Optimization on Weak Riemannian Manifolds

TL;DR

This paper develops a framework for gradient descent optimization on weak Riemannian manifolds, especially in infinite-dimensional shape analysis contexts, introducing the concept of a Hesse manifold.

Contribution

It introduces the notion of a Hesse manifold and establishes foundational properties for optimization on weak Riemannian manifolds in shape analysis.

Findings

Framework for gradient descent on weak Riemannian manifolds

Introduction of Hesse manifold concept

Foundational properties for optimization in shape analysis

Abstract

Riemannian structures on infinite-dimensional manifolds arise naturally in shape analysis and shape optimization. These applications lead to optimization problems on manifolds which are not modeled on Banach spaces. The present article develops the basic framework for optimization via gradient descent on weak Riemannian manifolds leading to the notion of a Hesse manifold. Further, foundational properties for optimization are established for several classes of weak Riemannian manifolds connected to shape analysis and shape optimization.