Optimal Column-Based Low-Rank Matrix Reconstruction

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 r+1 r−k+1 -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.

• Type: Journal
• Year: 2011
• Where: CORR
• Authors: Venkatesan Guruswami, Ali Kemal Sinop
• Keywords: Arithmetic Operations, CORR 2011, Education, Faster Randomized Algorithm, Lower Order Terms