Schatten Norms in Matrix Streams: Hello Sparsity, Goodbye Dimension

TL;DR

This paper introduces space-efficient algorithms for approximating Schatten norms of large, sparse matrices in streaming settings, enabling spectral analysis with memory independent of matrix size.

Contribution

It presents the first algorithms with dimension-independent space complexity for Schatten norm approximation in doubly-sparse matrix streams.

Findings

Algorithms achieve space complexity independent of matrix dimension.

Validated performance on real-world social network matrices.

Multiple passes are proven necessary for the streaming approximation.

Abstract

Spectral functions of large matrices contains important structural information about the underlying data, and is thus becoming increasingly important. Many times, large matrices representing real-world data are \emph{sparse} or \emph{doubly sparse} (i.e., sparse in both rows and columns), and are accessed as a \emph{stream} of updates, typically organized in \emph{row-order}. In this setting, where space (memory) is the limiting resource, all known algorithms require space that is polynomial in the dimension of the matrix, even for sparse matrices. We address this challenge by providing the first algorithms whose space requirement is \emph{independent of the matrix dimension}, assuming the matrix is doubly-sparse and presented in row-order. Our algorithms approximate the Schatten $p$-norms, which we use in turn to approximate other spectral functions, such as logarithm of the determinant, trace of matrix inverse, and Estrada index. We validate these theoretical performance bounds by numerical experiments on real-world matrices representing social networks. We further prove that multiple passes are unavoidable in this setting, and show extensions of our primary technique, including a trade-off between space requirements and number of passes.