On Polar Polytopes and the Recovery of Sparse Representations

Suppose we have a signal ${\mmb y}$ which we wish to represent using a linear combination of a number of basis atoms $\break {\mmb a}\!\!\!$ $_i,{\mmb y}$ $=\!\!\!\sum_i x_i {\mmb a}_i ={\mmb A}{\mmb x}$. The problem of finding the minimum $\ell_0$ norm representation for ${\mmb y}$ is a hard problem. The Basis Pursuit (BP) approach proposes to find the minimum $\ell_1$ 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 $\ell_1$ norm) and sparse (minimum $\ell_0$ 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 ${\pm}{\mmb a}$ $_i$ 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 geome