Robust Sparse Covariance Estimation by Thresholding Tyler's M-estimator
Talkat Robust and High-Dimensional Statistics (Simons Institute workshop, 2018) · 2018-10-29..11-02·
summary
Proposes that thresholded Tyler's M-estimator and its regularized variant recover an approximately sparse shape matrix for elliptical heavy‑tailed data at the same minimax spectral‑norm rate as sub‑Gaussian methods when p/n → c < 1, subject to a bounded top eigenvalue; shows the estimator’s implicit sample weights concentrate around R/N, uses random‑matrix and concentration tools, proves linear convergence with log(1/ε) iterations for large enough regularization, demonstrates success on Laplace/Cauchy simulations and weight‑based outlier removal, and lists data‑driven thresholding and stronger contamination guarantees as open problems.
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chapters (17)
- 0:00Start
- 1:09Intro
- 2:04Sample / Empirical Estimates
- 2:26The good old days
- 3:32Sparsity and High-Dimensions
- 5:37Prior Work: Sparse Covariance Estimation
- 9:17Outlier/Heavy Tail breakdown of sample covariance
- 12:09Elliptical Distribution, sparse shape matrix
- 13:50Removing the unknown mean / symmetrization
- 19:51Our Results
- 23:42Proof Outline
- 34:58Simulation Results, Model 1
- 35:46TME with Elliptical outliers
- 37:15Simulation Results, Model 2
- 38:27Open Questions
- 39:48Summary
- 46:37Computational Complexity
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