A Unified Framework for the Mechanics of Information in Convolutional Neural Network Image Space

2026-08-26

AI Summary

本文提出一个统一的数学框架，通过分析卷积神经网络中信息传播与物理空间和信息空间的联系，利用对称性和非对称性滤波器模拟能量-动量关系，揭示了图像处理中的尺度不变特征。

Abstract

This paper introduces a unified mathematical framework for modeling information propagation through convolutional neural networks (CNNs), with the aim of connecting descriptions of physical space and information space. A correspondence is presented linking discrete filter symmetry and the relativistic energy--momentum relation under the widely used nonlinear rectified convolution operation. Specifically, symmetric filter components (e.g. the sum $Σ= [1,1]$) operate analogously to rest energy $mc^2$ in preserving the image centre of mass (e.g. isotropic diffusion), whereas antisymmetric components (e.g. the gradient $\nabla = [-1,1]$) operate analogously to the momentum term $pc$ in generally inducing a displacement (e.g. vibration or translation). For typical small discrete filters, this displacement is determined by the ratio of antisymmetric to total filter energy, analogously to how the displacement of a relativistic particle relates to a Lorentz transform with beta parameter $β= \frac{v}{c}=\frac{pc}{E}$ equal to the ratio of momentum $pc$ to total energy $E$. Repeated filtering leads to the Gaussian scale-space and emergent scale-invariant features. These constructions share a Laplacian-driven structure with the classical heat (diffusion) equation and, via standard mathematical correspondences, with the Schrödinger equation and aspects of the Friedmann equations, together with emergent Morse topological structure. Demonstrations in 3D images reveal blob-like, scale-invariant Morse critical points in images spanning a wide range of physical scales, including organic sugar molecules and inorganic silicon crystals, human and primate brains in magnetic resonance images (MRI), galaxies and the cosmic microwave background (CMB).

Problem

• Convolutional Neural Networks
• Information Propagation
• Physical Space
• Information Space
• Symmetric Filter

Innovation

• Unified Mathematical Framework
• Information Propagation
• Discrete Filter Symmetry
• Relativistic Energy-Momentum Relation
• Gaussian Scale-Space