Smoothness of Schatten Norms and Sliding-Window Matrix Streams

TL;DR

This paper introduces a method for approximating Schatten p-norms in sliding-window matrix streams, leveraging the smoothness property of these norms to enable efficient computation in time-sensitive data scenarios.

Contribution

It proves that Schatten p-norms are smooth in row-order streams, enabling the use of smooth-histograms for $(1+psilon)$-approximation in sliding-window models.

Findings

First $(1+psilon)$-approximation algorithm for Schatten p-norms in sliding-window streams.

Shows Schatten p-norms are smooth, allowing application of smooth-histograms.

Extends streaming algorithms to handle time-sensitive, row-order matrix data.

Abstract

Large matrices are often accessed as a row-order stream. We consider the setting where rows are time-sensitive (i.e. they expire), which can be described by the sliding-window row-order model, and provide the first $(1+\epsilon)$-approximation of Schatten $p$-norms in this setting. Our main technical contribution is a proof that Schatten $p$-norms in row-order streams are smooth, and thus fit the smooth-histograms technique of Braverman and Ostrovsky (FOCS 2007) for sliding-window streams.