{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2026:MROX4JOZ3VZN64NDX2BM35HGIR","short_pith_number":"pith:MROX4JOZ","schema_version":"1.0","canonical_sha256":"645d7e25d9dd72df71a3be82cdf4e644595de28abd223f7b4c6a04ab826bc17a","source":{"kind":"arxiv","id":"2607.06696","version":1},"attestation_state":"computed","paper":{"title":"Heat-Kernel Entropy Profiles and Geometric Effective Sample Size for Weighted Measures on Manifolds","license":"http://creativecommons.org/licenses/by-nc-nd/4.0/","headline":"","cross_cats":["cs.LG","stat.ME"],"primary_cat":"stat.ML","authors_text":"Kisung You","submitted_at":"2026-07-07T18:15:09Z","abstract_excerpt":"Weighted empirical measures on compact manifolds arise in importance sampling, particle approximations, posterior summaries, quadrature, and representation learning. Standard weight-only summaries, such as ordinary effective sample size, ignore the geometry of the support. We introduce heat-kernel entropy profiles, a multiscale summary that diffuses weighted atoms by intrinsic heat flow and tracks nonuniformity across scales. For order-two R\\'enyi entropy, the profile is computable from pairwise heat-kernel overlaps and yields a geometric effective sample size that discounts nearby or duplicat"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"2607.06696","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by-nc-nd/4.0/","primary_cat":"stat.ML","submitted_at":"2026-07-07T18:15:09Z","cross_cats_sorted":["cs.LG","stat.ME"],"title_canon_sha256":"7f769c586dc2f774cbb049465f2bf98d6f452eb00b4a1260565664dc1b177ad9","abstract_canon_sha256":"cee2f304bd5587a076cb63df0fde6b2e1f7116b2393b8099e6af8eec4468f4a9"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-09T00:19:23.737137Z","signature_b64":"UCpIxziFokIMCJqDlyF28E/bAq588V57evgyaGJx8JUL9TfQrqMvQj4cYbPE6Gx7DWXzNevilNUAEn7czsh0CQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"645d7e25d9dd72df71a3be82cdf4e644595de28abd223f7b4c6a04ab826bc17a","last_reissued_at":"2026-07-09T00:19:23.736683Z","signature_status":"signed_v1","first_computed_at":"2026-07-09T00:19:23.736683Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Heat-Kernel Entropy Profiles and Geometric Effective Sample Size for Weighted Measures on Manifolds","license":"http://creativecommons.org/licenses/by-nc-nd/4.0/","headline":"","cross_cats":["cs.LG","stat.ME"],"primary_cat":"stat.ML","authors_text":"Kisung You","submitted_at":"2026-07-07T18:15:09Z","abstract_excerpt":"Weighted empirical measures on compact manifolds arise in importance sampling, particle approximations, posterior summaries, quadrature, and representation learning. Standard weight-only summaries, such as ordinary effective sample size, ignore the geometry of the support. We introduce heat-kernel entropy profiles, a multiscale summary that diffuses weighted atoms by intrinsic heat flow and tracks nonuniformity across scales. For order-two R\\'enyi entropy, the profile is computable from pairwise heat-kernel overlaps and yields a geometric effective sample size that discounts nearby or duplicat"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2607.06696","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/2607.06696/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"},"aliases":[{"alias_kind":"arxiv","alias_value":"2607.06696","created_at":"2026-07-09T00:19:23.736747+00:00"},{"alias_kind":"arxiv_version","alias_value":"2607.06696v1","created_at":"2026-07-09T00:19:23.736747+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2607.06696","created_at":"2026-07-09T00:19:23.736747+00:00"},{"alias_kind":"pith_short_12","alias_value":"MROX4JOZ3VZN","created_at":"2026-07-09T00:19:23.736747+00:00"},{"alias_kind":"pith_short_16","alias_value":"MROX4JOZ3VZN64ND","created_at":"2026-07-09T00:19:23.736747+00:00"},{"alias_kind":"pith_short_8","alias_value":"MROX4JOZ","created_at":"2026-07-09T00:19:23.736747+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/MROX4JOZ3VZN64NDX2BM35HGIR","json":"https://pith.science/pith/MROX4JOZ3VZN64NDX2BM35HGIR.json","graph_json":"https://pith.science/api/pith-number/MROX4JOZ3VZN64NDX2BM35HGIR/graph.json","events_json":"https://pith.science/api/pith-number/MROX4JOZ3VZN64NDX2BM35HGIR/events.json","paper":"https://pith.science/paper/MROX4JOZ"},"agent_actions":{"view_html":"https://pith.science/pith/MROX4JOZ3VZN64NDX2BM35HGIR","download_json":"https://pith.science/pith/MROX4JOZ3VZN64NDX2BM35HGIR.json","view_paper":"https://pith.science/paper/MROX4JOZ","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2607.06696&json=true","fetch_graph":"https://pith.science/api/pith-number/MROX4JOZ3VZN64NDX2BM35HGIR/graph.json","fetch_events":"https://pith.science/api/pith-number/MROX4JOZ3VZN64NDX2BM35HGIR/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/MROX4JOZ3VZN64NDX2BM35HGIR/action/timestamp_anchor","attest_storage":"https://pith.science/pith/MROX4JOZ3VZN64NDX2BM35HGIR/action/storage_attestation","attest_author":"https://pith.science/pith/MROX4JOZ3VZN64NDX2BM35HGIR/action/author_attestation","sign_citation":"https://pith.science/pith/MROX4JOZ3VZN64NDX2BM35HGIR/action/citation_signature","submit_replication":"https://pith.science/pith/MROX4JOZ3VZN64NDX2BM35HGIR/action/replication_record"}},"created_at":"2026-07-09T00:19:23.736747+00:00","updated_at":"2026-07-09T00:19:23.736747+00:00"}