{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2015:MK4YR2GYHU7NYEOPY5222Z5JWL","short_pith_number":"pith:MK4YR2GY","canonical_record":{"source":{"id":"1505.04413","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"stat.ML","submitted_at":"2015-05-17T16:27:14Z","cross_cats_sorted":[],"title_canon_sha256":"3de83a121d95ac9ce49461fd84430d2d19001d46c8ff20a3491ccc803b15c877","abstract_canon_sha256":"8248e10bc6df1ef61682d5699e894cad06d39c4312dc4c0192b615912ac2df49"},"schema_version":"1.0"},"canonical_sha256":"62b988e8d83d3edc11cfc775ad67a9b2e19e7e08ef07dd06ac351f3ee624d53b","source":{"kind":"arxiv","id":"1505.04413","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1505.04413","created_at":"2026-05-17T23:48:07Z"},{"alias_kind":"arxiv_version","alias_value":"1505.04413v2","created_at":"2026-05-17T23:48:07Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1505.04413","created_at":"2026-05-17T23:48:07Z"},{"alias_kind":"pith_short_12","alias_value":"MK4YR2GYHU7N","created_at":"2026-05-18T12:29:32Z"},{"alias_kind":"pith_short_16","alias_value":"MK4YR2GYHU7NYEOP","created_at":"2026-05-18T12:29:32Z"},{"alias_kind":"pith_short_8","alias_value":"MK4YR2GY","created_at":"2026-05-18T12:29:32Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2015:MK4YR2GYHU7NYEOPY5222Z5JWL","target":"record","payload":{"canonical_record":{"source":{"id":"1505.04413","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"stat.ML","submitted_at":"2015-05-17T16:27:14Z","cross_cats_sorted":[],"title_canon_sha256":"3de83a121d95ac9ce49461fd84430d2d19001d46c8ff20a3491ccc803b15c877","abstract_canon_sha256":"8248e10bc6df1ef61682d5699e894cad06d39c4312dc4c0192b615912ac2df49"},"schema_version":"1.0"},"canonical_sha256":"62b988e8d83d3edc11cfc775ad67a9b2e19e7e08ef07dd06ac351f3ee624d53b","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-17T23:48:07.112628Z","signature_b64":"WmDRFBK/fZYwg7xFcXNXGPNrpi1y2pQnwiG4mWvK1MXyjZgVHZ/ZdKyah8uG9AFqHz4QntOyHZrOZMO9LMgUAQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"62b988e8d83d3edc11cfc775ad67a9b2e19e7e08ef07dd06ac351f3ee624d53b","last_reissued_at":"2026-05-17T23:48:07.112083Z","signature_status":"signed_v1","first_computed_at":"2026-05-17T23:48:07.112083Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1505.04413","source_version":2,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-17T23:48:07Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"OuoExCyuHTTPqWSyBcnp83SYxMPGYOmALXunpVNSqOW4qhAjgz0ByXgaaQ1XV7Om0T936sz3x5CL7d7OKmYnBw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-05T09:05:20.317337Z"},"content_sha256":"832abf5606b57f6767b6f3aba5ae137af417f0c64cfde22f4321bd7a4e66612e","schema_version":"1.0","event_id":"sha256:832abf5606b57f6767b6f3aba5ae137af417f0c64cfde22f4321bd7a4e66612e"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2015:MK4YR2GYHU7NYEOPY5222Z5JWL","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Harmonic Exponential Families on Manifolds","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"stat.ML","authors_text":"Max Welling, Taco S. Cohen","submitted_at":"2015-05-17T16:27:14Z","abstract_excerpt":"In a range of fields including the geosciences, molecular biology, robotics and computer vision, one encounters problems that involve random variables on manifolds. Currently, there is a lack of flexible probabilistic models on manifolds that are fast and easy to train. We define an extremely flexible class of exponential family distributions on manifolds such as the torus, sphere, and rotation groups, and show that for these distributions the gradient of the log-likelihood can be computed efficiently using a non-commutative generalization of the Fast Fourier Transform (FFT). We discuss applic"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1505.04413","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"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-17T23:48:07Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"6NQ52ZZCkVjQoFstCYvkn4jRWm/qsrKA8dNDwkHGy9azup6+X4pMsXpTwmxqMqGk1XBBOegvc9J5fURHGG1uAg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-05T09:05:20.317957Z"},"content_sha256":"d4c3e4f3bcf2198ed0b624b6cab0667274d3f53fffe83ecd1e69319be89b7cfa","schema_version":"1.0","event_id":"sha256:d4c3e4f3bcf2198ed0b624b6cab0667274d3f53fffe83ecd1e69319be89b7cfa"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/MK4YR2GYHU7NYEOPY5222Z5JWL/bundle.json","state_url":"https://pith.science/pith/MK4YR2GYHU7NYEOPY5222Z5JWL/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/MK4YR2GYHU7NYEOPY5222Z5JWL/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":"2026-06-05T09:05:20Z","links":{"resolver":"https://pith.science/pith/MK4YR2GYHU7NYEOPY5222Z5JWL","bundle":"https://pith.science/pith/MK4YR2GYHU7NYEOPY5222Z5JWL/bundle.json","state":"https://pith.science/pith/MK4YR2GYHU7NYEOPY5222Z5JWL/state.json","well_known_bundle":"https://pith.science/.well-known/pith/MK4YR2GYHU7NYEOPY5222Z5JWL/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2015:MK4YR2GYHU7NYEOPY5222Z5JWL","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":"8248e10bc6df1ef61682d5699e894cad06d39c4312dc4c0192b615912ac2df49","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"stat.ML","submitted_at":"2015-05-17T16:27:14Z","title_canon_sha256":"3de83a121d95ac9ce49461fd84430d2d19001d46c8ff20a3491ccc803b15c877"},"schema_version":"1.0","source":{"id":"1505.04413","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1505.04413","created_at":"2026-05-17T23:48:07Z"},{"alias_kind":"arxiv_version","alias_value":"1505.04413v2","created_at":"2026-05-17T23:48:07Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1505.04413","created_at":"2026-05-17T23:48:07Z"},{"alias_kind":"pith_short_12","alias_value":"MK4YR2GYHU7N","created_at":"2026-05-18T12:29:32Z"},{"alias_kind":"pith_short_16","alias_value":"MK4YR2GYHU7NYEOP","created_at":"2026-05-18T12:29:32Z"},{"alias_kind":"pith_short_8","alias_value":"MK4YR2GY","created_at":"2026-05-18T12:29:32Z"}],"graph_snapshots":[{"event_id":"sha256:d4c3e4f3bcf2198ed0b624b6cab0667274d3f53fffe83ecd1e69319be89b7cfa","target":"graph","created_at":"2026-05-17T23:48:07Z","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":"In a range of fields including the geosciences, molecular biology, robotics and computer vision, one encounters problems that involve random variables on manifolds. Currently, there is a lack of flexible probabilistic models on manifolds that are fast and easy to train. We define an extremely flexible class of exponential family distributions on manifolds such as the torus, sphere, and rotation groups, and show that for these distributions the gradient of the log-likelihood can be computed efficiently using a non-commutative generalization of the Fast Fourier Transform (FFT). We discuss applic","authors_text":"Max Welling, Taco S. Cohen","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"stat.ML","submitted_at":"2015-05-17T16:27:14Z","title":"Harmonic Exponential Families on Manifolds"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1505.04413","kind":"arxiv","version":2},"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":"sha256:832abf5606b57f6767b6f3aba5ae137af417f0c64cfde22f4321bd7a4e66612e","target":"record","created_at":"2026-05-17T23:48:07Z","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":"8248e10bc6df1ef61682d5699e894cad06d39c4312dc4c0192b615912ac2df49","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"stat.ML","submitted_at":"2015-05-17T16:27:14Z","title_canon_sha256":"3de83a121d95ac9ce49461fd84430d2d19001d46c8ff20a3491ccc803b15c877"},"schema_version":"1.0","source":{"id":"1505.04413","kind":"arxiv","version":2}},"canonical_sha256":"62b988e8d83d3edc11cfc775ad67a9b2e19e7e08ef07dd06ac351f3ee624d53b","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"62b988e8d83d3edc11cfc775ad67a9b2e19e7e08ef07dd06ac351f3ee624d53b","first_computed_at":"2026-05-17T23:48:07.112083Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-17T23:48:07.112083Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"WmDRFBK/fZYwg7xFcXNXGPNrpi1y2pQnwiG4mWvK1MXyjZgVHZ/ZdKyah8uG9AFqHz4QntOyHZrOZMO9LMgUAQ==","signature_status":"signed_v1","signed_at":"2026-05-17T23:48:07.112628Z","signed_message":"canonical_sha256_bytes"},"source_id":"1505.04413","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:832abf5606b57f6767b6f3aba5ae137af417f0c64cfde22f4321bd7a4e66612e","sha256:d4c3e4f3bcf2198ed0b624b6cab0667274d3f53fffe83ecd1e69319be89b7cfa"],"state_sha256":"9835839f0a5a21e8a6c20de384eb35d0c7d9d72856970feddeb8bc648ea97c99"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"dZg7QT71qCh0ruJ0rRTS7//x0yIQ0DFJFOOolWNcfMGPz4qTxUaw59iTS1IFR76aZIIRNSNcPbnkjBxHhJMMAw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-05T09:05:20.321047Z","bundle_sha256":"b710855f35493bf353e45cb24d66e1bb985b50278cda2419ba09583f29eea059"}}