A Riemannian Primal-dual Algorithm Based on Proximal Operator and its Application in Metric Learning
This work addresses optimization challenges in Riemannian spaces for applications like metric learning, but it appears incremental as it builds on existing primal-dual and proximal methods.
The paper tackles the problem of optimizing smooth convex functions with constraints in Riemannian space by proposing a primal-dual algorithm using proximal operators, proving its convergence and non-asymptotic rate, and applies it to metric learning for feature transformation in positive definite matrices, showing efficacy in a fund selection task.
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.