Mismatch in the Classification of Linear Subspaces: Sufficient Conditions for Reliable Classification

TL;DR

This paper derives conditions under which linear subspace classifiers with mismatched parameters can reliably classify signals in noisy environments, predicting error behavior based on geometric relationships between true and mismatched subspaces.

Contribution

It introduces asymptotic low-noise error bounds for mismatched classifiers and provides geometric conditions for reliable classification, including the absence of an error floor.

Findings

Conditions accurately predict classifier performance in synthetic data

Conditions successfully applied to motion segmentation tasks

Conditions explain classification behavior in handwritten digit recognition

Abstract

This paper considers the classification of linear subspaces with mismatched classifiers. In particular, we assume a model where one observes signals in the presence of isotropic Gaussian noise and the distribution of the signals conditioned on a given class is Gaussian with a zero mean and a low-rank covariance matrix. We also assume that the classifier knows only a mismatched version of the parameters of input distribution in lieu of the true parameters. By constructing an asymptotic low-noise expansion of an upper bound to the error probability of such a mismatched classifier, we provide sufficient conditions for reliable classification in the low-noise regime that are able to sharply predict the absence of a classification error floor. Such conditions are a function of the geometry of the true signal distribution, the geometry of the mismatched signal distributions as well as the interplay between such geometries, namely, the principal angles and the overlap between the true and the mismatched signal subspaces. Numerical results demonstrate that our conditions for reliable classification can sharply predict the behavior of a mismatched classifier both with synthetic data and in a motion segmentation and a hand-written digit classification applications.