A Riemannian Primal-dual Algorithm Based on Proximal Operator and its Application in Metric Learning

TL;DR

This paper introduces a Riemannian primal-dual algorithm utilizing proximal operators for constrained convex optimization, with applications in metric learning and demonstrated effectiveness in fund selection tasks.

Contribution

It proposes a novel primal-dual optimization algorithm on Riemannian manifolds with convergence guarantees and applies it to metric learning for positive definite matrices.

Findings

Proven convergence and non-asymptotic rate of the algorithm.

Effective metric learning algorithm for positive definite matrices.

Successful application in fund of funds management.

Abstract

In this paper, we consider optimizing a smooth, convex, lower semicontinuous function in Riemannian space with constraints. To solve the problem, we first convert it to a dual problem and then propose a general primal-dual algorithm to optimize the primal and dual variables iteratively. In each optimization iteration, we employ a proximal operator to search optimal solution in the primal space. We prove convergence of the proposed algorithm and show its non-asymptotic convergence rate. By utilizing the proposed primal-dual optimization technique, we propose a novel metric learning algorithm which learns an optimal feature transformation matrix in the Riemannian space of positive definite matrices. Preliminary experimental results on an optimal fund selection problem in fund of funds (FOF) management for quantitative investment showed its efficacy.