Core Concepts
Investigating the optimization of morphological layers using the Bouligand derivative and chain rule, highlighting challenges and insights for training.
Abstract
The content explores the challenges in training morphological neural networks with gradient descent. It delves into theoretical insights, including the representation of complete lattice operators, difficulties in training architectures with morphological layers, and the potential of differentiation-based approaches. The paper discusses the Bouligand derivative, initialization, optimization with gradient descent, message passing issues, and practical consequences for dense and convolutional layers.
Introduction
- Morphological neural networks introduced in late 1980s.
- Revisited in recent years with new perspectives.
Optimization Challenges
- Training difficulties due to non-smoothness of morphological layers.
- Comparison with state-of-the-art networks for image analysis.
- Exploration of differentiation-based algorithms.
Bouligand Derivative
- Introduction to the concept in nonsmooth analysis.
- Directional derivative providing first-order approximation.
- Properties similar to FrĀ“echet derivative.
Parameter Update
- Propositions for updating parameters based on Bouligand derivative.
- Challenges in finding optimal update directions.
Message Passing Issues
- Difficulties in ensuring message passing optimality.
- Heuristic solutions proposed for updating input variables.
Practical Consequences
- Positioning of morphological layers affects performance.
- Importance of initialization and learning rates for convergence.
Quotes
"Despite several contributions, architectures with morphological layers are often shallow." - Blusseau et al. (2024)
"Morphological layers act as noisy message transmitters in the chain rule paradigm." - Blusseau et al. (2024)