Approximations of Schatten Norms via Taylor Expansions

TL;DR

This paper introduces two simple algorithms for approximating the Schatten p-norm of symmetric positive semidefinite matrices, improving computational efficiency and extending applicability compared to previous methods.

Contribution

The paper presents two novel algorithms for Schatten norm approximation that are simple, extendable to general matrices, and improve upon or match existing methods' efficiency.

Findings

Algorithms work for all SPSD matrices and non-singular cases.

Improved computational complexity over recent methods.

Avoids Chebyshev polynomial coefficient computations.

Abstract

In this paper we consider symmetric, positive semidefinite (SPSD) matrix $A$ and present two algorithms for computing the $p$-Schatten norm $\|A\|_p$. The first algorithm works for any SPSD matrix $A$. The second algorithm works for non-singular SPSD matrices and runs in time that depends on $\kappa = {\lambda_1(A)\over \lambda_n(A)}$, where $\lambda_i(A)$ is the $i$-th eigenvalue of $A$. 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) and match the running time of the methods by Han, Malioutov, Avron, and Shin (SISC 2017) while avoiding computations of coefficients of Chebyshev polynomials.