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Machine Learning seminars at the Cambridge University Engineering Department

Group Theory and Machine Learning

author: Imre Risi Kondor, UCL

Description

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 machine learning. The purpose of this tutorial is to give an entertaining but informative introduction to the background to these developments and sketch some of the many possible applications, including multi-object tracking, learning rankings, and constructing translation and rotation invariant features for image recognition. The tutorial is intended to be palatable by a non-specialist audience with no prior background in abstract algebra.

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Slides
0:00 Group theoretical methods in Machine Learning
1:09 Group theoretical methods in Machine Learning
2:03 Motive
2:36 (G, · ) is a group if:
4:43 Symmetry
5:24 g : X -> X
6:13 g : X -> X
7:16 For any g 1, g 2 E G , g 2 g 1 E G .
7:42 For any g 1 , g 2 , g 3 E G , g 3 ( g 2 g 1 ) = ( g 3 g 2 ) g 1 .
8:31 There exists an identity e E G such that eg = ge = g for any g E G .
8:45 For any g E G there is a g-1 E G such that g −1g = e .
9:15 A set G endowed with multiplication is a group if
11:28 Example 1
12:04 Example 2
12:59 Example 3
13:53 Example 4
14:05 Example 5
15:29 Example 6
16:07 Example 7
16:27 Example 8
17:29 Example 9
18:37 Example 10
18:47 Example 11
19:15 Example 12
19:59 Example 13
21:24 Finite groups, Infinite groups
22:56 Groups
23:45 Representations
25:55 Example
31:16 The idea is to “model” groups by...
31:31 Example
31:49 What are the “fundamental” representations?
32:13 Two representations are equivalent if
35:00 Maschke/Wedderburn theorem
35:49 The Clebsch-Gordan decomposition
37:38 Harmonic analysis
38:17 The Fourier transform on a group is
41:01 Classical properties
41:37 The Fourier transform on a group is
42:27 Classical properties
42:35 How about right translation f ( z ) ( x ) = f ( x z ? −1)
47:27 f(z)(p) = f(p) p(z) so right-translation preserves the subspaces spanned by the rows of f(p).
50:23 The Fourier transform F : f -> f is an isomorphism
51:28 Application:
53:11 The problem: translation and rotation invariance (1)
55:39 The problem: translation and rotation invariance (2)
58:06 The classical bispectrum
58:39 The problem
59:32 Start with Fourier Transform
59:41 The Power Spectrum
59:43 Start with Fourier Transform
60:46 The Power Spectrum
62:15 Note translation property
62:20 Problem: We lose an awful lot of information
62:21 The Bispectrum
63:32 It is the Fourier transform of the triple correlation
63:34 Applications of the classical bispectrum:
64:58 Want to do the same for I S O +(2) ...
65:00 Non-commutative bispectra
65:03 Under translations
65:41 Therefore, the spectrum
68:28 Now try and couple different components
71:10 In general, p1(x) ! p2(x) decomposes in the form
72:03 The bispectrum on a compact group is defined
73:36 Kakarala
80:33 X is a homogeneous space of G if G acts on x and x = g(x0) sweeps out the whole of X - Example (1)
81:46 X is a homogeneous space of G if G acts on x and x = g(x0) sweeps out the whole of X - Example (2)
81:48 X is a homogeneous space of G if G acts on x and x = g(x0) sweeps out the whole of X - Example (3)
81:49 X is a homogeneous space of G if G acts on x and x = g(x0) sweeps out the whole of X - Example (4)
81:53 X is a homogeneous space of G if G acts on x and x = g(x0) sweeps out the whole of X - Example (5)
81:55 Bispectral invariants for ISO+(2)
82:09 We have:
82:22 We want: (1)
83:11 We want: (2)
83:36 We want: (3)
83:48 We want: (4)
83:51 We want: (5)
84:31 ... locally the action of ISO+(2) on R2 is isomorpic to the action of SO(3) on S2.
84:33 ... the algorithm
84:35 The Projection
85:45 The Fourier transform (1)
86:07 The Fourier transform (2)
86:15 The bispectrum
89:58 Results (1)
90:00 Results (2)
93:08 Results (3)

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Reviews and comments:

Comment1 Hassan, July 10, 2008 at 9:46 p.m.:

I think you definitely need to know some group theory before you see this, as well as some harmonic analysis. The ideas are very interesting, but difficult, especially for a machine learning researcher with the usual background in mainly statistics and linear algebra.

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