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Feature Discovery Using Group Theory

Sridhar Mahadevan

The representation theory of groups has played a pivotal role in many scientific and engineering disciplines, from particle physics and molecular chemistry to signal processing, providing a powerful formalism for understanding the behavior of complex interacting systems by analyzing their underlying symmetries. In this talk, I will discuss the application of group representation theory to problems in AI, in particular feature discovery.

Group characters play a central role in this framework: these are basis-independent representations that reveal the unique structure of a group. In Abelian groups, characters are one-dimensional: an example is classical Fourier analysis. In the richer setting of non-Abelian groups, characters are matrix entries of irreducible representations, generalizing Fourier analysis to non-commutative harmonic analysis.

Many classical methods for feature discovery (e.g. PCA, SVD) and recent graph-based methods (ISOMAP, LLE, Laplacian eigenmaps) involve eigenvalue computations at their core. I will illustrate how group representation theory provides a principled approach to scaling these methods. Other applications of group representation theory to AI such as clustering, solution of Markov decision processes, ranking, and robotics will also be outlined.