IMPROVED BOUNDS ON RESTRICTED ISOMETRY CONSTANTS FOR GAUSSIAN MATRICES

The restricted isometry constant ( RIC) of a matrix A measures how close to an isometry is the action of A on vectors with few nonzero entries, measured in the l(2) norm. Specifically, the upper and lower RICs of a matrix A of size nxN are the maximum and the minimum deviation from unity (one) of the largest and smallest, respectively, square of singular values of all ((N)(k)) matrices formed by taking k columns from A. Calculation of the RIC is intractable for most matrices due to its combinatorial nature; however, many random matrices typically have bounded RIC in some range of problem sizes (k, n, N). We provide the best known bound on the RIC for Gaussian matrices, which is also the smallest known bound on the RIC for any large rectangular matrix. Our results are built on the prior bounds of Blanchard, Cartis, and Tanner [SIAM Rev., to appear], with improvements achieved by grouping submatrices that share a substantial number of columns.