Advances in the training, pruning and enforcement of shape constraints of Morphological Neural Networks using Tropical Algebra
This work addresses efficiency and performance issues in neural networks for researchers in mathematical morphology and tropical geometry, though it is incremental in extending existing methods.
The paper tackles the training, pruning, and shape constraint enforcement of Morphological Neural Networks using tropical algebra, showing that these networks outperform linear counterparts in compression under heavy pruning and improve training convergence with softened architectures.
In this paper we study an emerging class of neural networks based on the morphological operators of dilation and erosion. We explore these networks mathematically from a tropical geometry perspective as well as mathematical morphology. Our contributions are threefold. First, we examine the training of morphological networks via Difference-of-Convex programming methods and extend a binary morphological classifier to multiclass tasks. Second, we focus on the sparsity of dense morphological networks trained via gradient descent algorithms and compare their performance to their linear counterparts under heavy pruning, showing that the morphological networks cope far better and are characterized with superior compression capabilities. Our approach incorporates the effect of the training optimizer used and offers quantitative and qualitative explanations. Finally, we study how the architectural structure of a morphological network can affect shape constraints, focusing on monotonicity. Via Maslov Dequantization, we obtain a softened version of a known architecture and show how this approach can improve training convergence and performance.