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Robust regularized covariance matrix estimation: well-posedness and convergent algorithm

Mengxi Yi, David Tyler

Abstract

In this paper, we study properties of penalized and structured M-estimators of multivariate scatter, based on geodesically convex but not necessarily smooth penalty functions. Existence and uniqueness conditions for these penalized and structured estimators are given. However, we show that the standard fixed-point algorithm which is usually applied to an M-estimation problem does not necessarily converge for penalized M-estimation problems. Hence, we develop a new but simple re-weighting algorithm and prove that it has monotone convergence for a broad class of penalized and structured M-estimators of multivariate scatter.

Robust regularized covariance matrix estimation: well-posedness and convergent algorithm

Abstract

In this paper, we study properties of penalized and structured M-estimators of multivariate scatter, based on geodesically convex but not necessarily smooth penalty functions. Existence and uniqueness conditions for these penalized and structured estimators are given. However, we show that the standard fixed-point algorithm which is usually applied to an M-estimation problem does not necessarily converge for penalized M-estimation problems. Hence, we develop a new but simple re-weighting algorithm and prove that it has monotone convergence for a broad class of penalized and structured M-estimators of multivariate scatter.

Paper Structure

This paper contains 10 sections, 14 theorems, 36 equations.

Table of Contents

  1. Introduction and Motivation
  2. Regularized M-estimators of multivariate scatter
  3. Review of elliptical distributions and the M-estimators of scatter
  4. Penalized M-estimation of Scatter
  5. Bayesian interpretation
  6. Reweighting algorithms
  7. Constrained optimization
  8. Concluding remarks
  9. Geodesic convexity
  10. Proofs

Key Result

Lemma 1

If $\rho(s)$ and $\Pi(\Sigma)$ are g-convex, then $L_{\rho}(\Sigma; \eta)$ is g-convex and the solution set is g-convex. Furthermore, if either $\rho(s)$ or $\Pi(\Sigma)$ is strictly g-convex, then $L_\rho(\Sigma; \eta)$ is strictly g-convex and hence the solution set $\mathcal{A}_{\rho}$ is either empty or contains a single element.

Theorems & Definitions (17)

  • Lemma 1
  • Lemma 2
  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Theorem 4
  • Example 1
  • Theorem 5
  • Theorem 6
  • Example 2
  • ...and 7 more