# Rigorous Guarantees for Tyler's M-estimator via quantum expansion

arXiv: https://arxiv.org/abs/2002.00071 · HTML: https://mlsanity.com/paper/2002.00071/

## Key facts

- Cited by 10 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 31 Jan 2020.

## Built on by (most cited first, 10 of 10)

- [A Subspace-Constrained Tyler's Estimator and its Applications to Structure from Motion](https://mlsanity.com/paper/2404.11590/) (2024)
- [No-go Theorem for Acceleration in the Hyperbolic Plane](https://mlsanity.com/paper/2101.05657/) (2021)
- [Near optimal sample complexity for matrix and tensor normal models via geodesic convexity](https://mlsanity.com/paper/2110.07583/) (2021)
- [Frank-Wolfe-based Algorithms for Approximating Tyler's M-estimator](https://mlsanity.com/paper/2206.09370/) (2022)
- [On a class of geodesically convex optimization problems solved via Euclidean MM methods](https://mlsanity.com/paper/2206.11426/) (2022)
- [Optimal Bounds for Tyler's M-Estimator for Elliptical Distributions](https://mlsanity.com/paper/2510.13751/) (2025)
- [Cubic-Regularized Newton for Spectral Constrained Matrix Optimization and its Application to Fairness](https://mlsanity.com/paper/2209.01229/) (2022)
- [Improved Convergence Factor of Windowed Anderson Acceleration for Symmetric Fixed-Point Iterations](https://mlsanity.com/paper/2311.02490/) (2023)
- [T-Rex: Fitting a Robust Factor Model via Expectation-Maximization](https://mlsanity.com/paper/2505.12117/) (2025)
- [The Sharp Phase Transition of Tyler's M-Estimator for Robust Subspace Recovery](https://mlsanity.com/paper/2606.06782/) (2026)

## Abstract (arXiv)

Estimating the shape of an elliptical distribution is a fundamental problem in statistics. One estimator for the shape matrix, Tyler's M-estimator, has been shown to have many appealing asymptotic properties. It performs well in numerical experiments and can be quickly computed in practice by a simple iterative procedure. Despite the many years the estimator has been studied in the statistics community, there was neither a tight non-asymptotic bound on the rate of the estimator nor a proof that the iterative procedure converges in polynomially many steps. Here we observe a surprising connection between Tyler's M-estimator and operator scaling, which has been intensively studied in recent years in part because of its connections to the Brascamp-Lieb inequality in analysis. We use this connection, together with novel results on quantum expanders, to show that Tyler's M-estimator has the optimal rate up to factors logarithmic in the dimension, and that in the generative model the iterative procedure has a linear convergence rate even without regularization.