title: LSV-Based Tail Inequalities for Sums of Random Matrices.
abstract: The techniques of random matrices have played an important role in many machine learning models. In this letter, we present a new method to study the tail inequalities for sums of random matrices. Different from other work (Ahlswede & Winter, 2002 ; Tropp, 2012 ; Hsu, Kakade, & Zhang, 2012 ), our tail results are based on the largest singular value (LSV) and independent of the matrix dimension. Since the LSV operation and the expectation are noncommutative, we introduce a diagonalization method to convert the LSV operation into the trace operation of an infinitely dimensional diagonal matrix. In this way, we obtain another version of Laplace-transform bounds and then achieve the LSV-based tail inequalities for sums of random matrices.
authors.affiliation: School of Mathematical Sciences, Dalian University of Technology, Dalian, Liaoning, 116024, P.R.C. chao.zhang@dlut.edu.cn.
authors.forename: Chao
authors.initials: C
authors.lastname: Zhang
authors.affiliation: School of Mathematical Sciences, Dalian University of Technology, Dalian, Liaoning, 116024, P.R.C. dulei@dlut.edu.cn.
authors.forename: Lei
authors.initials: L
authors.lastname: Du
authors.affiliation: Centre for Artificial Intelligence, Faculty of Engineering and Information Technology, University of Technology Sydney Ultimo, NSW 2007, Australia. dacheng.tao@uts.edu.au.
authors.forename: Dacheng
authors.initials: D
authors.lastname: Tao