Pursuit of a Discriminative Representation for Multiple Subspaces via Sequential Games

TL;DR

This paper introduces a game-theoretic approach within the CTRL framework to learn discriminative representations for data supported on multiple linear subspaces, providing theoretical guarantees and empirical validation.

Contribution

It presents a novel sequential game formulation for subspace learning that unifies classical and modern representation learning methods, with theoretical proof of correctness.

Findings

Equilibrium solutions yield correct subspace representations.

The approach unifies classical subspace learning with deep learning.

Empirical results support the theoretical claims.

Abstract

We consider the problem of learning discriminative representations for data in a high-dimensional space with distribution supported on or around multiple low-dimensional linear subspaces. That is, we wish to compute a linear injective map of the data such that the features lie on multiple orthogonal subspaces. Instead of treating this learning problem using multiple PCAs, we cast it as a sequential game using the closed-loop transcription (CTRL) framework recently proposed for learning discriminative and generative representations for general low-dimensional submanifolds. We prove that the equilibrium solutions to the game indeed give correct representations. Our approach unifies classical methods of learning subspaces with modern deep learning practice, by showing that subspace learning problems may be provably solved using the modern toolkit of representation learning. In addition, our work provides the first theoretical justification for the CTRL framework, in the important case of linear subspaces. We support our theoretical findings with compelling empirical evidence. We also generalize the sequential game formulation to more general representation learning problems. Our code, including methods for easy reproduction of experimental results, is publically available on GitHub.