Marčenko–Pastur law for Tyler’s M-estimator

Abstract

This paper studies the limiting behavior of Tyler's M-estimator for the scatter matrix, in the regime that the number of samples n and their dimension p both go to infinity, and p/n converges to a constant y with 0 < y < 1. We prove that when the data samples x1, . . . , xn are identically and independently generated from the Gaussian distribution N (0, I), the operator norm of the difference between a properly scaled Tyler's Mestimator and n i=1 xix ⊤ i /n tends to zero. As a result, the spectral distribution of Tyler's M-estimator converges weakly to the Marčenko-Pastur distribution.

Comparative Analysis of Covariance Matrix Estimation for Anomaly Detection in Hyperspectral Images

“…Note that the normalization by the trace in the right side of (11) is not mandatory but it is often used in Tyler based estimation to make easier the comparison and analysis of its spectral properties without any decrement in the detection performance. Recently, a similar M-P law to (6) for the empirical eigenvalues of (11) has been shown in [30], [31].…”

Robust sparse covariance estimation by thresholding Tyler’s M-estimator

“…The lemmas, proven in Appendices A.6-A.8, assume the setting of Theorem 3, and their generic constants depend only on γ, α and s max . Our analysis of the weights w i follows the pioneering works of Silverstein (2014, 2015), who proved that the weights in Maronna's M-estimators converge to suitable constants, and Zhang, Cheng and Singer (2016), who derived concentration results for the weights of Tyler's M-estimator as p, n → ∞ with p/n → γ < 1.…”

Large-Dimensional Behavior of Regularized Maronna's M-Estimators of Covariance Matrices

“…However, Theorem 1 excludes the "under-sampled" case N ≥ n. Regularized versions of Maronna's M-estimators have been proposed to alleviate this issue, in most cases considering regularized versions of Tyler's estimator (u(x) = 1/x) [1,2,15,28], the behavior of which has been studied in [17,18]. Recently, a regularized M-estimator which accounts for a wider class of u functions has been introduced in [22], but its large-dimensional behavior remains unknown.…”

“…Their structure is non-trivial, involving matrix fixedpoint equations, and their analysis challenging. Nonetheless, significant progress towards understanding these estimators has been made in large-dimensional settings [15][16][17][18][19], motivated by the increasing number of applications where N, n are both large and comparable. Salient messages of these works are: (i) outliers or impulsive data can be handled by these estimators, if appropriately designed (the choice of the specific form of the estimator is important to handle different types of outliers) [19]; (ii) in the absence of outliers, robust M-estimators essentially behave as the SCM and, therefore, are still subject to the data scarcity issue [16].…”

“…Assume the same contaminated data model as in Theorem 3. Let Assumptions 1-2 hold and letĈ n N (ρ) be the unique solution to(18). Then, as N, n → ∞, for all…”