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

arXiv: https://arxiv.org/abs/1908.02022 · HTML: https://mlsanity.com/paper/1908.02022/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 6 Aug 2019.

## Built on by (most cited first, 2 of 2)

- [Pulling back symmetric Riemannian geometry for data analysis](https://mlsanity.com/paper/2403.06612/) (2024)
- [Iso-Riemannian Optimization on Learned Data Manifolds](https://mlsanity.com/paper/2510.21033/) (2025)

## Abstract (arXiv)

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.