Finding stationary points on bounded-rank matrices: A geometric hurdle and a smooth remedy

TL;DR

This paper investigates the challenges of finding stationary points on bounded-rank matrix varieties, identifies a geometric obstacle called apocalypses, and proposes a trust-region method to reliably find stationary points.

Contribution

It introduces the concept of apocalypses as a geometric obstacle and develops a trust-region algorithm that guarantees convergence to stationary points on bounded-rank matrices.

Findings

Identifies apocalypses as a geometric obstacle in stationarity.

Proposes a trust-region method that overcomes this obstacle.

Generalizes the approach to other Clarke-regular sets.

Abstract

We consider the problem of provably finding a stationary point of a smooth function to be minimized on the variety of bounded-rank matrices. This turns out to be unexpectedly delicate. We trace the difficulty back to a geometric obstacle: On a nonsmooth set, there may be sequences of points along which standard measures of stationarity tend to zero, but whose limit points are not stationary. We name such events apocalypses, as they can cause optimization algorithms to converge to non-stationary points. We illustrate this explicitly for an existing optimization algorithm on bounded-rank matrices. To provably find stationary points, we modify a trust-region method on a standard smooth parameterization of the variety. The method relies on the known fact that second-order stationary points on the parameter space map to stationary points on the variety. Our geometric observations and proposed algorithm generalize beyond bounded-rank matrices. We give a geometric characterization of apocalypses on general constraint sets, which implies that Clarke-regular sets do not admit apocalypses. Such sets include smooth manifolds, manifolds with boundaries, and convex sets. Our trust-region method supports parameterization by any complete Riemannian manifold.