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. Ex- periments 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 com- plexities 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 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 and peak memory usage. This experimentally confirms the provably effective theoretical bounds we show for our compressed-matrix approach.

Improving Matrix-vector Multiplication via Lossless Grammar-Compressed Matrices

Ferragina P.;Manzini G.;Tosoni F.
2022-01-01

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. Ex- periments 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 com- plexities 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 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 and peak memory usage. This experimentally confirms the provably effective theoretical bounds we show for our compressed-matrix approach.
Ferragina, P.; Manzini, G.; Gagie, T.; Koppl, D.; Navarro, G.; Striani, M.; Tosoni, F.
File in questo prodotto:
File Dimensione Formato  
Improving matrix vector multiplication.pdf

accesso aperto

Tipologia: Versione finale editoriale
Licenza: Creative commons
Dimensione 1.16 MB
Formato Adobe PDF
1.16 MB Adobe PDF Visualizza/Apri

I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.

Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11568/1157854
Citazioni
  • ???jsp.display-item.citation.pmc??? ND
  • Scopus 0
  • ???jsp.display-item.citation.isi??? ND
social impact