**3**Author(s)

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 *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 lscr_{1} -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)/isin^{3}) . The relaxation given in (*) can be solved in polynomial time using semi-definite programming.

- Page(s):
- 3075 - 3085
- ISSN :
- 1053-587X
- INSPEC Accession Number:
- 10764088
- Digital Object Identifier :
- 10.1109/TSP.2009.2020754

- Date of Publication :
- 10 April 2009
- Date of Current Version :
- 14 July 2009
- Issue Date :
- Aug. 2009
- Sponsored by :
- IEEE Signal Processing Society