{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2022:7TCWQOULKWE36IZOXSBSCCRP3J","short_pith_number":"pith:7TCWQOUL","canonical_record":{"source":{"id":"2209.03416","kind":"arxiv","version":5},"metadata":{"license":"http://creativecommons.org/licenses/by-nc-nd/4.0/","primary_cat":"cs.LG","submitted_at":"2022-09-07T18:34:48Z","cross_cats_sorted":[],"title_canon_sha256":"e1a341092d3a5a9c9614bc05806a750d5cb737eb05d4bf20de7f72b1eeea1838","abstract_canon_sha256":"c69f2c55ee576b259ccb04c8da9ae6cc0b503d2095d7650270d6c41a8882881a"},"schema_version":"1.0"},"canonical_sha256":"fcc5683a8b5589bf232ebc83210a2fda52546a3797edb618fce003277c9dbdee","source":{"kind":"arxiv","id":"2209.03416","version":5},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2209.03416","created_at":"2026-07-05T06:11:56Z"},{"alias_kind":"arxiv_version","alias_value":"2209.03416v5","created_at":"2026-07-05T06:11:56Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2209.03416","created_at":"2026-07-05T06:11:56Z"},{"alias_kind":"pith_short_12","alias_value":"7TCWQOULKWE3","created_at":"2026-07-05T06:11:56Z"},{"alias_kind":"pith_short_16","alias_value":"7TCWQOULKWE36IZO","created_at":"2026-07-05T06:11:56Z"},{"alias_kind":"pith_short_8","alias_value":"7TCWQOUL","created_at":"2026-07-05T06:11:56Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2022:7TCWQOULKWE36IZOXSBSCCRP3J","target":"record","payload":{"canonical_record":{"source":{"id":"2209.03416","kind":"arxiv","version":5},"metadata":{"license":"http://creativecommons.org/licenses/by-nc-nd/4.0/","primary_cat":"cs.LG","submitted_at":"2022-09-07T18:34:48Z","cross_cats_sorted":[],"title_canon_sha256":"e1a341092d3a5a9c9614bc05806a750d5cb737eb05d4bf20de7f72b1eeea1838","abstract_canon_sha256":"c69f2c55ee576b259ccb04c8da9ae6cc0b503d2095d7650270d6c41a8882881a"},"schema_version":"1.0"},"canonical_sha256":"fcc5683a8b5589bf232ebc83210a2fda52546a3797edb618fce003277c9dbdee","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T06:11:56.971883Z","signature_b64":"KVop0IyZcexqE6JK6CEYdV21WdmlHp4OFVQ6J1ZE235zH81HzRvr8AplmCY9x3nZfFa1Mn7nWaPo8HJ/C29MAg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"fcc5683a8b5589bf232ebc83210a2fda52546a3797edb618fce003277c9dbdee","last_reissued_at":"2026-07-05T06:11:56.971529Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T06:11:56.971529Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2209.03416","source_version":5,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-07-05T06:11:56Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"pLip2OAc33FHHf6aLjfjwG6OMQkvV2mV820cg2yBt0swUD+vHJeNOw3Dq0fftdG1iGRILbRKeYKmeYpH80PiBA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-14T06:51:04.839564Z"},"content_sha256":"3b3a1a2caf10d7ee0313ecddcdd8c6534526b8e18b4c74e2f76c34701b6910ee","schema_version":"1.0","event_id":"sha256:3b3a1a2caf10d7ee0313ecddcdd8c6534526b8e18b4c74e2f76c34701b6910ee"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2022:7TCWQOULKWE36IZOXSBSCCRP3J","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Bispectral Neural Networks","license":"http://creativecommons.org/licenses/by-nc-nd/4.0/","headline":"","cross_cats":[],"primary_cat":"cs.LG","authors_text":"Bruno Olshausen, Christian Shewmake, Christopher Hillar, Sophia Sanborn","submitted_at":"2022-09-07T18:34:48Z","abstract_excerpt":"We present a neural network architecture, Bispectral Neural Networks (BNNs) for learning representations that are invariant to the actions of compact commutative groups on the space over which a signal is defined. The model incorporates the ansatz of the bispectrum, an analytically defined group invariant that is complete -- that is, it preserves all signal structure while removing only the variation due to group actions. Here, we demonstrate that BNNs are able to simultaneously learn groups, their irreducible representations, and corresponding equivariant and complete-invariant maps purely fr"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2209.03416","kind":"arxiv","version":5},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2209.03416/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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":"2026-07-05T06:11:56Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"Zo1w4FVnVVnrAmzeyWLlz5qSOxLqh6AMmd8Yk/DmNIcw7XSosykhqNYpTrszax6Q1Qp7PxC9QaJJaeEYj5X0Bw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-14T06:51:04.840074Z"},"content_sha256":"4f4501ac2c3c3f9b13765bdca6c4151496edadae464937e30fa2cc3e20be6739","schema_version":"1.0","event_id":"sha256:4f4501ac2c3c3f9b13765bdca6c4151496edadae464937e30fa2cc3e20be6739"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/7TCWQOULKWE36IZOXSBSCCRP3J/bundle.json","state_url":"https://pith.science/pith/7TCWQOULKWE36IZOXSBSCCRP3J/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/7TCWQOULKWE36IZOXSBSCCRP3J/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":"2026-08-14T06:51:04Z","links":{"resolver":"https://pith.science/pith/7TCWQOULKWE36IZOXSBSCCRP3J","bundle":"https://pith.science/pith/7TCWQOULKWE36IZOXSBSCCRP3J/bundle.json","state":"https://pith.science/pith/7TCWQOULKWE36IZOXSBSCCRP3J/state.json","well_known_bundle":"https://pith.science/.well-known/pith/7TCWQOULKWE36IZOXSBSCCRP3J/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2022:7TCWQOULKWE36IZOXSBSCCRP3J","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":"c69f2c55ee576b259ccb04c8da9ae6cc0b503d2095d7650270d6c41a8882881a","cross_cats_sorted":[],"license":"http://creativecommons.org/licenses/by-nc-nd/4.0/","primary_cat":"cs.LG","submitted_at":"2022-09-07T18:34:48Z","title_canon_sha256":"e1a341092d3a5a9c9614bc05806a750d5cb737eb05d4bf20de7f72b1eeea1838"},"schema_version":"1.0","source":{"id":"2209.03416","kind":"arxiv","version":5}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2209.03416","created_at":"2026-07-05T06:11:56Z"},{"alias_kind":"arxiv_version","alias_value":"2209.03416v5","created_at":"2026-07-05T06:11:56Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2209.03416","created_at":"2026-07-05T06:11:56Z"},{"alias_kind":"pith_short_12","alias_value":"7TCWQOULKWE3","created_at":"2026-07-05T06:11:56Z"},{"alias_kind":"pith_short_16","alias_value":"7TCWQOULKWE36IZO","created_at":"2026-07-05T06:11:56Z"},{"alias_kind":"pith_short_8","alias_value":"7TCWQOUL","created_at":"2026-07-05T06:11:56Z"}],"graph_snapshots":[{"event_id":"sha256:4f4501ac2c3c3f9b13765bdca6c4151496edadae464937e30fa2cc3e20be6739","target":"graph","created_at":"2026-07-05T06:11:56Z","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"},"integrity":{"available":true,"clean":true,"detectors_run":[],"endpoint":"/pith/2209.03416/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"We present a neural network architecture, Bispectral Neural Networks (BNNs) for learning representations that are invariant to the actions of compact commutative groups on the space over which a signal is defined. The model incorporates the ansatz of the bispectrum, an analytically defined group invariant that is complete -- that is, it preserves all signal structure while removing only the variation due to group actions. Here, we demonstrate that BNNs are able to simultaneously learn groups, their irreducible representations, and corresponding equivariant and complete-invariant maps purely fr","authors_text":"Bruno Olshausen, Christian Shewmake, Christopher Hillar, Sophia Sanborn","cross_cats":[],"headline":"","license":"http://creativecommons.org/licenses/by-nc-nd/4.0/","primary_cat":"cs.LG","submitted_at":"2022-09-07T18:34:48Z","title":"Bispectral Neural Networks"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2209.03416","kind":"arxiv","version":5},"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":"sha256:3b3a1a2caf10d7ee0313ecddcdd8c6534526b8e18b4c74e2f76c34701b6910ee","target":"record","created_at":"2026-07-05T06:11:56Z","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":"c69f2c55ee576b259ccb04c8da9ae6cc0b503d2095d7650270d6c41a8882881a","cross_cats_sorted":[],"license":"http://creativecommons.org/licenses/by-nc-nd/4.0/","primary_cat":"cs.LG","submitted_at":"2022-09-07T18:34:48Z","title_canon_sha256":"e1a341092d3a5a9c9614bc05806a750d5cb737eb05d4bf20de7f72b1eeea1838"},"schema_version":"1.0","source":{"id":"2209.03416","kind":"arxiv","version":5}},"canonical_sha256":"fcc5683a8b5589bf232ebc83210a2fda52546a3797edb618fce003277c9dbdee","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"fcc5683a8b5589bf232ebc83210a2fda52546a3797edb618fce003277c9dbdee","first_computed_at":"2026-07-05T06:11:56.971529Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-05T06:11:56.971529Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"KVop0IyZcexqE6JK6CEYdV21WdmlHp4OFVQ6J1ZE235zH81HzRvr8AplmCY9x3nZfFa1Mn7nWaPo8HJ/C29MAg==","signature_status":"signed_v1","signed_at":"2026-07-05T06:11:56.971883Z","signed_message":"canonical_sha256_bytes"},"source_id":"2209.03416","source_kind":"arxiv","source_version":5}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:3b3a1a2caf10d7ee0313ecddcdd8c6534526b8e18b4c74e2f76c34701b6910ee","sha256:4f4501ac2c3c3f9b13765bdca6c4151496edadae464937e30fa2cc3e20be6739"],"state_sha256":"3fb58652ed46bafa3d7bfffa8245df08407770e1168a3c5b4706a072f37ce55b"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"JowwUpnaxvWk6WYHLxLKCzKQgXQXWDMy7JGgbddKXzf+8G8PLhuETiEameHjGX6Z7lVyNyX2d5ttE//9JmI7Dw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-08-14T06:51:04.844536Z","bundle_sha256":"3a73b118608a2d37a1c2956451f06780766834b5cc1a6014673a95c4ddbede81"}}