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RANDOM
DOI: 10.4230/LIPIcs.APPROX/RANDOM.2026.40
Output-Sparse Matrix Multiplication Using Compressed Sensing

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

Published in: LIPIcs, Volume 392, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2026)

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)

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@InProceedings{bennett_et_al:LIPIcs.APPROX/RANDOM.2026.40,
  author =	{Bennett, Huck and Gajulapalli, Karthik and Golovnev, Alexander and Warton, Evelyn},
  title =	{{Output-Sparse Matrix Multiplication Using Compressed Sensing}},
  booktitle =	{Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2026)},
  pages =	{40:1--40:20},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-449-9},
  ISSN =	{1868-8969},
  year =	{2026},
  volume =	{392},
  editor =	{Singh, Mohit and Gur, Tom},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2026.40},
  URN =		{urn:nbn:de:0030-drops-277570},
  doi =		{10.4230/LIPIcs.APPROX/RANDOM.2026.40},
  annote =	{Keywords: Matrix Multiplication, Sparse Matrices, Compressed Sensing}
}
Document
RANDOM
DOI: 10.4230/LIPIcs.APPROX/RANDOM.2024.42
Matrix Multiplication Verification Using Coding Theory

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

Published in: LIPIcs, Volume 317, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2024)

Abstract
We study the Matrix Multiplication Verification Problem (MMV) where the goal is, given three n × n matrices A, B, and C as input, to decide whether AB = C. A classic randomized algorithm by Freivalds (MFCS, 1979) solves MMV in Õ(n²) time, and a longstanding challenge is to (partially) derandomize it while still running in faster than matrix multiplication time (i.e., in o(n^ω) time). To that end, we give two algorithms for MMV in the case where AB - C is sparse. Specifically, when AB - C has at most O(n^δ) non-zero entries for a constant 0 ≤ δ < 2, we give (1) a deterministic O(n^(ω-ε))-time algorithm for constant ε = ε(δ) > 0, and (2) a randomized Õ(n²)-time algorithm using δ/2 ⋅ log₂ n + O(1) random bits. The former algorithm is faster than the deterministic algorithm of Künnemann (ESA, 2018) when δ ≥ 1.056, and the latter algorithm uses fewer random bits than the algorithm of Kimbrel and Sinha (IPL, 1993), which runs in the same time and uses log₂ n + O(1) random bits (in turn fewer than Freivalds’s algorithm). Our algorithms are simple and use techniques from coding theory. Let H be a parity-check matrix of a Maximum Distance Separable (MDS) code, and let G = (I | G') be a generator matrix of a (possibly different) MDS code in systematic form. Our deterministic algorithm uses fast rectangular matrix multiplication to check whether HAB = HC and H(AB)^T = H(C^T), and our randomized algorithm samples a uniformly random row g' from G' and checks whether g'AB = g'C and g'(AB)^T = g'C^T. We additionally study the complexity of MMV. We first show that all algorithms in a natural class of deterministic linear algebraic algorithms for MMV (including ours) require Ω(n^ω) time. We also show a barrier to proving a super-quadratic running time lower bound for matrix multiplication (and hence MMV) under the Strong Exponential Time Hypothesis (SETH). Finally, we study relationships between natural variants and special cases of MMV (with respect to deterministic Õ(n²)-time reductions).
Cite as

Huck Bennett, Karthik Gajulapalli, Alexander Golovnev, and Evelyn Warton. Matrix Multiplication Verification Using Coding Theory. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 317, pp. 42:1-42:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{bennett_et_al:LIPIcs.APPROX/RANDOM.2024.42,
  author =	{Bennett, Huck and Gajulapalli, Karthik and Golovnev, Alexander and Warton, Evelyn},
  title =	{{Matrix Multiplication Verification Using Coding Theory}},
  booktitle =	{Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2024)},
  pages =	{42:1--42:13},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-348-5},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{317},
  editor =	{Kumar, Amit and Ron-Zewi, Noga},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2024.42},
  URN =		{urn:nbn:de:0030-drops-210352},
  doi =		{10.4230/LIPIcs.APPROX/RANDOM.2024.42},
  annote =	{Keywords: Matrix Multiplication Verification, Derandomization, Sparse Matrices, Error-Correcting Codes, Hardness Barriers, Reductions}
}

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