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Sparsity inducing regularization is an important part for learningover-complete visual representations. Despite the popularity of $\ell_1$regularization, in this paper, we investigate the usage of non-convexregularizations in this problem. Our contribution consists of three parts.First, we propose the leaky capped norm regularization (LCNR), which allowsmodel weights below a certain threshold to be regularized more strongly asopposed to those above, therefore imposes strong sparsity and only introducescontrollable estimation bias. We propose a majorization-minimization algorithmto optimize the joint objective function. Second, our study over monocular 3Dshape recovery and neural networks with LCNR outperforms $\ell_1$ and othernon-convex regularizations, achieving state-of-the-art performance and fasterconvergence. Third, we prove a theoretical global convergence speed on the 3Drecovery problem. To the best of our knowledge, this is the first convergenceanalysis of the 3D recovery problem.

Learning Sparse Visual Representations with Leaky Capped Norm Regularizers