{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2021:VUG4UZF3TXA3VKMMZ7OD2BAOVJ","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":"2201fd98dcd76c8cbcefdd2f287a691d617debacadb27398a406db6aaeca4c78","cross_cats_sorted":["cs.CV","hep-th"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.LG","submitted_at":"2021-05-28T15:41:52Z","title_canon_sha256":"0ed1397fc9c037759a493ea08b269bab3ab07f0304c6da652fc718b11a317655"},"schema_version":"1.0","source":{"id":"2105.13926","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2105.13926","created_at":"2026-07-05T02:44:12Z"},{"alias_kind":"arxiv_version","alias_value":"2105.13926v1","created_at":"2026-07-05T02:44:12Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2105.13926","created_at":"2026-07-05T02:44:12Z"},{"alias_kind":"pith_short_12","alias_value":"VUG4UZF3TXA3","created_at":"2026-07-05T02:44:12Z"},{"alias_kind":"pith_short_16","alias_value":"VUG4UZF3TXA3VKMM","created_at":"2026-07-05T02:44:12Z"},{"alias_kind":"pith_short_8","alias_value":"VUG4UZF3","created_at":"2026-07-05T02:44:12Z"}],"graph_snapshots":[{"event_id":"sha256:2f581c0b05233570c517311a8af48684bdf58f370f507dd13d938300a189c493","target":"graph","created_at":"2026-07-05T02:44:12Z","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/2105.13926/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"We survey the mathematical foundations of geometric deep learning, focusing on group equivariant and gauge equivariant neural networks. We develop gauge equivariant convolutional neural networks on arbitrary manifolds $\\mathcal{M}$ using principal bundles with structure group $K$ and equivariant maps between sections of associated vector bundles. We also discuss group equivariant neural networks for homogeneous spaces $\\mathcal{M}=G/K$, which are instead equivariant with respect to the global symmetry $G$ on $\\mathcal{M}$. Group equivariant layers can be interpreted as intertwiners between ind","authors_text":"Christoffer Petersson, Daniel Persson, Fredrik Ohlsson, Hampus Linander, Jan E. Gerken, Jimmy Aronsson, Oscar Carlsson","cross_cats":["cs.CV","hep-th"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.LG","submitted_at":"2021-05-28T15:41:52Z","title":"Geometric Deep Learning and Equivariant Neural Networks"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2105.13926","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":"sha256:967178c7e0581416203aac996136d5221b63e89994c6c0d50334685bfded61f8","target":"record","created_at":"2026-07-05T02:44:12Z","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":"2201fd98dcd76c8cbcefdd2f287a691d617debacadb27398a406db6aaeca4c78","cross_cats_sorted":["cs.CV","hep-th"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.LG","submitted_at":"2021-05-28T15:41:52Z","title_canon_sha256":"0ed1397fc9c037759a493ea08b269bab3ab07f0304c6da652fc718b11a317655"},"schema_version":"1.0","source":{"id":"2105.13926","kind":"arxiv","version":1}},"canonical_sha256":"ad0dca64bb9dc1baa98ccfdc3d040eaa59f9152eb23b52e9d4269aaaf0c9a6e0","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"ad0dca64bb9dc1baa98ccfdc3d040eaa59f9152eb23b52e9d4269aaaf0c9a6e0","first_computed_at":"2026-07-05T02:44:12.644881Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-05T02:44:12.644881Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"BOlMJg3A9q2swp317N1JQ6eYOe7YReCpmB/L8UrOoi0pivoR2WLwqNSehGxyImMs+wvDAh6ApDM8V5oI3qL4AA==","signature_status":"signed_v1","signed_at":"2026-07-05T02:44:12.645280Z","signed_message":"canonical_sha256_bytes"},"source_id":"2105.13926","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:967178c7e0581416203aac996136d5221b63e89994c6c0d50334685bfded61f8","sha256:2f581c0b05233570c517311a8af48684bdf58f370f507dd13d938300a189c493"],"state_sha256":"74202becf74c5ec6725c39f51874002d45efc5dd1f82103c2df80792a511f8d3"}