On Low Rank Matrix Approximations with Applications to Synthesis Problem in Compressed Sensing

We consider the synthesis problem of Compressed Sensing –given s and an M×n matrix A, extract from it an m × n submatrix Am, certified to be s-good, with m as small as possible. Starting from the verifiable sufficient conditions of s-goodness, we express the synthesis problem as the problem of approximating a given matrix by a matrix of specified low rank in the uniform norm. We propose randomized algorithms for efficient construction of rank k approximation of matrices of size m × n achieving accuracy bounds O(1) √ ln(mn) k which hold in expectation or with high probability. We also supply derandomized versions of the approximation algorithms which does not require random sampling of matrices and attains the same accuracy bounds. We further demonstrate that our algorithms are optimal up to the logarithmic in m,n factor, i.e. the accuracy of such an approximation for the identity matrix In cannot be better than O(1)k − 1 2 ). We provide preliminary numerical results on the performance of our algorithms for the synthesis problem.