A Unified Framework for the Mechanics of Information in Convolutional Neural Network Image Space
Authors: Aryan Shukla, Matthew Toews
Organizations: Department of Systems Engineering, ´Ecole de technologie sup´erieure, 1100 R. Notre Dame O., Montr´eal, QC H3C 1K3, Canada
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 mc2 in preserving the image centre of mass (e.g. isotropic diffusion), whereas antisymmetric components (e.g. the gradient ∇=[−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=cv=Epc 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).
We develop a rigorous algebraic framework for deep convolutional architectures, CNNs, ResNets, and encoder--decoder networks such as UNet, grounded in lattice theory and mathematical morphology. The central tool is the Matheron--Maragos--Banon--Barrera (MMBB) universal representation theory for translation-invariant operators, which we apply systematically to every layer of a standard deep network. The principal finding is that the standard CNN pipeline (linear convolution~+ ReLU~+ flat max-pooling) is a cross-lattice operator: the convolution is an erosion in the Fourier inf-semilattice while ReLU is a lattice-join closing and max-pooling is a dilation in the pointwise max-plus lattice, and their composition is a morphological opening in neither. A second finding is that the upper adjoint of ReLU in the pointwise lattice is a global (non-local) operator, the identity on globally non-negative functions and −∞ otherwise, so no local morphological erosion can form an adjunction pair with ReLU. These two results together provide the precise algebraic reason why depth in standard CNNs introduces genuine representational power: the composed layer is not idempotent. Three layer designs that are genuine idempotent openings are identified and fully characterised: the pure max-plus morphological layer (pointwise lattice), the spectral Wiener layer (Fourier lattice), and the self-dual morphological layer. We establish a complete fixed-point and convergence theory. The framework also unifies max-pooling, strided convolution, and the Laplacian pyramid under the Goutsias--Heijmans adjoint pyramid theory, and gives the Activation--Pooling Dilation (APD) factorisation with its correct adjoint.
In physical systems, whenever a continuous symmetry is spontaneously broken, the system possesses excitations called Goldstone modes, which allow coherent information propagation over long distances and times. In this work, we study deep neural networks whose internal layers are equivariant under a continuous symmetry and may therefore support analogous Goldstone-like degrees of freedom. We demonstrate, both analytically and empirically, that these degrees of freedom enable coherent signal propagation across depth and recurrent iterations, providing a mechanism for stable information flow without relying on architectural stabilizers such as residual connections or normalization. In feedforward networks, this results in improved trainability and representational diversity across layers. In recurrent settings, we demonstrate the same mechanism is valuable for long-term memory by propagating information over recurrent iterations, thereby improving performance of RNNs and GRUs on long-sequence modeling tasks.
While researchers continue to find new and improved network structures for CNNs, most of the newly invented architectures still rely on the traditional pattern of stacking convolutional blocks and separating them with pointwise activation functions. However, there are drawbacks to a network purely building on pointwise nonlinearities. One alternative is to introduce a pairwise connection between two filters of a network. Typical connection functions use multiplications or the minimum operation to realize logical AND connections. In this paper, we go one step further by demonstrating that CNNs can benefit from more general connections, which include parameters that are learned. With such parameters, the network is able to implement different connections in different network layers and better adapt the connection function to the task at hand.