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Kantorovich--Kernel Neural Operators: Approximation Theory, Asymptotics, and Neural Network Interpretation

Tian-Xiao He

Abstract

This paper studies a class of multivariate Kantorovich-kernel neural network operators, including the deep Kantorovich-type neural network operators studied by Sharma and Singh. We prove density results, establish quantitative convergence estimates, derive Voronovskaya-type theorems, analyze the limits of partial differential equations for deep composite operators, prove Korovkin-type theorems, and propose inversion theorems. This paper studies a class of multivariate Kantorovich-kernel neural network operators, including the deep Kantorovich-type neural network operators studied by Sharma and Singh. We prove density results, establish quantitative convergence estimates, derive Voronovskaya-type theorems, analyze the limits of partial differential equations for deep composite operators, prove Korovkin-type theorems, and propose inversion theorems. Furthermore, this paper discusses the connection between neural network architectures and the classical positive operators proposed by Chui, Hsu, He, Lorentz, and Korovkin.

Kantorovich--Kernel Neural Operators: Approximation Theory, Asymptotics, and Neural Network Interpretation

Abstract

This paper studies a class of multivariate Kantorovich-kernel neural network operators, including the deep Kantorovich-type neural network operators studied by Sharma and Singh. We prove density results, establish quantitative convergence estimates, derive Voronovskaya-type theorems, analyze the limits of partial differential equations for deep composite operators, prove Korovkin-type theorems, and propose inversion theorems. This paper studies a class of multivariate Kantorovich-kernel neural network operators, including the deep Kantorovich-type neural network operators studied by Sharma and Singh. We prove density results, establish quantitative convergence estimates, derive Voronovskaya-type theorems, analyze the limits of partial differential equations for deep composite operators, prove Korovkin-type theorems, and propose inversion theorems. Furthermore, this paper discusses the connection between neural network architectures and the classical positive operators proposed by Chui, Hsu, He, Lorentz, and Korovkin.

Paper Structure

This paper contains 13 sections, 7 theorems, 67 equations.

Table of Contents

  1. Introduction
  2. Kantorovich--Kernel Neural Operators (KKNO)
  3. Density of Kantorovich--Kernel Neural Operators
  4. Quantitative Convergence
  5. Voronovskaya-Type Asymptotics
  6. Diffusion PDE Limit of Deep Compositions
  7. Korovkin-Type Theorem
  8. Inverse Theorem
  9. Neural Network Interpretation and Comparisons
  10. KKNO as Controllable Filtering Layers
  11. Comparison with Classical Positive Operators
  12. Implications for Deep Learning Design
  13. 3. Explicit error and convergence control.

Key Result

Proposition 2.8

Let $\mathcal{L}_n$ be a KKNO defined in Definition . Then, we have (i) (Linearity and positivity) For all $f,g \in C_b(\mathbb R^d)$ and $\alpha,\beta \in \mathbb R$, where positivity follows from $K_n\ge0$. Here, the positivity ensures stability under noise. (ii) (Preservation of constants) Let $f(x)\equiv 1$. Then This shows that KKNO operators preserve constants, a standard property of posi

Theorems & Definitions (26)

  • Definition 2.1
  • Definition 2.2
  • Remark 2.3
  • Remark 2.4
  • Remark 2.5
  • Example 2.6
  • Example 2.7
  • Proposition 2.8
  • Theorem 3.1
  • Remark 3.2
  • ...and 16 more