{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2015:W53KHS433DVD2ON7KTOW6LBSA5","short_pith_number":"pith:W53KHS43","schema_version":"1.0","canonical_sha256":"b776a3cb9bd8ea3d39bf54dd6f2c320779f8976e1fa9c04a46992db799bf5ad6","source":{"kind":"arxiv","id":"1511.00871","version":1},"attestation_state":"computed","paper":{"title":"Properties of the Sample Mean in Graph Spaces and the Majorize-Minimize-Mean Algorithm","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.LG","stat.ML"],"primary_cat":"cs.CV","authors_text":"Brijnesh J. Jain","submitted_at":"2015-11-03T12:09:26Z","abstract_excerpt":"One of the most fundamental concepts in statistics is the concept of sample mean. Properties of the sample mean that are well-defined in Euclidean spaces become unwieldy or even unclear in graph spaces. Open problems related to the sample mean of graphs include: non-existence, non-uniqueness, statistical inconsistency, lack of convergence results of mean algorithms, non-existence of midpoints, and disparity to midpoints. We present conditions to resolve all six problems and propose a Majorize-Minimize-Mean (MMM) Algorithm. Experiments on graph datasets representing images and molecules show th"},"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":"1511.00871","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.CV","submitted_at":"2015-11-03T12:09:26Z","cross_cats_sorted":["cs.LG","stat.ML"],"title_canon_sha256":"34b101e88232a859047c60dd100b109169ad2463b9323686185269c681621b38","abstract_canon_sha256":"9600664637f6d4427dc5af1ecfcaf975bd18e20c6bf4ec4cb953b99a71ef6f6d"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:27:55.988347Z","signature_b64":"teQtwCqqTosUdtVcMuTf9bTII88DOARmXBNfKFE+WB/ACVjdqxhJJQGzYWrvEQ6Pc81ckzXc1O32VRaz3LXCBw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"b776a3cb9bd8ea3d39bf54dd6f2c320779f8976e1fa9c04a46992db799bf5ad6","last_reissued_at":"2026-05-18T01:27:55.987607Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:27:55.987607Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Properties of the Sample Mean in Graph Spaces and the Majorize-Minimize-Mean Algorithm","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.LG","stat.ML"],"primary_cat":"cs.CV","authors_text":"Brijnesh J. Jain","submitted_at":"2015-11-03T12:09:26Z","abstract_excerpt":"One of the most fundamental concepts in statistics is the concept of sample mean. Properties of the sample mean that are well-defined in Euclidean spaces become unwieldy or even unclear in graph spaces. Open problems related to the sample mean of graphs include: non-existence, non-uniqueness, statistical inconsistency, lack of convergence results of mean algorithms, non-existence of midpoints, and disparity to midpoints. We present conditions to resolve all six problems and propose a Majorize-Minimize-Mean (MMM) Algorithm. Experiments on graph datasets representing images and molecules show th"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1511.00871","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":""},"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":"1511.00871","created_at":"2026-05-18T01:27:55.987718+00:00"},{"alias_kind":"arxiv_version","alias_value":"1511.00871v1","created_at":"2026-05-18T01:27:55.987718+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1511.00871","created_at":"2026-05-18T01:27:55.987718+00:00"},{"alias_kind":"pith_short_12","alias_value":"W53KHS433DVD","created_at":"2026-05-18T12:29:47.479230+00:00"},{"alias_kind":"pith_short_16","alias_value":"W53KHS433DVD2ON7","created_at":"2026-05-18T12:29:47.479230+00:00"},{"alias_kind":"pith_short_8","alias_value":"W53KHS43","created_at":"2026-05-18T12:29:47.479230+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/W53KHS433DVD2ON7KTOW6LBSA5","json":"https://pith.science/pith/W53KHS433DVD2ON7KTOW6LBSA5.json","graph_json":"https://pith.science/api/pith-number/W53KHS433DVD2ON7KTOW6LBSA5/graph.json","events_json":"https://pith.science/api/pith-number/W53KHS433DVD2ON7KTOW6LBSA5/events.json","paper":"https://pith.science/paper/W53KHS43"},"agent_actions":{"view_html":"https://pith.science/pith/W53KHS433DVD2ON7KTOW6LBSA5","download_json":"https://pith.science/pith/W53KHS433DVD2ON7KTOW6LBSA5.json","view_paper":"https://pith.science/paper/W53KHS43","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1511.00871&json=true","fetch_graph":"https://pith.science/api/pith-number/W53KHS433DVD2ON7KTOW6LBSA5/graph.json","fetch_events":"https://pith.science/api/pith-number/W53KHS433DVD2ON7KTOW6LBSA5/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/W53KHS433DVD2ON7KTOW6LBSA5/action/timestamp_anchor","attest_storage":"https://pith.science/pith/W53KHS433DVD2ON7KTOW6LBSA5/action/storage_attestation","attest_author":"https://pith.science/pith/W53KHS433DVD2ON7KTOW6LBSA5/action/author_attestation","sign_citation":"https://pith.science/pith/W53KHS433DVD2ON7KTOW6LBSA5/action/citation_signature","submit_replication":"https://pith.science/pith/W53KHS433DVD2ON7KTOW6LBSA5/action/replication_record"}},"created_at":"2026-05-18T01:27:55.987718+00:00","updated_at":"2026-05-18T01:27:55.987718+00:00"}