Abstract 

We study the -Low Rank Approximation Problem, where the goal is, given an m × n matrix A, to output a rank-k matrix for which is minimized. Here, for a matrix denotes the number of its non-zero entries. This NPhard variant of low rank approximation is natural for problems with no underlying metric, and its goal is to minimize the number of disagreeing data positions. We provide approximation algorithms which significantly improve the running time and approximation factor of previous work. For k > 1, we show how to find, in poly(mn) time for every k, a rank O(klog(n/k)) matrix A0 for which OPT. To the best of our knowledge, this is the first algorithm with provable guarantees for the -Low Rank Approximation Problem for k > 1, even for bicriteria algorithms. For the well-studied case when k = 1, we give a (2+)-approximation in sublinear time, which is impossible for other variants of low rank approximation such as for the Frobenius norm. We strengthen this for the well-studied case of binary matrices to obtain -approximation in sublinear time, where  For small , our approximation factor is 1 + o(1).