Identifying Topological Differences in Two Populations of Random Geometric Objects
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arXiv
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| Auteurs principaux: | , |
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| Format: | Preprint |
| Publié: |
2026
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| _version_ | 1866908889084067840 |
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| author | Kumar, Satish Dhar, Subhra Sankar |
| author_facet | Kumar, Satish Dhar, Subhra Sankar |
| contents | We propose a statistical framework to identify topological differences in two populations of random geometric objects. The proposed framework involves first associating a topological signature with random geometric objects and then performing a two-sample test using the observed topological signatures. We associate persistence barcodes, a topological signature from topological data analysis, with each observed random geometric object. This, in turn, yields a two-sample problem on the space of persistence barcodes. As the space of persistence barcodes is not suitable for standard statistical analysis, we translate the two-sample problem on a suitable subset of a Euclidean space. In the course of this study, we embed the topological signatures in an ordered convex cone in a Euclidean space using functions from tropical geometry. We show that the embedding is a sufficient statistic for the persistence barcodes. This fact leads to the proposal of a two-sample test based on this sufficient statistic, and its equivalence to the two-sample problem on the barcode space is established. Finally, the consistency of the proposed test is studied. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_15082 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Identifying Topological Differences in Two Populations of Random Geometric Objects Kumar, Satish Dhar, Subhra Sankar Methodology Statistics Theory We propose a statistical framework to identify topological differences in two populations of random geometric objects. The proposed framework involves first associating a topological signature with random geometric objects and then performing a two-sample test using the observed topological signatures. We associate persistence barcodes, a topological signature from topological data analysis, with each observed random geometric object. This, in turn, yields a two-sample problem on the space of persistence barcodes. As the space of persistence barcodes is not suitable for standard statistical analysis, we translate the two-sample problem on a suitable subset of a Euclidean space. In the course of this study, we embed the topological signatures in an ordered convex cone in a Euclidean space using functions from tropical geometry. We show that the embedding is a sufficient statistic for the persistence barcodes. This fact leads to the proposal of a two-sample test based on this sufficient statistic, and its equivalence to the two-sample problem on the barcode space is established. Finally, the consistency of the proposed test is studied. |
| title | Identifying Topological Differences in Two Populations of Random Geometric Objects |
| topic | Methodology Statistics Theory |
| url | https://arxiv.org/abs/2603.15082 |