{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2020:3HLABZ5MJO57N5KHP5ZNKH2IPK","short_pith_number":"pith:3HLABZ5M","canonical_record":{"source":{"id":"2006.03627","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.LG","submitted_at":"2020-06-05T18:42:12Z","cross_cats_sorted":["cs.CV","math.GR","stat.ML"],"title_canon_sha256":"e76953ccd7c716cec59c09acb9d3405b336adcda6255a099a415a2e01622df58","abstract_canon_sha256":"3f8a66b3831e6c30ae5cc8fb8894927631a34b36eee76e57bb7e76f55735ee0d"},"schema_version":"1.0"},"canonical_sha256":"d9d600e7ac4bbbf6f5477f72d51f487a9754c38265fe705cf3a4f18c02cf87ab","source":{"kind":"arxiv","id":"2006.03627","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2006.03627","created_at":"2026-07-05T01:53:51Z"},{"alias_kind":"arxiv_version","alias_value":"2006.03627v2","created_at":"2026-07-05T01:53:51Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2006.03627","created_at":"2026-07-05T01:53:51Z"},{"alias_kind":"pith_short_12","alias_value":"3HLABZ5MJO57","created_at":"2026-07-05T01:53:51Z"},{"alias_kind":"pith_short_16","alias_value":"3HLABZ5MJO57N5KH","created_at":"2026-07-05T01:53:51Z"},{"alias_kind":"pith_short_8","alias_value":"3HLABZ5M","created_at":"2026-07-05T01:53:51Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2020:3HLABZ5MJO57N5KHP5ZNKH2IPK","target":"record","payload":{"canonical_record":{"source":{"id":"2006.03627","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.LG","submitted_at":"2020-06-05T18:42:12Z","cross_cats_sorted":["cs.CV","math.GR","stat.ML"],"title_canon_sha256":"e76953ccd7c716cec59c09acb9d3405b336adcda6255a099a415a2e01622df58","abstract_canon_sha256":"3f8a66b3831e6c30ae5cc8fb8894927631a34b36eee76e57bb7e76f55735ee0d"},"schema_version":"1.0"},"canonical_sha256":"d9d600e7ac4bbbf6f5477f72d51f487a9754c38265fe705cf3a4f18c02cf87ab","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T01:53:51.508311Z","signature_b64":"m9LI1IYfI7S6E8JRNGGCTTiSFcE8yyiRtGRLAkxQrVezPrVEzaS0oM0kDHm6JBSVf3po6KQJ7cZZo8toSKLdAQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"d9d600e7ac4bbbf6f5477f72d51f487a9754c38265fe705cf3a4f18c02cf87ab","last_reissued_at":"2026-07-05T01:53:51.507899Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T01:53:51.507899Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2006.03627","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":"2026-07-05T01:53:51Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"jsFHYRzmeNJxdpz2RXJ464dVbOnYO41Kx+B77Oo5ET31E+vRPBYLCC3Wl13DL9ssQYn4OdzFOu1GSNjaULvzCw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-06T16:42:54.768421Z"},"content_sha256":"11291653b7fa60d94258822e0d5ebca7204e3e1ed53ba1fe472b27282a937d6a","schema_version":"1.0","event_id":"sha256:11291653b7fa60d94258822e0d5ebca7204e3e1ed53ba1fe472b27282a937d6a"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2020:3HLABZ5MJO57N5KHP5ZNKH2IPK","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Equivariant Maps for Hierarchical Structures","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.CV","math.GR","stat.ML"],"primary_cat":"cs.LG","authors_text":"Marjan Albooyeh, Renhao Wang, Siamak Ravanbakhsh","submitted_at":"2020-06-05T18:42:12Z","abstract_excerpt":"While using invariant and equivariant maps, it is possible to apply deep learning to a range of primitive data structures, a formalism for dealing with hierarchy is lacking. This is a significant issue because many practical structures are hierarchies of simple building blocks; some examples include sequences of sets, graphs of graphs, or multiresolution images. Observing that the symmetry of a hierarchical structure is the \"wreath product\" of symmetries of the building blocks, we express the equivariant map for the hierarchy using an intuitive combination of the equivariant linear layers of t"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2006.03627","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2006.03627/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":"2026-07-05T01:53:51Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"n9tHrL4yADJRqlEteil0Fxb3Wo+JVHIrB0AxgIlN7+Ju1xqWB3N05LJCH/j4PiVh/BNZ/ivXdHpgw4NcngJkAw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-06T16:42:54.769747Z"},"content_sha256":"9d5d1a227dac1aed1ec636664bae104d518ea2c77dc4a0698cde8f198e42088b","schema_version":"1.0","event_id":"sha256:9d5d1a227dac1aed1ec636664bae104d518ea2c77dc4a0698cde8f198e42088b"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/3HLABZ5MJO57N5KHP5ZNKH2IPK/bundle.json","state_url":"https://pith.science/pith/3HLABZ5MJO57N5KHP5ZNKH2IPK/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/3HLABZ5MJO57N5KHP5ZNKH2IPK/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":"2026-08-06T16:42:54Z","links":{"resolver":"https://pith.science/pith/3HLABZ5MJO57N5KHP5ZNKH2IPK","bundle":"https://pith.science/pith/3HLABZ5MJO57N5KHP5ZNKH2IPK/bundle.json","state":"https://pith.science/pith/3HLABZ5MJO57N5KHP5ZNKH2IPK/state.json","well_known_bundle":"https://pith.science/.well-known/pith/3HLABZ5MJO57N5KHP5ZNKH2IPK/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2020:3HLABZ5MJO57N5KHP5ZNKH2IPK","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":"3f8a66b3831e6c30ae5cc8fb8894927631a34b36eee76e57bb7e76f55735ee0d","cross_cats_sorted":["cs.CV","math.GR","stat.ML"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.LG","submitted_at":"2020-06-05T18:42:12Z","title_canon_sha256":"e76953ccd7c716cec59c09acb9d3405b336adcda6255a099a415a2e01622df58"},"schema_version":"1.0","source":{"id":"2006.03627","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2006.03627","created_at":"2026-07-05T01:53:51Z"},{"alias_kind":"arxiv_version","alias_value":"2006.03627v2","created_at":"2026-07-05T01:53:51Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2006.03627","created_at":"2026-07-05T01:53:51Z"},{"alias_kind":"pith_short_12","alias_value":"3HLABZ5MJO57","created_at":"2026-07-05T01:53:51Z"},{"alias_kind":"pith_short_16","alias_value":"3HLABZ5MJO57N5KH","created_at":"2026-07-05T01:53:51Z"},{"alias_kind":"pith_short_8","alias_value":"3HLABZ5M","created_at":"2026-07-05T01:53:51Z"}],"graph_snapshots":[{"event_id":"sha256:9d5d1a227dac1aed1ec636664bae104d518ea2c77dc4a0698cde8f198e42088b","target":"graph","created_at":"2026-07-05T01:53:51Z","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/2006.03627/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"While using invariant and equivariant maps, it is possible to apply deep learning to a range of primitive data structures, a formalism for dealing with hierarchy is lacking. This is a significant issue because many practical structures are hierarchies of simple building blocks; some examples include sequences of sets, graphs of graphs, or multiresolution images. Observing that the symmetry of a hierarchical structure is the \"wreath product\" of symmetries of the building blocks, we express the equivariant map for the hierarchy using an intuitive combination of the equivariant linear layers of t","authors_text":"Marjan Albooyeh, Renhao Wang, Siamak Ravanbakhsh","cross_cats":["cs.CV","math.GR","stat.ML"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.LG","submitted_at":"2020-06-05T18:42:12Z","title":"Equivariant Maps for Hierarchical Structures"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2006.03627","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":"sha256:11291653b7fa60d94258822e0d5ebca7204e3e1ed53ba1fe472b27282a937d6a","target":"record","created_at":"2026-07-05T01:53:51Z","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":"3f8a66b3831e6c30ae5cc8fb8894927631a34b36eee76e57bb7e76f55735ee0d","cross_cats_sorted":["cs.CV","math.GR","stat.ML"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.LG","submitted_at":"2020-06-05T18:42:12Z","title_canon_sha256":"e76953ccd7c716cec59c09acb9d3405b336adcda6255a099a415a2e01622df58"},"schema_version":"1.0","source":{"id":"2006.03627","kind":"arxiv","version":2}},"canonical_sha256":"d9d600e7ac4bbbf6f5477f72d51f487a9754c38265fe705cf3a4f18c02cf87ab","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"d9d600e7ac4bbbf6f5477f72d51f487a9754c38265fe705cf3a4f18c02cf87ab","first_computed_at":"2026-07-05T01:53:51.507899Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-05T01:53:51.507899Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"m9LI1IYfI7S6E8JRNGGCTTiSFcE8yyiRtGRLAkxQrVezPrVEzaS0oM0kDHm6JBSVf3po6KQJ7cZZo8toSKLdAQ==","signature_status":"signed_v1","signed_at":"2026-07-05T01:53:51.508311Z","signed_message":"canonical_sha256_bytes"},"source_id":"2006.03627","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:11291653b7fa60d94258822e0d5ebca7204e3e1ed53ba1fe472b27282a937d6a","sha256:9d5d1a227dac1aed1ec636664bae104d518ea2c77dc4a0698cde8f198e42088b"],"state_sha256":"7cccf6a4e008bf3de020bcb95c787297aa439df526833f31e82760877e81cb7a"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"iUfYEIRJF1I12AxcBfXG11TTFEC6suuQa6p5e97gM7Lsmou8yS4F6EUrXtL26A2HS1Sb5vEvivBB/4aG4uM6Cg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-08-06T16:42:54.779159Z","bundle_sha256":"a7e0711d1431f6d1ef616b919882c97b4900a525ff5990bf1956e4e54285d2e4"}}