Optimal Column-Based Low-Rank Matrix Reconstruction
Venkatesan Guruswami, Ali Kemal Sinop · 2012
We prove that for any real-valued matrix X ∊ ℝm×n, and positive integers r ≥ k, there is a subset of r columns of X such that projecting X onto their span gives a -approximation to best rank-k approximation of X in Frobenius norm. We show that the trade-off we achieve between the number of columns and the approximation ratio is optimal up to lower order terms. Furthermore, there is a deterministic algorithm to find such a subset of columns that runs in O(rnmω log m) arithmetic operations where ω is the exponent of matrix multiplication. We also give a faster randomized algorithm that runs in O(rnm2) arithmetic operations.