Nonconvex Sparse Representation With Slowly Vanishing Gradient Regularizers

Sparse representation has been widely used over the past decade in computer vision and signal processing to model a wide range of natural phenomena. For computational convenience and robustness against noises, the optimization problem for sparse representation is often relaxed using convex or nonconvex surrogates instead of using the l
₀
-norm, the ideal sparsity penalty function. In this paper, we pose the following question for nonconvex sparsity-promoting surrogates: What is a good sparsity surrogate for general nonconvex systems? As an answer to this question, we suggest that the difficulty of handling the l
₀
-norm does not only come from the nonconvexity but also from its gradient being zero or not well-defined. Accordingly, we propose desirable criteria to be a good nonconvex surrogate and suggest a corresponding family of surrogates. The proposed family of surrogates allows a simple regularizer, which enables efficient computation. The proposed surrogate embraces the benefits of both l
₀
and l
₁
-norms, and most importantly, its gradient vanishes slowly, which allows stable optimization. We apply the proposed surrogate to wellknown sparse representation problems and benchmark datasets to demonstrate its robustness and efficiency.