{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2016:G6IULBQRYH6ODTTJRI3JNQYPFG","short_pith_number":"pith:G6IULBQR","schema_version":"1.0","canonical_sha256":"3791458611c1fce1ce698a3696c30f2980413ebb2878c892568f0be14a1a830c","source":{"kind":"arxiv","id":"1603.07021","version":2},"attestation_state":"computed","paper":{"title":"On the Separability of Stochastic Geometric Objects, with Applications","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"cs.CG","authors_text":"Jie Xue, Ravi Janardan, Yuan Li","submitted_at":"2016-03-22T22:51:55Z","abstract_excerpt":"In this paper, we study the linear separability problem for stochastic geometric objects under the well-known unipoint/multipoint uncertainty models. Let $S=S_R \\cup S_B$ be a given set of stochastic bichromatic points, and define $n = \\min\\{|S_R|, |S_B|\\}$ and $N = \\max\\{|S_R|, |S_B|\\}$. We show that the separable-probability (SP) of $S$ can be computed in $O(nN^{d-1})$ time for $d \\geq 3$ and $O(\\min\\{nN \\log N, N^2\\})$ time for $d=2$, while the expected separation-margin (ESM) of $S$ can be computed in $O(nN^{d})$ time for $d \\geq 2$. In addition, we give an $\\Omega(nN^{d-1})$ witness-based"},"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":"1603.07021","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.CG","submitted_at":"2016-03-22T22:51:55Z","cross_cats_sorted":[],"title_canon_sha256":"37446b49e3da7433eb21a4e1df4da95206baed1d0f060a1363a21133d4935831","abstract_canon_sha256":"0b33995ed040dc040599ee42d6eb2e80d2b99e21651185647c8693441125119c"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:17:46.743103Z","signature_b64":"AfJs2kZ5ye6MfR6q0hadzmRZPMcHNLRMvh4t735jztfU9TRaiPWMADf+UWGX9VtfXNPOrHsfX+57EHzv1D5ZCw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"3791458611c1fce1ce698a3696c30f2980413ebb2878c892568f0be14a1a830c","last_reissued_at":"2026-05-18T01:17:46.742456Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:17:46.742456Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"On the Separability of Stochastic Geometric Objects, with Applications","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"cs.CG","authors_text":"Jie Xue, Ravi Janardan, Yuan Li","submitted_at":"2016-03-22T22:51:55Z","abstract_excerpt":"In this paper, we study the linear separability problem for stochastic geometric objects under the well-known unipoint/multipoint uncertainty models. Let $S=S_R \\cup S_B$ be a given set of stochastic bichromatic points, and define $n = \\min\\{|S_R|, |S_B|\\}$ and $N = \\max\\{|S_R|, |S_B|\\}$. We show that the separable-probability (SP) of $S$ can be computed in $O(nN^{d-1})$ time for $d \\geq 3$ and $O(\\min\\{nN \\log N, N^2\\})$ time for $d=2$, while the expected separation-margin (ESM) of $S$ can be computed in $O(nN^{d})$ time for $d \\geq 2$. In addition, we give an $\\Omega(nN^{d-1})$ witness-based"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1603.07021","kind":"arxiv","version":2},"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":"1603.07021","created_at":"2026-05-18T01:17:46.742544+00:00"},{"alias_kind":"arxiv_version","alias_value":"1603.07021v2","created_at":"2026-05-18T01:17:46.742544+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1603.07021","created_at":"2026-05-18T01:17:46.742544+00:00"},{"alias_kind":"pith_short_12","alias_value":"G6IULBQRYH6O","created_at":"2026-05-18T12:30:15.759754+00:00"},{"alias_kind":"pith_short_16","alias_value":"G6IULBQRYH6ODTTJ","created_at":"2026-05-18T12:30:15.759754+00:00"},{"alias_kind":"pith_short_8","alias_value":"G6IULBQR","created_at":"2026-05-18T12:30:15.759754+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/G6IULBQRYH6ODTTJRI3JNQYPFG","json":"https://pith.science/pith/G6IULBQRYH6ODTTJRI3JNQYPFG.json","graph_json":"https://pith.science/api/pith-number/G6IULBQRYH6ODTTJRI3JNQYPFG/graph.json","events_json":"https://pith.science/api/pith-number/G6IULBQRYH6ODTTJRI3JNQYPFG/events.json","paper":"https://pith.science/paper/G6IULBQR"},"agent_actions":{"view_html":"https://pith.science/pith/G6IULBQRYH6ODTTJRI3JNQYPFG","download_json":"https://pith.science/pith/G6IULBQRYH6ODTTJRI3JNQYPFG.json","view_paper":"https://pith.science/paper/G6IULBQR","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1603.07021&json=true","fetch_graph":"https://pith.science/api/pith-number/G6IULBQRYH6ODTTJRI3JNQYPFG/graph.json","fetch_events":"https://pith.science/api/pith-number/G6IULBQRYH6ODTTJRI3JNQYPFG/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/G6IULBQRYH6ODTTJRI3JNQYPFG/action/timestamp_anchor","attest_storage":"https://pith.science/pith/G6IULBQRYH6ODTTJRI3JNQYPFG/action/storage_attestation","attest_author":"https://pith.science/pith/G6IULBQRYH6ODTTJRI3JNQYPFG/action/author_attestation","sign_citation":"https://pith.science/pith/G6IULBQRYH6ODTTJRI3JNQYPFG/action/citation_signature","submit_replication":"https://pith.science/pith/G6IULBQRYH6ODTTJRI3JNQYPFG/action/replication_record"}},"created_at":"2026-05-18T01:17:46.742544+00:00","updated_at":"2026-05-18T01:17:46.742544+00:00"}