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ACM Transactions on Computation Theory (TOCT)
, Volume 5 (3) – Aug 1, 2013

/lp/association-for-computing-machinery/compressed-matrix-multiplication-lnJBeAPkcH

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- Publisher
- Association for Computing Machinery
- Copyright
- Copyright © 2013 by ACM Inc.
- ISSN
- 1942-3454
- DOI
- 10.1145/2493252.2493254
- Publisher site
- See Article on Publisher Site

Compressed Matrix Multiplication RASMUS PAGH, IT University of Copenhagen We present a simple algorithm that approximates the product of n-by-n real matrices A and B. Let ||AB|| F denote the Frobenius norm of AB, and b be a parameter determining the time/accuracy trade-off. Given 2-wise independent hash functions h1 , h2 : [n] [b], and s1 , s2 : [n] {-1, +1} the algorithm works by first "compressing" the matrix product into the polynomial n n p(x) = k=1 i=1 Aiks1 (i) x h1 (i) n Bkj s2 ( j) x h2 ( j) . j=1 Using the fast Fourier transform to compute polynomial multiplication, we can compute c0 , . . . , cb-1 such ~ that i ci xi = ( p(x) mod x b ) + ( p(x) div x b ) in time O(n2 + nb). An unbiased estimator of (AB)i j with variance at most ||AB||2 /b can then be computed as: F Ci j = s1 (i) s2 ( j) c(h1 (i)+h2 ( j)) mod b . ~ Our approach also leads to an algorithm for computing AB exactly, with high probability, in time O(N + nb) in the case where A

ACM Transactions on Computation Theory (TOCT) – Association for Computing Machinery

**Published: ** Aug 1, 2013

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