Moment inequalities for matrix-valued U-statistics of order 2

Abstract

We present Rosenthal-type moment inequalities for matrix-valued U-statistics of order 2. As a corollary, we obtain new matrix concentration inequalities for U-statistics. One of our main technical tools, a version of the non-commutative Khintchine inequality for the spectral norm of the Rademacher chaos, could be of independent interest. Introduction.Since being introduced by W. Hoeffding [16], U-statistics have become an active topic of research. Many classical results in estimation and testing are related to U-statistics; detailed treatment of the subject can be found in excellent monographs [7,20,30,21]. A large body of research has been devoted to understanding the asymptotic behavior of real-valued U-statistics. Such asymptotic results, as well as moment and concentration inequalities, are discussed in the works [8,7,12,13,18,11,17], among others. The case of vector-valued and matrix-valued U-statistics received less attention; natural examples of matrix-valued U-statistics include various estimators of covariance matrices, such as the usual sample covariance matrix and the estimators based on Kendall's tau [37,15].Exponential and moment inequalities for Hilbert space-valued U-statistics have been developed in [2]. The goal of the present work is to obtain moment and concentration inequalities for generalized degenerate U-statistics of order 2 with values in the set of matrices with complexvalued entries equipped with the operator (spectral) norm. The emphasis is made on expressing the upper bounds in terms of computable parameters. Our results extend the matrix Rosenthal's inequality for the sums of independent random matrices due to Chen, Gittens and Tropp [5] (see also [19,25]) to the framework of U-statistics. As a corollary of our bounds, we deduce a variant of the Matrix Bernstein inequality for U-statistics of order 2.We also discuss connections of our bounds with general moment inequalities for Banach spacevalued U-statistics due to R. Adamczak [1], and leverage Adamczak's inequalities to obtain additional refinements and improvements of the results.We note that U-statistics with values in the set of self-adjoint matrices have been considered in [6], however, most results in that work deal with the element-wise sup-norm, while we are primarily interested in results about the moments and tail behavior of the spectral norm of Ustatistics. Another recent work [26] investigates robust estimators of covariance matrices based on U-statistics, but deals only with the case of non-degenerate U-statitistics that can be reduced to the study of independent sums.The key technical tool used in our arguments is the extension of the non-commutative Khintchine's inequality (Lemma 3.3) which could be of independent interest. Notation and background material.Given A P C d 1ˆd2 , A˚P C d 2ˆd1 will denote the Hermitian adjoint of A. H d Ă C dˆd stands for the set of all self-adjoint matrices. If A " A˚, we will write λ max pAq and λ min pAq for the largest and smallest eigenvalues of

Deviation inequalities for Banach space valued martingales differences sequences and random fields

“…• In [36], a Rosenthal type inequality for Hermitian matrix-valued U -statistics was obtained. The presented bound for the moments of such a U -statistic has explicit constant, and the terms in the spirit of second the right hand side of (1.36) involves the moment of order q of the maximum over i instead of the sum of moments.…”