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.

翻译：鲁棒子空间恢复（RSR）旨在从被离群点严重污染的数据集中识别潜在d维子空间。计算复杂性理论结果基于维度归一化信噪比（DS-SNR）建立了该问题计算难度的阈值：当DS-SNR严格小于1时，问题具有SSE难度；而当其大于1时，在一般位置假设下可通过实用算法求解。然而，实用算法在临界边界DS-SNR=1时的精确行为此前尚不明确。本研究揭示了泰勒M估计（TME）在该临界边界处的行为，从而建立了尖锐相变。具体而言，我们证明在一种比先前文献中使用的一般位置假设限制更小的新稳定性条件下，当DS-SNR≥1时，TME精确收敛到真实子空间。我们的分析利用了大化-小化框架中TME迭代的分解。