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| Format: | Preprint |
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2026
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| Online Access: | https://arxiv.org/abs/2604.19450 |
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| _version_ | 1866915947449679872 |
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| author | Zhang, Yifan |
| author_facet | Zhang, Yifan |
| contents | We study a family of local depth-based corrections to maxmin landmark selection for lazy witness persistence. Starting from maxmin seeds, we partition the cloud into nearest-seed cells and replace or move each seed toward a deep representative of its cell. The principal implemented variant, \emph{support-weighted partial recentering}, scales the amount of movement by cell support.
The contributions are both mathematical and algorithmic. On the mathematical side, we prove local geometric guarantees for these corrections: a convex-core robustness lemma derived from halfspace depth, a $2r$ cover bound for subset recentering, and projected cover bounds for the implemented partial-recentering rules. On the algorithmic side, we identify a practically effective variant through a layered empirical study consisting of planar synthetic benchmarks, a parameter-sensitivity study, and an MPEG-7 silhouette benchmark, together with a modest three-dimensional torus extension. The main planar experiments show that support-weighted partial recentering gives a consistent geometric improvement over maxmin while preserving the thresholded $H_1$ summary used in the study. The three-dimensional experiment shows the same geometric tendency but only mixed topological behavior. The paper should therefore be read as a controlled study of a local depth-based alternative to maxmin, rather than as a global witness-approximation theorem or a claim of uniform empirical superiority. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_19450 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Local Depth-Based Corrections to Maxmin Landmark Selection for Lazy Witness Persistence Zhang, Yifan Computational Geometry Combinatorics Primary 55N31, Secondary 52A35, 68U05 We study a family of local depth-based corrections to maxmin landmark selection for lazy witness persistence. Starting from maxmin seeds, we partition the cloud into nearest-seed cells and replace or move each seed toward a deep representative of its cell. The principal implemented variant, \emph{support-weighted partial recentering}, scales the amount of movement by cell support. The contributions are both mathematical and algorithmic. On the mathematical side, we prove local geometric guarantees for these corrections: a convex-core robustness lemma derived from halfspace depth, a $2r$ cover bound for subset recentering, and projected cover bounds for the implemented partial-recentering rules. On the algorithmic side, we identify a practically effective variant through a layered empirical study consisting of planar synthetic benchmarks, a parameter-sensitivity study, and an MPEG-7 silhouette benchmark, together with a modest three-dimensional torus extension. The main planar experiments show that support-weighted partial recentering gives a consistent geometric improvement over maxmin while preserving the thresholded $H_1$ summary used in the study. The three-dimensional experiment shows the same geometric tendency but only mixed topological behavior. The paper should therefore be read as a controlled study of a local depth-based alternative to maxmin, rather than as a global witness-approximation theorem or a claim of uniform empirical superiority. |
| title | Local Depth-Based Corrections to Maxmin Landmark Selection for Lazy Witness Persistence |
| topic | Computational Geometry Combinatorics Primary 55N31, Secondary 52A35, 68U05 |
| url | https://arxiv.org/abs/2604.19450 |