Optimization over the intersection of manifolds

2026-05-21

AI Summary

This work addresses the challenging problem of optimization over the intersection of two manifolds by proposing a novel geometric optimization method. The approach requires retractions only on one of the manifolds and performs simultaneous updates along two orthogonal directions: one asymptotically approaching the other manifold and the other descending the objective function. A key theoretical contribution is the proof of equivalence between “clean intersection” and “intrinsic transversality,” which yields an explicitly computable projection onto the tangent space of the intersection, thereby establishing a rigorous foundation for algorithm design. The proposed algorithm demonstrates strong empirical performance on sparse and low-rank optimization tasks—including spherical fitting, hyperbolic embedding, and compressed mode computation—and is provably convergent with guaranteed first-order stationarity.

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.

Problem

• manifold intersection
• constrained optimization
• intrinsic transversality
• feasible region geometry

Innovation

• manifold intersection
• intrinsic transversality
• geometric optimization
• retraction
• orthogonal update directions