{"paper":{"title":"Kan Approximations of the Persistent Homology Transform","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":["cs.CG","math.CT"],"primary_cat":"math.AT","authors_text":"Justin Curry, Shreya Arya","submitted_at":"2025-07-30T16:32:33Z","abstract_excerpt":"The persistent homology transform (PHT) of a subset $M \\subset \\mathbb{R}^d$ is a map $\\text{PHT}(M):\\mathbb{S}^{d-1} \\to \\mathbf{Dgm}$ from the unit sphere to the space of persistence diagrams. This map assigns to each direction $v\\in \\mathbb{S}^{d-1}$ the persistent homology of the filtration of $M$ in direction $v$. In practice, one can only sample the map $\\text{PHT}(M)$ at a finite set of directions $A \\subset \\mathbb{S}^{d-1}$. This suggests two natural questions: (1) Can we interpolate the PHT from this finite sample of directions to the entire sphere? If so, (2) can we prove that the r"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2507.22816","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/2507.22816/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"}