Towards optimization techniques on diffeological spaces by generalizing Riemannian concepts

TL;DR

This paper extends optimization techniques to diffeological spaces by defining tangent spaces, Riemannian structures, gradients, retractions, and connections, enabling the application of optimization algorithms beyond traditional manifolds.

Contribution

It introduces a new framework for optimization on diffeological spaces, including definitions of tangent spaces, Riemannian structures, and algorithms for optimization.

Findings

Defined tangent spaces suitable for optimization

Developed a diffeological Riemannian space and gradient

Applied the algorithm to a specific optimization problem

Abstract

Diffeological spaces firstly introduced by J.M. Souriau in the 1980s are a natural generalization of smooth manifolds. However, optimization techniques are only known on manifolds so far. Generalizing these techniques to diffeological spaces is very challenging because of several reasons. One of the main reasons is that there are various definitions of tangent spaces which do not coincide. Additionally, one needs to deal with a generalization of a Riemannian space in order to define gradients which are indispensable for optimization methods. This paper is devoted to an optimization technique on diffeological spaces. Thus, one main aim of this paper is a suitable definition of a tangent space in view to optimization methods. Based on this definition, we present a diffeological Riemannian space and a diffeological gradient, which we need to formulate an optimization algorithm on diffeological spaces. Moreover, in order to be able to update the iterates in an optimization algorithm on diffeological spaces, we present a diffeological retraction and the Levi-Civita connection on diffeological spaces. We give examples for the novel objects and apply the presented diffeological algorithm to an optimization problem.