Sufficient Conditions for the Shrinking Wellness Lemma
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arXiv
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| Format: | Preprint |
| Publié: |
2025
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| _version_ | 1866915712452263936 |
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| author | Bannwart, Clemens |
| author_facet | Bannwart, Clemens |
| contents | The well groups were introduced by Edelsbrunner, Morozov, and Patel to measure the robustness of geometric features of a function with respect to perturbations. Roughly speaking, the $r$-th well group measures the number of features that cannot be removed by perturbing the function by at most $r$. The Shrinking Wellness Lemma states that the rank of these groups decreases as $r$ increases. In the generality originally stated, it is wrong. We present a counterexample and give conditions under which the result holds. These conditions are general enough to cover most cases in which the well groups have been applied. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2601_03275 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Sufficient Conditions for the Shrinking Wellness Lemma Bannwart, Clemens General Topology Computational Geometry Group Theory 55N31 The well groups were introduced by Edelsbrunner, Morozov, and Patel to measure the robustness of geometric features of a function with respect to perturbations. Roughly speaking, the $r$-th well group measures the number of features that cannot be removed by perturbing the function by at most $r$. The Shrinking Wellness Lemma states that the rank of these groups decreases as $r$ increases. In the generality originally stated, it is wrong. We present a counterexample and give conditions under which the result holds. These conditions are general enough to cover most cases in which the well groups have been applied. |
| title | Sufficient Conditions for the Shrinking Wellness Lemma |
| topic | General Topology Computational Geometry Group Theory 55N31 |
| url | https://arxiv.org/abs/2601.03275 |