# The role of parameter Jacobians in the stability of network outputs

arXiv: https://arxiv.org/abs/2608.27748 · HTML: https://mlsanity.com/paper/2608.27748/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 18 of its 50 references that are in the map.
- First posted on arXiv 27 Aug 2026.

## Builds on (most cited first, 18 of 18)

- [Neural Tangent Kernel: Convergence and Generalization in Neural Networks](https://mlsanity.com/paper/1806.07572/) (2018)
- [Wide Neural Networks of Any Depth Evolve as Linear Models Under Gradient Descent](https://mlsanity.com/paper/1902.06720/) (2019)
- [On Lazy Training in Differentiable Programming](https://mlsanity.com/paper/1812.07956/) (2018)
- [Intrinsic Dimensionality Explains the Effectiveness of Language Model Fine-Tuning](https://mlsanity.com/paper/2012.13255/) (2020)
- [Neural Networks as Kernel Learners: The Silent Alignment Effect](https://mlsanity.com/paper/2111.00034/) (2021)
- [Dynamics of Deep Neural Networks and Neural Tangent Hierarchy](https://mlsanity.com/paper/1909.08156/) (2019)
- [Limitations of the NTK for Understanding Generalization in Deep Learning](https://mlsanity.com/paper/2206.10012/) (2022)
- [Analyzing Finite Neural Networks: Can We Trust Neural Tangent Kernel Theory?](https://mlsanity.com/paper/2012.04477/) (2020)
- [Spectral Analysis of the Neural Tangent Kernel for Deep Residual Networks](https://mlsanity.com/paper/2104.03093/) (2021)
- [Evolution of Neural Tangent Kernels under Benign and Adversarial Training](https://mlsanity.com/paper/2210.12030/) (2022)
- [Binary Iterative Hard Thresholding Converges with Optimal Number of Measurements for 1-Bit Compressed Sensing](https://mlsanity.com/paper/2207.03427/) (2022)
- [An Exact Kernel Equivalence for Finite Classification Models](https://mlsanity.com/paper/2308.00824/) (2023)
- [Robust 1-bit Compressed Sensing with Iterative Hard Thresholding](https://mlsanity.com/paper/2310.08019/) (2023)
- [On the Convergence Analysis of Over-Parameterized Variational Autoencoders: A Neural Tangent Kernel Perspective](https://mlsanity.com/paper/2409.05349/) (2024)
- [Operator theory, kernels, and Feedforward Neural Networks](https://mlsanity.com/paper/2301.01327/) (2023)
- [Infinite-Dimensional Operator/Block Kaczmarz Algorithms: Regret Bounds and λ-Effectiveness](https://mlsanity.com/paper/2511.07604/) (2025)
- [Tight conditions for when the NTK approximation is valid](https://mlsanity.com/paper/2305.13141/) (2023)
- [Use of operator defect identities in multi-channel signal plus residual-analysis via iterated products and telescoping energy-residuals: Applications to kernels in machine learning](https://mlsanity.com/paper/2601.18080/) (2026)

## Abstract (arXiv)

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, Ces`aro-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.