Riemann-Oracle: A general-purpose Riemannian optimizer to solve nearness problems in matrix theory

TL;DR

This paper introduces Riemann-Oracle, a versatile Riemannian optimization framework that efficiently solves a wide range of matrix nearness problems, often outperforming existing specialized algorithms.

Contribution

The paper presents a general Riemannian optimization approach for matrix nearness problems, unifying many problems under a common framework and demonstrating superior practical performance.

Findings

Outperforms existing algorithms on various matrix nearness problems

Applicable to problems with additional linear constraints

Handles discontinuous objective functions with regularization

Abstract

We propose an extremely versatile approach to address a large family of matrix nearness problems, possibly with additional linear constraints. Our method is based on splitting a matrix nearness problem into two nested optimization problems, of which the inner one can be solved either exactly or cheaply, while the outer one can be recast as an unconstrained optimization task over a smooth real Riemannian manifold. We observe that this paradigm applies to many matrix nearness problems of practical interest appearing in the literature, thus revealing that they are equivalent in this sense to a Riemannian optimization problem. We also show that the objective function to be minimized on the Riemannian manifold can be discontinuous, thus requiring regularization techniques, and we give conditions for this to happen. Finally, we demonstrate the practical applicability of our method by implementing it for a number of matrix nearness problems that are relevant for applications and are currently considered very demanding in practice. Extensive numerical experiments demonstrate that our method often greatly outperforms its predecessors, including algorithms specifically designed for those particular problems.