Modulus of conically averaged mappings and its applications to angles between two subspaces

Abstract

Conically averaged mappings, a generalization of averaged mappings, are important in a wide range of Optimization Algorithms. In this paper, we propose the modulus of conical averagedness to classify conical averaged mappings. Introducing the monotone and comonotone values of generalized monotone mappings, we investigate their connections to the modulus of conical averagedness. In the linear setting, we completely characterize conically averaged matrices, and derive explicit and pleasing formulae for computing their modulus of averagedness. As applications, we compute the Dixmier and Friedrichs angles between two subspaces. Nonlinear results are established as extensions of the linear case. Conical averagedness of proximal and reflection mappings of hypoconvex functions are also studied.