{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2016:HVDMKQWVCYIBTUVOKEWIBKGGSW","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":"b825a5ba7ff603bdb1a6305499a31bd18cbbd4709dd200616a0e2c978439ce6d","cross_cats_sorted":["cs.NE"],"license":"http://creativecommons.org/publicdomain/zero/1.0/","primary_cat":"cs.LG","submitted_at":"2016-10-13T16:34:30Z","title_canon_sha256":"6181a9743297cd5be3b34a4d45416e141482126ee8b9da675ba02571ccbc1431"},"schema_version":"1.0","source":{"id":"1610.04161","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1610.04161","created_at":"2026-05-18T00:49:31Z"},{"alias_kind":"arxiv_version","alias_value":"1610.04161v2","created_at":"2026-05-18T00:49:31Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1610.04161","created_at":"2026-05-18T00:49:31Z"},{"alias_kind":"pith_short_12","alias_value":"HVDMKQWVCYIB","created_at":"2026-05-18T12:30:22Z"},{"alias_kind":"pith_short_16","alias_value":"HVDMKQWVCYIBTUVO","created_at":"2026-05-18T12:30:22Z"},{"alias_kind":"pith_short_8","alias_value":"HVDMKQWV","created_at":"2026-05-18T12:30:22Z"}],"graph_snapshots":[{"event_id":"sha256:02b7a749b3c47e656d10c16f8c252df7b808e7d8e054fc95df917c4258ff58e2","target":"graph","created_at":"2026-05-18T00:49:31Z","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"},"paper":{"abstract_excerpt":"Recently there has been much interest in understanding why deep neural networks are preferred to shallow networks. We show that, for a large class of piecewise smooth functions, the number of neurons needed by a shallow network to approximate a function is exponentially larger than the corresponding number of neurons needed by a deep network for a given degree of function approximation. First, we consider univariate functions on a bounded interval and require a neural network to achieve an approximation error of $\\varepsilon$ uniformly over the interval. We show that shallow networks (i.e., ne","authors_text":"R. Srikant, Shiyu Liang","cross_cats":["cs.NE"],"headline":"","license":"http://creativecommons.org/publicdomain/zero/1.0/","primary_cat":"cs.LG","submitted_at":"2016-10-13T16:34:30Z","title":"Why Deep Neural Networks for Function Approximation?"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1610.04161","kind":"arxiv","version":2},"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:dd2b8bfd280f95c00c654af8c93ada87714ff44069ce8795ae375014e7f191df","target":"record","created_at":"2026-05-18T00:49:31Z","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":"b825a5ba7ff603bdb1a6305499a31bd18cbbd4709dd200616a0e2c978439ce6d","cross_cats_sorted":["cs.NE"],"license":"http://creativecommons.org/publicdomain/zero/1.0/","primary_cat":"cs.LG","submitted_at":"2016-10-13T16:34:30Z","title_canon_sha256":"6181a9743297cd5be3b34a4d45416e141482126ee8b9da675ba02571ccbc1431"},"schema_version":"1.0","source":{"id":"1610.04161","kind":"arxiv","version":2}},"canonical_sha256":"3d46c542d5161019d2ae512c80a8c6959fef405fbe32b67740d1edb8a38669bb","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"3d46c542d5161019d2ae512c80a8c6959fef405fbe32b67740d1edb8a38669bb","first_computed_at":"2026-05-18T00:49:31.986073Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:49:31.986073Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"1i4Tqr6SQAJ1x5FUq6yY+U5bVlnKAqODNp0B8lbWWHNnqQPB8n7b2fAJ9ElSbiOpYfXdX6YYv7o4VLVgDzCaCQ==","signature_status":"signed_v1","signed_at":"2026-05-18T00:49:31.986624Z","signed_message":"canonical_sha256_bytes"},"source_id":"1610.04161","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:dd2b8bfd280f95c00c654af8c93ada87714ff44069ce8795ae375014e7f191df","sha256:02b7a749b3c47e656d10c16f8c252df7b808e7d8e054fc95df917c4258ff58e2"],"state_sha256":"664519692ef3ad287283eca9feded4c746145ab42b51039b9bf3810a7ccaf1e4"}