Alpha shapes and optimal transport on the sphere
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arXiv
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| Auteurs principaux: | , |
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866909417554837504 |
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| author | Carlsson, Erik DePaul, Greg |
| author_facet | Carlsson, Erik DePaul, Greg |
| contents | In [3], the authors used the Legendre transform to give a tractable method for studying Topological Data Analysis (TDA) in terms of sums of Gaussian kernels. In this paper, we prove a variant for sums of cosine similarity-based kernel functions, which requires considering the more general "$c$-transform" from optimal transport theory [16]. We then apply these methods to a point cloud arising from a recent breakthrough study, which exhibits a toroidal structure in the brain activity of rats [11]. A key part of this application is that the transport map and transformed density function arising from the theorem replace certain delicate preprocessing steps related to density-based denoising and subsampling. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2412_04286 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Alpha shapes and optimal transport on the sphere Carlsson, Erik DePaul, Greg Statistics Theory Algebraic Topology 55N31, 62R40, 62G07, 49Q22 In [3], the authors used the Legendre transform to give a tractable method for studying Topological Data Analysis (TDA) in terms of sums of Gaussian kernels. In this paper, we prove a variant for sums of cosine similarity-based kernel functions, which requires considering the more general "$c$-transform" from optimal transport theory [16]. We then apply these methods to a point cloud arising from a recent breakthrough study, which exhibits a toroidal structure in the brain activity of rats [11]. A key part of this application is that the transport map and transformed density function arising from the theorem replace certain delicate preprocessing steps related to density-based denoising and subsampling. |
| title | Alpha shapes and optimal transport on the sphere |
| topic | Statistics Theory Algebraic Topology 55N31, 62R40, 62G07, 49Q22 |
| url | https://arxiv.org/abs/2412.04286 |