{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2025:OVOPLXPEP4EHDTIO67HPZMOVAK","short_pith_number":"pith:OVOPLXPE","schema_version":"1.0","canonical_sha256":"755cf5dde47f0871cd0ef7cefcb1d502b418046b59862b056438ac9d0a36bc96","source":{"kind":"arxiv","id":"2503.08502","version":1},"attestation_state":"computed","paper":{"title":"The Space Between: On Folding, Symmetries and Sampling","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":["cs.NE"],"primary_cat":"cs.LG","authors_text":"Bernhard A.Moser, Bernhard Heinzl, Michal Lewandowski, Raphael Pisoni","submitted_at":"2025-03-11T14:54:25Z","abstract_excerpt":"Recent findings suggest that consecutive layers of neural networks with the ReLU activation function \\emph{fold} the input space during the learning process. While many works hint at this phenomenon, an approach to quantify the folding was only recently proposed by means of a space folding measure based on Hamming distance in the ReLU activation space. We generalize this measure to a wider class of activation functions through introduction of equivalence classes of input data, analyse its mathematical and computational properties and come up with an efficient sampling strategy for its implemen"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"2503.08502","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"cs.LG","submitted_at":"2025-03-11T14:54:25Z","cross_cats_sorted":["cs.NE"],"title_canon_sha256":"883f5413680f832f3b0508cda22213e75fdfe2f967a9847b46a80c7acd231bef","abstract_canon_sha256":"b51cd63f93bb397691fa75b0327a4b104849c4b4699cc072bd343ff3bccf6416"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T10:29:00.807159Z","signature_b64":"rgcmxINSvX0Y7AxzKkkmuk0dlRmB87JMIm+5vp9EbHn3geom6BUBcGtNvpVOmhyiLi2dz6PJfHysq1TjhnFLCQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"755cf5dde47f0871cd0ef7cefcb1d502b418046b59862b056438ac9d0a36bc96","last_reissued_at":"2026-07-05T10:29:00.806620Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T10:29:00.806620Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"The Space Between: On Folding, Symmetries and Sampling","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":["cs.NE"],"primary_cat":"cs.LG","authors_text":"Bernhard A.Moser, Bernhard Heinzl, Michal Lewandowski, Raphael Pisoni","submitted_at":"2025-03-11T14:54:25Z","abstract_excerpt":"Recent findings suggest that consecutive layers of neural networks with the ReLU activation function \\emph{fold} the input space during the learning process. While many works hint at this phenomenon, an approach to quantify the folding was only recently proposed by means of a space folding measure based on Hamming distance in the ReLU activation space. We generalize this measure to a wider class of activation functions through introduction of equivalence classes of input data, analyse its mathematical and computational properties and come up with an efficient sampling strategy for its implemen"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2503.08502","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2503.08502/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"},"aliases":[{"alias_kind":"arxiv","alias_value":"2503.08502","created_at":"2026-07-05T10:29:00.806687+00:00"},{"alias_kind":"arxiv_version","alias_value":"2503.08502v1","created_at":"2026-07-05T10:29:00.806687+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2503.08502","created_at":"2026-07-05T10:29:00.806687+00:00"},{"alias_kind":"pith_short_12","alias_value":"OVOPLXPEP4EH","created_at":"2026-07-05T10:29:00.806687+00:00"},{"alias_kind":"pith_short_16","alias_value":"OVOPLXPEP4EHDTIO","created_at":"2026-07-05T10:29:00.806687+00:00"},{"alias_kind":"pith_short_8","alias_value":"OVOPLXPE","created_at":"2026-07-05T10:29:00.806687+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/OVOPLXPEP4EHDTIO67HPZMOVAK","json":"https://pith.science/pith/OVOPLXPEP4EHDTIO67HPZMOVAK.json","graph_json":"https://pith.science/api/pith-number/OVOPLXPEP4EHDTIO67HPZMOVAK/graph.json","events_json":"https://pith.science/api/pith-number/OVOPLXPEP4EHDTIO67HPZMOVAK/events.json","paper":"https://pith.science/paper/OVOPLXPE"},"agent_actions":{"view_html":"https://pith.science/pith/OVOPLXPEP4EHDTIO67HPZMOVAK","download_json":"https://pith.science/pith/OVOPLXPEP4EHDTIO67HPZMOVAK.json","view_paper":"https://pith.science/paper/OVOPLXPE","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2503.08502&json=true","fetch_graph":"https://pith.science/api/pith-number/OVOPLXPEP4EHDTIO67HPZMOVAK/graph.json","fetch_events":"https://pith.science/api/pith-number/OVOPLXPEP4EHDTIO67HPZMOVAK/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/OVOPLXPEP4EHDTIO67HPZMOVAK/action/timestamp_anchor","attest_storage":"https://pith.science/pith/OVOPLXPEP4EHDTIO67HPZMOVAK/action/storage_attestation","attest_author":"https://pith.science/pith/OVOPLXPEP4EHDTIO67HPZMOVAK/action/author_attestation","sign_citation":"https://pith.science/pith/OVOPLXPEP4EHDTIO67HPZMOVAK/action/citation_signature","submit_replication":"https://pith.science/pith/OVOPLXPEP4EHDTIO67HPZMOVAK/action/replication_record"}},"created_at":"2026-07-05T10:29:00.806687+00:00","updated_at":"2026-07-05T10:29:00.806687+00:00"}