High-order Accurate Inference on Manifolds
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arXiv
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| Format: | Preprint |
| Published: |
2025
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| _version_ | 1866918316147212288 |
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| author | Huang, Chengzhu Zhang, Anru R. |
| author_facet | Huang, Chengzhu Zhang, Anru R. |
| contents | We present a new framework for statistical inference on Riemannian manifolds that achieves high-order accuracy, addressing the challenges posed by non-Euclidean parameter spaces frequently encountered in modern data science. Our approach leverages a novel and computationally efficient procedure to reach higher-order asymptotic precision. In particular, we develop a bootstrap algorithm on Riemannian manifolds that is both computationally efficient and accurate for hypothesis testing and confidence region construction. Although locational hypothesis testing can be reformulated as a standard Euclidean problem, constructing high-order accurate confidence regions necessitates careful treatment of manifold geometry. To this end, we establish high-order asymptotics under an appropriate coordinate representation induced by a second-order retraction, thereby enabling precise expansions that incorporate curvature effects. We demonstrate the versatility of this framework across various manifold settings, including spheres, the Stiefel manifold, fixed-rank matrix manifolds, and rank-one tensor manifolds; for Euclidean submanifolds, we also introduce a class of projection-like coordinate charts with strong consistency properties. Finally, numerical studies confirm the practical merits of the proposed procedure. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2501_06652 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | High-order Accurate Inference on Manifolds Huang, Chengzhu Zhang, Anru R. Statistics Theory Numerical Analysis Optimization and Control Methodology We present a new framework for statistical inference on Riemannian manifolds that achieves high-order accuracy, addressing the challenges posed by non-Euclidean parameter spaces frequently encountered in modern data science. Our approach leverages a novel and computationally efficient procedure to reach higher-order asymptotic precision. In particular, we develop a bootstrap algorithm on Riemannian manifolds that is both computationally efficient and accurate for hypothesis testing and confidence region construction. Although locational hypothesis testing can be reformulated as a standard Euclidean problem, constructing high-order accurate confidence regions necessitates careful treatment of manifold geometry. To this end, we establish high-order asymptotics under an appropriate coordinate representation induced by a second-order retraction, thereby enabling precise expansions that incorporate curvature effects. We demonstrate the versatility of this framework across various manifold settings, including spheres, the Stiefel manifold, fixed-rank matrix manifolds, and rank-one tensor manifolds; for Euclidean submanifolds, we also introduce a class of projection-like coordinate charts with strong consistency properties. Finally, numerical studies confirm the practical merits of the proposed procedure. |
| title | High-order Accurate Inference on Manifolds |
| topic | Statistics Theory Numerical Analysis Optimization and Control Methodology |
| url | https://arxiv.org/abs/2501.06652 |