Critical initialization destabilizes higher input derivatives in wide scalar-input networks — Paper

Abstract

The edge-of-chaos condition preserves first-order input perturbations in wide randomly initialized networks, but physics-informed losses, score matching and derivative regularization depend on higher input derivatives. For smooth scalar-input fully connected networks, using a joint Gaussianity of the finite derivative jet that holds in the infinite-width limit at each fixed depth, we derive mean-field recursions through third order that are exact at the variance fixed point, with finite-depth corrections that decay geometrically. At criticality, the first-derivative variance is depth-invariant, whereas the second-derivative variance grows linearly whenever the activation has nonzero curvature. The resulting third-order system closes on mean-field susceptibilities. For residual networks with branch scale L^{-1/2}, we prove that every fixed finite derivative order has uniformly bounded variance under explicit regularity assumptions. Simulations verify the critical growth laws, the residual bound, and the closed recursion. The results concern initialization, not trained-network performance.

Authors 2

Prashant Singh, Pranav Singh

Specification

Official page: arxiv.org/abs/2609.09244
Source:arXiv (Atom API + RSS)T1observed 2 h agohigh
Arxiv announce type: cross
Source:arXiv (Atom API + RSS)T1observed 2 h agohigh
arXiv id: 2609.09244
Source:arXiv (Atom API + RSS)T1observed 2 h agohigh
Categories: stat.ML, cs.AI, cs.LG
Source:arXiv (Atom API + RSS)T1observed 2 h agohigh
PDF: arxiv.org/pdf/2609.09244
Source:arXiv (Atom API + RSS)T1observed 2 h agohigh
Primary category: stat.ML
Source:arXiv (Atom API + RSS)T1observed 2 h agohigh
Published: 11 Sept 2026
Source:arXiv (Atom API + RSS)T1observed 2 h agohigh

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9

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T19

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2 h ago

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Authors: Prashant Singh, Pranav Singh

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Claim history for Official page
Value	Valid from → to	Status	Source	Confidence	Extractor
https://arxiv.org/abs/2609.09244	11 Sept 2026 21:03 UTC → current	current	arXiv (Atom API + RSS)T1	high	deterministic

Claims are temporal and append-only: a new observation closes the previous claim (valid_to) instead of overwriting it. Conflicting claims from different sources are kept side by side and flagged — never averaged. Methodology →

11 Sept 2026
New paperPaperCritical initialization destabilizes higher input derivatives in wide scalar-input networks
New paper: Critical initialization destabilizes higher input derivatives in wide scalar-input networks
arxiv
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Source	Document	Type	Tier	Last observed	Snapshots
arXiv (Atom API + RSS)	rss.arxiv.org/rss/cs.AI	feed	T1· Official	2 h ago	1

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