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The role of parameter Jacobians in the stability of network outputs
Preprint   Open access

The role of parameter Jacobians in the stability of network outputs

Halyun Jeong, Palle E. T Jorgensen, Hyun-Kyoung Kwon, Myung-Sin Song and James Tian
arXiv
arXiv
08/27/2026
DOI: 10.48550/arxiv.2608.27748
url
https://doi.org/10.48550/arxiv.2608.27748View
Preprint (Author's original) This preprint has not been evaluated by subject experts through peer review. Preprints may undergo extensive changes and/or become peer-reviewed journal articles. Open Access

Abstract

In the framework of network dynamics, learning models, and neural tangent kernels (NTK), we show that the corresponding linearized dynamics leads naturally to a semigroup formulation. More precisely, in our analysis of input/output models, the time-dynamics is presented via special semigroups of linear operators on Hilbert spaces, together with an associated class of semigroup perturbations. In this context, we then present new and explicit a priori perturbation-bound results: for the fixed-kernel linearization constructions arising in the NTK setting, we prove norm-bounds on the corresponding semigroup perturbations, in the form of explicit finite-time perturbation estimates. We further present refinements on prescribed task spaces, Cesàro-averaged (ergodic) comparisons estimates, and versions in which the lower spectral edge assumption is replaced by a spectral-distribution condition. We also extend the comparison to nonautonomous NTK evolutions through piecewise-frozen approximations, record a corresponding discrete Euler specialization, and offer worked examples in order to illustrate our perturbation-bound estimates.
 Mathematics - Functional Analysis Statistics - Machine Learning

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