The Role of Principal Angles in Subspace Classification

TL;DR

This paper investigates how the geometry of subspaces, specifically principal angles, affects classification accuracy, and introduces a new transform (TRAIT) to optimize these angles for improved signal classification.

Contribution

The paper introduces TRAIT, a linear transform that optimizes principal angles to enhance subspace classification, complementing existing methods like LRT.

Findings

TRAIT improves classification accuracy on synthetic data.

Larger principal angles correlate with lower misclassification probability.

TRAIT outperforms existing transforms in the presence of model mismatch.

Abstract

Subspace models play an important role in a wide range of signal processing tasks, and this paper explores how the pairwise geometry of subspaces influences the probability of misclassification. When the mismatch between the signal and the model is vanishingly small, the probability of misclassification is determined by the product of the sines of the principal angles between subspaces. When the mismatch is more significant, the probability of misclassification is determined by the sum of the squares of the sines of the principal angles. Reliability of classification is derived in terms of the distribution of signal energy across principal vectors. Larger principal angles lead to smaller classification error, motivating a linear transform that optimizes principal angles. The transform presented here (TRAIT) preserves some specific characteristic of each individual class, and this approach is shown to be complementary to a previously developed transform (LRT) that enlarges inter-class distance while suppressing intra-class dispersion. Theoretical results are supported by demonstration of superior classification accuracy on synthetic and measured data even in the presence of significant model mismatch.