# Optimization over the intersection of manifolds

arXiv: https://arxiv.org/abs/2605.22736 · HTML: https://mlsanity.com/paper/2605.22736/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 30 of its 60 references that are in the map.
- First posted on arXiv 21 May 2026.

## Builds on (most cited first, 30 of 30)

- [Poincar'e Embeddings for Learning Hierarchical Representations](https://mlsanity.com/paper/1705.08039/) (2017)
- [Manopt, a Matlab toolbox for optimization on manifolds](https://mlsanity.com/paper/1308.5200/) (2013)
- [Global rates of convergence for nonconvex optimization on manifolds](https://mlsanity.com/paper/1605.08101/) (2016)
- [Fast and accurate optimization on the orthogonal manifold without retraction](https://mlsanity.com/paper/2102.07432/) (2021)
- [The effect of smooth parametrizations on nonconvex optimization landscapes](https://mlsanity.com/paper/2207.03512/) (2022)
- [Infeasible Deterministic, Stochastic, and Variance-Reduction Algorithms for Optimization under Orthogonality Constraints](https://mlsanity.com/paper/2303.16510/) (2023)
- [Simple algorithms for optimization on Riemannian manifolds with constraints](https://mlsanity.com/paper/1901.10000/) (2019)
- [Orthogonal Directions Constrained Gradient Method: from non-linear equality constraints to Stiefel manifold](https://mlsanity.com/paper/2303.09261/) (2023)
- [Optimization flows landing on the Stiefel manifold](https://mlsanity.com/paper/2202.09058/) (2022)
- [A Penalty-free Infeasible Approach for a Class of Nonsmooth Optimization Problems over the Stiefel Manifold](https://mlsanity.com/paper/2103.03514/) (2021)
- [Optimization without Retraction on the Random Generalized Stiefel Manifold](https://mlsanity.com/paper/2405.01702/) (2024)
- [A Semismooth Newton based Augmented Lagrangian Method for Nonsmooth Optimization on Matrix Manifolds](https://mlsanity.com/paper/2103.02855/) (2021)
- [An Exact Penalty Approach for Equality Constrained Optimization over a Convex Set](https://mlsanity.com/paper/2505.02495/) (2025)
- [Transversality and alternating projections for nonconvex sets](https://mlsanity.com/paper/1401.7569/) (2014)
- [Low-rank approximations of hyperbolic embeddings](https://mlsanity.com/paper/1903.07307/) (2019)
- [Normal Cones Intersection Rule and Optimality Analysis for Low-Rank Matrix Optimization with Affine Manifolds](https://mlsanity.com/paper/2202.08464/) (2022)
- [Riemannian Interior Point Methods for Constrained Optimization on Manifolds](https://mlsanity.com/paper/2203.09762/) (2022)
- [Distributed Retraction-Free and Communication-Efficient Optimization on the Stiefel Manifold](https://mlsanity.com/paper/2506.02879/) (2025)
- [On local convergence of the method of alternating projections](https://mlsanity.com/paper/1312.5681/) (2013)
- [An Augmented Lagrangian Method for Optimization Problems with Structured Geometric Constraints](https://mlsanity.com/paper/2105.08317/) (2021)
- [Quasioptimal alternating projections and their use in low-rank approximation of matrices and tensors](https://mlsanity.com/paper/2308.16097/) (2023)
- [A space-decoupling framework for optimization on bounded-rank matrices with orthogonally invariant constraints](https://mlsanity.com/paper/2501.13830/) (2025)
- [The tangent cone to the real determinantal variety: various expressions and a proof](https://mlsanity.com/paper/2504.11382/) (2025)
- [Normalized tensor train decomposition](https://mlsanity.com/paper/2511.04369/) (2025)
- [The Riemannian Landing Method: From projected gradient flows to SQP](https://mlsanity.com/paper/2603.24309/) (2026)
- [Sequential Quadratic Optimization for Nonlinear Optimization Problems on Riemannian Manifolds](https://mlsanity.com/paper/2009.07153/) (2020)
- [Constraint qualifications and strong global convergence properties of an augmented Lagrangian method on Riemannian manifolds](https://mlsanity.com/paper/2306.14345/) (2023)
- [Variational analysis of determinantal varieties](https://mlsanity.com/paper/2511.22613/) (2025)
- [A Quadratically Convergent Alternating Projection Method for Nonconvex Sets](https://mlsanity.com/paper/2511.22916/) (2025)
- [Retractions by Alternating Projections](https://mlsanity.com/paper/2605.17384/) (2026)

## Abstract (arXiv)

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.