Training Deep Morphological Neural Networks as Universal Approximators (2505.09710v1)

Abstract: 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.

• The paper proves that integrating linear layers into Deep Morphological Neural Networks (DMNNs) is indispensable for achieving universal approximation capabilities.
• The study introduces novel constrained and hybrid architectures, empirically showing hybrid DMNNs achieve rapid convergence and competitive accuracy on datasets like MNIST.
• This work establishes a framework blending morphology and deep learning, laying a foundation for future research to enhance DMNN expressiveness and trainability.

Insights into Deep Morphological Neural Networks as Universal Approximators

The paper "Training Deep Morphological Neural Networks as Universal Approximators" presents significant advancements in the domain of deep learning by exploring the integration of mathematical morphology into neural network architectures. Authored by Konstantinos Fotopoulos and Petros Maragos, the work investigates the potential of Deep Morphological Neural Networks (DMNNs) to act as universal approximators, aiming to bridge the gap between traditional deep learning methods centered on linear operations and morphological techniques renowned for their geometric processing capabilities.

Mathematical morphology is a well-established technique in image and signal processing, traditionally applied to tasks such as edge detection and feature extraction. It operates using algebraic structures known as max-plus and min-plus operations, aligning with tropical algebra, wherein addition is defined as the maximum/minimum operator and multiplication is replaced by addition. Thus, morphological operations diverge fundamentally from the linear transformations prevalent in deep learning.

Novel Architectures and Constraints

The paper introduces several morphological network architectures, each subject to distinct parameter constraints. The architectures can be categorized as follows:

1. Constrained Parameter Architectures:
   • Majority of parameters allocated to morphological operations with limited learnable parameters.
   • Emphasis on compressibility and efficiency during training, drawing parallels to pruning methods in linear networks.
2. Hybrid Architectures:
   • Linear and morphological layers combined to leverage the strengths of both mathematical constructs.
   • Demonstration of accelerated convergence in training, particularly when utilizing large batch sizes.

The central contribution lies in proving that, despite the intrinsic non-linearity of DMNNs, learnable linear activations between layers are indispensable for achieving universal approximation capabilities. Earlier attempts, such as Morphological Perceptrons (MPs) and Dilation-Erosion Perceptrons (DEPs), faced limitations due to their axis-aligned decision boundaries and difficulties in training non-differentiable operations via backpropagation. The paper resolves these challenges by integrating linear transformations within DMNNs and showcasing their potential to learn complex representations.

Theoretical Implications

The work provides theoretical insights through theorems stating that pure max-plus/min-plus networks are not universal approximators. This limitation arises primarily due to the restrictive gradient propagation inherent in such architectures, unable to model gradients beyond elementary directions.

Additionally, the paper introduces a mathematical framework based on tropical algebra and lattice theory to construct DMNNs that overcome traditional morphological networks' limitations. Embedding linear layers alongside morphological ones establishes the architectures as universal approximators, effectively leveraging the power of deep learning to learn representations.

Empirical Validation

Extensive experiments on benchmark datasets such as MNIST and Fashion-MNIST reveal the practical benefits of the proposed DMNNs. The findings indicate improved trainability and generalization in achieving competitive accuracy levels while maintaining high parameter prunability. However, morphological networks still underperform compared to standard MLPs in terms of generalization, suggesting further refinements are necessary.

The Hybrid-MLP, owing to its large batch requirements, demonstrates noteworthy potential in distributed training scenarios by exhibiting rapid convergence rates. This insight paves the way for schema optimizations in distributed systems where computational constraints are prevalent.

Future Research Directions

The paper creates a foundation for subsequent explorations into the enhancement of morphological networks. Potential avenues include exploring alternate forms of activation parameterization, implementing advanced gradient estimation techniques, and refining hybrid architectures to improve expressive capabilities and trainability further.

In summary, the research delineates new directions for integrating mathematical morphology into deep learning, promising advancements in the development of efficient and expressive neural network architectures that adhere closely to the principles recognized within natural systems. The proposed networks' universal approximation capability holds vast implications for future developments in artificial intelligence, presenting a profound confluence of theoretical and practical achievements.