On Sketching Matrix Norms and the Top Singular Vector Yi Li Huy L. Nguy˜ên University of Michigan, Ann Arbor Princeton University
[email protected] [email protected] David P. Woodruff IBM Research, Almaden
[email protected] Abstract 1 Introduction Sketching is a prominent algorithmic tool for processing Sketching is an algorithmic tool for handling big data. A large data. In this paper, we study the problem of sketching matrix norms. We consider two sketching models. The first linear skech is a distribution over k × n matrices S, k is bilinear sketching, in which there is a distribution over n, so that for any fixed vector v, one can approximate a pairs of r × n matrices S and n × s matrices T such that for function f(v) from Sv with high probability. The goal any fixed n×n matrix A, from S ·A·T one can approximate is to minimize the dimension k. kAkp up to an approximation factor α ≥ 1 with constant probability, where kAkp is a matrix norm. The second is The sketch Sv thus provides a compression of v, and general linear sketching, in which there is a distribution over is useful for compressed sensing and for achieving low- n2 k linear maps L : R → R , such that for any fixed n × n communication protocols in distributed models. One n2 matrix A, interpreting it as a vector in R , from L(A) one can perform complex procedures on the sketch which can approximate kAkp up to a factor α. We study some of the most frequently occurring matrix would be too expensive to perform on v itself, and this norms, which correspond to Schatten p-norms for p ∈ has led to the fastest known approximation algorithms {0, 1, 2, ∞}.