{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2015:GVULU7DYJDC5UNAZTEIX4MQCCS","short_pith_number":"pith:GVULU7DY","canonical_record":{"source":{"id":"1505.03840","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.CV","submitted_at":"2015-05-14T19:11:27Z","cross_cats_sorted":["cs.DS","math.OC"],"title_canon_sha256":"10335f2bf398472be775cbba955559d0d827f542f9d8271ef52dfe25c0f9bec0","abstract_canon_sha256":"43f125b9622250502d38f11c8d2d2f66f7e461725914e39a0db27839464b5aa0"},"schema_version":"1.0"},"canonical_sha256":"3568ba7c7848c5da341999117e32021484eb487b494f7bba59ea34202de9a7e5","source":{"kind":"arxiv","id":"1505.03840","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1505.03840","created_at":"2026-05-18T02:09:52Z"},{"alias_kind":"arxiv_version","alias_value":"1505.03840v1","created_at":"2026-05-18T02:09:52Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1505.03840","created_at":"2026-05-18T02:09:52Z"},{"alias_kind":"pith_short_12","alias_value":"GVULU7DYJDC5","created_at":"2026-05-18T12:29:22Z"},{"alias_kind":"pith_short_16","alias_value":"GVULU7DYJDC5UNAZ","created_at":"2026-05-18T12:29:22Z"},{"alias_kind":"pith_short_8","alias_value":"GVULU7DY","created_at":"2026-05-18T12:29:22Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2015:GVULU7DYJDC5UNAZTEIX4MQCCS","target":"record","payload":{"canonical_record":{"source":{"id":"1505.03840","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.CV","submitted_at":"2015-05-14T19:11:27Z","cross_cats_sorted":["cs.DS","math.OC"],"title_canon_sha256":"10335f2bf398472be775cbba955559d0d827f542f9d8271ef52dfe25c0f9bec0","abstract_canon_sha256":"43f125b9622250502d38f11c8d2d2f66f7e461725914e39a0db27839464b5aa0"},"schema_version":"1.0"},"canonical_sha256":"3568ba7c7848c5da341999117e32021484eb487b494f7bba59ea34202de9a7e5","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:09:52.883358Z","signature_b64":"6vJBMAaB+4eVCZODAg1UcVGMYXQSCMTEeYw5CjqZ6p0sI4askVLB2zbJJkIAZLdxbGz2R2ZIbqz0DsAaMgVgAg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"3568ba7c7848c5da341999117e32021484eb487b494f7bba59ea34202de9a7e5","last_reissued_at":"2026-05-18T02:09:52.882699Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:09:52.882699Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1505.03840","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":"2026-05-18T02:09:52Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"WZh/wI37j/c1+2eSXqeG8kBstPssHLgiCQwIdTfxmF+y/fh1c3vc3DkqvvNfb0JZemOY1tuDef1/rwFg6i1/Dg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-28T16:05:23.819826Z"},"content_sha256":"d641c1ef372afd77eb7d538ba45fc0430519a451194a5568b90a13af60ba5c69","schema_version":"1.0","event_id":"sha256:d641c1ef372afd77eb7d538ba45fc0430519a451194a5568b90a13af60ba5c69"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2015:GVULU7DYJDC5UNAZTEIX4MQCCS","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Non-unique games over compact groups and orientation estimation in cryo-EM","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.DS","math.OC"],"primary_cat":"cs.CV","authors_text":"Afonso S. Bandeira, Amit Singer, Yutong Chen","submitted_at":"2015-05-14T19:11:27Z","abstract_excerpt":"Let $\\mathcal{G}$ be a compact group and let $f_{ij} \\in L^2(\\mathcal{G})$. We define the Non-Unique Games (NUG) problem as finding $g_1,\\dots,g_n \\in \\mathcal{G}$ to minimize $\\sum_{i,j=1}^n f_{ij} \\left( g_i g_j^{-1}\\right)$. We devise a relaxation of the NUG problem to a semidefinite program (SDP) by taking the Fourier transform of $f_{ij}$ over $\\mathcal{G}$, which can then be solved efficiently. The NUG framework can be seen as a generalization of the little Grothendieck problem over the orthogonal group and the Unique Games problem and includes many practically relevant problems, such as"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1505.03840","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":"2026-05-18T02:09:52Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"vywAqk4TtZ7SuKogNVS5RCEY5z6yF+LtV9hPFghv5bg+9rfuGW+PNeqPzt7tYY63tpwGtE2Ez9E8B7gSlgzHCg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-28T16:05:23.820491Z"},"content_sha256":"446a11960a51ffd4543d9bfbbb455d6daf37b29ae0b12c935b9c581a9a0e82fa","schema_version":"1.0","event_id":"sha256:446a11960a51ffd4543d9bfbbb455d6daf37b29ae0b12c935b9c581a9a0e82fa"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/GVULU7DYJDC5UNAZTEIX4MQCCS/bundle.json","state_url":"https://pith.science/pith/GVULU7DYJDC5UNAZTEIX4MQCCS/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/GVULU7DYJDC5UNAZTEIX4MQCCS/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":"2026-05-28T16:05:23Z","links":{"resolver":"https://pith.science/pith/GVULU7DYJDC5UNAZTEIX4MQCCS","bundle":"https://pith.science/pith/GVULU7DYJDC5UNAZTEIX4MQCCS/bundle.json","state":"https://pith.science/pith/GVULU7DYJDC5UNAZTEIX4MQCCS/state.json","well_known_bundle":"https://pith.science/.well-known/pith/GVULU7DYJDC5UNAZTEIX4MQCCS/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2015:GVULU7DYJDC5UNAZTEIX4MQCCS","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":"43f125b9622250502d38f11c8d2d2f66f7e461725914e39a0db27839464b5aa0","cross_cats_sorted":["cs.DS","math.OC"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.CV","submitted_at":"2015-05-14T19:11:27Z","title_canon_sha256":"10335f2bf398472be775cbba955559d0d827f542f9d8271ef52dfe25c0f9bec0"},"schema_version":"1.0","source":{"id":"1505.03840","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1505.03840","created_at":"2026-05-18T02:09:52Z"},{"alias_kind":"arxiv_version","alias_value":"1505.03840v1","created_at":"2026-05-18T02:09:52Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1505.03840","created_at":"2026-05-18T02:09:52Z"},{"alias_kind":"pith_short_12","alias_value":"GVULU7DYJDC5","created_at":"2026-05-18T12:29:22Z"},{"alias_kind":"pith_short_16","alias_value":"GVULU7DYJDC5UNAZ","created_at":"2026-05-18T12:29:22Z"},{"alias_kind":"pith_short_8","alias_value":"GVULU7DY","created_at":"2026-05-18T12:29:22Z"}],"graph_snapshots":[{"event_id":"sha256:446a11960a51ffd4543d9bfbbb455d6daf37b29ae0b12c935b9c581a9a0e82fa","target":"graph","created_at":"2026-05-18T02:09:52Z","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":"Let $\\mathcal{G}$ be a compact group and let $f_{ij} \\in L^2(\\mathcal{G})$. We define the Non-Unique Games (NUG) problem as finding $g_1,\\dots,g_n \\in \\mathcal{G}$ to minimize $\\sum_{i,j=1}^n f_{ij} \\left( g_i g_j^{-1}\\right)$. We devise a relaxation of the NUG problem to a semidefinite program (SDP) by taking the Fourier transform of $f_{ij}$ over $\\mathcal{G}$, which can then be solved efficiently. The NUG framework can be seen as a generalization of the little Grothendieck problem over the orthogonal group and the Unique Games problem and includes many practically relevant problems, such as","authors_text":"Afonso S. Bandeira, Amit Singer, Yutong Chen","cross_cats":["cs.DS","math.OC"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.CV","submitted_at":"2015-05-14T19:11:27Z","title":"Non-unique games over compact groups and orientation estimation in cryo-EM"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1505.03840","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:d641c1ef372afd77eb7d538ba45fc0430519a451194a5568b90a13af60ba5c69","target":"record","created_at":"2026-05-18T02:09:52Z","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":"43f125b9622250502d38f11c8d2d2f66f7e461725914e39a0db27839464b5aa0","cross_cats_sorted":["cs.DS","math.OC"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.CV","submitted_at":"2015-05-14T19:11:27Z","title_canon_sha256":"10335f2bf398472be775cbba955559d0d827f542f9d8271ef52dfe25c0f9bec0"},"schema_version":"1.0","source":{"id":"1505.03840","kind":"arxiv","version":1}},"canonical_sha256":"3568ba7c7848c5da341999117e32021484eb487b494f7bba59ea34202de9a7e5","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"3568ba7c7848c5da341999117e32021484eb487b494f7bba59ea34202de9a7e5","first_computed_at":"2026-05-18T02:09:52.882699Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T02:09:52.882699Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"6vJBMAaB+4eVCZODAg1UcVGMYXQSCMTEeYw5CjqZ6p0sI4askVLB2zbJJkIAZLdxbGz2R2ZIbqz0DsAaMgVgAg==","signature_status":"signed_v1","signed_at":"2026-05-18T02:09:52.883358Z","signed_message":"canonical_sha256_bytes"},"source_id":"1505.03840","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:d641c1ef372afd77eb7d538ba45fc0430519a451194a5568b90a13af60ba5c69","sha256:446a11960a51ffd4543d9bfbbb455d6daf37b29ae0b12c935b9c581a9a0e82fa"],"state_sha256":"344e1047aa9c2681905722ba661f15ca11bed712343ad5a60e0cc39a2f0dd3f1"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"34vvK/l1m7jImvGVs8icuYhJDqF0BlJoReRGxZWEjmrMVW9Bt9KzdITwVDopSpkdDLRagi5hhJ49rtPxpeocDA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-28T16:05:23.823540Z","bundle_sha256":"5af99b830225bb5ef82711d76e8d27343c8885f24fe2faa8053c9574f5c4dc16"}}