{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2022:WTAJHMXLTYK5BXZYW2JTXF4RQM","short_pith_number":"pith:WTAJHMXL","schema_version":"1.0","canonical_sha256":"b4c093b2eb9e15d0df38b6933b9791832a768dba9e6adbc34b33000fe7172244","source":{"kind":"arxiv","id":"2209.01173","version":1},"attestation_state":"computed","paper":{"title":"Optimal bump functions for shallow ReLU networks: Weight decay, depth separation and the curse of dimensionality","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.LG"],"primary_cat":"stat.ML","authors_text":"Stephan Wojtowytsch","submitted_at":"2022-09-02T17:14:36Z","abstract_excerpt":"In this note, we study how neural networks with a single hidden layer and ReLU activation interpolate data drawn from a radially symmetric distribution with target labels 1 at the origin and 0 outside the unit ball, if no labels are known inside the unit ball. With weight decay regularization and in the infinite neuron, infinite data limit, we prove that a unique radially symmetric minimizer exists, whose weight decay regularizer and Lipschitz constant grow as $d$ and $\\sqrt{d}$ respectively.\n  We furthermore show that the weight decay regularizer grows exponentially in $d$ if the label $1$ is"},"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":"2209.01173","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"stat.ML","submitted_at":"2022-09-02T17:14:36Z","cross_cats_sorted":["cs.LG"],"title_canon_sha256":"de16a9589f80c26a77a2dc36a3cff4ad52c70bd1ab9a02a5d4943143c7db1d68","abstract_canon_sha256":"e6dfe6f209352203d3f7450cf597da1a3e357bc104212ef67b50b528ee72c478"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T04:54:09.878586Z","signature_b64":"hzlnNIAXP6kA5mbSpDPD0pnln74g3yUe0SgC7/0YacnzmyCLxOdUVUrImAho8iWU+qqs3xb0Q+4WsxibJMEuBw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"b4c093b2eb9e15d0df38b6933b9791832a768dba9e6adbc34b33000fe7172244","last_reissued_at":"2026-07-05T04:54:09.878052Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T04:54:09.878052Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Optimal bump functions for shallow ReLU networks: Weight decay, depth separation and the curse of dimensionality","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.LG"],"primary_cat":"stat.ML","authors_text":"Stephan Wojtowytsch","submitted_at":"2022-09-02T17:14:36Z","abstract_excerpt":"In this note, we study how neural networks with a single hidden layer and ReLU activation interpolate data drawn from a radially symmetric distribution with target labels 1 at the origin and 0 outside the unit ball, if no labels are known inside the unit ball. With weight decay regularization and in the infinite neuron, infinite data limit, we prove that a unique radially symmetric minimizer exists, whose weight decay regularizer and Lipschitz constant grow as $d$ and $\\sqrt{d}$ respectively.\n  We furthermore show that the weight decay regularizer grows exponentially in $d$ if the label $1$ is"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2209.01173","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/2209.01173/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":"2209.01173","created_at":"2026-07-05T04:54:09.878107+00:00"},{"alias_kind":"arxiv_version","alias_value":"2209.01173v1","created_at":"2026-07-05T04:54:09.878107+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2209.01173","created_at":"2026-07-05T04:54:09.878107+00:00"},{"alias_kind":"pith_short_12","alias_value":"WTAJHMXLTYK5","created_at":"2026-07-05T04:54:09.878107+00:00"},{"alias_kind":"pith_short_16","alias_value":"WTAJHMXLTYK5BXZY","created_at":"2026-07-05T04:54:09.878107+00:00"},{"alias_kind":"pith_short_8","alias_value":"WTAJHMXL","created_at":"2026-07-05T04:54:09.878107+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/WTAJHMXLTYK5BXZYW2JTXF4RQM","json":"https://pith.science/pith/WTAJHMXLTYK5BXZYW2JTXF4RQM.json","graph_json":"https://pith.science/api/pith-number/WTAJHMXLTYK5BXZYW2JTXF4RQM/graph.json","events_json":"https://pith.science/api/pith-number/WTAJHMXLTYK5BXZYW2JTXF4RQM/events.json","paper":"https://pith.science/paper/WTAJHMXL"},"agent_actions":{"view_html":"https://pith.science/pith/WTAJHMXLTYK5BXZYW2JTXF4RQM","download_json":"https://pith.science/pith/WTAJHMXLTYK5BXZYW2JTXF4RQM.json","view_paper":"https://pith.science/paper/WTAJHMXL","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2209.01173&json=true","fetch_graph":"https://pith.science/api/pith-number/WTAJHMXLTYK5BXZYW2JTXF4RQM/graph.json","fetch_events":"https://pith.science/api/pith-number/WTAJHMXLTYK5BXZYW2JTXF4RQM/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/WTAJHMXLTYK5BXZYW2JTXF4RQM/action/timestamp_anchor","attest_storage":"https://pith.science/pith/WTAJHMXLTYK5BXZYW2JTXF4RQM/action/storage_attestation","attest_author":"https://pith.science/pith/WTAJHMXLTYK5BXZYW2JTXF4RQM/action/author_attestation","sign_citation":"https://pith.science/pith/WTAJHMXLTYK5BXZYW2JTXF4RQM/action/citation_signature","submit_replication":"https://pith.science/pith/WTAJHMXLTYK5BXZYW2JTXF4RQM/action/replication_record"}},"created_at":"2026-07-05T04:54:09.878107+00:00","updated_at":"2026-07-05T04:54:09.878107+00:00"}