{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2015:XW37IRCXEVMEVO7CJ64SLYPITA","short_pith_number":"pith:XW37IRCX","schema_version":"1.0","canonical_sha256":"bdb7f4445725584abbe24fb925e1e898286112f4f58c959757ad09d981c218d3","source":{"kind":"arxiv","id":"1504.03391","version":2},"attestation_state":"computed","paper":{"title":"Tight Bounds on Low-degree Spectral Concentration of Submodular and XOS functions","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.LG"],"primary_cat":"cs.DS","authors_text":"Jan Vondrak, Vitaly Feldman","submitted_at":"2015-04-13T23:51:45Z","abstract_excerpt":"Submodular and fractionally subadditive (or equivalently XOS) functions play a fundamental role in combinatorial optimization, algorithmic game theory and machine learning. Motivated by learnability of these classes of functions from random examples, we consider the question of how well such functions can be approximated by low-degree polynomials in $\\ell_2$ norm over the uniform distribution. This question is equivalent to understanding of the concentration of Fourier weight on low-degree coefficients, a central concept in Fourier analysis. We show that\n  1. For any submodular function $f:\\{0"},"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":"1504.03391","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.DS","submitted_at":"2015-04-13T23:51:45Z","cross_cats_sorted":["cs.LG"],"title_canon_sha256":"9cd5f6e552a59f732b59c18cf6dada891b4c62c9291bb3ba8104012ff3f020e7","abstract_canon_sha256":"b6fd6f7444e41240b20fa201a1c2282fbfe561c5e4e6d2aa110a67b30c2917ff"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:35:59.755261Z","signature_b64":"w997JfQN39lpXEtQ5ie7w0ZVLt+bQDTwGakp4OqD9TOo0BJWwujnkCu7C5da2DhvB0hjqNBvJsuMDbaAm1d3Ag==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"bdb7f4445725584abbe24fb925e1e898286112f4f58c959757ad09d981c218d3","last_reissued_at":"2026-05-18T01:35:59.754792Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:35:59.754792Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Tight Bounds on Low-degree Spectral Concentration of Submodular and XOS functions","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.LG"],"primary_cat":"cs.DS","authors_text":"Jan Vondrak, Vitaly Feldman","submitted_at":"2015-04-13T23:51:45Z","abstract_excerpt":"Submodular and fractionally subadditive (or equivalently XOS) functions play a fundamental role in combinatorial optimization, algorithmic game theory and machine learning. Motivated by learnability of these classes of functions from random examples, we consider the question of how well such functions can be approximated by low-degree polynomials in $\\ell_2$ norm over the uniform distribution. This question is equivalent to understanding of the concentration of Fourier weight on low-degree coefficients, a central concept in Fourier analysis. We show that\n  1. For any submodular function $f:\\{0"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1504.03391","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":"1504.03391","created_at":"2026-05-18T01:35:59.754856+00:00"},{"alias_kind":"arxiv_version","alias_value":"1504.03391v2","created_at":"2026-05-18T01:35:59.754856+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1504.03391","created_at":"2026-05-18T01:35:59.754856+00:00"},{"alias_kind":"pith_short_12","alias_value":"XW37IRCXEVME","created_at":"2026-05-18T12:29:50.041715+00:00"},{"alias_kind":"pith_short_16","alias_value":"XW37IRCXEVMEVO7C","created_at":"2026-05-18T12:29:50.041715+00:00"},{"alias_kind":"pith_short_8","alias_value":"XW37IRCX","created_at":"2026-05-18T12:29:50.041715+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/XW37IRCXEVMEVO7CJ64SLYPITA","json":"https://pith.science/pith/XW37IRCXEVMEVO7CJ64SLYPITA.json","graph_json":"https://pith.science/api/pith-number/XW37IRCXEVMEVO7CJ64SLYPITA/graph.json","events_json":"https://pith.science/api/pith-number/XW37IRCXEVMEVO7CJ64SLYPITA/events.json","paper":"https://pith.science/paper/XW37IRCX"},"agent_actions":{"view_html":"https://pith.science/pith/XW37IRCXEVMEVO7CJ64SLYPITA","download_json":"https://pith.science/pith/XW37IRCXEVMEVO7CJ64SLYPITA.json","view_paper":"https://pith.science/paper/XW37IRCX","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1504.03391&json=true","fetch_graph":"https://pith.science/api/pith-number/XW37IRCXEVMEVO7CJ64SLYPITA/graph.json","fetch_events":"https://pith.science/api/pith-number/XW37IRCXEVMEVO7CJ64SLYPITA/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/XW37IRCXEVMEVO7CJ64SLYPITA/action/timestamp_anchor","attest_storage":"https://pith.science/pith/XW37IRCXEVMEVO7CJ64SLYPITA/action/storage_attestation","attest_author":"https://pith.science/pith/XW37IRCXEVMEVO7CJ64SLYPITA/action/author_attestation","sign_citation":"https://pith.science/pith/XW37IRCXEVMEVO7CJ64SLYPITA/action/citation_signature","submit_replication":"https://pith.science/pith/XW37IRCXEVMEVO7CJ64SLYPITA/action/replication_record"}},"created_at":"2026-05-18T01:35:59.754856+00:00","updated_at":"2026-05-18T01:35:59.754856+00:00"}