Compressed sensing of block-sparse positive vectors

In this paper we revisit one of the classical problems of compressed sensing. Namely, we consider linear under-determined systems with sparse solutions. A substantial success in mathematical characterization of an l1 optimization technique typically used for solving such systems has been achieved during the last decade. Seminal works [4, 18] showed that the l1 can recover a so-called linear sparsity (i.e. solve systems even when the solution has a sparsity linearly proportional to the length of the unknown vector). Later considerations [13, 14] (as well as our own ones [51, 55]) provided the precise characterization of this linearity. In this paper we consider the so-called structured version of the above sparsity driven problem. Namely, we view a special case of sparse solutions, the so-called block-sparse solutions. Typically one employs l2/l1-optimization as a variant of the standard l1 to handle block-sparse case of sparse solution systems. We considered systems with block-sparse solutions in a series of work [46, 52, 54, 58] where we were able to provide precise performance characterizations if the l2/l1-optimization similar to those obtained for the standard l1 optimization in [51,55]. Here we look at a similar class of systems where on top of being blocksparse the unknown vectors are also known to have components of the same sign. In this paper we slightly adjust l2/l1-optimization to account for the known signs and provide a precise performance characterization of such an adjustment.