On Polar Polytopes and the Recovery of Sparse Representations

Suppose we have a signal y which we wish to represent using a linear combination of a number of basis atoms a_i,y=Sigma_ix_ia_i=Ax. The problem of finding the minimum l₀ norm representation for y is a hard problem. The basis pursuit (BP) approach proposes to find the minimum l₁ norm representation instead, which corresponds to a linear program (LP) that can be solved using modern LP techniques, and several recent authors have given conditions for the BP (minimum l₁ norm) and sparse (minimum l₀ norm) representations to be identical. In this paper, we explore this sparse representation problem using the geometry of convex polytopes, as recently introduced into the field by Donoho. By considering the dual LP we find that the so-called polar polytope P* of the centrally symmetric polytope P whose vertices are the atom pairs plusmna_i is particularly helpful in providing us with geometrical insight into optimality conditions given by Fuchs and Tropp for non-unit-norm atom sets. In exploring this geometry, we are able to tighten some of these earlier results, showing for example that a condition due to Fuchs is both necessary and sufficient for l₁-unique-optimality, and there are cases where orthogonal matching pursuit (OMP) can eventually find all l₁-unique-optimal solutions with m nonzeros even if the exact recover condition (ERC) fails for m.