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The $\\Phi$-maximum range is $A^* = \\arg \\max_{A \\in (X,\\c{A})} \\Phi(A)$. Our goal is to find some $\\hat{A}$ such that $|\\Phi(\\hat{A}) - \\Phi(A^*)| \\leq \\varepsilon.$ We develop algorithms for this problem for range spaces with bounded VC-dimension, as well as significant improvements for those def"},"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":"1804.11287","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.CG","submitted_at":"2018-04-30T16:02:12Z","cross_cats_sorted":[],"title_canon_sha256":"7bebb118afb9a86d4c0786b72261b2ed23a0ba69974aed78247813e4df7f2bd8","abstract_canon_sha256":"6b1ee6e388a8b272d096993c7a5fb4245f97560ebac9deef69242d0e8d38dbf2"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:04:37.434626Z","signature_b64":"JC9EyqUvcxeM8PSGJBd29uKqIHsUaudVayBUyMPDbFCo80jmAiu+ql3xmVcb3YR0JDaJdWtki2sEgntUHexXCA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"11d9d6f6cf69a4ae9354c03fc1e3505c7594c3df1460c168319682b347435cd8","last_reissued_at":"2026-05-18T00:04:37.434144Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:04:37.434144Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Computing Approximate Statistical Discrepancy","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"cs.CG","authors_text":"Jeff M. Phillips, Michael Matheny","submitted_at":"2018-04-30T16:02:12Z","abstract_excerpt":"Consider a geometric range space $(X,\\c{A})$ where each data point $x \\in X$ has two or more values (say $r(x)$ and $b(x)$). Also consider a function $\\Phi(A)$ defined on any subset $A \\in (X,\\c{A})$ on the sum of values in that range e.g., $r_A = \\sum_{x \\in A} r(x)$ and $b_A = \\sum_{x \\in A} b(x)$. The $\\Phi$-maximum range is $A^* = \\arg \\max_{A \\in (X,\\c{A})} \\Phi(A)$. Our goal is to find some $\\hat{A}$ such that $|\\Phi(\\hat{A}) - \\Phi(A^*)| \\leq \\varepsilon.$ We develop algorithms for this problem for range spaces with bounded VC-dimension, as well as significant improvements for those def"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1804.11287","kind":"arxiv","version":3},"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":"1804.11287","created_at":"2026-05-18T00:04:37.434221+00:00"},{"alias_kind":"arxiv_version","alias_value":"1804.11287v3","created_at":"2026-05-18T00:04:37.434221+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1804.11287","created_at":"2026-05-18T00:04:37.434221+00:00"},{"alias_kind":"pith_short_12","alias_value":"CHM5N5WPNGSK","created_at":"2026-05-18T12:32:16.446611+00:00"},{"alias_kind":"pith_short_16","alias_value":"CHM5N5WPNGSK5E2U","created_at":"2026-05-18T12:32:16.446611+00:00"},{"alias_kind":"pith_short_8","alias_value":"CHM5N5WP","created_at":"2026-05-18T12:32:16.446611+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/CHM5N5WPNGSK5E2UYA74DY2QLR","json":"https://pith.science/pith/CHM5N5WPNGSK5E2UYA74DY2QLR.json","graph_json":"https://pith.science/api/pith-number/CHM5N5WPNGSK5E2UYA74DY2QLR/graph.json","events_json":"https://pith.science/api/pith-number/CHM5N5WPNGSK5E2UYA74DY2QLR/events.json","paper":"https://pith.science/paper/CHM5N5WP"},"agent_actions":{"view_html":"https://pith.science/pith/CHM5N5WPNGSK5E2UYA74DY2QLR","download_json":"https://pith.science/pith/CHM5N5WPNGSK5E2UYA74DY2QLR.json","view_paper":"https://pith.science/paper/CHM5N5WP","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1804.11287&json=true","fetch_graph":"https://pith.science/api/pith-number/CHM5N5WPNGSK5E2UYA74DY2QLR/graph.json","fetch_events":"https://pith.science/api/pith-number/CHM5N5WPNGSK5E2UYA74DY2QLR/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/CHM5N5WPNGSK5E2UYA74DY2QLR/action/timestamp_anchor","attest_storage":"https://pith.science/pith/CHM5N5WPNGSK5E2UYA74DY2QLR/action/storage_attestation","attest_author":"https://pith.science/pith/CHM5N5WPNGSK5E2UYA74DY2QLR/action/author_attestation","sign_citation":"https://pith.science/pith/CHM5N5WPNGSK5E2UYA74DY2QLR/action/citation_signature","submit_replication":"https://pith.science/pith/CHM5N5WPNGSK5E2UYA74DY2QLR/action/replication_record"}},"created_at":"2026-05-18T00:04:37.434221+00:00","updated_at":"2026-05-18T00:04:37.434221+00:00"}