Fast ℓ₁-minimization algorithms for robust face recognition

ℓ₁-minimization refers to finding the minimum ℓ₁-norm solution to an underdetermined linear system b=Ax. Under certain conditions as described in compressive sensing theory, the minimum ℓ₁-norm solution is also the sparsest solution. In this paper, we study the speed and scalability of its algorithms. In particular, we focus on the numerical implementation of a sparsity-based classification framework in robust face recognition, where sparse representation is sought to recover human identities from high-dimensional facial images that may be corrupted by illumination, facial disguise, and pose variation. Although the underlying numerical problem is a linear program, traditional algorithms are known to suffer poor scalability for large-scale applications. We investigate a new solution based on a classical convex optimization framework, known as augmented Lagrangian methods. We conduct extensive experiments to validate and compare its performance against several popular ℓ₁-minimization solvers, including interior-point method, Homotopy, FISTA, SESOP-PCD, approximate message passing, and TFOCS. To aid peer evaluation, the code for all the algorithms has been made publicly available.