Communication Lower Bounds and Algorithms for Sketching with Random Dense Matrices

TL;DR

This paper establishes fundamental limits on communication for dense matrix sketching in distributed systems, and presents optimal algorithms that are scalable on modern supercomputers with CPU and GPU architectures.

Contribution

It derives tight communication lower bounds for dense matrix sketching and introduces near-optimal parallel algorithms that are scalable on supercomputing infrastructures.

Findings

Communication lower bounds are tight and determine data movement requirements.

Proposed algorithms achieve near-optimal communication costs.

Algorithms demonstrate good scalability on CPU and GPU supercomputing systems.

Abstract

Sketching is widely used in randomized linear algebra for low-rank matrix approximation, column subset selection, and many other problems, and it has gained significant traction in machine learning applications. However, sketching large matrices often necessitates distributed memory algorithms, where communication overhead becomes a critical bottleneck on modern supercomputing clusters. Despite its growing relevance, distributed-memory parallel strategies for sketching remain largely unexplored. In this work, we establish communication lower bounds for sketching using dense matrices that determine how much data movement is required to perform it in parallel. One important observation of our lower bounds is that no communication is required for a small number of processors. We show that our lower bounds are tight by presenting communication optimal algorithms. Furthermore, we extend our approach to determine communication lower bounds for computations of Nystr\"om approximation where sketching is applied twice. We also introduce novel parallel algorithms whose communication costs are close to the lower bounds. Finally, we implement our algorithms on modern state-of-the-art supercomputing infrastructures which have both CPU- and GPU-equipped systems and demonstrate their parallel scalability.