On the reconstruction of block-sparse signals with an optimal number of measurements

Abstract

Let A be an M by N(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 Ax = 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 block sparse, i.e., x consists of n = N/d blocks where each block is of length d and is either a zero vector or a nonzero vector (under nonzero vector we consider a vector that can have both, zero and nonzero components). Instead of ℓ1-norm relaxation, we consider the following relaxation: min ∥ X1 ∥2 + ∥X2∥2 + ⋯ + ∥ Xn ∥2, subject to A x = y (*) where Xi = (x(i-1)d+1,x (i-1)d+2, ⋯ xid) T i = 1,2, ⋯ N. Our main result is that as n → ∞, (z.ast;) finds the sparsest solution to A x = y, with overwhelming probability in A, for any x whose sparsity is k/n < (1/2) - O(ε), provided m/n > 1 - 1/d, and d = Ω(log(1/ε)/ ε 3. The relaxation given in (z.ast;) can be solved in polynomial time using semi-definite programming. © 2009 IEEE.