Recovery of Sparse Representations by Polytope Faces Pursuit

Abstract

We introduce a new greedy algorithm to find approximate sparse representations $\vec s$ of $\vec x={\vec A}\vec s$ by finding the Basis Pursuit (BP) solution of the linear program $\min\{{\|s\|}_{\vec 1} \mid {\vec x}={\vec A}{\vec s}\}$. The proposed algorithm is based on the geometry of the polar polytope $P^* = \{{\vec c} \mid {\tilde{\vec A}}^T{\vec c} \le {\vec 1} \}$ where ${\tilde{\vec A}} = [{\vec A},-{\vec A}]$ and searches for the vertex ${\vec c}^*\in P^*$ which maximizes ${\vec x}^{T}{\vec c}$ using a path following method. The resulting algorithm is in the style of Matching Pursuits (MP), in that it adds new basis vectors one at a time, but it uses a different correlation criterion to determine which basis vector to add and can switch out basis vectors as necessary. The algorithm complexity is of a similar order to Orthogonal Matching Pursuits (OMP). Experimental results show that this algorithm, which we call Polytope Faces Pursuit, produces good results on examples that are known to be hard for MP, and it is faster than the interior point method for BP on the experiments presented.