  The non-negative solution to an underdetermined linear system can be uniquely
recovered sometimes, even without imposing any additional sparsity constraints.
In this paper, we derive conditions under which a unique non-negative solution
for such a system can exist, based on the theory of polytopes. Furthermore, we
develop the paradigm of combined sparse representations, where only a part of
the coefficient vector is constrained to be non-negative, and the rest is
unconstrained (general). We analyze the recovery of the unique, sparsest
solution, for combined representations, under three different cases of
coefficient support knowledge: (a) the non-zero supports of non-negative and
general coefficients are known, (b) the non-zero support of general
coefficients alone is known, and (c) both the non-zero supports are unknown.
For case (c), we propose the combined orthogonal matching pursuit algorithm for
coefficient recovery and derive the deterministic sparsity threshold under
which recovery of the unique, sparsest coefficient vector is possible. We
quantify the order complexity of the algorithms, and examine their performance
in exact and approximate recovery of coefficients under various conditions of
noise. Furthermore, we also obtain their empirical phase transition
characteristics. We show that the basis pursuit algorithm, with partial
non-negative constraints, and the proposed greedy algorithm perform better in
recovering the unique sparse representation when compared to their
unconstrained counterparts. Finally, we demonstrate the utility of the proposed
methods in recovering images corrupted by saturation noise.
