# Training Deep Morphological Neural Networks as Universal Approximators

arXiv: https://arxiv.org/abs/2505.09710 · HTML: https://mlsanity.com/paper/2505.09710/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 11 of its 52 references that are in the map.
- First posted on arXiv 14 May 2025.

## Builds on (most cited first, 11 of 11)

- [Fashion-MNIST: a Novel Image Dataset for Benchmarking Machine Learning Algorithms](https://mlsanity.com/paper/1708.07747/) (2017)
- [Delving Deep into Rectifiers: Surpassing Human-Level Performance on ImageNet Classification](https://mlsanity.com/paper/1502.01852/) (2015)
- [SNIP: Single-shot Network Pruning based on Connection Sensitivity](https://mlsanity.com/paper/1810.02340/) (2018)
- [Maxout Networks](https://mlsanity.com/paper/1302.4389/) (2013)
- [A comprehensive review of Binary Neural Network](https://mlsanity.com/paper/2110.06804/) (2021)
- [Deep Morphological Neural Networks](https://mlsanity.com/paper/1909.01532/) (2019)
- [Min-Max-Plus Neural Networks](https://mlsanity.com/paper/2102.06358/) (2021)
- [Max-plus Operators Applied to Filter Selection and Model Pruning in Neural Networks](https://mlsanity.com/paper/1903.08072/) (2019)
- [Reduced Dilation-Erosion Perceptron for Binary Classification](https://mlsanity.com/paper/2003.02306/) (2020)
- [Learning Deep Morphological Networks with Neural Architecture Search](https://mlsanity.com/paper/2106.07714/) (2021)
- [MorphoActivation: Generalizing ReLU activation function by mathematical morphology](https://mlsanity.com/paper/2207.06413/) (2022)

## Built on by (most cited first, 2 of 2)

- [Lattice theory and algebraic models for deep convolutional learning based on mathematical morphology](https://mlsanity.com/paper/2605.24608/) (2026)
- [Correlational Training of Morphological Neural Networks](https://mlsanity.com/paper/2610.11740/) (2026)

## Abstract (arXiv)

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.