Pith Number

pith:CP2AQAEN

pith:2026:CP2AQAENCJKOBCEIK7T72NCQ2V

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Transformers Can Implement Preconditioned Richardson Iteration for In-Context Gaussian Kernel Regression

Charles Kulick, Dongyang Li, Mingsong Yan, Sui Tang

A standard softmax-attention transformer can run preconditioned Richardson iteration to approximate in-context Gaussian kernel ridge regression.

arxiv:2605.08475 v2 · 2026-05-08 · cs.LG · cs.AI · cs.NA · math.NA · math.OC

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Claims

C1strongest claim

a standard softmax-attention transformer can approximate the KRR predictor during its forward pass by implementing preconditioned Richardson iteration on the associated kernel linear system. Under bounded-data assumptions, we construct a single-head transformer with O(log(1/ε)) blocks and MLP width O(√(N/ε)) that achieves ε-accurate prediction for prompts of length N.

C2weakest assumption

Under bounded-data assumptions we construct... (the bounded-data assumptions that enable the approximation guarantee and error bounds for the transformer implementation).

C3one line summary

Standard softmax-attention transformers can approximate the Gaussian kernel ridge regression predictor by implementing preconditioned Richardson iteration during their forward pass.

Receipt and verification

First computed	2026-05-20T00:02:12.653063Z
Builder	pith-number-builder-2026-05-17-v1
Signature	Pith Ed25519 (`pith-v1-2026-05`) · public key
Schema	pith-number/v1.0

Canonical hash

13f408008d1254e0888857e7fd3450d54638ecf51340700a56044a7e0cd8216e

Aliases

arxiv: 2605.08475 · arxiv_version: 2605.08475v2 · doi: 10.48550/arxiv.2605.08475 · pith_short_12: CP2AQAENCJKO · pith_short_16: CP2AQAENCJKOBCEI · pith_short_8: CP2AQAEN

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Verify this Pith Number yourself

curl -sH 'Accept: application/ld+json' https://pith.science/pith/CP2AQAENCJKOBCEIK7T72NCQ2V \
  | 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: 13f408008d1254e0888857e7fd3450d54638ecf51340700a56044a7e0cd8216e

Canonical record JSON

{
  "metadata": {
    "abstract_canon_sha256": "edffada1f687293db60b61f9efe64d358fab6c4671e670e3721f8f888fcbeb84",
    "cross_cats_sorted": [
      "cs.AI",
      "cs.NA",
      "math.NA",
      "math.OC"
    ],
    "license": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
    "primary_cat": "cs.LG",
    "submitted_at": "2026-05-08T20:50:58Z",
    "title_canon_sha256": "5e714626cc7f3312c7528de9b6aa3936c5b20016ba954268020980a28c7e4415"
  },
  "schema_version": "1.0",
  "source": {
    "id": "2605.08475",
    "kind": "arxiv",
    "version": 2
  }
}