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

arXiv: https://arxiv.org/abs/2606.06782 · HTML: https://mlsanity.com/paper/2606.06782/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 5 of its 21 references that are in the map.
- First posted on arXiv 4 Jun 2026.

## Builds on (most cited first, 5 of 5)

- [An Overview of Robust Subspace Recovery](https://mlsanity.com/paper/1803.01013/) (2018)
- [A Well-Tempered Landscape for Non-convex Robust Subspace Recovery](https://mlsanity.com/paper/1706.03896/) (2017)
- [Rigorous Guarantees for Tyler's M-estimator via quantum expansion](https://mlsanity.com/paper/2002.00071/) (2020)
- [Robust Subspace Recovery with Adversarial Outliers](https://mlsanity.com/paper/1904.03275/) (2019)
- [A Subspace-Constrained Tyler's Estimator and its Applications to Structure from Motion](https://mlsanity.com/paper/2404.11590/) (2024)

## Abstract (arXiv)

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 ≥ 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.