Embeddings of Schatten Norms with Applications to Data Streams

Given an n×d matrix A, its Schatten-p norm, p ≥ 1, is defined as ‖A‖p = (∑rank(A) i=1 σi(A) p )1/p , where σi(A) is the i-th largest singular value of A. These norms have been studied in functional analysis in the context of non-commutative `p-spaces, and recently in data stream and linear sketching models of computation. Basic questions on the relations between these norms, such as their embeddability, are still open. Specifically, given a set of matrices A1, . . . , Apoly(nd) ∈ Rn×d, suppose we want to construct a linear map L such that L(A) ∈ Rn×d for each i, where n′ ≤ n and d′ ≤ d, and further, ‖A‖p ≤ ‖L(A)‖q ≤ Dp,q‖A‖p for a given approximation factor Dp,q and real number q ≥ 1. Then how large do n′ and d′ need to be as a function of Dp,q? We nearly resolve this question for every p, q ≥ 1, for the case where L(A) can be expressed as R ·A ·S, where R and S are arbitrary matrices that are allowed to depend on A1, . . . , A, that is, L(A) can be implemented by left and right matrix multiplication. Namely, for every p, q ≥ 1, we provide nearly matching upper and lower bounds on the size of n′ and d′ as a function of Dp,q. Importantly, our upper bounds are oblivious, meaning that R and S do not depend on the A, while our lower bounds hold even if R and S depend on the A. As an application of our upper bounds, we answer a recent open question of Blasiok et al. about space-approximation trade-offs for the Schatten 1-norm, showing in a data stream it is possible to estimate the Schatten-1 norm up to a factor of D ≥ 1 using Õ(min(n, d)2/D4) space. 1998 ACM Subject Classification G. Mathematics of Computing