Nonuniform Sparse Recovery with Subgaussian Matrices

Compressive sensing predicts that sufficiently sparse vectors can be recovered from highly incomplete information using efficient recovery methods such as `1-minimization. Random matrices have become a popular choice for the measurement matrix. Indeed, near-optimal uniform recovery results have been shown for such matrices. In this note we focus on nonuniform recovery using subgaussian random matrices and `1minimization. We provide conditions on the number of samples in terms of the sparsity and the signal length which guarantees that a fixed sparse signal can be recovered with a random draw of the matrix using `1-minimization. Our proofs are short and provide explicit and good constants.