Non-convex optimization for linear system with pregaussian matrices and recovery from multiple measurements

The extremal singular values of random matrices in ell_{2}-norm, including Gaussian random matrices, Bernoulli random matrices, subgaussian random matrices, etc, have attracted major research interest in recent years. In this thesis, we study the q-singular values, defined in terms of the ell_{q}-quasinorm, of pregaussian random matrices. We give the upper tail probability estimate on the largest q-singular value of pregaussian random matrices for 01, and some numerical-experimental results as well.Compressed sensing, a technique for recovering sparse signals, has also been an active research topic recently. The extremal singular values of random matrices have applications in compressed sensing, mainly because the restricted isometry constant of sensing matrices depends on them. We prove that pregaussian random matrices with m much less than N but much larger than N^{q/2} have the q-modified restricted isometry property for 0