Fenchel Duality Theory and A Primal-Dual Algorithm on Riemannian Manifolds

TL;DR

This paper extends Fenchel duality to Riemannian manifolds, develops a primal-dual algorithm with convergence guarantees, and demonstrates its effectiveness through numerical experiments on various curved manifolds.

Contribution

It introduces a new Fenchel conjugate for functions on Riemannian manifolds and develops a primal-dual algorithm with proven convergence on Hadamard and positive curvature manifolds.

Findings

Algorithm converges on Hadamard manifolds.

Performance is comparable to Douglas--Rachford algorithm.

Converges even on manifolds of positive curvature.

Abstract

This paper introduces a new notion of a Fenchel conjugate, which generalizes the classical Fenchel conjugation to functions defined on Riemannian manifolds. We investigate its properties, e.g.,~the Fenchel--Young inequality and the characterization of the convex subdifferential using the analogue of the Fenchel--Moreau Theorem. These properties of the Fenchel conjugate are employed to derive a Riemannian primal-dual optimization algorithm, and to prove its convergence for the case of Hadamard manifolds under appropriate assumptions. Numerical results illustrate the performance of the algorithm, which competes with the recently derived Douglas--Rachford algorithm on manifolds of nonpositive curvature. Furthermore, we show numerically that our novel algorithm even converges on manifolds of positive curvature.