Pith Number

pith:OWPK4BFY

pith:2026:OWPK4BFY47CJHQVEV4BYDH6Z7D

not attested not anchored not stored refs pending

Neural Networks With Dense Weights Are Not Universal Approximators

Levi Rauchwerger, Ron Levie, Stefanie Jegelka

Dense neural networks with constraints on weights and dimensions cannot approximate all Lipschitz continuous functions.

arxiv:2602.07618 v6 · 2026-02-07 · cs.LG · stat.ML

Open paper page JSON Open Graph Bundle Merged state Verified badge What is a Pith Number?

Add to your LaTeX paper

\usepackage{pith}
\pithnumber{OWPK4BFY47CJHQVEV4BYDH6Z7D}

Prints a linked badge after your title and injects PDF metadata. Compiles on arXiv. Learn more · Embed verified badge

Record completeness

1 Bitcoin timestamp

2 Internet Archive

3 Author claim open · sign in to claim

4 Citations open

5 Replications open

✓ Portable graph bundle live · download bundle · merged state

The bundle contains the canonical record plus signed events. A mirror can host it anywhere and recompute the same current state with the deterministic merge algorithm.

Claims

C1strongest claim

We demonstrate the existence of Lipschitz continuous functions that cannot be approximated by such networks.

C2weakest assumption

The chosen constraints on weights and dimensions accurately model a notion of dense connectivity, and feedforward networks can be interpreted as message passing graph neural networks without loss of generality for the approximation question.

C3one line summary

Dense neural networks subject to constraints on weights and dimensions cannot approximate arbitrary Lipschitz continuous functions.

Formal links

2 machine-checked theorem links

Receipt and verification

First computed	2026-05-20T00:05:41.790668Z
Builder	pith-number-builder-2026-05-17-v1
Signature	Pith Ed25519 (`pith-v1-2026-05`) · public key
Schema	pith-number/v1.0

Canonical hash

759eae04b8e7c493c2a4af03819fd9f8e792acd6b7cdd40d4aea29745088d97d

Aliases

arxiv: 2602.07618 · arxiv_version: 2602.07618v6 · doi: 10.48550/arxiv.2602.07618 · pith_short_12: OWPK4BFY47CJ · pith_short_16: OWPK4BFY47CJHQVE · pith_short_8: OWPK4BFY

Agent API

Resolver JSON Graph JSON Events JSON Schema Signing key

Verify this Pith Number yourself

curl -sH 'Accept: application/ld+json' https://pith.science/pith/OWPK4BFY47CJHQVEV4BYDH6Z7D \
  | jq -c '.canonical_record' \
  | python3 -c "import sys,json,hashlib; b=json.dumps(json.loads(sys.stdin.read()), sort_keys=True, separators=(',',':'), ensure_ascii=False).encode(); print(hashlib.sha256(b).hexdigest())"
# expect: 759eae04b8e7c493c2a4af03819fd9f8e792acd6b7cdd40d4aea29745088d97d

Canonical record JSON

{
  "metadata": {
    "abstract_canon_sha256": "d401ffe1572aa2b0b91d614c54be15dab4df12a69967d262d644a3b69113f001",
    "cross_cats_sorted": [
      "stat.ML"
    ],
    "license": "http://creativecommons.org/licenses/by/4.0/",
    "primary_cat": "cs.LG",
    "submitted_at": "2026-02-07T16:52:38Z",
    "title_canon_sha256": "400bd5230f4c4ac176391aeb223bfe9a66d02d7e527c151b9cddd0ace5081e39"
  },
  "schema_version": "1.0",
  "source": {
    "id": "2602.07618",
    "kind": "arxiv",
    "version": 6
  }
}