Optimal l1 column subset selection and a fast PTAS for low rank approximation

Abstract

We study the problem of entrywise l1 low rank approximation. We give the first polynomial time column subset selection-based l1 low rank approximation algorithm sampling Oe(k) columns and achieving an Oe(k1/2)approximation for any k, improving upon the previous best Oe(k)-approximation and matching a prior lower bound for column subset selection-based l1-low rank approximation which holds for any poly(k) number of columns. We extend our results to obtain tight upper and lower bounds for column subset selection-based lp low rank approximation for any 1 < p < 2, closing a long line of work on this problem. We next give a (1 + ε)-approximation algorithm for entrywise lp low rank approximation of an n×d matrix, for 1 ≤ p < 2, that is not a column subset selection algorithm. First, we obtain an algorithm which, given a matrix A ∈ Rn×d, returns a rank-k matrix Ab in 2poly(k/ε) + poly(nd) running time that achieves the following guarantee: kA− Abkp ≤ (1 + ε) · OPT + poly(εk)kAkp where OPT = minAkrank k kA−Akkp. Using this algorithm, in the same running time we give an algorithm which obtains error at most (1 + ε) · OPT and outputs a matrix of rank at most 3k - these algorithms significantly improve upon all previous (1 + ε)- and O(1)-approximation algorithms for the lp low rank approximation problem, which required at least npoly(k/ε) or npoly(k) running time, and either required strong bit complexity assumptions (our algorithms do not) or had bicriteria rank 3k. Finally, we show hardness results which nearly match our 2poly(k)+poly(nd) running time and the above additive error guarantee.