参数雅可比矩阵在网络输出稳定性中的作用
The role of parameter Jacobians in the stability of network outputs
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中文总结 AI 辅助
该研究在网络动力学、NTK框架下，分析参数雅可比矩阵相关线性化动力学的半群性质，推导了半群扰动的显式范数界，还扩展了相关估计至非自治NTK演化并给出示例。
中文摘要 AI 辅助
在网络动力学、学习模型及神经正切核（NTK）的框架下，我们证明了对应的线性化动力学自然导出半群公式。更确切地说，在对输入/输出模型的分析中，时间动力学通过希尔伯特空间上线性算子的特殊半群以及相关的半群扰动类来呈现。在此背景下，我们给出新的显式先验扰动界结果：针对NTK设置中产生的固定核线性化构造，我们以显式有限时间扰动估计的形式证明了对应半群扰动的范数界。我们还对指定任务空间给出了改进、切萨罗平均（遍历）比较估计，以及将下谱边假设替换为谱分布条件的版本。我们进一步通过分段冻结近似将比较扩展到非自治NTK演化，记录了对应的离散欧拉特例，并提供了示例以说明我们的扰动界估计。
英文摘要
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.
