Improving Matrix-vector Multiplication via Lossless Grammar-Compressed Matrices

TL;DR

This paper introduces a lossless compression scheme for real-valued matrices that enables efficient storage and matrix-vector multiplication, outperforming existing tools like gzip and xz, and approaching theoretical entropy limits.

Contribution

The authors propose a novel lossless compression method for matrices that supports fast matrix-vector multiplication and introduces column-reordering algorithms to further optimize performance.

Findings

Outperforms gzip and is close to xz in compression ratio.

Supports matrix-vector multiplication proportional to compressed size.

Runs at least twice as fast as state-of-the-art CLA methods.

Abstract

As nowadays Machine Learning (ML) techniques are generating huge data collections, the problem of how to efficiently engineer their storage and operations is becoming of paramount importance. In this article we propose a new lossless compression scheme for real-valued matrices which achieves efficient performance in terms of compression ratio and time for linear-algebra operations. Experiments show that, as a compressor, our tool is clearly superior to gzip and it is usually within 20% of xz in terms of compression ratio. In addition, our compressed format supports matrix-vector multiplications in time and space proportional to the size of the compressed representation, unlike gzip and xz that require the full decompression of the compressed matrix. To our knowledge our lossless compressor is the first one achieving time and space complexities which match the theoretical limit expressed by the $k$-th order statistical entropy of the input. To achieve further time/space reductions, we propose column-reordering algorithms hinging on a novel column-similarity score. Our experiments on various data sets of ML matrices show that, with a modest preprocessing time, our column reordering can yield a further reduction of up to 16% in the peak memory usage during matrix-vector multiplication. Finally, we compare our proposal against the state-of-the-art Compressed Linear Algebra (CLA) approach showing that ours runs always at least twice faster (in a multi-thread setting) and achieves better compressed space occupancy for most of the tested data sets. This experimentally confirms the provably effective theoretical bounds we show for our compressed-matrix approach.