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Group Theory and Machine Learning
Published on Mar 03, 200835526 Views
**Machine Learning Tutorial Lecture** The use of algebraic methods—specifically group theory, representation theory, and even some concepts from algebraic geometry—is an emerging new direction in mach
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Chapter list
Group theoretical methods in Machine Learning00:00
Motive02:03
(G, · ) is a group if:02:36
Symmetry04:43
g : X -> X05:24
g : X -> X06:13
For any g 1, g 2 E G , g 2 g 1 E G .07:16
For any g 1 , g 2 , g 3 E G , g 3 ( g 2 g 1 ) = ( g 3 g 2 ) g 1 .07:42
There exists an identity e E G such that eg = ge = g for any g E G .08:31
For any g E G there is a g-1 E G such that g −1g = e .08:45
A set G endowed with multiplication is a group if09:15
Example 111:28
Example 212:04
Example 312:59
Example 413:53
Example 514:05
Example 615:29
Example 716:07
Example 816:27
Example 917:29
Example 1018:37
Example 1118:47
Example 1219:15
Example 1319:59
Finite groups, Infinite groups21:24
Groups22:56
Representations23:45
Example25:55
The idea is to “model” groups by...31:16
Example31:31
What are the “fundamental” representations?31:49
Two representations are equivalent if32:13
Maschke/Wedderburn theorem35:00
The Clebsch-Gordan decomposition35:49
Harmonic analysis37:38
The Fourier transform on a group is38:17
Classical properties41:01
How about right translation f ( z ) ( x ) = f ( x z ? −1)42:35
f(z)(p) = f(p) p(z) so right-translation preserves the subspaces spanned by the rows of f(p).47:27
The Fourier transform F : f -> f is an isomorphism50:23
Application:51:28
The problem: translation and rotation invariance (1)53:11
The problem: translation and rotation invariance (2)55:39
The classical bispectrum58:06
The problem58:39
Start with Fourier Transform59:32
The Power Spectrum59:41
Note translation property01:02:15
Problem: We lose an awful lot of information01:02:20
The Bispectrum01:02:21
It is the Fourier transform of the triple correlation01:03:32
Applications of the classical bispectrum:01:03:34
Want to do the same for I S O +(2) ... 01:04:58
Non-commutative bispectra01:05:00
Under translations01:05:03
Therefore, the spectrum01:05:41
Now try and couple different components01:08:28
In general, p1(x) ! p2(x) decomposes in the form01:11:10
The bispectrum on a compact group is defined01:12:03
Kakarala01:13:36
X is a homogeneous space of G if G acts on x and x = g(x0) sweeps out the whole of X - Example (1)01:20:33
X is a homogeneous space of G if G acts on x and x = g(x0) sweeps out the whole of X - Example (2)01:21:46
X is a homogeneous space of G if G acts on x and x = g(x0) sweeps out the whole of X - Example (3)01:21:48
X is a homogeneous space of G if G acts on x and x = g(x0) sweeps out the whole of X - Example (4)01:21:49
X is a homogeneous space of G if G acts on x and x = g(x0) sweeps out the whole of X - Example (5)01:21:53
Bispectral invariants for ISO+(2)01:21:55
We have:01:22:09
We want: (1)01:22:22
We want: (2)01:23:11
We want: (3)01:23:36
We want: (4)01:23:48
We want: (5)01:23:51
... locally the action of ISO+(2) on R2 is isomorpic to the action of SO(3) on S2.01:24:31
... the algorithm01:24:33
The Projection01:24:35
The Fourier transform (1)01:25:45
The Fourier transform (2)01:26:07
The bispectrum01:26:15
Results (1)01:29:58
Results (2)01:30:00
Results (3)01:33:08