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

卷积神经网络图像空间信息力学的统一框架

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. 摘要： 本文引入了一个统一的数学框架，用于建模卷积神经网络（CNN）中的信息传播，旨在连接物理空间与信息空间的描述。

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 $\Sigma = [1,1]$) 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). 文中提出了一种对应关系，将离散滤波器对称性与广泛使用的非线性修正卷积运算下的相对论能量-动量关系联系起来。具体而言，对称滤波器分量（例如求和 $\Sigma = [1,1]$）的作用类似于静止能量 $mc^2$，用于保持图像质心（例如各向同性扩散）；而反对称分量（例如梯度 $\nabla = [-1,1]$）的作用则类似于动量项 $pc$，通常会引起位移（例如振动或平移）。

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 $\beta = \frac{v}{c}=\frac{pc}{E}$ equal to the ratio of momentum $pc$ to total energy $E$. 对于典型的小型离散滤波器，这种位移由反对称能量与总滤波器能量之比决定，这类似于相对论粒子的位移与洛伦兹变换的关系，其中 $\beta$ 参数 $\beta = \frac{v}{c}=\frac{pc}{E}$ 等于动量 $pc$ 与总能量 $E$ 的比值。

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. 重复滤波会导致高斯尺度空间和涌现的尺度不变特征。这些结构与经典热（扩散）方程共享拉普拉斯驱动的结构，并通过标准的数学对应关系，与薛定谔方程及弗里德曼方程的某些方面相联系，同时伴随着涌现的莫尔斯（Morse）拓扑结构。

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). 在 3D 图像中的演示揭示了跨越广泛物理尺度的图像中存在类斑点状、尺度不变的莫尔斯临界点，这些尺度包括有机糖分子和无机硅晶体、磁共振成像（MRI）中的人类和灵长类动物大脑、星系以及宇宙微波背景（CMB）。