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Output-Sparse Matrix Multiplication Using Compressed Sensing

Authors Huck Bennett , Karthik Gajulapalli , Alexander Golovnev , Evelyn Warton

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LIPIcs.APPROX-RANDOM.2026.40.pdf
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Subject Classification

ACM Subject Classification
  • Theory of computation → Design and analysis of algorithms
  • Theory of computation → Pseudorandomness and derandomization
  • Theory of computation → Expander graphs and randomness extractors
  • Theory of computation → Error-correcting codes
Keywords
  • Matrix Multiplication
  • Sparse Matrices
  • Compressed Sensing

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Abstract

We give two algorithms for output-sparse matrix multiplication (OSMM), the problem of multiplying two n × n matrices A, B when their product AB is promised to have at most O(n^δ) many non-zero entries for a given value δ ∈ [0, 2]. We then show how to speed up these algorithms in the fully sparse matrix multiplication (FSMM) setting, where the input matrices A, B are themselves sparse. All of our algorithms work over arbitrary rings.
Our first, deterministic algorithm for OSMM works via a two-pass reduction to compressed sensing. It runs in roughly n^{ω(δ/2, 1, 1)} time, where ω(⋅, ⋅, ⋅) is the rectangular matrix multiplication exponent. This substantially improves on prior deterministic algorithms for output-sparse matrix multiplication.
Our second, randomized algorithm for OSMM works via a reduction to compressed sensing and a variant of matrix multiplication verification, and runs in roughly n^{ω(δ-1, 1, 1)} time. This algorithm and its extension to the fully sparse setting have running times that match those of the (randomized) algorithms for OSMM and FSMM, respectively, in recent work of Abboud, Bringmann, Fischer, and Künnemann (SODA, 2024). Our algorithm is quite simple when taking a suitable compressed sensing scheme as a black box.

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Huck Bennett, Karthik Gajulapalli, Alexander Golovnev, and Evelyn Warton. Output-Sparse Matrix Multiplication Using Compressed Sensing. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 392, pp. 40:1-40:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026) https://doi.org/10.4230/LIPIcs.APPROX/RANDOM.2026.40

Author Details

Huck Bennett
  • University of Colorado Boulder, CO, USA
Karthik Gajulapalli
  • Georgetown University, Washington, DC, USA
Alexander Golovnev
  • Georgetown University, Washington, DC, USA
Evelyn Warton
  • Oregon State University, Corvallis, OR, USA

Funding

  • Bennett, Huck: Supported by NSF Grant CCF-2432132.
  • Gajulapalli, Karthik: Supported by NSF grant CCF-2338730.
  • Golovnev, Alexander: Supported by NSF grant CCF-2338730.

Acknowledgements

HB and AG would like to thank Divesh Aggarwal, the National University of Singapore (NUS), and the Centre for Quantum Technologies (CQT). Some of this work was completed while visiting them. The authors would also like to thank Thatchaphol Saranurak for suggesting the use of the expander construction in [Guruswami et al., 2009], and Noah Stephens-Davidowitz for helpful comments. Finally, the authors would like to thank anonymous reviewers for providing helpful references and comments (in particular, for noting the connection between [Abboud et al., 2024] and our work).

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