{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2022:E27G7UIVBCGDU622ZGWM2VCNJD","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":"51843f367f0b202a85d4015c42b3b540be4ac6f5a63cef40277d9bf8bc88483a","cross_cats_sorted":["cs.LG","cs.NE"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.CC","submitted_at":"2022-04-04T10:28:11Z","title_canon_sha256":"e7577955370e0fffb73e8cb23903ffb7986e3ff01a9b29e879a126c86d749880"},"schema_version":"1.0","source":{"id":"2204.01368","kind":"arxiv","version":3}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2204.01368","created_at":"2026-07-05T07:59:12Z"},{"alias_kind":"arxiv_version","alias_value":"2204.01368v3","created_at":"2026-07-05T07:59:12Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2204.01368","created_at":"2026-07-05T07:59:12Z"},{"alias_kind":"pith_short_12","alias_value":"E27G7UIVBCGD","created_at":"2026-07-05T07:59:12Z"},{"alias_kind":"pith_short_16","alias_value":"E27G7UIVBCGDU622","created_at":"2026-07-05T07:59:12Z"},{"alias_kind":"pith_short_8","alias_value":"E27G7UIV","created_at":"2026-07-05T07:59:12Z"}],"graph_snapshots":[{"event_id":"sha256:da4d88e0014131f089ffebf824be20de6a36c549237f6737fa06f1dc56fcbacf","target":"graph","created_at":"2026-07-05T07:59:12Z","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/2204.01368/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"We consider the problem of finding weights and biases for a two-layer fully connected neural network to fit a given set of data points as well as possible, also known as EmpiricalRiskMinimization. Our main result is that the associated decision problem is $\\exists\\mathbb{R}$-complete, that is, polynomial-time equivalent to determining whether a multivariate polynomial with integer coefficients has any real roots. Furthermore, we prove that algebraic numbers of arbitrarily large degree are required as weights to be able to train some instances to optimality, even if all data points are rational","authors_text":"Christoph Hertrich, Daniel Bertschinger, Paul Jungeblut, Simon Weber, Tillmann Miltzow","cross_cats":["cs.LG","cs.NE"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.CC","submitted_at":"2022-04-04T10:28:11Z","title":"Training Fully Connected Neural Networks is $\\exists\\mathbb{R}$-Complete"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2204.01368","kind":"arxiv","version":3},"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":"sha256:78cefa993f00f47f537bc9cfde7a7abe1e1146740130108fedbf03799472ebd3","target":"record","created_at":"2026-07-05T07:59:12Z","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":"51843f367f0b202a85d4015c42b3b540be4ac6f5a63cef40277d9bf8bc88483a","cross_cats_sorted":["cs.LG","cs.NE"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.CC","submitted_at":"2022-04-04T10:28:11Z","title_canon_sha256":"e7577955370e0fffb73e8cb23903ffb7986e3ff01a9b29e879a126c86d749880"},"schema_version":"1.0","source":{"id":"2204.01368","kind":"arxiv","version":3}},"canonical_sha256":"26be6fd115088c3a7b5ac9accd544d48f3c272bc7aa3728034bae030a7afffc6","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"26be6fd115088c3a7b5ac9accd544d48f3c272bc7aa3728034bae030a7afffc6","first_computed_at":"2026-07-05T07:59:12.808866Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-05T07:59:12.808866Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"jGkHlQxzsqA0ICy0ZsXlGU4iZpC63jP0q3o9EJt4/8ebgTRXidCWFkVtjTF4rHXqftSXIXIELfWodf1DeZ26Dw==","signature_status":"signed_v1","signed_at":"2026-07-05T07:59:12.809273Z","signed_message":"canonical_sha256_bytes"},"source_id":"2204.01368","source_kind":"arxiv","source_version":3}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:78cefa993f00f47f537bc9cfde7a7abe1e1146740130108fedbf03799472ebd3","sha256:da4d88e0014131f089ffebf824be20de6a36c549237f6737fa06f1dc56fcbacf"],"state_sha256":"f7f935d5b855890fb865cfcdf2fe705a36cc005b03103c7e121bcede3f44d486"}