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

A new arXiv paper (2608.26363v1) proposes a unified mathematical framework linking information propagation in convolutional neural networks (CNNs) to relativistic physics, showing that symmetric filter components act like rest energy and antisymmetric components like momentum, with repeated filtering producing scale-invariant features. The authors demonstrate emergent Morse critical points in 3D images across scales, from sugar molecules to the cosmic microwave background.

arXiv:2608.26363v1 Announce Type: new 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 $\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 . 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 $E$. 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 Schr\"odinger 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 .