A theory of stratification learning
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arXiv
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| Format: | Preprint |
| Published: |
2024
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| _version_ | 1866929366304292864 |
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| author | Aamari, Eddie Berenfeld, Clément |
| author_facet | Aamari, Eddie Berenfeld, Clément |
| contents | Given i.i.d. sample from a stratified mixture of immersed manifolds of different dimensions, we study the minimax estimation of the underlying stratified structure. We provide a constructive algorithm allowing to estimate each mixture component at its optimal dimension-specific rate adaptively. The method is based on an ascending hierarchical co-detection of points belonging to different layers, which also identifies the number of layers and their dimensions, assigns each data point to a layer accurately, and estimates tangent spaces optimally. These results hold regardless of any ambient assumption on the manifolds or on their intersection configurations. They open the way to a broad clustering framework, where each mixture component models a cluster emanating from a specific nonlinear correlation phenomenon. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2405_20066 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | A theory of stratification learning Aamari, Eddie Berenfeld, Clément Statistics Theory Differential Geometry 62G05, 57R42, 54E20 Given i.i.d. sample from a stratified mixture of immersed manifolds of different dimensions, we study the minimax estimation of the underlying stratified structure. We provide a constructive algorithm allowing to estimate each mixture component at its optimal dimension-specific rate adaptively. The method is based on an ascending hierarchical co-detection of points belonging to different layers, which also identifies the number of layers and their dimensions, assigns each data point to a layer accurately, and estimates tangent spaces optimally. These results hold regardless of any ambient assumption on the manifolds or on their intersection configurations. They open the way to a broad clustering framework, where each mixture component models a cluster emanating from a specific nonlinear correlation phenomenon. |
| title | A theory of stratification learning |
| topic | Statistics Theory Differential Geometry 62G05, 57R42, 54E20 |
| url | https://arxiv.org/abs/2405.20066 |