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

arXiv: https://arxiv.org/abs/2608.26363 · HTML: https://mlsanity.com/paper/2608.26363/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 7 of its 56 references that are in the map.
- First posted on arXiv 26 Aug 2026.

## Builds on (most cited first, 7 of 7)

- [Attention Is All You Need](https://mlsanity.com/paper/1706.03762/) (2017)
- [Deep Unsupervised Learning using Nonequilibrium Thermodynamics](https://mlsanity.com/paper/1503.03585/) (2015)
- [E(n) Equivariant Graph Neural Networks](https://mlsanity.com/paper/2102.09844/) (2021)
- [E(3)-Equivariant Graph Neural Networks for Data-Efficient and Accurate Interatomic Potentials](https://mlsanity.com/paper/2101.03164/) (2021)
- [Lorentz Group Equivariant Neural Network for Particle Physics](https://mlsanity.com/paper/2006.04780/) (2020)
- [Efficient Pairwise Neuroimage Analysis using the Soft Jaccard Index and 3D Keypoint Sets](https://mlsanity.com/paper/2103.06966/) (2021)
- [The Mechanics of CNN Filtering with Rectification](https://mlsanity.com/paper/2512.24338/) (2025)

## Abstract (arXiv)

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 mc2 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 = fracvc=fracpcE 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).