Optimization over the intersection of manifolds

TL;DR

This paper introduces a geometric optimization method for intersecting manifolds that ensures convergence and efficiency by leveraging intrinsic transversality and tangent space projections.

Contribution

It establishes the equivalence of regularity conditions and proposes a tractable projection-based method for optimization over intersecting manifolds.

Findings

Proves equivalence of regularity conditions for manifold intersections

Derives convergence rates under intrinsic transversality

Demonstrates effectiveness on sparse and low-rank optimization problems

Abstract

Optimization over the intersection of two manifolds arises in a broad range of applications, but is hindered by the coupled geometry of the feasible region. In this paper, we prove that the regularities -- clean intersection and intrinsic transversality -- are equivalent, which yields a tractable projection onto the tangent space of the intersection. Therefore, we propose a geometric method that employs a retraction on only one manifold and updates the iterate along two orthogonal directions. Specifically, the iterates stay on one manifold, and the two directions are responsible for asymptotically approaching the other manifold and decreasing the objective function, respectively. Under intrinsic transversality, we derive the convergence rate for both the feasibility and optimality measures, and show that every accumulation point is first-order stationary. Numerical experiments on problems stemming from sparse and low-rank optimization, including fitting spherical data, approximating hyperbolic embeddings on real data, and computing compressed modes, demonstrate the effectiveness of the proposed method.