Pith Number

pith:6QDFC2Y4

pith:2026:6QDFC2Y4U7PG646RJLUYHTFPOH

not attested not anchored not stored refs resolved

Training Infinitely Deep and Wide Transformers

Gabriel Peyr\'e, Maarten V. de Hoop, Rapha\"el Barboni, Takashi Furuya

Gradient flow in the conditional Wasserstein metric converges to global minima for infinitely deep and wide transformers when the initial loss is small enough and the attention NTK is injective.

arxiv:2605.17660 v1 · 2026-05-17 · math.OC · cs.AI · 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{6QDFC2Y4U7PG646RJLUYHTFPOH}

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

Under the NTK injectivity assumption (linear independence of log-sum-exp functions modulo affine functions), gradient flow in the conditional Wasserstein metric converges to global minima whenever the initial loss is sufficiently small.

C2weakest assumption

The mean-field limit and the NTK injectivity condition remain valid approximations for the finite but large transformers that are actually trained; this premise enters when the convergence theorem is stated after the injectivity characterization.

C3one line summary

Develops a mean-field neural PDE model for transformer training, proves forward-pass well-posedness via function-space ODEs, derives conditional Wasserstein gradients, and shows global convergence of gradient flow under an NTK injectivity condition equivalent to linear independence of log-sum-exp fu

References

53 extracted · 53 resolved · 0 Pith anchors

[1] Transformers learn to imple- ment preconditioned gradient descent for in-context learning 2023

[2] What learning algorithm is in-context learning? investigations with linear models 2023

[3] A convergence theory for deep learning via over-parameterization 2019

[4] Transport equation and Cauchy problem for non-smooth vector fields 2008

[5] A user’s guide to optimal transport 2062

Formal links

2 machine-checked theorem links

Receipt and verification

First computed	2026-05-20T00:04:51.429856Z
Builder	pith-number-builder-2026-05-17-v1
Signature	Pith Ed25519 (`pith-v1-2026-05`) · public key
Schema	pith-number/v1.0

Canonical hash

f406516b1ca7de6f73d14ae983ccaf71ffa71e1ba8407e18af9cc750415bf815

Aliases

arxiv: 2605.17660 · arxiv_version: 2605.17660v1 · doi: 10.48550/arxiv.2605.17660 · pith_short_12: 6QDFC2Y4U7PG · pith_short_16: 6QDFC2Y4U7PG646R · pith_short_8: 6QDFC2Y4

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/6QDFC2Y4U7PG646RJLUYHTFPOH \
  | 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: f406516b1ca7de6f73d14ae983ccaf71ffa71e1ba8407e18af9cc750415bf815

Canonical record JSON

{
  "metadata": {
    "abstract_canon_sha256": "62f80cc2f7e1bf01bdc441db144ff1c58a541f85b00010ed393581da69da6f36",
    "cross_cats_sorted": [
      "cs.AI",
      "cs.LG",
      "stat.ML"
    ],
    "license": "http://creativecommons.org/licenses/by/4.0/",
    "primary_cat": "math.OC",
    "submitted_at": "2026-05-17T21:30:13Z",
    "title_canon_sha256": "8e2ad1d8fba0ed6441fbfa822016d407e90e873fd5e608931be87cddd501d7fe"
  },
  "schema_version": "1.0",
  "source": {
    "id": "2605.17660",
    "kind": "arxiv",
    "version": 1
  }
}