Towards optimization techniques on diffeological spaces by generalizing Riemannian concepts

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 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 in an optimization algorithm on diffeological spaces. We give examples for these novel objects. 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.

On the geometry of diffeological vector pseudo-bundles and infinite dimensional vector bundles: automorphisms, connections and covariant derivatives

“…This was performed in [9] via mild considerations on colimits in categories. From another viewpoint, the external tangent space is spanned by derivations, and one can define the cone of derivations which are defined by germs of paths [32] as well as its vector space completion in the space of derivations following [18]. For finite dimensional manifolds these tangent spaces coincide.…”

Parameter-Free Shape Optimization: Various Shape Updates for Engineering Applications

“…Metrics for diffeological spaces have been researched to a lesser extent. However, most concepts can be transferred and in [66] a Riemannian metric is defined for a diffeological space, which yields a Riemannian diffeological space. Additionally, the Riemannian gradient and a steepest descent method on diffeological spaces are defined, assuming a Riemannian metric is available.…”

“…Therefore, the main objective for formulating optimization algorithms on a shape space, i.e., the generalization of concepts such as the definition of a gradient, a distance measure and optimality conditions, is not yet reached for the novel space B 1 2 (Γ 0 ). However, the necessary objects for the steepest descent method on a diffeological space are established and the corresponding algorithm is formulated in [66]. It is nevertheless worth mentioning that various numerical experiments, e.g., [31,[91][92][93], have shown that shape updates obtained from the Steklov-Poincaré metric can also be applied to problems involving non-smooth shapes.…”

“…As explained in [65], these plots do not necessarily have to be of the same dimension as the underlying diffeological space and the mappings between plots do not necessarily have to be reversible. In contrast to shape spaces as Riemannian manifolds, research for diffeological spaces as shape spaces has just begun, see, e.g., [30,66]. Therefore, for the following section, we focus on Riemannian manifolds first and then briefly consider diffeological spaces.…”