Matrix Norms in Data Streams: Faster, Multi-Pass and Row-Order

TL;DR

This paper introduces faster, multi-pass, and row-order algorithms for estimating matrix norms in data streams, providing new insights into memory efficiency and computational speed compared to previous methods.

Contribution

It presents novel algorithms for matrix norm estimation in data streams that are multi-pass, row-order, and more time-efficient, along with new lower bounds and separations between models.

Findings

Multi-pass algorithms use less memory than single-pass models.

Row-order algorithms require less memory than entrywise-update models.

Algorithms are significantly faster than previous approaches.

Abstract

A central problem in data streams is to characterize which functions of an underlying frequency vector can be approximated efficiently. Recently there has been considerable effort in extending this problem to that of estimating functions of a matrix that is presented as a data-stream. This setting generalizes classical problems to the analogous ones for matrices. For example, instead of estimating frequent-item counts, we now wish to estimate "frequent-direction" counts. A related example is to estimate norms, which now correspond to estimating a vector norm on the singular values of the matrix. Despite recent efforts, the current understanding for such matrix problems is considerably weaker than that for vector problems. We study a number of aspects of estimating matrix norms in a stream that have not previously been considered: (1) multi-pass algorithms, (2) algorithms that see the underlying matrix one row at a time, and (3) time-efficient algorithms. Our multi-pass and row-order algorithms use less memory than what is provably required in the single-pass and entrywise-update models, and thus give separations between these models (in terms of memory). Moreover, all of our algorithms are considerably faster than previous ones. We also prove a number of lower bounds, and obtain for instance, a near-complete characterization of the memory required of row-order algorithms for estimating Schatten $p$-norms of sparse matrices.