Lattice theory and algebraic models for deep convolutional learning based on mathematical morphology

Abstract

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 $-\infty$ 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.