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Optimal Oblivious Subspace Embeddings with Near-Optimal Sparsity

Authors Shabarish Chenakkod , Michał Dereziński , Xiaoyu Dong

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ACM Subject Classification
  • Theory of computation → Sketching and sampling
  • Theory of computation → Random projections and metric embeddings
Keywords
  • Randomized linear algebra
  • matrix sketching
  • subspace embeddings

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Abstract

An oblivious subspace embedding is a random m× n matrix Π such that, for any d-dimensional subspace, with high probability Π preserves the norms of all vectors in that subspace within a 1±ε factor. In this work, we give an oblivious subspace embedding with the optimal dimension m = Θ(d/ε²) that has a near-optimal sparsity of Õ(1/ε) non-zero entries per column of Π. This is the first result to nearly match the conjecture of Nelson and Nguyen [FOCS 2013] in terms of the best sparsity attainable by an optimal oblivious subspace embedding, improving on a prior bound of Õ(1/ε⁶) non-zeros per column [Chenakkod et al., STOC 2024]. We further extend our approach to the non-oblivious setting, proposing a new family of Leverage Score Sparsified embeddings with Independent Columns, which yield faster runtimes for matrix approximation and regression tasks.
In our analysis, we develop a new method which uses a decoupling argument together with the cumulant method for bounding the edge universality error of isotropic random matrices. To achieve near-optimal sparsity, we combine this general-purpose approach with new trace inequalities that leverage the specific structure of our subspace embedding construction.

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Shabarish Chenakkod, Michał Dereziński, and Xiaoyu Dong. Optimal Oblivious Subspace Embeddings with Near-Optimal Sparsity. In 52nd International Colloquium on Automata, Languages, and Programming (ICALP 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 334, pp. 55:1-55:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025) https://doi.org/10.4230/LIPIcs.ICALP.2025.55

Author Details

Shabarish Chenakkod
  • University of Michigan, Ann Arbor, MI, USA
Michał Dereziński
  • University of Michigan, Ann Arbor, MI, USA
Xiaoyu Dong
  • National University of Singapore, Singapore

Funding

  • Chenakkod, Shabarish: NSF grant DMS 20544.
  • Dereziński, Michał: NSF grant CCF 2338655.

Acknowledgements

The authors are grateful for the generous help and support of Mark Rudelson throughout the duration of this work.

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