Approximating Iterated Multiplication of Stochastic Matrices in Small Space

Gil Cohen, Dean Doron, Ori Sberlo, Amnon Ta-Shma

Research output: Chapter in Book/Report/Conference proceedingConference contributionpeer-review

Abstract

Matrix powering, and more generally iterated matrix multiplication, is a fundamental linear algebraic primitive with myriad applications in computer science. Of particular interest is the problem's space complexity as it constitutes the main route towards resolving the BPL vs. L problem. The seminal work by Saks and Zhou [JCSS '99] gives a deterministic algorithm for approximating the product of n stochastic matrices of dimension w × w in space O(log3/2n + logn · logw). The first improvement upon Saks-Zhou was achieved by Hoza [RANDOM '21] who gave a logarithmic improvement in the n=poly(w) regime, attaining O(1/loglogn · log3/2n) space. We give the first polynomial improvement over Saks and Zhou's algorithm. Our algorithm achieves space complexity of O(logn + logn· logw). In particular, in the regime logn > log2 w, our algorithm runs in nearly-optimal O(logn) space, improving upon the previous best O(log3/2n). To obtain our result for the special case of matrix powering, we harness recent machinery from time-and space-bounded Laplacian solvers to the Saks-Zhou framework and devise an intricate precision-alternating recursive scheme. This enables us to bypass the bottleneck of paying logn-space per recursion level. The general case of iterated matrix multiplication poses several additional challenges, the substantial of which is handled by devising an improved shift and truncate mechanism. The new mechanism is made possible by a novel use of the Richardson iteration.

Original languageEnglish
Title of host publicationSTOC 2023 - Proceedings of the 55th Annual ACM Symposium on Theory of Computing
EditorsBarna Saha, Rocco A. Servedio
PublisherAssociation for Computing Machinery
Pages35-45
Number of pages11
ISBN (Electronic)9781450399135
DOIs
StatePublished - 2 Jun 2023
Event55th Annual ACM Symposium on Theory of Computing, STOC 2023 - Orlando, United States
Duration: 20 Jun 202323 Jun 2023

Publication series

NameProceedings of the Annual ACM Symposium on Theory of Computing
ISSN (Print)0737-8017

Conference

Conference55th Annual ACM Symposium on Theory of Computing, STOC 2023
Country/TerritoryUnited States
CityOrlando
Period20/06/2323/06/23

Keywords

  • derandomization
  • iterated matrix multiplication
  • matrix powering
  • space-bounded computation

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