Abstract
We provide theory for the computation of convex envelopes of non-convex functionals including an ell ^2-term and use these to suggest a method for regularizing a more general set of problems. The applications are particularly aimed at compressed sensing and low-rank recovery problems, but the theory relies on results which potentially could be useful also for other types of non-convex problems. For optimization problems where the ell ^2-term contains a singular matrix, we prove that the regularizations never move the global minima. This result in turn relies on a theorem concerning the structure of convex envelopes, which is interesting in its own right. It says that at any point where the convex envelope does not touch the non-convex functional, we necessarily have a direction in which the convex envelope is affine.
Highlights
The present work is the extension of a chain of ideas with its roots in compressed sensing. 1 − 2-minimization tricks have a long history and got renewed attention with the work of Donoho, Candés and Tao among others [1,2,3]
We provide theory for the computation of convex envelopes of non-convex functionals including an 2-term and use these to suggest a method for regularizing a more general set of problems
For such choice of γ, we prove that the functional (10) is a convex functional below (9) and minimization of (10) will produce a minimizer which, not necessarily equal to the minimizer of the original problem, potentially is closer than that obtained by other convex relaxation methods
Summary
The present work is the extension of a chain of ideas with its roots in compressed sensing. 1 − 2-minimization tricks have a long history and got renewed attention with the work of Donoho, Candés and Tao among others [1,2,3]. Journal of Optimization Theory and Applications (2019) 183:66–84 severe) bias They are slow since one needs to find an appropriate value of involved penalty parameters. Due to such issues, there is a wealth of non-convex variations to replace 1/nuclear norm in the area of compressed sensing, we refer to [6] for a survey. We find a unifying framework and show that all these penalties are particular cases of the so-called proximal hull or quadratic envelope We systematically study this as a regularizer, and in particular, we lift the result of Aubert et al to a general context. We show that whenever a l.s.c. convex envelope is not in touch with the function that generates it, it necessarily has a direction in which it is affine linear
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