{"paper":{"title":"The Stability of Persistence Diagrams Under Non-Uniform Scaling","license":"http://creativecommons.org/licenses/by-nc-sa/4.0/","headline":"","cross_cats":["math.GT","math.MG"],"primary_cat":"math.AT","authors_text":"Mehmet Dik, Vu-Anh Le","submitted_at":"2024-11-25T06:25:36Z","abstract_excerpt":"We investigate the stability of persistence diagrams \\( D \\) under non-uniform scaling transformations \\( S \\) in \\( \\mathbb{R}^n \\). Given a finite metric space \\( X \\subset \\mathbb{R}^n \\) with Euclidean distance \\( d_X \\), and scaling factors \\( s_1, s_2, \\ldots, s_n > 0 \\) applied to each coordinate, we derive explicit bounds on the bottleneck distance \\( d_B(D, D_S) \\) between the persistence diagrams of \\( X \\) and its scaled version \\( S(X) \\). Specifically, we show that \\[ d_B(D, D_S) \\leq \\frac{1}{2} (s_{\\max} - s_{\\min}) \\cdot \\operatorname{diam}(X), \\] where \\( s_{\\min} \\) and \\( s_"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2411.16126","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2411.16126/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}