{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2014:45SJHEYOHAIEXNYZIMYAIWW3ID","short_pith_number":"pith:45SJHEYO","canonical_record":{"source":{"id":"1410.7940","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.OC","submitted_at":"2014-10-29T11:29:07Z","cross_cats_sorted":[],"title_canon_sha256":"b3b4d2196e15b660a6d19b21a81fc2b80d1b5752ecb5da86819d85c7ed3090f1","abstract_canon_sha256":"5edd3516ebe930c3df0fa984bcbd6d3c7f0d897bb40e57450248326f3a194daa"},"schema_version":"1.0"},"canonical_sha256":"e76493930e38104bb7194330045adb40fa2fbd8b079930097ca6cdaaf6abb1ad","source":{"kind":"arxiv","id":"1410.7940","version":3},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1410.7940","created_at":"2026-05-18T01:36:20Z"},{"alias_kind":"arxiv_version","alias_value":"1410.7940v3","created_at":"2026-05-18T01:36:20Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1410.7940","created_at":"2026-05-18T01:36:20Z"},{"alias_kind":"pith_short_12","alias_value":"45SJHEYOHAIE","created_at":"2026-05-18T12:28:14Z"},{"alias_kind":"pith_short_16","alias_value":"45SJHEYOHAIEXNYZ","created_at":"2026-05-18T12:28:14Z"},{"alias_kind":"pith_short_8","alias_value":"45SJHEYO","created_at":"2026-05-18T12:28:14Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2014:45SJHEYOHAIEXNYZIMYAIWW3ID","target":"record","payload":{"canonical_record":{"source":{"id":"1410.7940","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.OC","submitted_at":"2014-10-29T11:29:07Z","cross_cats_sorted":[],"title_canon_sha256":"b3b4d2196e15b660a6d19b21a81fc2b80d1b5752ecb5da86819d85c7ed3090f1","abstract_canon_sha256":"5edd3516ebe930c3df0fa984bcbd6d3c7f0d897bb40e57450248326f3a194daa"},"schema_version":"1.0"},"canonical_sha256":"e76493930e38104bb7194330045adb40fa2fbd8b079930097ca6cdaaf6abb1ad","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:36:20.411813Z","signature_b64":"z6Ox5sMQrLrmBpAxjODFDFHoDjbKJT5o4dD48omBai+2me0lilFV5lcaiU+E3Y1WibFcK66sJQfyS8CZVxhPCw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"e76493930e38104bb7194330045adb40fa2fbd8b079930097ca6cdaaf6abb1ad","last_reissued_at":"2026-05-18T01:36:20.411273Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:36:20.411273Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1410.7940","source_version":3,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T01:36:20Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"6gVLQ86MSIX5E0P8X6ibZjTDw7Q17RkzUwwNd3GcyAW6MRUMRq+iXsPK8YqEQh1CganL+5IPOKugo/98ZpFACQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-31T22:37:39.604079Z"},"content_sha256":"1c3b5e05f6e50403feac65eff743f6b10bbdab02261c980e5cfe4b8e3465ed8d","schema_version":"1.0","event_id":"sha256:1c3b5e05f6e50403feac65eff743f6b10bbdab02261c980e5cfe4b8e3465ed8d"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2014:45SJHEYOHAIEXNYZIMYAIWW3ID","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Orbital Geometry in Optimisation","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.OC","authors_text":"Andrew Eberhard, Vera Roshchina","submitted_at":"2014-10-29T11:29:07Z","abstract_excerpt":"We discuss the use of group symmetries in optimisation, in particular with respect to the structure of subdifferential and projection operators. This allows us to generalise a classic result of Adrian Lewis regarding the characterisation of the subdifferential of a permutation invariant convex function to the characterisation of the proximal subdifferential of a Schur convex function that is invariant with respect to a finite reflection group. We are also able to simplify and generalise results on projections onto symmetric sets, in particular, we study projections on sparsity constraints used"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1410.7940","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"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T01:36:20Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"GGfQJqn6LX/cR4q1ectrXjJtJuT5X2GdiU5Lj5kscubseqOlXsOwXZc5ByWtysyi1PiBwZZW/ADqXG3YQS0uDQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-31T22:37:39.604726Z"},"content_sha256":"33225741e288980eae29553cafdd9c095a4b45725311afc71658ca0d039e2bcb","schema_version":"1.0","event_id":"sha256:33225741e288980eae29553cafdd9c095a4b45725311afc71658ca0d039e2bcb"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/45SJHEYOHAIEXNYZIMYAIWW3ID/bundle.json","state_url":"https://pith.science/pith/45SJHEYOHAIEXNYZIMYAIWW3ID/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/45SJHEYOHAIEXNYZIMYAIWW3ID/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":"2026-05-31T22:37:39Z","links":{"resolver":"https://pith.science/pith/45SJHEYOHAIEXNYZIMYAIWW3ID","bundle":"https://pith.science/pith/45SJHEYOHAIEXNYZIMYAIWW3ID/bundle.json","state":"https://pith.science/pith/45SJHEYOHAIEXNYZIMYAIWW3ID/state.json","well_known_bundle":"https://pith.science/.well-known/pith/45SJHEYOHAIEXNYZIMYAIWW3ID/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2014:45SJHEYOHAIEXNYZIMYAIWW3ID","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":"5edd3516ebe930c3df0fa984bcbd6d3c7f0d897bb40e57450248326f3a194daa","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.OC","submitted_at":"2014-10-29T11:29:07Z","title_canon_sha256":"b3b4d2196e15b660a6d19b21a81fc2b80d1b5752ecb5da86819d85c7ed3090f1"},"schema_version":"1.0","source":{"id":"1410.7940","kind":"arxiv","version":3}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1410.7940","created_at":"2026-05-18T01:36:20Z"},{"alias_kind":"arxiv_version","alias_value":"1410.7940v3","created_at":"2026-05-18T01:36:20Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1410.7940","created_at":"2026-05-18T01:36:20Z"},{"alias_kind":"pith_short_12","alias_value":"45SJHEYOHAIE","created_at":"2026-05-18T12:28:14Z"},{"alias_kind":"pith_short_16","alias_value":"45SJHEYOHAIEXNYZ","created_at":"2026-05-18T12:28:14Z"},{"alias_kind":"pith_short_8","alias_value":"45SJHEYO","created_at":"2026-05-18T12:28:14Z"}],"graph_snapshots":[{"event_id":"sha256:33225741e288980eae29553cafdd9c095a4b45725311afc71658ca0d039e2bcb","target":"graph","created_at":"2026-05-18T01:36:20Z","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":"We discuss the use of group symmetries in optimisation, in particular with respect to the structure of subdifferential and projection operators. This allows us to generalise a classic result of Adrian Lewis regarding the characterisation of the subdifferential of a permutation invariant convex function to the characterisation of the proximal subdifferential of a Schur convex function that is invariant with respect to a finite reflection group. We are also able to simplify and generalise results on projections onto symmetric sets, in particular, we study projections on sparsity constraints used","authors_text":"Andrew Eberhard, Vera Roshchina","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.OC","submitted_at":"2014-10-29T11:29:07Z","title":"Orbital Geometry in Optimisation"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1410.7940","kind":"arxiv","version":3},"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":"sha256:1c3b5e05f6e50403feac65eff743f6b10bbdab02261c980e5cfe4b8e3465ed8d","target":"record","created_at":"2026-05-18T01:36:20Z","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":"5edd3516ebe930c3df0fa984bcbd6d3c7f0d897bb40e57450248326f3a194daa","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.OC","submitted_at":"2014-10-29T11:29:07Z","title_canon_sha256":"b3b4d2196e15b660a6d19b21a81fc2b80d1b5752ecb5da86819d85c7ed3090f1"},"schema_version":"1.0","source":{"id":"1410.7940","kind":"arxiv","version":3}},"canonical_sha256":"e76493930e38104bb7194330045adb40fa2fbd8b079930097ca6cdaaf6abb1ad","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"e76493930e38104bb7194330045adb40fa2fbd8b079930097ca6cdaaf6abb1ad","first_computed_at":"2026-05-18T01:36:20.411273Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:36:20.411273Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"z6Ox5sMQrLrmBpAxjODFDFHoDjbKJT5o4dD48omBai+2me0lilFV5lcaiU+E3Y1WibFcK66sJQfyS8CZVxhPCw==","signature_status":"signed_v1","signed_at":"2026-05-18T01:36:20.411813Z","signed_message":"canonical_sha256_bytes"},"source_id":"1410.7940","source_kind":"arxiv","source_version":3}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:1c3b5e05f6e50403feac65eff743f6b10bbdab02261c980e5cfe4b8e3465ed8d","sha256:33225741e288980eae29553cafdd9c095a4b45725311afc71658ca0d039e2bcb"],"state_sha256":"e3eaec2e91cc4e555f9507be8c298d8769aa0b0cb994a3e863088e5b4c35bae4"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"4LEbIapzN2xQwbf0NGQcNTe7nzdbvPN27hA40EKZi38Z4ZP2rWRG/EplglVAL1+QPksZJ+cn9QSTW4en8zXPCQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-31T22:37:39.608380Z","bundle_sha256":"53ce87227bb21afe7e63086cac1c4f802cd973f2c2f19c48dd916a9c311627b8"}}