Embeddings of Schatten Norms with Applications to Data Streams

Abstract: Given an $n \times d$ matrix $A$, its Schatten-$p$ norm, $p \geq 1$, is defined as $$|A|<em>p = \left (\sum</em>{i=1}^{{\textrm{rank}(A)}\sigma_i(A)^p} \right )<sup>{1/p}$$, where $\sigma_i(A)$ is the $i$-th largest singular value of $A$. These norms have been studied in functional analysis in the context of non-commutative $\ell_p$-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 $$A^1, \ldots, A^{{\operatorname{poly}(nd)}} \in \mathbb{R}^{n \times d}$$, suppose we want to construct a linear map $L$ such that $L(Aⁱ⁾ \in \mathbb{R}^{n&#39; \times d&#39;}$ for each $i$, where $n&#39; \leq n$ and $d&#39; \leq d$, and further, $$|A^i|_p \leq |L(A^i)|_q \leq D_{p,q} |A<sup>i|_p$$ for a given approximation factor $D_{p,q}$ and real number $q \geq 1$. Then how large do $n&#39;$ and $d&#39;$ need to be as a function of $D_{p,q}$? We nearly resolve this question for every $p, q \geq 1$, for the case where $L(A<sup>i)$ can be expressed as $R \cdot Aⁱ \cdot S$, where $R$ and $S$ are arbitrary matrices that are allowed to depend on $A^1, \ldots, A<sup>t$, that is, $L(A<sup>i)$ can be implemented by left and right matrix multiplication. Namely, for every $p, q \geq 1$, we provide nearly matching upper and lower bounds on the size of $n&#39;$ and $d&#39;$ as a function of $D_{p,q}$. Importantly, our upper bounds are {\it oblivious}, meaning that $R$ and $S$ do not depend on the $A<sup>i$, while our lower bounds hold even if $R$ and $S$ depend on the $A<sup>i$. 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 $D \geq 1$ using $\tilde{O}(\min(n,d)<sup>2/D<sup>4)$ space.