Roots of the identity operator and proximal mappings: (classical and phantom) cycles and gap vectors

TL;DR

This paper extends a geometric lemma to convex functions, linking it to duality theory, and uses it to characterize cycles and gap vectors of proximal mappings in Hilbert spaces.

Contribution

It generalizes Simons's lemma to convex functions and establishes new characterizations of cycles and gap vectors in proximal mappings.

Findings

Extended Simons's lemma to convex functions

Connected the lemma to Attouch-Thera duality

Characterized classical and phantom cycles and gap vectors

Abstract

Recently, Simons provided a lemma for a support function of a closed convex set in a general Hilbert space and used it to prove the geometry conjecture on cycles of projections. In this paper, we extend Simons's lemma to closed convex functions, show its connections to Attouch-Thera duality, and use it to characterize (classical and phantom) cycles and gap vectors of proximal mappings.