| Summary: | Non-convex sparsity-inducing penalties outperform their convex counterparts, but generally sacrifice the cost function convexity. As a middle ground, we propose the <italic>sharpening sparse regularizers</italic> (SSR) framework to design non-separable non-convex penalties that induce sparsity more effectively than convex penalties such as <inline-formula><tex-math notation="LaTeX">$\ell _1$</tex-math></inline-formula> and nuclear norms, but without sacrificing the cost function convexity. The overall problem convexity is preserved by exploiting the data fidelity <italic>relative</italic> strong convexity. The framework constructs penalties as the <italic>difference of convex functions</italic>, namely the difference between convex sparsity-inducing penalties and their smoothed versions. We propose a generalized infimal convolution smoothing technique to obtain the smoothed versions. Furthermore, SSR recovers and generalizes several non-convex penalties in the literature as special cases. The SSR framework is applicable to any sparsity regularized least squares ill-posed linear inverse problem. Beyond regularized least squares, the SSR framework can be extended to accommodate Bregman divergence, and other sparsity structures such as low-rankness. The SSR optimization problem can be formulated as a saddle point problem, and solved by a scalable forward-backward splitting algorithm. The effectiveness of the SSR framework is demonstrated by numerical experiments in different applications.
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