On Convex Envelopes and Regularization of Non-convex Functionals Without Moving Global Minima

• We provide theory for the computation of convex envelopes of non-convex functionals including an ℓ2-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 ℓ2-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.

Ämnesord

NATURVETENSKAP  -- Matematik (hsv//swe)

NATURAL SCIENCES  -- Mathematics (hsv//eng)

Nyckelord

Convex envelope

Fenchel conjugate

Non-convex/non-smooth optimization

Proximal hull

Regularization

Publikations- och innehållstyp

art (ämneskategori)

ref (ämneskategori)