From Sparse Solutions of Systems of Equations to Sparse Modeling of Signals and Images

A full-rank matrix A ∈ IR n×m with n < m generates an underdetermined system of linear equations Ax = b having infinitely many solutions. Suppose we seek the sparsest solution, i.e., the one with the fewest nonzero entries: can it ever be unique? If so, when? As optimization of sparsity is combinatorial in nature, are there efficient methods for finding the sparsest solution? These questions have been answered positively and constructively in recent years, exposing a wide variety of surprising phenomena; in particular, the existence of easily-verifiable conditions under which optimally-sparse solutions can be found by concrete, effective computational methods. Such theoretical results inspire a bold…