用物理能量动量模型解释卷积神经网络的过滤机制。
The Mechanics of CNN Filtering with Rectification
- 将卷积核分解为偶/奇分量，类比势能与动能。
- 信息传播速度与奇部能量占比线性相关。
- 首次建立CNN信息处理与相对论能量关系的联系。
本文提出一种基于基本信息力学的新模型，用于理解带整流的卷积滤波的机械特性，灵感来自狭义相对论和量子力学的物理理论。我们把卷积核分解为正交的偶分量与奇分量：偶分量使图像内容各向同性扩散且保持质心不变，类比于无净动量的静止或势能；奇分量则引起质心的定向位移，类比于具有非零动量的动能。信息传播速度与奇部能量占总能量的比例呈线性关系。通过离散余弦变换（DCT）在频域分析偶-奇性质，发现小卷积核（如3×3像素）的结构主要由低频基组成，特别是直流分量Σ和梯度分量∇，它们定义了信息传播的基本模式。据我们所知，这是首个揭示通用卷积神经网络中信息处理与现代相对论物理学基石——能量-动量关系之间关联的工作。
原文摘要 · Abstract (English)
This paper proposes elementary information mechanics as a new model for understanding the mechanical properties of convolutional filtering with rectification, inspired by physical theories of special relativity and quantum mechanics. We consider kernels decomposed into orthogonal even and odd components. Even components cause image content to diffuse isotropically while preserving the center of mass, analogously to rest or potential energy with zero net momentum. Odd kernels cause directional displacement of the center of mass, analogously to kinetic energy with non-zero momentum. The speed of information displacement is linearly related to the ratio of odd vs total kernel energy. Even-Odd properties are analyzed in the spectral domain via the discrete cosine transform (DCT), where the structure of small convolutional filters (e.g. $3 \times 3$ pixels) is dominated by low-frequency bases, specifically the DC $Σ$ and gradient components $\nabla$, which define the fundamental modes of information propagation. To our knowledge, this is the first work demonstrating the link between information processing in generic CNNs and the energy-momentum relation, a cornerstone of modern relativistic physics.
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