{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2012:JUVVSPCB6BKAAUBHFV4ZE6EZWI","short_pith_number":"pith:JUVVSPCB","schema_version":"1.0","canonical_sha256":"4d2b593c41f0540050272d79927899b2288ae7f028d84271b28fb9ee3ac2da44","source":{"kind":"arxiv","id":"1205.4484","version":3},"attestation_state":"computed","paper":{"title":"Hypercontractivity, Sum-of-Squares Proofs, and their Applications","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.DS","quant-ph"],"primary_cat":"cs.CC","authors_text":"Aram W. Harrow, Boaz Barak, David Steurer, Fernando G.S.L. Brand\\~ao, Jonathan A. Kelner, Yuan Zhou","submitted_at":"2012-05-21T04:00:32Z","abstract_excerpt":"We study the computational complexity of approximating the 2->q norm of linear operators (defined as ||A||_{2->q} = sup_v ||Av||_q/||v||_2), as well as connections between this question and issues arising in quantum information theory and the study of Khot's Unique Games Conjecture (UGC). We show the following:\n  1. For any constant even integer q>=4, a graph $G$ is a \"small-set expander\" if and only if the projector into the span of the top eigenvectors of G's adjacency matrix has bounded 2->q norm. As a corollary, a good approximation to the 2->q norm will refute the Small-Set Expansion Conj"},"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":"1205.4484","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.CC","submitted_at":"2012-05-21T04:00:32Z","cross_cats_sorted":["cs.DS","quant-ph"],"title_canon_sha256":"e9760e90a9a4e4a0ebd416cf07d733aac2e24dc60ece04c6d7b3be774c4d0ece","abstract_canon_sha256":"ef17c210285b5c84ff963100b457b80c57b9b8e0e47be12852d30534c039a087"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:35:52.763852Z","signature_b64":"KnQhZMiae0yOARkerja6YXNpOWci0kCIZL4sD+ERheudrjSJHWEj2K8y6cBL/Ypr4P47Xc1XW5kECWagYsVKBA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"4d2b593c41f0540050272d79927899b2288ae7f028d84271b28fb9ee3ac2da44","last_reissued_at":"2026-05-18T02:35:52.763207Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:35:52.763207Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Hypercontractivity, Sum-of-Squares Proofs, and their Applications","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.DS","quant-ph"],"primary_cat":"cs.CC","authors_text":"Aram W. Harrow, Boaz Barak, David Steurer, Fernando G.S.L. Brand\\~ao, Jonathan A. Kelner, Yuan Zhou","submitted_at":"2012-05-21T04:00:32Z","abstract_excerpt":"We study the computational complexity of approximating the 2->q norm of linear operators (defined as ||A||_{2->q} = sup_v ||Av||_q/||v||_2), as well as connections between this question and issues arising in quantum information theory and the study of Khot's Unique Games Conjecture (UGC). We show the following:\n  1. For any constant even integer q>=4, a graph $G$ is a \"small-set expander\" if and only if the projector into the span of the top eigenvectors of G's adjacency matrix has bounded 2->q norm. As a corollary, a good approximation to the 2->q norm will refute the Small-Set Expansion Conj"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1205.4484","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":"1205.4484","created_at":"2026-05-18T02:35:52.763300+00:00"},{"alias_kind":"arxiv_version","alias_value":"1205.4484v3","created_at":"2026-05-18T02:35:52.763300+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1205.4484","created_at":"2026-05-18T02:35:52.763300+00:00"},{"alias_kind":"pith_short_12","alias_value":"JUVVSPCB6BKA","created_at":"2026-05-18T12:27:11.947152+00:00"},{"alias_kind":"pith_short_16","alias_value":"JUVVSPCB6BKAAUBH","created_at":"2026-05-18T12:27:11.947152+00:00"},{"alias_kind":"pith_short_8","alias_value":"JUVVSPCB","created_at":"2026-05-18T12:27:11.947152+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/JUVVSPCB6BKAAUBHFV4ZE6EZWI","json":"https://pith.science/pith/JUVVSPCB6BKAAUBHFV4ZE6EZWI.json","graph_json":"https://pith.science/api/pith-number/JUVVSPCB6BKAAUBHFV4ZE6EZWI/graph.json","events_json":"https://pith.science/api/pith-number/JUVVSPCB6BKAAUBHFV4ZE6EZWI/events.json","paper":"https://pith.science/paper/JUVVSPCB"},"agent_actions":{"view_html":"https://pith.science/pith/JUVVSPCB6BKAAUBHFV4ZE6EZWI","download_json":"https://pith.science/pith/JUVVSPCB6BKAAUBHFV4ZE6EZWI.json","view_paper":"https://pith.science/paper/JUVVSPCB","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1205.4484&json=true","fetch_graph":"https://pith.science/api/pith-number/JUVVSPCB6BKAAUBHFV4ZE6EZWI/graph.json","fetch_events":"https://pith.science/api/pith-number/JUVVSPCB6BKAAUBHFV4ZE6EZWI/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/JUVVSPCB6BKAAUBHFV4ZE6EZWI/action/timestamp_anchor","attest_storage":"https://pith.science/pith/JUVVSPCB6BKAAUBHFV4ZE6EZWI/action/storage_attestation","attest_author":"https://pith.science/pith/JUVVSPCB6BKAAUBHFV4ZE6EZWI/action/author_attestation","sign_citation":"https://pith.science/pith/JUVVSPCB6BKAAUBHFV4ZE6EZWI/action/citation_signature","submit_replication":"https://pith.science/pith/JUVVSPCB6BKAAUBHFV4ZE6EZWI/action/replication_record"}},"created_at":"2026-05-18T02:35:52.763300+00:00","updated_at":"2026-05-18T02:35:52.763300+00:00"}