2026-08-06T19:46:27Z https://drops.dagstuhl.de/oai

oai:drops-oai.dagstuhl.de:19541 2024-03-06T11:04:16Z ddc:004 open_access

Universal Matrix Sparsifiers and Fast Deterministic Algorithms for Linear Algebra Bhattacharjee, Rajarshi Dexter, Gregory Musco, Cameron Ray, Archan Sachdeva, Sushant Woodruff, David P. sublinear algorithms randomized linear algebra spectral sparsification expanders Let S ∈ ℝ^{n × n} be any matrix satisfying ‖1-S‖₂ ≤ εn, where 1 is the all ones matrix and ‖⋅‖₂ is the spectral norm. It is well-known that there exists S with just O(n/ε²) non-zero entries achieving this guarantee: we can let 𝐒 be the scaled adjacency matrix of a Ramanujan expander graph. We show that, beyond giving a sparse approximation to the all ones matrix, 𝐒 yields a universal sparsifier for any positive semidefinite (PSD) matrix. In particular, for any PSD A ∈ ℝ^{n×n} which is normalized so that its entries are bounded in magnitude by 1, we show that ‖A-A∘S‖₂ ≤ ε n, where ∘ denotes the entrywise (Hadamard) product. Our techniques also yield universal sparsifiers for non-PSD matrices. In this case, we show that if S satisfies ‖1-S‖₂ ≤ (ε²n)/(c log²(1/ε)) for some sufficiently large constant c, then ‖A-A∘S‖₂ ≤ ε⋅max(n,‖ A‖₁), where ‖A‖₁ is the nuclear norm. Again letting 𝐒 be a scaled Ramanujan graph adjacency matrix, this yields a sparsifier with Õ(n/ε⁴) entries. We prove that the above universal sparsification bounds for both PSD and non-PSD matrices are tight up to logarithmic factors. Since 𝐀∘𝐒 can be constructed deterministically without reading all of A, our result for PSD matrices derandomizes and improves upon established results for randomized matrix sparsification, which require sampling a random subset of O((n log n)/ε²) entries and only give an approximation to any fixed A with high probability. We further show that any randomized algorithm must read at least Ω(n/ε²) entries to spectrally approximate general A to error εn, thus proving that these existing randomized algorithms are optimal up to logarithmic factors. We leverage our deterministic sparsification results to give the first {deterministic algorithms} for several problems, including singular value and singular vector approximation and positive semidefiniteness testing, that run in faster than matrix multiplication time. This partially addresses a significant gap between randomized and deterministic algorithms for fast linear algebraic computation. Finally, if A ∈ {-1,0,1}^{n × n} is PSD, we show that a spectral approximation à with ‖A-Ã‖₂ ≤ ε n can be obtained by deterministically reading Õ(n/ε) entries of A. This improves the 1/ε dependence on our result for general PSD matrices by a quadratic factor and is information-theoretically optimal up to a logarithmic factor. Schloss Dagstuhl – Leibniz-Zentrum für Informatik Rajarshi Bhattacharjee and Gregory Dexter and Cameron Musco and Archan Ray and Sushant Sachdeva and David P. Woodruff 2024 Is Part Of LIPIcs, Volume 287, 15th Innovations in Theoretical Computer Science Conference (ITCS 2024) InProceedings Text doc-type:ResearchArticle publishedVersion application/pdf doi:10.4230/LIPIcs.ITCS.2024.13 urn:nbn:de:0030-drops-195415 https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2024.13 eng https://creativecommons.org/licenses/by/4.0/legalcode