Pith Number

pith:3VKFXRQM

pith:2026:3VKFXRQM6RXARY7J43WUMGYQRO

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A Theory of Saddle Escape in Deep Nonlinear Networks

Divit Rawal, Michael R. DeWeese

An exact identity on Frobenius norm imbalances in layer weights reduces deep nonlinear training to a scalar ODE whose escape time scales as ε to the power of minus (r minus 2), where r is the number of bottleneck layers.

arxiv:2605.01288 v3 · 2026-05-02 · cs.LG · cond-mat.dis-nn · stat.ML

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\pithnumber{3VKFXRQM6RXARY7J43WUMGYQRO}

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2 Internet Archive

3 Author claim open · sign in to claim

4 Citations open

5 Replications open

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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 derive an exact identity for the imbalance of Frobenius norms of layer weight matrices that holds for any smooth activation and any differentiable loss [...] giving a critical-depth escape time law τ★ = Θ(ε^{-(r-2)}) governed by the number r of layers at the bottleneck scale rather than the total depth L.

C2weakest assumption

The reduction to a scalar ODE relies on an approximate balance law on the permutation-symmetric submanifold whose accuracy and range of validity are not derived from first principles but stated as holding approximately.

C3one line summary

An exact norm-imbalance identity classifies activations into four classes and reduces deep nonlinear training flow to a scalar ODE that predicts saddle escape time scaling as ε to the power of minus (r-2) for r bottleneck layers.

Formal links

2 machine-checked theorem links

Receipt and verification

First computed	2026-06-24T01:15:03.268859Z
Builder	pith-number-builder-2026-05-17-v1
Signature	Pith Ed25519 (`pith-v1-2026-05`) · public key
Schema	pith-number/v1.0

Canonical hash

dd545bc60cf46e08e3e9e6ed461b108b829581fa3e9cacc5ccac9f69c012f4bf

Aliases

arxiv: 2605.01288 · arxiv_version: 2605.01288v3 · doi: 10.48550/arxiv.2605.01288 · pith_short_12: 3VKFXRQM6RXA · pith_short_16: 3VKFXRQM6RXARY7J · pith_short_8: 3VKFXRQM

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Verify this Pith Number yourself

curl -sH 'Accept: application/ld+json' https://pith.science/pith/3VKFXRQM6RXARY7J43WUMGYQRO \
  | 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: dd545bc60cf46e08e3e9e6ed461b108b829581fa3e9cacc5ccac9f69c012f4bf

Canonical record JSON

{
  "metadata": {
    "abstract_canon_sha256": "8635d78054aeba55bcc86b93793aeb1cb6e41a0c5a13564e20ba2274e366a0c6",
    "cross_cats_sorted": [
      "cond-mat.dis-nn",
      "stat.ML"
    ],
    "license": "http://creativecommons.org/licenses/by/4.0/",
    "primary_cat": "cs.LG",
    "submitted_at": "2026-05-02T06:55:15Z",
    "title_canon_sha256": "52163603e99629cbdd0bb6cb17146be35167df9f9963c970d5f3e49cf163a903"
  },
  "schema_version": "1.0",
  "source": {
    "id": "2605.01288",
    "kind": "arxiv",
    "version": 3
  }
}