On the reconstruction of block-sparse signals with an optimal number of measurements
Abstract
Let A be an M by N matrix (M < N) which is an instance of a real random Gaussian ensemble. In compressed sensing we are interested in finding the sparsest solution to the system of equations A x = y for a given y. In general, whenever the sparsity of x is smaller than half the dimension of y then with overwhelming probability over A the sparsest solution is unique and can be found by an exhaustive search over x with an exponential time complexity for any y. The recent work of Candés, Donoho, and Tao shows that minimization of the L1 norm of x subject to A x = y results in the sparsest solution provided the sparsity of x, say K, is smaller than a certain threshold for a given number of measurements. Specifically, if the dimension of y approaches the dimension of x, the sparsity of x should be K < 0.239 N. Here, we consider the case where x is d-block sparse, i.e., x consists of n = N / d blocks where each block is either a zero vector or a nonzero vector. Instead of L1-norm relaxation, we consider the following relaxation min x \| X1 \|2 + \| X2 \|2 + ... + \| Xn \|2, subject to A x = y where Xi = (x(i-1)d+1, x(i-1)d+2, ..., xi d) for i = 1,2, ..., N. Our main result is that as n -> ∞, the minimization finds the sparsest solution to Ax = y, with overwhelming probability in A, for any x whose block sparsity is k/n < 1/2 - O(ε), provided M/N > 1 - 1/d, and d = Ω((1/ε)/ε). The relaxation can be solved in polynomial time using semi-definite programming.