Group Theory for Machine Learning
Group Theory for Machine Learning
Lectures : At 12:00 noon on all working days
Internal - 60%
1. Quiz- 20% (Four quizzes will be taken and best three will be considered)
2. Project - 20%
3. Viva-Voce -15%
4. Presentations of solutions of Practice problems- 5%
End Semester Examination- 40%
Topics to be covered:
Introduction to groups for Machine Learning (6 lecture): Groups as systems of transformations, Group axioms, Classes of groups: finite, countable groupsGroup representations (8 lecture): Equivalence and reducibility of group representations, isomorphism, homomorphism, normal subgroups, direct product and semi-direct product of groups, classification of finite abelian groups, group action, groups in the real world.Harmonic analysis on the symmetric group (10 lecture): Permutation cycle notation and cycle type, Partial rankings, Fourier Transforms, convolution theorem and Plancherel’s theorem, Decomposition of the group into isotypal components, Young diagrams, Young tableaux and Young’s orthogonal representation. Application of group theory (10 lecture): Spectral analysis of ranking data, Fast Fourier transforms, the Cooley-Tukey algorithm and its interpretation in terms of subgroups, Clausen’s FFT for permutation and multi-object tracking.Group theory in Deep learning (8 lecture) Lie groups and invariance (if time permits): Definition of Lie groups, Generators, the exponential map and Lie algebra, the rotation groups: parametrization and representations, connection to spherical harmonics Homogeneous spaces, the Euclidean motion groups, the classical spectrum and bispectrum and their generalization to non-commutative groups, Kakarala’s completeness results, application to fast pattern matching, rotation and translation invariant features in image processing.References:
R. Kondor, (2008) Doctoral Thesis, https://people.cs.uchicago.edu/~risi/papers/KondorThesis.pdf
M. A. Armstrong, (1997) Groups and symmetry, Springer
H. Edelsbrunner, J. Harer, (2009) Computational Topology: An Introduction. AMS Press.
Chirikjian, G. S., Kyatkin, A. B., (2001) Engineering applications of noncommutative harmonic analysis. CRC Press.
Diaconis, P., (1988) Group Representation in Probability and Statistics. Volume 11 of IMS Lecture Series, Institute of Mathematical Statistics.
Kanatani, K., (1990) Group theoretical methods in image understanding, Springer-Verlag.
Symmetry and Platonic solids [Ankit]
Error correcting codes and linear codes [Nammi]
Cyclic Codes [Naveen]
Symmetry and Wallpaper groups [Pushpendra]
Ruler and Compass constructions [Rohan]
Group Theory in Cryptography [Vaibhav]
Cryptanalysis based on symmetric group representations [Vishal]