Theoretical Guarantees for the Subspace-Constrained Tyler's Estimator

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. To demonstrate the usefulness of this result, we use a common setting in the theoretical computer science community, whose special case is when the inliers are sampled from a Gaussian distribution and there are no restrictions on the outliers. We show that in this setting, with high probability, STE initialized by a polynomial algorithm exactly recovers the underlying subspace with linear convergence. 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.