Fast and RIP-Optimal Transforms

Abstract

We study constructions of (Formula presented.) 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 (Formula presented.) elements on the diagonal, followed by any (Formula presented.) matrix of orthonormal rows (e.g. selection of k coordinates). We provide guarantees (1) and (2) for a regime of parameters that is comparable with previous constructions, but using a construction that uses Fourier transforms and diagonal matrices only. Our main result can be interpreted as a rate of convergence to a random matrix of a random walk in the orthogonal group, in which each step is obtained by a Fourier transform H followed by a random sign change matrix D. After a few number of steps, the resulting matrix is random enough in the sense that any arbitrary selection of rows gives rise to an RIP matrix for, sparsity as high as slightly below (Formula presented.), with high probability. The proof uses a bootstrapping technique that, roughly speaking, says that if a matrix A has some suboptimal RIP parameters, then the action of two steps in this random walk on this matrix has improved parameters. This idea is interesting in its own right, and may be used to strengthen other constructions.