Testing the variety hypothesis
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arXiv
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| Main Authors: | , , , |
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| Format: | Preprint |
| Published: |
2025
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| Subjects: | |
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| _version_ | 1866912497302241280 |
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| author | Lerario, A. Hoefgeest, P. Roos Scolamiero, M. Tamai, A. |
| author_facet | Lerario, A. Hoefgeest, P. Roos Scolamiero, M. Tamai, A. |
| contents | Given a probability measure on the unit disk, we study the problem of deciding whether, for some threshold probability, this measure is supported near a real algebraic variety of given dimension and bounded degree. We call this "testing the variety hypothesis". We prove an upper bound on the so-called "sample complexity" of this problem and show how it can be reduced to a semialgebraic decision problem. This is done by studying in a quantitative way the Hausdorff geometry of the space of real algebraic varieties of a given dimension and degree. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2507_16705 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Testing the variety hypothesis Lerario, A. Hoefgeest, P. Roos Scolamiero, M. Tamai, A. Algebraic Geometry Metric Geometry Statistics Theory Machine Learning Given a probability measure on the unit disk, we study the problem of deciding whether, for some threshold probability, this measure is supported near a real algebraic variety of given dimension and bounded degree. We call this "testing the variety hypothesis". We prove an upper bound on the so-called "sample complexity" of this problem and show how it can be reduced to a semialgebraic decision problem. This is done by studying in a quantitative way the Hausdorff geometry of the space of real algebraic varieties of a given dimension and degree. |
| title | Testing the variety hypothesis |
| topic | Algebraic Geometry Metric Geometry Statistics Theory Machine Learning |
| url | https://arxiv.org/abs/2507.16705 |