Pith Number

pith:DRZCBV6T

pith:2026:DRZCBV6TMWSASLJFFPKMO4MD3D

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Spectral Dynamics in Deep Networks: Feature Learning, Outlier Escape, and Learning Rate Transfer

Blake Bordelon, Cengiz Pehlevan, Clarissa Lauditi

Spectral outliers in wide neural networks evolve predictably during gradient descent, with one scaling regime producing width-independent dynamics and hyperparameter transfer.

arxiv:2605.07870 v2 · 2026-05-08 · cond-mat.dis-nn · cs.AI · stat.ML

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4 Citations open

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Claims

C1strongest claim

Our theory predicts how outliers evolve with training time, width, output scale, and initialization variance. In deep linear networks, μP yields width-consistent outlier dynamics and hyperparameter transfer, including width-stable growth of the leading NTK mode toward the edge of stability (EoS).

C2weakest assumption

The two-level DMFT remains accurate when spike directions stay statistically dependent on the random bulk and when the infinite-width or proportional limits faithfully represent finite practical networks.

C3one line summary

A two-level DMFT predicts width-consistent outlier escape and hyperparameter transfer under μP in deep networks, with bulk restructuring dominating for tasks with many outputs.

Formal links

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Cited by

1 paper in Pith

The Spectral Dynamics and Noise Geometry of Muon

Receipt and verification

First computed	2026-05-22T02:04:42.104887Z
Builder	pith-number-builder-2026-05-17-v1
Signature	Pith Ed25519 (`pith-v1-2026-05`) · public key
Schema	pith-number/v1.0

Canonical hash

1c7220d7d365a4092d252bd4c77183d8d9d9d7058f82c1aabdf475f6a6080f40

Aliases

arxiv: 2605.07870 · arxiv_version: 2605.07870v2 · doi: 10.48550/arxiv.2605.07870 · pith_short_12: DRZCBV6TMWSA · pith_short_16: DRZCBV6TMWSASLJF · pith_short_8: DRZCBV6T

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Verify this Pith Number yourself

curl -sH 'Accept: application/ld+json' https://pith.science/pith/DRZCBV6TMWSASLJFFPKMO4MD3D \
  | 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: 1c7220d7d365a4092d252bd4c77183d8d9d9d7058f82c1aabdf475f6a6080f40

Canonical record JSON

{
  "metadata": {
    "abstract_canon_sha256": "1195290f85da72d5befc2514984ab931c348b856f297218316e0b992724caa03",
    "cross_cats_sorted": [
      "cs.AI",
      "stat.ML"
    ],
    "license": "http://creativecommons.org/licenses/by/4.0/",
    "primary_cat": "cond-mat.dis-nn",
    "submitted_at": "2026-05-08T15:28:01Z",
    "title_canon_sha256": "e9e8536eb24aa02729327bbd50058e0684b82ee83b3b12a8a7e8992606ae7358"
  },
  "schema_version": "1.0",
  "source": {
    "id": "2605.07870",
    "kind": "arxiv",
    "version": 2
  }
}