Leveraging Well-Conditioned Bases: Streaming \& Distributed Summaries in Minkowski $p$-Norms

TL;DR

This paper introduces deterministic, versatile algorithms for streaming and distributed environments that efficiently handle a range of Minkowski p-norms, extending approximate linear algebra techniques beyond the Euclidean case.

Contribution

It provides the first deterministic algorithms that work for all p ≥ 1, including p = ∞, in both distributed and streaming settings, for various linear algebra problems.

Findings

Algorithms are deterministic and p-agnostic for all p ≥ 1.

Applicable to multiple problems: regression, low rank approximation, matrix multiplication.

Effective in distributed and streaming environments.

Abstract

Work on approximate linear algebra has led to efficient distributed and streaming algorithms for problems such as approximate matrix multiplication, low rank approximation, and regression, primarily for the Euclidean norm $\ell_2$. We study other $\ell_p$ norms, which are more robust for $p < 2$, and can be used to find outliers for $p > 2$. Unlike previous algorithms for such norms, we give algorithms that are (1) deterministic, (2) work simultaneously for every $p \geq 1$, including $p = \infty$, and (3) can be implemented in both distributed and streaming environments. We apply our results to $\ell_p$-regression, entrywise $\ell_1$-low rank approximation, and approximate matrix multiplication.