Approximations of schatten norms via taylor expansions

Abstract

In many applications of data science and machine learning data is represented by large matrices. Fast and accurate analysis of such matrices is a challenging task that is of paramount importance for the aforementioned applications. Randomized numerical linear algebra (RNLA) is an popular area of research that often provides such fast and accurate algorithmic methods for massive matrix computations. Many critical problems in RNLA boil down to approximating spectral functions and one of the most fundamental examples of such spectral functions is Schatten p norm. The p-th Schatten norm for matrix (formula presented) is defined as the lp norm of a vector comprised of singular values of matrix A, i.e., (formula presented), where σi(A) is the i-th singular value of A. In this paper we consider symmetric, positive semidefinite (SPSD) matrix A and present an algorithm for computing the p-Schatten norm (formula presented). Our methods are simple and easy to implement and can be extended to general matrices. Our algorithms improve, for a range of parameters, recent results of Musco, Netrapalli, Sidford, Ubaru and Woodruff (ITCS 2018), e.g., for p> 2 and sufficiently small values of ϵ.