Efficient reconstruction of block-sparse signals

In many sparse reconstruction problems, M observations are used to estimate K components in an N dimensional basis, where N >; M ≫ K. The exact basis vectors, however, are not known a priori and must be chosen from an M × N matrix. Such under-determined problems can be solved using an ℓ
₂
optimization with an ℓ
₁
penalty on the sparsity of the solution. There are practical applications in which multiple measurements can be grouped together, so that K × P data must be estimated from M × P observations, where the ℓ
₁
sparsity penalty is taken with respect to the vector formed using the ℓ
₂
norms of the rows of the data matrix. In this paper we develop a computationally efficient block partitioned homotopy method for reconstructing K × P data from M × P observations using a grouped sparsity constraint, and compare its performance to other block reconstruction algorithms.