Faster Algorithms for Binary Matrix Factorization

We give faster approximation algorithms for wellstudied variants of Binary Matrix Factorization (BMF), where we are given a binarym×nmatrix A and would like to find binary rank-k matrices U, V to minimize the Frobenius norm ofU ·V −A. In the first setting, U · V denotes multiplication over Z, and we give a constantfactor approximation algorithm that runs in 2 2 log poly(mn) time, improving upon the previous min(2 k , 2)poly(mn) time. Our techniques generalize to minimizing ‖U · V − A‖p for p ≥ 1, in 2O(k log poly(mn) time. For p = 1, this has a graph-theoretic consequence, namely, a 2 2)poly(mn)-time algorithm to approximate a graph as a union of disjoint bicliques. In the second setting, U · V is over GF(2), and we give a bicriteria constant-factor approximation algorithm that runs in 2 3)poly(mn) time to find binary rankO(k logm) matrices U , V whose cost is as good as the best rank-k approximation, improving upon min(2 k mn,min(m,n) O(1) poly(mn)) time.

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