Fast and RIP-optimal transforms

We study constructions of k×n matrices A that both (1) satisfy the restricted isometry property (RIP) at sparsity s with optimal parameters, and (2) are efficient in the sense that only O(n log n) operations are required to compute Ax given a vector x. Our construction is based on repeated application of independent transformations of the form DH , where H is a Hadamard or Fourier transform and D is a diagonal matrix with random {+1,−1} elements on the diagonal, followed by any k×n matrix of orthonormal rows (e.g. selection of k coordinates). We provide guarantees (1) and (2) for a larger regime of parameters for which such constructions were previously unknown. Additionally, our construction does not suffer from the extra poly-logarithmic factor multiplying the number of observations k as a function of the sparsity s, as present in the currently best known RIP estimates for partial random Fourier matrices and other classes of structured random matrices.