By Topic

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

Sign In

Cookies must be enabled to login.After enabling cookies , please use refresh or reload or ctrl+f5 on the browser for the login options.

Formats Non-Member Member
$33 $13
Learn how you can qualify for the best price for this item!
Become an IEEE Member or Subscribe to
IEEE Xplore for exclusive pricing!
close button

puzzle piece

IEEE membership options for an individual and IEEE Xplore subscriptions for an organization offer the most affordable access to essential journal articles, conference papers, standards, eBooks, and eLearning courses.

Learn more about:

IEEE membership

IEEE Xplore subscriptions

3 Author(s)
Stojnic, M. ; Sch. of Ind. Eng., Purdue Univ., West Lafayette, IN, USA ; Parvaresh, F. ; Hassibi, B.

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.

Published in:

Signal Processing, IEEE Transactions on  (Volume:57 ,  Issue: 8 )