Minimizing movements with general costs, and nonnegative cross-curvature in infinite dimensions
speaker: Flavien Léger (INRIA, Paris)abstract: I will present a class of minimizing movement schemes which iteratively minimize a given energy in infinite dimensions using a general cost function c(x,y). This class includes implicit, proximal-like methods, as well as explicit schemes when the energy is c-concave. These minimizing movement schemes are all formulated as alternating minimizations. I will introduce an EVI for general alternating minimizations schemes, from which may be deduced rates of convergence.
The EVI will be shown to hold when the energy is convex along variational c-segments, an extension of generalized geodesics developed within the framework of spaces with nonnegative cross-curvature (NNCC spaces). NNCC spaces are based on a synthetic formulation of nonnegative cross-curvature applicable to infinite dimensions. They generalize most previously known properties of nonnegative cross-curvature, and interesting examples include Wasserstein and Gromov-Wasserstein spaces, Hellinger–Kantorovich costs, the Fisher-Rao squared distance and the relative entropy.
timetable:
Tue 3 Dec, 17:00 - 17:30, Aula Dini
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