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

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

2008-03-31

arxiv.org/abs/0804.0041v1

Computer Science

Information Theory

Scientific paper

10.1016/j.jmr.2011.02.015

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 Cand\'es, Donoho, and Tao shows that minimization of the L_1 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 L_1-norm relaxation, 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, with overwhelming 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-definite programming.

Affiliated with

Also associated with

No associations

Rating

On the reconstruction of block-sparse signals with an optimal number of measurements does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.

If you have personal experience with On the reconstruction of block-sparse signals with an optimal number of measurements, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and On the reconstruction of block-sparse signals with an optimal number of measurements will most certainly appreciate the feedback.

Rate now

Comments { 0 }

Profile ID: LFWR-SCP-O-216268