On the Reconstruction of Block-Sparse Signals With an Optimal Number of Measurements

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 Candes, Donoho, and Tao shows that minimization of the lscr ₁ norm of x subject to Ax = 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 lscr₁ -norm relaxation, we consider the following relaxation: _times ^min ||X ₁||₂ + ||X ₂||₂ + ldrldrldr + ||X n ||₂, subject to A x = y (*) where X i = (x ₍ i-1)d+1, x ₍ i-1)d+2, ldrldrldr , x i d)T for i = 1, 2, ldrldrldr , N. Our main result is that as n rarr infin, (*) finds the sparsest solution to A=x = y, with overwhelming probability in A, for any x whose sparsity is k/n < (1/2) - O (isi- - n), provided m /n > 1 - 1/d, and d = Omega(log(1/isin)/isin³) . The relaxation given in (*) can be solved in polynomial time using semi-definite programming.