Optimal CUR matrix decompositions

Abstract

The CUR decomposition of an m×n matrix A finds an m×c matrix C with a small subset of c n columns of A, together with an r×n matrix R with a small subset of r m rows of A, as well as a c×r low rank matrix U such that the matrix CUR approximates the input matrix A, that is, ||A --- CUR||2F ≤ (1 + e)||A --- Ak||2F, where ||.||F denotes the Frobenius norm, 0 e Ak is the best m × n matrix of rank k constructed via the SVD of A. We present input-sparsity-time and deterministic algorithms for constructing such a CUR matrix decomposition of A where c = O(k/e) and r = O(k/e) and rank(U) = k. Up to constant factors, our construction is simultaneously optimal in c, r, and rank(U).