Theoretical Guarantees for the Subspace-Constrained Tyler's Estimator

TL;DR

This paper provides theoretical guarantees for the subspace-constrained Tyler's estimator (STE), showing it can effectively recover low-dimensional subspaces under challenging conditions with outliers and noise, given proper initialization.

Contribution

It establishes recovery guarantees for STE under weak inlier-outlier models and demonstrates its effectiveness with Tyler's M-estimator initialization in low inlier scenarios.

Findings

STE can recover subspaces with weak inlier-outlier models

Proper initialization ensures effective recovery by STE

STE with TME initialization succeeds even with very low inlier fractions

Abstract

This work analyzes the subspace-constrained Tyler's estimator (STE), a method designed to recover a low-dimensional subspace from a dataset that may be heavily corrupted by outliers. The STE has previously been shown to be competitive for fundamental computer vision problems. We assume a weak inlier-outlier model and allow the inlier fraction to fall below the threshold at which robust subspace recovery becomes computationally hard. We show that, in this setting, if the initialization of STE satisfies a certain condition, then STE-which is computationally efficient-can effectively recover the underlying subspace. Furthermore, we establish approximate recovery guarantees for STE in the presence of noisy inliers. Finally, under the asymptotic generalized haystack model, we demonstrate that STE initialized with Tyler's M-estimator (TME) recovers the subspace even when the inlier fraction is too small for TME to succeed on its own.