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Efficient Block Approximate Matrix Multiplication

Authors Chuhan Yang, Christopher Musco

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Author Details

Chuhan Yang
  • Division of Engineering, New York University Abu Dhabi, UAE
  • Tandon School of Engineering, New York University, NY, USA
Christopher Musco
  • Tandon School of Engineering, New York University, NY, USA

Funding

This work was partially supported by National Science Foundation Award #2045590.

Cite AsGet BibTex

Chuhan Yang and Christopher Musco. Efficient Block Approximate Matrix Multiplication. In 31st Annual European Symposium on Algorithms (ESA 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 274, pp. 103:1-103:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)
https://doi.org/10.4230/LIPIcs.ESA.2023.103

Abstract

Randomized matrix algorithms have had significant recent impact on numerical linear algebra. One especially powerful class of methods are algorithms for approximate matrix multiplication based on sampling. Such methods typically sample individual matrix rows and columns using carefully chosen importance sampling probabilities. However, due to practical considerations like memory locality and the preservation of matrix structure, it is often preferable to sample contiguous blocks of rows and columns all together. Recently, (Wu, 2018) addressed this setting by developing an approximate matrix multiplication method based on block sampling. However, the method is inefficient, as it requires knowledge of optimal importance sampling probabilities that are expensive to compute. We address this issue by showing that the method of Wu can be accelerated through the use of a randomized implicit trace estimation method. Doing so allows us to provably reduce the cost of sampling to near-linear in the size of the matrices being multiplied, without impacting the accuracy of the final approximate matrix multiplication. Overall, this yields a fast practical algorithm, which we test on a number of synthetic and real-world data sets. We complement our algorithmic contribution with the first extensive empirical comparison of block algorithms for randomized matrix multiplication. Our method offers a significant runtime advantage over the method of (Wu, 2018) and also outperforms basic uniform sampling of blocks. However, we find another recent method of (Charalambides, 2021) which uses sub-optimal but efficiently computable sampling probabilities often (but not always) offers the best trade-off between speed and accuracy.

Subject Classification

ACM Subject Classification
  • Theory of computation → Sketching and sampling
Keywords
  • Approximate matrix multiplication
  • randomized numerical linear algebra
  • trace estimation

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