{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2015:47OQUCPBYXIUMKLUXHIC2YZSB6","short_pith_number":"pith:47OQUCPB","canonical_record":{"source":{"id":"1510.08577","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.OC","submitted_at":"2015-10-29T06:38:00Z","cross_cats_sorted":[],"title_canon_sha256":"212db9375ecaae972a400b19c2f9582154b48c98a69085716b1284d285eb37b2","abstract_canon_sha256":"366caeabd1d5e30647be3f91827f6143b0b906188fe753b7ae0f0f83511f432d"},"schema_version":"1.0"},"canonical_sha256":"e7dd0a09e1c5d1462974b9d02d63320f91a6d1f9b52c31dab3a6d1b745f851df","source":{"kind":"arxiv","id":"1510.08577","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1510.08577","created_at":"2026-05-18T01:28:28Z"},{"alias_kind":"arxiv_version","alias_value":"1510.08577v1","created_at":"2026-05-18T01:28:28Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1510.08577","created_at":"2026-05-18T01:28:28Z"},{"alias_kind":"pith_short_12","alias_value":"47OQUCPBYXIU","created_at":"2026-05-18T12:29:05Z"},{"alias_kind":"pith_short_16","alias_value":"47OQUCPBYXIUMKLU","created_at":"2026-05-18T12:29:05Z"},{"alias_kind":"pith_short_8","alias_value":"47OQUCPB","created_at":"2026-05-18T12:29:05Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2015:47OQUCPBYXIUMKLUXHIC2YZSB6","target":"record","payload":{"canonical_record":{"source":{"id":"1510.08577","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.OC","submitted_at":"2015-10-29T06:38:00Z","cross_cats_sorted":[],"title_canon_sha256":"212db9375ecaae972a400b19c2f9582154b48c98a69085716b1284d285eb37b2","abstract_canon_sha256":"366caeabd1d5e30647be3f91827f6143b0b906188fe753b7ae0f0f83511f432d"},"schema_version":"1.0"},"canonical_sha256":"e7dd0a09e1c5d1462974b9d02d63320f91a6d1f9b52c31dab3a6d1b745f851df","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:28:28.439573Z","signature_b64":"6bisPyxtQXyxWWWlzXRRxvBEHfPemZAfJFe54zPVgZBVmQjYVBlOw1TrY+5bPpU+NOe1t1BSBy1mGZ9mjsyTDQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"e7dd0a09e1c5d1462974b9d02d63320f91a6d1f9b52c31dab3a6d1b745f851df","last_reissued_at":"2026-05-18T01:28:28.438969Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:28:28.438969Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1510.08577","source_version":1,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T01:28:28Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"cNh4KFXEwGIzBy8oaFnJFAMAkVoMgw5eYDmMatFk1LvAuLa8XKsh77xMDBot+BJmZgkmcHmsBEISGYO60rFMDA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-31T15:49:51.156355Z"},"content_sha256":"29bf54932cb2471d03941adda9022c1821016e78e2b29e4837107fd7ba09d226","schema_version":"1.0","event_id":"sha256:29bf54932cb2471d03941adda9022c1821016e78e2b29e4837107fd7ba09d226"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2015:47OQUCPBYXIUMKLUXHIC2YZSB6","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"The U-Lagrangian of a prox-regular function","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.OC","authors_text":"Andrew Eberhard, Shuai Liu, Yousong Luo","submitted_at":"2015-10-29T06:38:00Z","abstract_excerpt":"When restricted to a subspace, a nonsmooth function can be differentiable. It is known that for a nonsmooth convex function f and a point x, the Euclidean space can be decomposed into two subspaces: U, over which a special Lagrangian can be defined and has nice smooth properties and V, the orthogonal complement subspace of U. In this paper we generalize the definition of UV-decomposition and U-Lagrangian to the context of nonconvex functions, specifically that of a prox-regular function."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1510.08577","kind":"arxiv","version":1},"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"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T01:28:28Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"hO/skyLZE2oJ5zuSqTyjW8cvHVe6bEPY13gR72snngzntn22RuHWLiIXIFlmSWu4iQOAo/Bi4gdvq9VP7wgXCA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-31T15:49:51.157246Z"},"content_sha256":"bb04b9d563b68cea86452062e19d7bdd961d07cd204d21ef895413e3f68ba01b","schema_version":"1.0","event_id":"sha256:bb04b9d563b68cea86452062e19d7bdd961d07cd204d21ef895413e3f68ba01b"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/47OQUCPBYXIUMKLUXHIC2YZSB6/bundle.json","state_url":"https://pith.science/pith/47OQUCPBYXIUMKLUXHIC2YZSB6/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/47OQUCPBYXIUMKLUXHIC2YZSB6/bundle.json","status":"primary"}],"public_keys":[{"key_id":"pith-v1-2026-05","algorithm":"ed25519","format":"raw","public_key_b64":"stVStoiQhXFxp4s2pdzPNoqVNBMojDU/fJ2db5S3CbM=","public_key_hex":"b2d552b68890857171a78b36a5dccf368a953413288c353f7c9d9d6f94b709b3","fingerprint_sha256_b32_first128bits":"RVFV5Z2OI2J3ZUO7ERDEBCYNKS","fingerprint_sha256_hex":"8d4b5ee74e4693bcd1df2446408b0d54","rotates_at":null,"url":"https://pith.science/pith-signing-key.json","notes":"Pith uses this Ed25519 key to sign canonical record SHA-256 digests. Verify with: ed25519_verify(public_key, message=canonical_sha256_bytes, signature=base64decode(signature_b64))."}],"merge_version":"pith-open-graph-merge-v1","built_at":"2026-05-31T15:49:51Z","links":{"resolver":"https://pith.science/pith/47OQUCPBYXIUMKLUXHIC2YZSB6","bundle":"https://pith.science/pith/47OQUCPBYXIUMKLUXHIC2YZSB6/bundle.json","state":"https://pith.science/pith/47OQUCPBYXIUMKLUXHIC2YZSB6/state.json","well_known_bundle":"https://pith.science/.well-known/pith/47OQUCPBYXIUMKLUXHIC2YZSB6/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2015:47OQUCPBYXIUMKLUXHIC2YZSB6","merge_version":"pith-open-graph-merge-v1","event_count":2,"valid_event_count":2,"invalid_event_count":0,"equivocation_count":0,"current":{"canonical_record":{"metadata":{"abstract_canon_sha256":"366caeabd1d5e30647be3f91827f6143b0b906188fe753b7ae0f0f83511f432d","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.OC","submitted_at":"2015-10-29T06:38:00Z","title_canon_sha256":"212db9375ecaae972a400b19c2f9582154b48c98a69085716b1284d285eb37b2"},"schema_version":"1.0","source":{"id":"1510.08577","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1510.08577","created_at":"2026-05-18T01:28:28Z"},{"alias_kind":"arxiv_version","alias_value":"1510.08577v1","created_at":"2026-05-18T01:28:28Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1510.08577","created_at":"2026-05-18T01:28:28Z"},{"alias_kind":"pith_short_12","alias_value":"47OQUCPBYXIU","created_at":"2026-05-18T12:29:05Z"},{"alias_kind":"pith_short_16","alias_value":"47OQUCPBYXIUMKLU","created_at":"2026-05-18T12:29:05Z"},{"alias_kind":"pith_short_8","alias_value":"47OQUCPB","created_at":"2026-05-18T12:29:05Z"}],"graph_snapshots":[{"event_id":"sha256:bb04b9d563b68cea86452062e19d7bdd961d07cd204d21ef895413e3f68ba01b","target":"graph","created_at":"2026-05-18T01:28:28Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"graph_snapshot":{"author_claims":{"count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","strong_count":0},"builder_version":"pith-number-builder-2026-05-17-v1","claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"paper":{"abstract_excerpt":"When restricted to a subspace, a nonsmooth function can be differentiable. It is known that for a nonsmooth convex function f and a point x, the Euclidean space can be decomposed into two subspaces: U, over which a special Lagrangian can be defined and has nice smooth properties and V, the orthogonal complement subspace of U. In this paper we generalize the definition of UV-decomposition and U-Lagrangian to the context of nonconvex functions, specifically that of a prox-regular function.","authors_text":"Andrew Eberhard, Shuai Liu, Yousong Luo","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.OC","submitted_at":"2015-10-29T06:38:00Z","title":"The U-Lagrangian of a prox-regular function"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1510.08577","kind":"arxiv","version":1},"verdict":{"created_at":null,"id":null,"model_set":{},"one_line_summary":"","pipeline_version":null,"pith_extraction_headline":"","strongest_claim":"","weakest_assumption":""}},"verdict_id":null}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:29bf54932cb2471d03941adda9022c1821016e78e2b29e4837107fd7ba09d226","target":"record","created_at":"2026-05-18T01:28:28Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"attestation_state":"computed","canonical_record":{"metadata":{"abstract_canon_sha256":"366caeabd1d5e30647be3f91827f6143b0b906188fe753b7ae0f0f83511f432d","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.OC","submitted_at":"2015-10-29T06:38:00Z","title_canon_sha256":"212db9375ecaae972a400b19c2f9582154b48c98a69085716b1284d285eb37b2"},"schema_version":"1.0","source":{"id":"1510.08577","kind":"arxiv","version":1}},"canonical_sha256":"e7dd0a09e1c5d1462974b9d02d63320f91a6d1f9b52c31dab3a6d1b745f851df","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"e7dd0a09e1c5d1462974b9d02d63320f91a6d1f9b52c31dab3a6d1b745f851df","first_computed_at":"2026-05-18T01:28:28.438969Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:28:28.438969Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"6bisPyxtQXyxWWWlzXRRxvBEHfPemZAfJFe54zPVgZBVmQjYVBlOw1TrY+5bPpU+NOe1t1BSBy1mGZ9mjsyTDQ==","signature_status":"signed_v1","signed_at":"2026-05-18T01:28:28.439573Z","signed_message":"canonical_sha256_bytes"},"source_id":"1510.08577","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:29bf54932cb2471d03941adda9022c1821016e78e2b29e4837107fd7ba09d226","sha256:bb04b9d563b68cea86452062e19d7bdd961d07cd204d21ef895413e3f68ba01b"],"state_sha256":"476d34239b8341ed2b40b30b188bba8da5c3eb527999f10b0c986c2b8fc6da56"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"4I8CHLQUXY0/8PWpmjYs8pGRH7Rqd+RWNrtGSoerf9gulQUMuyTARjBolkzBXS4hjGFEEHH8ChB1lILK69JaDA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-31T15:49:51.162385Z","bundle_sha256":"7459bd477672a94d473ad0ef1b7cf857ce53882453e461bb4a118b8a260559f8"}}