A Riemannian ADMM

TL;DR

This paper introduces a novel Riemannian ADMM algorithm for solving nonconvex optimization problems involving nonsmooth objectives on manifolds, with applications in machine learning and statistics.

Contribution

It presents the first ADMM algorithm capable of minimizing nonsmooth functions over nonconvex Riemannian manifolds, with proven iteration complexity.

Findings

Algorithm effectively handles nonsmooth objectives on manifolds.

Numerical experiments show advantages over existing methods.

Iteration complexity analyzed under mild assumptions.

Abstract

We consider a class of Riemannian optimization problems where the objective is the sum of a smooth function and a nonsmooth function, considered in the ambient space. This class of problems finds important applications in machine learning and statistics such as the sparse principal component analysis, sparse spectral clustering, and orthogonal dictionary learning. We propose a Riemannian alternating direction method of multipliers (ADMM) to solve this class of problems. Our algorithm adopts easily computable steps in each iteration. The iteration complexity of the proposed algorithm for obtaining an $\epsilon$-stationary point is analyzed under mild assumptions. Existing ADMM for solving nonconvex problems either does not allow nonconvex constraint set, or does not allow nonsmooth objective function. Our algorithm is the first ADMM type algorithm that minimizes a nonsmooth objective over manifold -- a particular nonconvex set. Numerical experiments are conducted to demonstrate the advantage of the proposed method.