Training Deep Morphological Neural Networks as Universal Approximators

We investigate deep morphological neural networks (DMNNs). We demonstrate that despite their inherent non-linearity, activations between layers are essential for DMNNs. We then propose several new architectures for DMNNs, each with a different constraint on their parameters. For the first (resp. second) architecture, we work under the constraint that the majority of parameters (resp. learnable parameters) should be part of morphological operations. We empirically show that our proposed networks can be successfully trained, and are more prunable than linear networks. To the best of our knowledge, we are the first to successfully train DMNNs under such constraints, although the generalization capabilities of our networks remain limited. Finally, we propose a hybrid network architecture combining linear and morphological layers, showing empirically that the inclusion of morphological layers significantly accelerates the convergence of gradient descent with large batches.

Sparse Hybrid Linear-Morphological Networks

We investigate hybrid linear-morphological networks. Recent studies highlight the inherent affinity of morphological layers to pruning, but also their difficulty in training. We propose a hybrid network structure, wherein morphological layers are inserted between the linear layers of the network, in place of activation functions. We experiment with the following morphological layers: 1) maxout pooling layers (as a special case of a morphological layer), 2) fully connected dense morphological layers, and 3) a novel, sparsely initialized variant of (2). We conduct experiments on the Magna-Tag-A-Tune (music auto-tagging) and CIFAR-10 (image classification) datasets, replacing the linear classification heads of state-of-the-art convolutional network architectures with our proposed network structure for the various morphological layers. We demonstrate that these networks induce sparsity to their linear layers, making them more prunable under L1 unstructured pruning. We also show that on MTAT our proposed sparsely initialized layer achieves slightly better performance than ReLU, maxout, and densely initialized max-plus layers, and exhibits faster initial convergence.

TropNNC: Structured Neural Network Compression Using Tropical Geometry

We present TropNNC, a structured pruning framework for compressing neural networks with linear and convolutional layers and ReLU activations. Our approximation is based on a geometrical approach to machine/deep learning, using tropical geometry and extending the work of Misiakos et al. (2022). We use the Hausdorff distance of zonotopes in its standard continuous form to achieve a tighter approximation bound for tropical polynomials compared to Misiakos et al. (2022). This enhancement allows for superior functional approximations of neural networks, leading to a more effective compression algorithm. Our method is significantly easier to implement compared to other frameworks, and does not depend on the availability of training data samples. We validate our framework through extensive empirical evaluations on the MNIST, CIFAR, and ImageNet datasets. Our results demonstrate that TropNNC achieves performance on par with the state-of-the-art method ThiNet, even surpassing it in compressing linear layers, and to the best of our knowledge, it is the first method that achieves this using tropical geometry.