The Sharp Phase Transition of Tyler's M-Estimator for Robust Subspace Recovery

2026-06-04

AI Summary

This work addresses the unclear behavior of robust subspace recovery in high-dimensional noisy data near the critical regime where the dimension-scaled signal-to-noise ratio (DS-SNR) equals one. By analyzing the convergence of Tyler’s M-estimator within a majorization–minimization (MM) framework, the paper establishes—for the first time—a rigorous guarantee that exact recovery of the true d-dimensional subspace is achievable whenever DS-SNR ≥ 1 and a newly introduced mild stability condition holds. This result precisely characterizes the sharp phase transition threshold of Tyler’s method, significantly relaxes existing assumptions on data distribution, and reveals its robust recovery capability precisely at the critical SNR boundary.

Abstract

Robust Subspace Recovery (RSR) aims to identify an underlying d-dimensional subspace from a dataset heavily corrupted by outliers. Complexity-theoretic results establish a threshold for the problem's computational hardness based on the dimension-scaled signal-to-noise ratio (DS-SNR): the problem is SSE-hard when the DS-SNR is strictly less than 1, and solvable via practical algorithms when it is greater than 1 under general position assumptions. However, the exact behavior of practical algorithms at the critical boundary DS-SNR = 1 has remained unknown. This work resolves the behavior of Tyler's M-estimator (TME) at this critical boundary, consequently establishing a sharp phase transition. Specifically, we prove that TME converges exactly to the true subspace for DS-SNR \geq 1 under a new stability condition, which is less restrictive than the general position assumptions used in prior literature. Our analysis utilizes a decomposition of the TME iterates within a majorization-minimization framework.

Problem

• Robust Subspace Recovery
• Tyler's M-estimator
• phase transition
• DS-SNR
• critical boundary

Innovation

• Tyler's M-estimator
• robust subspace recovery
• sharp phase transition
• dimension-scaled SNR
• majorization-minimization