Schoenberg characterization of continuous non-stationary isotropic positive definite kernels

2025/06/27 by Felix Benning, Max David Schölpple, Benning, Felix +2
Computer Science · Mathematics · #Artificial neural network #Characterization (materials science) #Invariant (physics) #Isotropy #Neural Networks and Applications #Positive-definite matrix #Product (mathematics) #Stochastic Gradient Optimization Techniques #Tensor decomposition and applications

paper · pdf · doi:10.48550/arxiv.2506.22048

published in arXiv (Cornell University) (Cornell University)

openalex publication_date 2025/06/27 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05

Abstract

We characterize the continuous isotropic positive definite kernels on ℝ^d, where isotropy refers to invariance under the orthogonal group O(d) but not necessarily stationarity. Furthermore, we characterize strict positive definiteness for such kernels. The class of isotropic kernels is fairly general as it unifies stationary isotropic and dot product kernels, and includes neural network kernels that arise from infinite-width limits of neural networks. As an application, we further characterize the continuous isotropic Gaussian random functions in terms of a series representation.

Citations

• Tensor Programs I: Wide Feedforward or Recurrent Neural Networks of Any Architecture are Gaussian Processes
• Handbook of Mathematical Functions With Formulas, Graphs and Mathematical Tables (National Bureau of Standards Applied Mathematics Series No. 55)
• Metric spaces and positive definite functions

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