{"paper":{"title":"Empirical Hodge Laplacians, Cohomology Ring, and Manifold Learning","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.AT","math.PR","math.ST","stat.TH"],"primary_cat":"math.DG","authors_text":"H\\^ong V\\^an L\\^e","submitted_at":"2026-05-21T10:09:14Z","abstract_excerpt":"Let $M^n$ be a compact orientable Riemannian smooth submanifold of dimension $n \\ge 2$ in $\\mathbf R^d$. We construct a family of deformed Hodge Laplacians $\\Delta ^*_t, t \\in \\mathbf R_{+},$ acting on differential forms using the extrinsic geometry of $M^n$ and prove their uniform convergence to the Hodge Laplacian $\\Delta^*$ as $t \\to 0^+$. Given a point cloud $S_m \\subset M^n$, we define symmetrized empirical operators $\\Delta^*_{sym, t, S_m}$ and establish their spectral convergence in probability to $\\Delta^*$, as $t \\to 0^+$, under suitable scaling regimes. This extends the scalar framew"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2605.22265","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/2605.22265/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"}