TY - GEN

T1 - Approximating Iterated Multiplication of Stochastic Matrices in Small Space

AU - Cohen, Gil

AU - Doron, Dean

AU - Sberlo, Ori

AU - Ta-Shma, Amnon

N1 - Funding Information:
Gil Cohen was funded by ERC starting grant 949499 and by ISF grant 1569/18. Ori Sberlo was funded by ERC starting grant 949499 and by ISF grant 952/18. Amnon Ta-Shma was funded by ISF grant 952/18.
Publisher Copyright:
© 2023 Owner/Author.

PY - 2023/6/2

Y1 - 2023/6/2

N2 - 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.

AB - 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.

KW - derandomization

KW - iterated matrix multiplication

KW - matrix powering

KW - space-bounded computation

UR - http://www.scopus.com/inward/record.url?scp=85163057419&partnerID=8YFLogxK

U2 - 10.1145/3564246.3585181

DO - 10.1145/3564246.3585181

M3 - Conference contribution

AN - SCOPUS:85163057419

T3 - Proceedings of the Annual ACM Symposium on Theory of Computing

SP - 35

EP - 45

BT - STOC 2023 - Proceedings of the 55th Annual ACM Symposium on Theory of Computing

A2 - Saha, Barna

A2 - Servedio, Rocco A.

PB - Association for Computing Machinery

T2 - 55th Annual ACM Symposium on Theory of Computing, STOC 2023

Y2 - 20 June 2023 through 23 June 2023

ER -