Analysis of the restricted isometry property for Gaussian random matrices

In the context of compressed sensing, we provide a new approach to the analysis of the symmetric and asymmetric restricted isometry property for Gaussian measurement matrices. The proposed method relies on the exact distribution of the extreme eigenvalues for Wishart matrices, or on its approximation based on the Tracy-Widom law, which in turn can be approximated by means of properly shifted and scaled Gamma distributions. The resulting probability that the measurement submatrix is ill conditioned is compared with the known concentration of measure inequality bound, which has been originally adopted to prove that Gaussian matrices satisfy the restricted isometry property with overwhelming probability. The new analytical approach gives an accurate prediction of such probability, tighter than the concentration of measure bound by many orders of magnitude. Thus, the proposed method leads to an improved estimation of the minimum number of measurements required for perfect signal recovery.