On Polar Polytopes and the Recovery of Sparse Representations

Abstract

<para xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> Suppose we have a signal <emphasis><formula formulatype="inline"><tex>${\mmb y}$</tex></formula></emphasis> which we wish to represent using a linear combination of a number of basis atoms <formula><tex>$\break {\mmb a}\!\!\!$ </tex></formula><emphasis><formula> <tex>$_i,{\mmb y}$</tex></formula></emphasis><emphasis><formula formulatype="inline"> <tex>$=\!\!\!\sum_i x_i {\mmb a}_i ={\mmb A}{\mmb x}$</tex></formula></emphasis>. The problem of finding the minimum <emphasis><formula formulatype="inline"> <tex>$\ell_0$</tex></formula></emphasis> norm representation for <formula formulatype="inline"><tex>${\mmb y}$</tex></formula> is a hard problem. The Basis Pursuit (BP) approach proposes to find the minimum <emphasis><formula formulatype="inline"><tex>$\ell_1$</tex></formula></emphasis> 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 <emphasis><formula formulatype="inline"><tex>$\ell_1$</tex> </formula></emphasis> norm) and sparse (minimum <emphasis><formula formulatype="inline"> <tex>$\ell_0$</tex></formula></emphasis> norm) representations to be identical. In this paper, we explore this <emphasis emphasistype="boldital">sparse representation problem</emphasis> 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 <emphasis><formula formulatype="inline"><tex>$P^*$</tex></formula></emphasis> of the centrally symmetric polytope <emphasis><formula formulatype="inline"> <tex>$P$</tex></formula></emphasis> whose vertices are the atom pairs <formula formulatype="inline"><tex>${\pm}{\mmb a}$</tex></formula><emphasis><formula> <tex>$_i$</tex></formula></emphasis> 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</para>