{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2011:JECIZJMXTPWP77UULZLTFBZDPJ","short_pith_number":"pith:JECIZJMX","schema_version":"1.0","canonical_sha256":"49048ca5979becfffe945e573287237a63d7104597a3aca274257bf9cc6477cc","source":{"kind":"arxiv","id":"1111.6453","version":2},"attestation_state":"computed","paper":{"title":"Learning with Submodular Functions: A Convex Optimization Perspective","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.OC"],"primary_cat":"cs.LG","authors_text":"Francis Bach (LIENS, INRIA Paris - Rocquencourt)","submitted_at":"2011-11-28T14:45:01Z","abstract_excerpt":"Submodular functions are relevant to machine learning for at least two reasons: (1) some problems may be expressed directly as the optimization of submodular functions and (2) the lovasz extension of submodular functions provides a useful set of regularization functions for supervised and unsupervised learning. In this monograph, we present the theory of submodular functions from a convex analysis perspective, presenting tight links between certain polyhedra, combinatorial optimization and convex optimization problems. In particular, we show how submodular function minimization is equivalent t"},"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":"1111.6453","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.LG","submitted_at":"2011-11-28T14:45:01Z","cross_cats_sorted":["math.OC"],"title_canon_sha256":"e64fdf6a69abf41d39e67450dcc875b80ca255f959a554d247654b5fee512ee4","abstract_canon_sha256":"ae4107e9429c8ff05a2079e03c6c7bc802d0704b350e1b783749207913ca48ec"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:11:09.775355Z","signature_b64":"5vv0CIfCLLJEGOxc9cSMbP942P+HcumVKf1Ex+Uk7A/XnLCkd3PjY364ugQ+krJED3hGrMZmxehhLBqQJXouAg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"49048ca5979becfffe945e573287237a63d7104597a3aca274257bf9cc6477cc","last_reissued_at":"2026-05-18T03:11:09.774536Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:11:09.774536Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Learning with Submodular Functions: A Convex Optimization Perspective","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.OC"],"primary_cat":"cs.LG","authors_text":"Francis Bach (LIENS, INRIA Paris - Rocquencourt)","submitted_at":"2011-11-28T14:45:01Z","abstract_excerpt":"Submodular functions are relevant to machine learning for at least two reasons: (1) some problems may be expressed directly as the optimization of submodular functions and (2) the lovasz extension of submodular functions provides a useful set of regularization functions for supervised and unsupervised learning. In this monograph, we present the theory of submodular functions from a convex analysis perspective, presenting tight links between certain polyhedra, combinatorial optimization and convex optimization problems. In particular, we show how submodular function minimization is equivalent t"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1111.6453","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":"1111.6453","created_at":"2026-05-18T03:11:09.774663+00:00"},{"alias_kind":"arxiv_version","alias_value":"1111.6453v2","created_at":"2026-05-18T03:11:09.774663+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1111.6453","created_at":"2026-05-18T03:11:09.774663+00:00"},{"alias_kind":"pith_short_12","alias_value":"JECIZJMXTPWP","created_at":"2026-05-18T12:26:32.869790+00:00"},{"alias_kind":"pith_short_16","alias_value":"JECIZJMXTPWP77UU","created_at":"2026-05-18T12:26:32.869790+00:00"},{"alias_kind":"pith_short_8","alias_value":"JECIZJMX","created_at":"2026-05-18T12:26:32.869790+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":2,"internal_anchor_count":1,"sample":[{"citing_arxiv_id":"2604.03419","citing_title":"Adaptive Threshold-Driven Continuous Greedy Method for Scalable Submodular Optimization","ref_index":4,"is_internal_anchor":true},{"citing_arxiv_id":"2604.03419","citing_title":"Adaptive Threshold-Driven Continuous Greedy Method for Scalable Submodular Optimization","ref_index":4,"is_internal_anchor":false}]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/JECIZJMXTPWP77UULZLTFBZDPJ","json":"https://pith.science/pith/JECIZJMXTPWP77UULZLTFBZDPJ.json","graph_json":"https://pith.science/api/pith-number/JECIZJMXTPWP77UULZLTFBZDPJ/graph.json","events_json":"https://pith.science/api/pith-number/JECIZJMXTPWP77UULZLTFBZDPJ/events.json","paper":"https://pith.science/paper/JECIZJMX"},"agent_actions":{"view_html":"https://pith.science/pith/JECIZJMXTPWP77UULZLTFBZDPJ","download_json":"https://pith.science/pith/JECIZJMXTPWP77UULZLTFBZDPJ.json","view_paper":"https://pith.science/paper/JECIZJMX","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1111.6453&json=true","fetch_graph":"https://pith.science/api/pith-number/JECIZJMXTPWP77UULZLTFBZDPJ/graph.json","fetch_events":"https://pith.science/api/pith-number/JECIZJMXTPWP77UULZLTFBZDPJ/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/JECIZJMXTPWP77UULZLTFBZDPJ/action/timestamp_anchor","attest_storage":"https://pith.science/pith/JECIZJMXTPWP77UULZLTFBZDPJ/action/storage_attestation","attest_author":"https://pith.science/pith/JECIZJMXTPWP77UULZLTFBZDPJ/action/author_attestation","sign_citation":"https://pith.science/pith/JECIZJMXTPWP77UULZLTFBZDPJ/action/citation_signature","submit_replication":"https://pith.science/pith/JECIZJMXTPWP77UULZLTFBZDPJ/action/replication_record"}},"created_at":"2026-05-18T03:11:09.774663+00:00","updated_at":"2026-05-18T03:11:09.774663+00:00"}