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Let A be an M by N matrix (M < N) which is an instance of a real randomGaussian ensemble. In compressed sensing we are interested in finding thesparsest 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 withoverwhelming probability over A the sparsest solution is unique and can befound by an exhaustive search over x with an exponential time complexity forany y. The recent work of Cand\'es, Donoho, and Tao shows that minimization ofthe L_1 norm of x subject to A x = y results in the sparsest solution providedthe sparsity of x, say K, is smaller than a certain threshold for a givennumber of measurements. Specifically, if the dimension of y approaches thedimension of x, the sparsity of x should be K < 0.239 N. Here, we consider thecase where x is d-block sparse, i.e., x consists of n = N / d blocks where eachblock is either a zero vector or a nonzero vector. Instead of L_1-normrelaxation, we consider the following relaxation min x \| X_1 \|_2 + \| X_2\|_2 + ... + \| X_n \|_2, subject to A x = y where X_i = (x_{(i-1)d+1},x_{(i-1)d+2}, ..., x_{i d}) for i = 1,2, ..., N. Our main result is that as n-> \infty, the minimization finds the sparsest solution to Ax = y, withoverwhelming probability in A, for any x whose block sparsity is k/n < 1/2 -O(\epsilon), provided M/N > 1 - 1/d, and d = \Omega(\log(1/\epsilon)/\epsilon).The relaxation can be solved in polynomial time using semi-definiteprogramming.

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