Riemannian cubics on the group of diffeomorphisms and the Fisher-Rao metric

Rabah Tahraoui (1) , François-Xavier Vialard (2, 3)

1 UNIROUEN - Université de Rouen Normandie
2 CEREMADE - CEntre de REcherches en MAthématiques de la DEcision
3 MOKAPLAN - Méthodes numériques pour le problème de Monge-Kantorovich et Applications en sciences sociales

We study a second-order variational problem on the group of diffeomorphisms of the interval [0, 1] endowed with a right-invariant Sobolev metric of order 2, which consists in the minimization of the acceleration. We compute the relaxation of the problem which involves the so-called Fisher-Rao functional a convex functional on the space of measures. This relaxation enables the derivation of several optimality conditions and, in particular, a sufficient condition which guarantees that a given path of the initial problem is also a minimizer of the relaxed one. This sufficient condition is related to the existence of a solution to a Riccati equation involving the path acceleration.

Mots clés: Group of Diffeomorphisms; Fisher-Rao; Right-invariant metric; Riemannian splines

Domaines: Optimisation et contrôle [math.OC]

Rabah Tahraoui, François-Xavier Vialard. Riemannian cubics on the group of diffeomorphisms and the Fisher-Rao metric. 2016. ⟨hal-01331110v3⟩