  We consider list versions of sparse approximation problems, where unlike the
existing results in sparse approximation that consider situations with unique
solutions, we are interested in multiple solutions. We introduce these problems
and present the first combinatorial results on the output list size. These
generalize and enhance some of the existing results on threshold phenomenon and
uncertainty principles in sparse approximations. Our definitions and results
are inspired by similar results in list decoding. We also present lower bound
examples that bolster our results and show they are of the appropriate size.
