Discrimination on the Grassmann Manifold: Fundamental Limits of Subspace Classifiers

TL;DR

This paper establishes fundamental theoretical limits for classifying subspaces in noisy environments, using information-theoretic measures inspired by wireless communication, and validates findings with numerical experiments including face recognition.

Contribution

It introduces the concepts of classification capacity and diversity-discrimination tradeoff for subspace classification, providing tight bounds and practical validation.

Findings

Derived bounds on classification capacity and diversity-discrimination tradeoff.

Established analogies between subspace discrimination and wireless channel communication.

Validated theoretical results with numerical experiments, including face recognition.

Abstract

We present fundamental limits on the reliable classification of linear and affine subspaces from noisy, linear features. Drawing an analogy between discrimination among subspaces and communication over vector wireless channels, we propose two Shannon-inspired measures to characterize asymptotic classifier performance. First, we define the classification capacity, which characterizes necessary and sufficient conditions for the misclassification probability to vanish as the signal dimension, the number of features, and the number of subspaces to be discerned all approach infinity. Second, we define the diversity-discrimination tradeoff which, by analogy with the diversity-multiplexing tradeoff of fading vector channels, characterizes relationships between the number of discernible subspaces and the misclassification probability as the noise power approaches zero. We derive upper and lower bounds on these measures which are tight in many regimes. Numerical results, including a face recognition application, validate the results in practice.