 13, 2023
|
Alan Jo
|
Alan Jo
|
Apr 13, 2023
|
๊ฐ์ ํ๋ฉด ์ ์กด์ฌ
|
e32a7186c1e44fdfabe9af64705f4a1f
|
|
Normal Vector
|
Plane Notion
|
Aug 31, 2021
|
Alan Jo
|
Alan Jo
|
Apr 6, 2023
|
84412bdb0b8c43e3a206cd54bc9901f7
|
||
Hyperplane
|
Planes
|
Apr 6, 2023
|
Alan Jo
|
Alan Jo
|
Apr 18, 2023
|
๋ค์ฐจ์ํ๋ฉด ์ผ๋ฐํ
$$w^Tx + b = 0$$
w is [Normal Vector](https://texonom.com/normal-vector-84412bdb0b8c43e3a206cd54bc9901f7) so
ํ๋ฉด์์ 1๋งํผ ๋ฉ์ด์ง๋ค๊ณ ์๊ฐํ๋ฉด w๋ฒกํฐ 1๋งํผ ๋ํด์ง๊ฑฐ๋ ๊ฒฐ๊ณผ๊ฐ์ด 1๋งํผ ๋์ด๋๋ค
๊ทธ๋ฌ๋ w์ฌ์ด์ฆ๋งํผ ๋๋ ์ค์ผ ์ค์ ๊ฑฐ๋ฆฌ ๋ณด์ฌ์ค
$$Margin = (x^Tx + b) / ||w||$$
b is called as bias
|
cce08b97fc4a461d806aee1ce68a1911
|
|
R**egular Polyhedron**
|
Polyhedrons
|
Jun 20, 2022
|
Alan Jo
|
Alan Jo
|
Sep 3, 2023
|
[Penrose tiling](https://texonom.com/penrose-tiling-1726e61c22aa4d388ba15acbb02fba5f) [Bravais lattice](https://texonom.com/bravais-lattice-e0bad58061dd4df2bcfa8067878e1604)
|
### R**egular Polyhedrons**
|Title|
|:-:|
|[Platonic solid](https://texonom.com/platonic-solid-e15de9735a2a48f7a7b1ad1d0c38ee96)|
> [Regular polyhedron](https://en.wikipedia.org/wiki/Regular_polyhedron)
> [๋ฌดํํ์ง๋ง ๋ฐ๋ณต๋์ง ์๋ ํจํด (ํ๋ก์ฆ ํ์ผ)](https://www.youtube.com/watch?v=lkms0YlFVAw)
|
90f91a9623f645c8906d4a904d754ff9
|
Euler's Formula
|
Sinusoid Notion
|
Aug 29, 2021
|
Alan Jo
|
Alan Jo
|
Jun 12, 2023
|
## Euler's identity
### Including complex number
1714๋
๋ก์ ์ฝ์ธ ๊ฐ ๋ค์๊ณผ ๊ฐ์ ํํ๋ก ์ฒ์ ๋ฐ๊ฒฌ
$e^{ix}$๋ ์ง์ํจ์๋ค
$$\ln(cosx+isinx) = ix$$
$$e^{ix} = cos(x) + i*sin(x)$$
$$e^{i\pi} + 1 = 0$$
> [Euler's formula with introductory group theory](https://youtu.be/mvmuCPvRoWQ)
### Visualization
> [์ธ์์์ ๊ฐ์ฅ ์๋ฆ๋ค์ด ์์์ ์ดํดํด๋ณด์ (์ด๊ณผ์ฉ)](https://www.youtube.com/watch?v=kgTSUZjVqas&t=2334s)
|
71f5337221904ba5b26e1398af5de0ab
|
|
Sinc Function
|
Sinusoid Notion
|
Mar 21, 2023
|
Alan Jo
|
Alan Jo
|
Sep 20, 2023
|
[Fourier Transform](https://texonom.com/fourier-transform-5ee1f9063a9743dd9846ddc600286d5a) [Box Filter](https://texonom.com/box-filter-0d975966a9474accb02603c75baafb0c)
|
$$sinc(x) = \begin{cases}\frac{sinx}{x}, & \text{if } x \ne 0\\1, & \text{if } x=0\end{cases}$$
$$sinc(\theta) = \frac{sin(\pi\theta)}{\pi\theta}$$
### [Fourier Transform](https://texonom.com/fourier-transform-5ee1f9063a9743dd9846ddc600286d5a) then [Rectangular function](https://texonom.com/rectangular-function-4d71deb4ffec44379df7cb97eb078dc9)
|
ecbf029e361a476184147df258961a3a
|
Cosine
|
Sinusoids
|
Mar 21, 2023
|
Alan Jo
|
Alan Jo
|
Mar 21, 2023
|
be8bdb6b9a714ee1bb748930e7cfccfe
|
||
Sine
|
Sinusoids
|
Mar 21, 2023
|
Alan Jo
|
Alan Jo
|
Mar 21, 2023
|
4c32ef7f18c84089a02d7c29f9d14e98
|
||
Circle
|
Spheres
|
Apr 8, 2023
|
Alan Jo
|
Alan Jo
|
Apr 8, 2023
|
### approximate by two line
> [์ ๋ฐํ ๊ทผ์ฌ (์ค์ฐจ 4% ์ดํ)](https://youtu.be/VfiADTUoxnM)
|
ee04d780f37b4376ae197ea28a8e9a1e
|
|
Hypersphere
|
Spheres
|
Apr 5, 2023
|
Alan Jo
|
Alan Jo
|
Apr 5, 2023
|
Generalized version of sphere
|
02397359500f4cd885654cbc1505c23b
|
|
Statistical Model
|
Statistics Notion
|
Mar 23, 2023
|
Alan Jo
|
Alan Jo
|
Mar 27, 2023
|
[Probability](https://texonom.com/probability-6b1e766c2c6442c7a599b558453300e4)
|
## Probability Model
### Statistical Model Notion
|Title|
|:-:|
|[Density Estimation](https://texonom.com/density-estimation-238e2a8e51f44a67a91c0567892ab643)|
|
2dcba57e60b44272a4bdf0d0dbbee68b
|
Statistics Field
|
Statistics Notion
|
Apr 3, 2022
|
Alan Jo
|
Alan Jo
|
Jun 25, 2023
|
### Statistics Fields
|Title|
|:-:|
|[Bayesian Statistics](https://texonom.com/bayesian-statistics-1f53546302c647ea9eac07139c762854)|
|[Probability theory](https://texonom.com/probability-theory-31161a5d9b9942dcb7b03bafd7659f31)|
|[Stochastic](https://texonom.com/stochastic-fc6c8f3295dd4a49a44086d393fd5b1a)|
|
29d4a41889a84f64beaa4906ab7123ec
|
|
Statistics Term
|
Statistics Notion
|
Jun 28, 2021
|
Alan Jo
|
Alan Jo
|
Jun 25, 2023
|
### Statistics Terms
|Title|
|:-:|
|[Statistical Population](https://texonom.com/statistical-population-f2a529ec04b646b1af86335b788a2030)|
|[Statistical parameter](https://texonom.com/statistical-parameter-0e888c61f6cd48b7a2df1217087f07d1)|
|[Bayesian](https://texonom.com/bayesian-abfa2017bb054f3a9dfcbc9a6c1042f6)|
|[Frequentist](https://texonom.com/frequentist-c7636520ad8d4ae3ba69b8c24f516ab1)|
|[Test statistic](https://texonom.com/test-statistic-11746f80eefb4309b516bc06160718d6)|
|[p-value](https://texonom.com/p-value-73c98ef557494f86a7ae0c1dba8f81df)|
|[Mean](https://texonom.com/mean-e903056370974e0082893cb8ea8952dd)|
|[Robust](https://texonom.com/robust-062dfadfcdd64fff9b828cca5442a684)|
|[Degrees of Freedom](https://texonom.com/degrees-of-freedom-35beaefd93fd46d58172097155eedab9)|
|[BP](https://texonom.com/bp-512f1e376cbc4448adec9a1f5acb66a6)|
|[Percent](https://texonom.com/percent-f4c541ab49fe4ffb92ec673995898108)|
|[Autoregressive](https://texonom.com/autoregressive-93cf5710b4b54730a7e7efcc6e0fc642)|
|
bfb83f1911a747d38855eb9b38180493
|
|
Density Estimation
|
Statistical Model Notion
|
Mar 23, 2023
|
Alan Jo
|
Alan Jo
|
Jun 14, 2023
|
[Probability Density Function](https://texonom.com/probability-density-function-8c4790cbb52e485586d767f4f499da49)
|
### Density Estimations
|Title|
|:-:|
|[Parameter Estimation](https://texonom.com/parameter-estimation-2c5667f2707b4d3082e96a5928464029)|
|[Non-parametric Estimation](https://texonom.com/non-parametric-estimation-88b1d8556a524a5cb1ed4792123ccb02)|
|[Semi-parametric Estimation](https://texonom.com/semi-parametric-estimation-aa0e7a6b8cc54d8d948385ac7206ba95)|
|[Explicit Density Estimation](https://texonom.com/explicit-density-estimation-b16730094f244ea9b1cee355b815e9a7)|
|[Implicit Density Estimation](https://texonom.com/implicit-density-estimation-7619316f81484428a2a571f55ca57c19)|
|
238e2a8e51f44a67a91c0567892ab643
|
Explicit Density Estimation
|
Density Estimations
|
Jun 1, 2023
|
Alan Jo
|
Alan Jo
|
Jun 14, 2023
|
๋ชจ๋ธ์ย ์ฌ์ ๋ถํฌ๋ฅผย ๊ฐ์ ํ์ฌ ๊ธฐ์กด ๊ฐ์ผ๋ก๋ถํฐ ๋ฐ์ดํฐ ๋ถํฌ๋ฅผ ์ถ์ ย (MLE, MAPย ๋ฑ)
- [VAE](https://texonom.com/vae-972dd73900a944c3840e1edf40f14f90) - Approximate **Explicit Density**
- [AutoEncoder](https://texonom.com/autoencoder-7de48b307d1347ac8da05f17f314caff) - Decoder
### Explicit Density Notion
|Title|
|:-:|
|
b16730094f244ea9b1cee355b815e9a7
|
|
Implicit Density Estimation
|
Density Estimations
|
Jun 1, 2023
|
Alan Jo
|
Alan Jo
|
Jun 14, 2023
|
๋ชจ๋ธ์ ๋ช
ํํ ์ ์ํ๋ ๋์ ์ํ๋ง์ ๋ฐ๋ณตํ์ฌ ํน์ ํ๋ฅ ๋ถํฌ์ ์๋ ด์ํดย (Markov Chain)
- [GAN](https://texonom.com/gan-66482b5f518d47f6b337eba9a30ff792) - **Implicit Density**
### Implicit Density Notion
|Title|
|:-:|
|
7619316f81484428a2a571f55ca57c19
|
|
Non-parametric Estimation
|
Density Estimations
|
Mar 23, 2023
|
Alan Jo
|
Alan Jo
|
Mar 23, 2023
|
### no specific function form
### Non-parametric Estimations
|Title|
|:-:|
|[Kernel Density Estimation](https://texonom.com/kernel-density-estimation-eb7b92dc9ce343ddb412d1cc7696ad98)|
|
88b1d8556a524a5cb1ed4792123ccb02
|
|
Parameter Estimation
|
Density Estimations
|
Mar 23, 2023
|
Alan Jo
|
Alan Jo
|
Jun 13, 2023
|
## Model Fitting
### finds the most likely parameter $\theta_*$ that explain the data
if $\Theta \subset R$, $Risk = bias^2 + variance$
statistical experiment be a sample $X_1$ โฆ , $X_n$ of i.i.d.random variables in some measurable space ฮฉ, usually ฮฉ โ โ
hyperparameter $\alpha$, $D$ is data set
ํ๋ฅ ๊ณ์ฐ์ ์กฐ๊ฑด์ ๋ฐ์ดํฐ์ด๊ณ
๊ฒฐ๊ณผ ํ๋ผ๋ฏธํฐ๋ ๋ฐ์ดํฐ๊ฐ์ผ๋ก ์ธํ ์์ผ๋ก ๋์์ผํ๋ค
### Parameter Estimations
|Title|
|:-:|
|[MLE](https://texonom.com/mle-79884be98a0447628b39ab3953780b4a)|
|[MAP](https://texonom.com/map-9ecf5e2396fb4901983a57761a3e7174)|
|[EM Algorithm](https://texonom.com/em-algorithm-1c9ae15421d24b00afc033e7ccc743a9)|
|[Bayesian inference](https://texonom.com/bayesian-inference-e3cabe7844954db097867a732143ddd8)|
|[Back Propagation](https://texonom.com/back-propagation-18f4493692ad43449d4271f1bb293781)|
> [Variational Inference ์์๋ณด๊ธฐ - MLE, MAP๋ถํฐ ELBO๊น์ง](https://modulabs.co.kr/blog/variational-inference-intro/)
|
2c5667f2707b4d3082e96a5928464029
|
|
Semi-parametric Estimation
|
Density Estimations
|
Mar 23, 2023
|
Alan Jo
|
Alan Jo
|
Mar 23, 2023
|
aa0e7a6b8cc54d8d948385ac7206ba95
|
||
Kernel Density Estimation
|
Non-parametric Estimations
|
Mar 23, 2023
|
Alan Jo
|
Alan Jo
|
Jun 13, 2023
|
eb7b92dc9ce343ddb412d1cc7696ad98
|
||
Back Propagation
|
Parameter Estimations
|
Oct 6, 2021
|
Alan Jo
|
Alan Jo
|
Jul 17, 2023
|
[Gradient Descent](https://texonom.com/gradient-descent-c1342b13182f4fb6959023a75b5e2ff8) [Parameter Estimation](https://texonom.com/parameter-estimation-2c5667f2707b4d3082e96a5928464029) [Neuroplasticity](https://texonom.com/neuroplasticity-63de0a67105e4c6ab8f6958b281c4e19)
|
### Calculate the partial Derivatives via chain-rule
$$\frac{\partial L}{\partial W_L} = \frac{\partial L}{\partial h_L} \frac{\partial h_L}{\partial W_L}$$
Compute the gradient of [Loss Function](https://texonom.com/loss-function-e8f6343914494828988137987cf459f9) and update weights [Chain Rule](https://texonom.com/chain-rule-cdc4287f5b2a4bcda97886907836d5d0)
[Computational Graph](https://texonom.com/computational-graph-51f54bce30ed44569be80f56d596340f), H is hidden layer
$$downstream \; gradient = upstream \; gradient \times local \; gradient$$
Not a same path of [Computational Graph](https://texonom.com/computational-graph-51f54bce30ed44569be80f56d596340f), no reuse, function part also has one derivative chain
> 
> 
### Back Propagation Notion
|Title|
|:-:|
|[Forward Propagation](https://texonom.com/forward-propagation-59704e8a904b464fabc23519ae4875f2)|
|[Vanishing Gradient](https://texonom.com/vanishing-gradient-6dc21442840841b3879d6502c5efd98a)|
|[Back Propagation History](https://texonom.com/back-propagation-history-caed7f53c8b546e394b18f81390f924d)|
|
18f4493692ad43449d4271f1bb293781
|
Bayesian inference
|
Parameter Estimations
|
Mar 23, 2023
|
Alan Jo
|
Alan Jo
|
Jun 25, 2023
|
[Statistical Inference](https://texonom.com/statistical-inference-168ef97c06d24b44bb49653d70516e73) [Bayes Theorem](https://texonom.com/bayes-theorem-e512b6c0308f4270aaae9ae53060ccab) [Bayesian Statistics](https://texonom.com/bayesian-statistics-1f53546302c647ea9eac07139c762854)
|
statistical method that uses Bayes' theorem to update the probability of a hypothesis as more evidence becomes available
do not directly estimate and choose exact $\theta$
Just compute for every $\theta$ [Joint Probability](https://texonom.com/joint-probability-ddefdc8d4f8d4b3eb3d8dafb5706b8e5) and [Marginal Probability](https://texonom.com/marginal-probability-ee22b391f7f44fab8b821400cf67e63d)
### Bayesian inference Notion
|Title|
|:-:|
|[Conjugate Prior](https://texonom.com/conjugate-prior-93ea7bec4f0c443eb67d383b6c9be6e3)|
> [Bayesian inference - Wikipedia](https://en.wikipedia.org/wiki/Bayesian_inference)
|
e3cabe7844954db097867a732143ddd8
|
EM Algorithm
|
Parameter Estimations
|
May 18, 2023
|
Alan Jo
|
Alan Jo
|
Sep 11, 2023
|
[Variational Inference](https://texonom.com/variational-inference-05563e34ddec467c95c9bda931770d50)
|
## **Expectation Maximization Method**
iterative algorithm that has two main steps
## Until Convergence
### E Step ([ELBO](https://texonom.com/elbo-bb7352bbce42480298d501c2ae1076d1), โฃ)
**calculates the expected complete log likelihood with posterior for fixed parameters**
**That is why Expectation maximization because It is same to maximize expectation **$**E[logp]**$
calculates the expected complete-data log-likelihood for fixed parameters A and ฮฃ
๊ฐ ๋ฐ์ดํฐ ํฌ์ธํธ๊ฐ ์ด๋ค ๋ถํฌ์์ ์์ฑ๋์๋์ง์ ๋ํ ํ๋ฅ ์ ์ถ์
$$l(\theta) \ge ELBO(x^{(i)};Q_i, \theta) = $$
### M Step
re-estimates next stepโs fixed parameters to maximizing the expected complete-data log-likelihood (derivative)
ํ๋ฅ ๊ณผ ํจ๊ป ๋ชจ๋ธ ํ๋ผ๋ฏธํฐ๋ฅผ ์
๋ฐ์ดํธ
$$\theta := argmax_\theta $$
> [ML simple works - A Step by Step Introduction to EM Algorithm](https://metamath1.github.io/blog/posts/em/em_algorithm.html)
|
1c9ae15421d24b00afc033e7ccc743a9
|
MAP
|
Parameter Estimations
|
Mar 23, 2023
|
Alan Jo
|
Alan Jo
|
Jun 6, 2023
|
[Bayes Theorem](https://texonom.com/bayes-theorem-e512b6c0308f4270aaae9ae53060ccab)
|
## Maximum A Posteriori
- priori mean โ**from the earlier**โ
- posteriori means โ**from the later**โ
finds the parameters $\tilde{\theta}_{MAP}$ maximizing a posteriori distribution
assume $\theta$ also has some distribution and find optimal $\theta$
We assume a zero-mean Gaussian prior with covariance ฮฃ for parameters $\theta$
|
9ecf5e2396fb4901983a57761a3e7174
|
MLE
|
Parameter Estimations
|
Mar 23, 2023
|
Alan Jo
|
Alan Jo
|
Jun 6, 2023
|
[KL Divergence](https://texonom.com/kl-divergence-c7964872f5184a7baaf312605405aef6) [Cost Function](https://texonom.com/cost-function-66c21423cc7347909016b3423f2ada7e)
|
## Maximum likelihood estimation
MLE๋ MAP์ Prior๊ฐ Uniform Distribution์ธ ๊ฒฝ์ฐ
We can also minimize $-logL(\theta)$
### Maximum likelihood estimation
|Title|
|:-:|
|[Likelihood function](https://texonom.com/likelihood-function-0750726af35a45bb86036dbc7507f725)|
|[Log-likelihood function](https://texonom.com/log-likelihood-function-c824711ae9454e53a4347add60060762)|
|[Complete Log Likelihood](https://texonom.com/complete-log-likelihood-30cc920673524e68838bd08c831b8090)|
### Maximum likelihood estimatiors
[Stochastic Gradient Descent](https://texonom.com/stochastic-gradient-descent-d8b8d008e0a34f4bb55175ffba21db44) (Ascent)
[Gradient Descent](https://texonom.com/gradient-descent-c1342b13182f4fb6959023a75b5e2ff8)
[NewtonโRaphson method](https://texonom.com/newtonraphson-method-0bc249bfb0fe4e498829110bf785eb6c)
> [1. Linear Basis Function Models](http://norman3.github.io/prml/docs/chapter03/1)
> [MLE์ MAP์ ์ฐจ์ด](https://niceguy1575.medium.com/mle์-map์-์ฐจ์ด-7d2cc0bee9c)
|
79884be98a0447628b39ab3953780b4a
|
Back Propagation History
|
Back Propagation Notion
|
Jun 7, 2023
|
Alan Jo
|
Alan Jo
|
Jun 7, 2023
|
1986 [Geoffrey Hinton](https://texonom.com/geoffrey-hinton-441d5ce2b78146d0935454042b4f06d9)
Rumelhart et al., 1986: First time back-propagation became popular
Hinton and Salakhutdinov 2006
> [0022 Back Propagation - Deepest Documentation](https://deepestdocs.readthedocs.io/en/latest/002_deep_learning_part_1/0022/)
|
caed7f53c8b546e394b18f81390f924d
|
|
Forward Propagation
|
Back Propagation Notion
|
Apr 27, 2023
|
Alan Jo
|
Alan Jo
|
Apr 27, 2023
|
์
๋ ฅ๊ฐ์ด ์
๋ ฅ์ธต์ ํต๊ณผํ์ฌ ์ถ๋ ฅ์ธต๊น์ง ์ ๋ฌ๋๋ฉด์ ์์ธก๊ฐ์ด ๊ณ์ฐ
|
59704e8a904b464fabc23519ae4875f2
|
|
Vanishing Gradient
|
Back Propagation Notion
|
May 23, 2023
|
Alan Jo
|
Alan Jo
|
May 23, 2023
|
Layer๋ฅผ ๋ง์ด ์์์๋ก ๋ฐ์ดํฐ ํํ๋ ฅ์ด ์ฆ๊ฐํ๊ธฐ ๋๋ฌธ์ ํ์ต์ด ์ ๋ ๊ฒ ๊ฐ์ง๋ง
Layer๊ฐ ๋ง์์ง์๋ก ํ์ต์ด ์ ๋์ง ์๋๋ค
์ถ๋ ฅ์ธต์์ ๋ฉ์ด์ง์๋ก Gradient ๊ฐ์ด ๋งค์ฐ ์์์ง๋ ํ์
[Activation Function](https://texonom.com/activation-function-8e52ee5f83a244d88abeeee3fb9497a8) ์ Gradient๋ Activation์ ์ค์ ๊ฐ๋ณด๋ค ํจ์ฌ ์์ ๋ ์๊น (Sigmoid๋ ๊ทธ๋ผ)
[Tanh Function](https://texonom.com/tanh-function-679abf35c4eb4fb0bb0e56140cac3d23) ๋ก ์ํํ [ReLU](https://texonom.com/relu-e582549804da48b893758895e446ffb9) ๋ก ์์ ํด๊ฒฐ
> [[๋ฅ๋ฌ๋] ๊ธฐ์ธ๊ธฐ ์์ค(Vanishing Gradient)์ ์๋ฏธ์ ํด๊ฒฐ๋ฐฉ๋ฒ](https://heytech.tistory.com/388)
|
6dc21442840841b3879d6502c5efd98a
|
|
Conjugate Prior
|
Bayesian inference Notion
|
Mar 28, 2023
|
Alan Jo
|
Alan Jo
|
Jun 6, 2023
|
## Conjugate Distribution
์ฌํํ๋ฅ ์ ๊ณ์ฐํจ์ ์์ด ์ฌํ ํ๋ฅ ์ด ์ฌ์ ํ๋ฅ ๋ถํฌ์ ๊ฐ์ ๋ถํฌ ๊ณ์ด์ ์ํ๋ ๊ฒฝ์ฐ ๊ทธย ์ฌ์ ํ๋ฅ ๋ถํฌ๋ฅผย **์ผค๋ ์ฌ์ ๋ถํฌ**(Conjugate Prior) ๋ผ๊ณ ๋ถ๋ฅธ๋ค. ์ผค๋ ์ฌ์ ๋ถํฌ๋ฅผ ์ด์ฉํ๋ฉด ์ฌ์ ํ๋ฅ ๋ถํฌ์ **ํ์ดํผํ๋ผ๋ฏธํฐ๋ฅผ ์
๋ฐ์ดํธํ๋ ๋ฐฉ์์ผ๋กย ์ฌํํ๋ฅ ์ ๊ณ์ฐํ ์ ์๊ฒ ๋์ด ๊ณ์ฐ์ด ๊ฐํธ**
์ฌ์ ๋ถํฌ๋ฅผ ์ด์ฉํ์ง ๋ชปํ ๊ฒฝ์ฐ ์์น์ ๋ถ์ ํด์ผ ํ๋ ๊ฒ๊ณผ ๋ฌ๋ฆฌ ํด์์ ์ ๋ถ์ผ๋ก ๊ณ์ฐ์ด ๊ฐ๋ฅ
> [์ผค๋ ์ฌ์ ๋ถํฌ](https://ko.wikipedia.org/wiki/์ผค๋ ์ฌ์ ๋ถํฌ)
|
93ea7bec4f0c443eb67d383b6c9be6e3
|
|
Complete Log Likelihood
|
Maximum likelihood estimation
|
May 30, 2023
|
Alan Jo
|
Alan Jo
|
Jun 1, 2023
|
๊ด์ฐฐ๋ ๋ฐ์ดํฐ๋ง์ ๊ณ ๋ คํ์ฌ ๋ชจ๋ธ์ ์ฐ๋๋ฅผ ๊ณ์ฐํ์ง ์๊ณ consider [Latent Variable](https://texonom.com/latent-variable-c2bb1127d53444018b2d0eaab231cc6c) too
[Conditional probability](https://texonom.com/conditional-probability-89f109f0347b4a68b7f31e27dcd0794d) ์์ ์ฐ์ธก์ ๊ด์ฐฐ๋ ๋ฐ์ดํฐ์ ํจ๊ป ๋ชจ๋ธ์ ๋ชจ๋ ํ๋ผ๋ฏธํฐ๋ฅผ ๊ณ ๋ คํ์ฌ ์ฐ๋๋ฅผ ๊ณ์ฐ
|
30cc920673524e68838bd08c831b8090
|
|
Likelihood function
|
Maximum likelihood estimation
|
Mar 9, 2023
|
Alan Jo
|
Alan Jo
|
Mar 27, 2023
|
## Probability so 0 < L < 1
$$L(\theta) = L(\theta;X; \vec{y}) = p(\vec{y} | X ; \theta)$$
|
0750726af35a45bb86036dbc7507f725
|
|
Log-likelihood function
|
Maximum likelihood estimation
|
Mar 23, 2023
|
Alan Jo
|
Alan Jo
|
Apr 18, 2023
|
### Becuz log monotonically increasing argmax easy
should maximize likelihood function, so negative value when it is logged
$l(\theta)$ is log likelihood and $-l(\theta)$ is negative log likelihood
$$L(\theta) = L(\theta; X, \vec{y}) = p(\vec{y}|X;\theta)$$
$$l(\theta) = log{L(\theta)} = \Sigma_{i=1}^np(y^{(i)} |x{(i)};\theta)$$
it measures how well the parameters fit the observed data. The notation used to represent the likelihood function is L(ฮธ), where ฮธ represents the parameters of the model, and X and y represent the data. The likelihood function is defined as the conditional probability of the observed data given the values of the parameters of the model: $L(\theta) = p(y|X;\theta)$.
j function usually means negative log likelihood
|
c824711ae9454e53a4347add60060762
|
|
Bayesian Statistics
| null | null | null | null | null | null |
### Bayesian Statistics Notion
|Title|
|:-:|
|[Bayes Theorem](https://texonom.com/bayes-theorem-e512b6c0308f4270aaae9ae53060ccab)|
|[Markovย Chain](https://texonom.com/markovchain-497b662ec8cd4bba84c4d947067d717f)|
|[Metropolis Algorithm](https://texonom.com/metropolis-algorithm-5af42c3e93a64159a2016705eb71ef4c)|
### Bayesian Statistics Usages
|Title|
|:-:|
|[Continuous Joint Probability Distribution](https://texonom.com/continuous-joint-probability-distribution-82734d0b70f04ebabef90d2e1395c980)|
|[MCMC](https://texonom.com/mcmc-80063d374ddd4a3381edc04748063826)|
> [Bayesian statistics - Wikipedia](https://en.wikipedia.org/wiki/Bayesian_statistics)
|
1f53546302c647ea9eac07139c762854
|
**Probability theory**
| null | null | null | null | null | null |
### **Probability theory Notion**
|Title|
|:-:|
|[Probability](https://texonom.com/probability-6b1e766c2c6442c7a599b558453300e4)|
|[Probability distribution](https://texonom.com/probability-distribution-2d33768bdc43438794829b004dd37804)|
|[Random Variable](https://texonom.com/random-variable-0bc19e0582784ec3a6e0de5b91296212)|
|[Expectation value](https://texonom.com/expectation-value-3e14abcd31734ddea3fa0d1c92e3d63f)|
|[Mean](https://texonom.com/mean-1ea2a9d53da24baab2bc057d4d781e41)|
|[Variance](https://texonom.com/variance-08c1eccc7dc84957afbb815ad6b41280)|
|[Covariance](https://texonom.com/covariance-f3b864d48bf14625a4415c8f0500e688)|
|[Correlation](https://texonom.com/correlation-d75ea902ec294bf19e1180b6bfd22137)|
|[Deviation](https://texonom.com/deviation-6637194ca1b046ac9b681c550e6b853b)|
|[Bias](https://texonom.com/bias-ba063cd622a54deb8a677e8fb87dfdc8)|
|[Bias-Variance Trade-off](https://texonom.com/bias-variance-trade-off-753e9b8d891f4f6e9e3e595024ac546e)|
### **Probability theory Usages**
|Title|
|:-:|
|[chi Square](https://texonom.com/chi-square-697c74917a0b4a9c83b1856d89bde6af)|
|[The Goodhart's Law](https://texonom.com/the-goodharts-law-ccece645e5e3484381fe0721d6ba2cd8)|
|
31161a5d9b9942dcb7b03bafd7659f31
|
Stochastic
|
Statistics Fields
|
Jul 11, 2023
|
Alan Jo
|
Alan Jo
|
Jul 11, 2023
|
### **์ถ์ธก ํต๊ณํ**
๋ชจ์ง๋จ์์ ์์๋ก ์ถ์ถํ ํ๋ณธ์ ๋ฐ๋ผ ๋ชจ์ง๋จ์ ์ํ๋ฅผ ์ถ์ธกํ๋ ํ๋ฌธ
> [Stochastic](https://en.wikipedia.org/wiki/Stochastic)
|
fc6c8f3295dd4a49a44086d393fd5b1a
|
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