Training Deep Morphological Neural Networks as Universal Approximators

Abstract:We investigate deep morphological neural networks (DMNNs), studying how changes in algebraic structure affect the expressivity and trainability of deep architectures. We show that despite the inherent non-linearity of morphological operations, existing deep morphological architectures fail to be universal approximators and exhibit optimization limitations related to sparse and uninformative gradients. To address these issues, we introduce architectures incorporating constrained "linear" activations between morphological layers and averaging max-plus and min-plus neurons. Only O(N) parameters (or learnable parameters) per layer of size N belong to the activations, with the remaining parameters constrained to morphological operations. We prove universal approximation results for the proposed architectures without requiring substantially larger parameter counts than comparable linear networks. Residual connections and weight dropout further improve generalization. Our experiments show that our networks are trainable and compact, despite the imposed architectural restrictions.

v4: Added 1st-order UA interpretation, Example 1, Proposition 7, and the min-max-plus network baseline. Corrected SNIP pruning experiments

Machine Learning (cs.LG)

arXiv:2505.09710 [cs.LG]

(or arXiv:2505.09710v4 [cs.LG] for this version)

https://doi.org/10.48550/arXiv.2505.09710 Focus to learn more arXiv-issued DOI via DataCite

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