{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2026:4ECQC54MDC5OWPYH6TFC2327A7","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":"aff58b3697b6b805f272ab94686d857f4d24c04ab65f9407af0c0487a59bb3ae","cross_cats_sorted":[],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"cs.LG","submitted_at":"2026-06-01T17:01:08Z","title_canon_sha256":"ec6d7831ce40f9a7864eb2e9fcf7d1fbe69585033e143bb55a91f1320f4e5982"},"schema_version":"1.0","source":{"id":"2606.02490","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2606.02490","created_at":"2026-06-02T03:05:07Z"},{"alias_kind":"arxiv_version","alias_value":"2606.02490v1","created_at":"2026-06-02T03:05:07Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2606.02490","created_at":"2026-06-02T03:05:07Z"},{"alias_kind":"pith_short_12","alias_value":"4ECQC54MDC5O","created_at":"2026-06-02T03:05:07Z"},{"alias_kind":"pith_short_16","alias_value":"4ECQC54MDC5OWPYH","created_at":"2026-06-02T03:05:07Z"},{"alias_kind":"pith_short_8","alias_value":"4ECQC54M","created_at":"2026-06-02T03:05:07Z"}],"graph_snapshots":[{"event_id":"sha256:98896fe1000a5cbf787504fa2494935fda68bab4672a2417db0c3cbeaabea38d","target":"graph","created_at":"2026-06-02T03:05:07Z","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"},"integrity":{"available":true,"clean":true,"detectors_run":[],"endpoint":"/pith/2606.02490/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"This work studies neural architectures for classifying symmetric positive-definite matrices, focusing on congruence-like layers, in which the input matrix is multiplied on the left and right by a (possibly rectangular) weight matrix $W$ and its transpose. Such layers lie at the core of the celebrated SPDNet and have also been employed independently for dimensionality reduction on positive-definite data. We show that the (semi)-orthogonality constraint commonly imposed on $W$ limits the expressivity of these layers: for certain activation functions, the resulting architecture collapses to a one","authors_text":"Antonin Oswald, Estelle Massart","cross_cats":[],"headline":"","license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"cs.LG","submitted_at":"2026-06-01T17:01:08Z","title":"Expressivity of congruence-based architectures for DNNs on positive-definite matrices"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2606.02490","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:f4d15ee39ba4655a4308888d422fd7f2744b2cece2c1ec8583278caf2e411ae8","target":"record","created_at":"2026-06-02T03:05:07Z","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":"aff58b3697b6b805f272ab94686d857f4d24c04ab65f9407af0c0487a59bb3ae","cross_cats_sorted":[],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"cs.LG","submitted_at":"2026-06-01T17:01:08Z","title_canon_sha256":"ec6d7831ce40f9a7864eb2e9fcf7d1fbe69585033e143bb55a91f1320f4e5982"},"schema_version":"1.0","source":{"id":"2606.02490","kind":"arxiv","version":1}},"canonical_sha256":"e10501778c18baeb3f07f4ca2d6f5f07cac00e393201bc43ccd26393bc3496b5","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"e10501778c18baeb3f07f4ca2d6f5f07cac00e393201bc43ccd26393bc3496b5","first_computed_at":"2026-06-02T03:05:07.654693Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-06-02T03:05:07.654693Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"f+8wbDG4zuuU6Y0p9wOjkUCFjIZXXUaLdIxzemi7aNrQNsJRUJxc941P0+cKGCWWERa6miFXKmezGasxRZfECw==","signature_status":"signed_v1","signed_at":"2026-06-02T03:05:07.655127Z","signed_message":"canonical_sha256_bytes"},"source_id":"2606.02490","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:f4d15ee39ba4655a4308888d422fd7f2744b2cece2c1ec8583278caf2e411ae8","sha256:98896fe1000a5cbf787504fa2494935fda68bab4672a2417db0c3cbeaabea38d"],"state_sha256":"5952d48ed0f92b35dbd15ac0baba266ca9c4dc0c8b96a2ea092c2286c911de54"}