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| Main Authors: | , , |
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| Format: | Preprint |
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2026
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2605.17384 |
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| _version_ | 1866913147119468544 |
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| author | Chen, Shixiang He, Yixiao Huang, Wen |
| author_facet | Chen, Shixiang He, Yixiao Huang, Wen |
| contents | Alternating projections and their variants are classical tools for computing points in intersections of sets. Existing analyses for smooth manifolds mainly focus on local convergence rates under transversality or related regularity conditions. In this work, we develop a unified framework for a broad class of (possibly inexact) alternating-projection-type methods on intersections of smooth manifolds. Specifically, under the assumption that two $C^{2,1}$ embedded submanifolds $\mathcal{M}_1, \mathcal{M}_2 \subset \mathbb{R}^n$ intersect cleanly, we show that the associated alternating mapping admits a well-defined local limiting map $ψ$ on the intersection manifold $\mathcal{M}=\mathcal{M}_1\cap \mathcal{M}_2$, and that $ψ$ is a retraction on $\mathcal{M}$. If, in addition, $\mathcal{M}_1$ and $\mathcal{M}_2$ are $C^{3,1}$, then $ψ$ is a second-order retraction. Furthermore, the standard NewtonSLRA scheme, which exhibits quadratic local behavior under transversality, can be understood as inducing a second-order retraction on \(\M\). This framework thus provides new retraction-based optimization tools for problems constrained to the intersection manifold. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_17384 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Retractions by Alternating Projections Chen, Shixiang He, Yixiao Huang, Wen Optimization and Control 65K10, 58C05, 49M37, 90C26 Alternating projections and their variants are classical tools for computing points in intersections of sets. Existing analyses for smooth manifolds mainly focus on local convergence rates under transversality or related regularity conditions. In this work, we develop a unified framework for a broad class of (possibly inexact) alternating-projection-type methods on intersections of smooth manifolds. Specifically, under the assumption that two $C^{2,1}$ embedded submanifolds $\mathcal{M}_1, \mathcal{M}_2 \subset \mathbb{R}^n$ intersect cleanly, we show that the associated alternating mapping admits a well-defined local limiting map $ψ$ on the intersection manifold $\mathcal{M}=\mathcal{M}_1\cap \mathcal{M}_2$, and that $ψ$ is a retraction on $\mathcal{M}$. If, in addition, $\mathcal{M}_1$ and $\mathcal{M}_2$ are $C^{3,1}$, then $ψ$ is a second-order retraction. Furthermore, the standard NewtonSLRA scheme, which exhibits quadratic local behavior under transversality, can be understood as inducing a second-order retraction on \(\M\). This framework thus provides new retraction-based optimization tools for problems constrained to the intersection manifold. |
| title | Retractions by Alternating Projections |
| topic | Optimization and Control 65K10, 58C05, 49M37, 90C26 |
| url | https://arxiv.org/abs/2605.17384 |