Fractal and Regular Geometry of Deep Neural Networks (2504.06250v1)

Abstract: We study the geometric properties of random neural networks by investigating the boundary volumes of their excursion sets for different activation functions, as the depth increases. More specifically, we show that, for activations which are not very regular (e.g., the Heaviside step function), the boundary volumes exhibit fractal behavior, with their Hausdorff dimension monotonically increasing with the depth. On the other hand, for activations which are more regular (e.g., ReLU, logistic and $\tanh$), as the depth increases, the expected boundary volumes can either converge to zero, remain constant or diverge exponentially, depending on a single spectral parameter which can be easily computed. Our theoretical results are confirmed in some numerical experiments based on Monte Carlo simulations.

• The paper reveals how neural network excursion sets exhibit fractal behavior for less regular activations and regular geometry for smoother ones.
• It introduces the Covariance Regularity Index (CRI) to classify networks into fractal and Kac-Rice classes based on measurable spectral properties.
• The findings offer practical insights for designing DNN architectures that balance complexity and stability for advanced pattern recognition tasks.

Fractal and Regular Geometry of Deep Neural Networks: A Technical Overview

The paper "Fractal and Regular Geometry of Deep Neural Networks" by Simmaco Di Lillo, Domenico Marinucci, Michele Salvi, and Stefano Vigogna offers an in-depth exploration of the geometric properties of deep neural networks (DNNs) through the lens of random field theory. Specifically, it investigates how the complexity of DNNs is influenced by their architecture, including activation functions and depth, as well as their correspondence to Gaussian processes.

Key Insights

1. Excursion Set Geometry:

The paper focuses on the boundary volumes of the excursion sets (the regions where the random field exceeds a certain threshold) of neural networks. It identifies a dichotomy in the geometric behavior based on the regularity of the activation functions:

• Fractal Geometry: For less regular activations, such as the Heaviside step function, the boundary volumes of excursion sets exhibit a fractal nature. Here, the Hausdorff dimension of these boundaries increases with depth, eventually approaching the dimension of the input space. This behavior includes a non-integer Hausdorff dimension indicating fractal characteristics.
• Regular Geometry: More regular activations like ReLU, logistic, and $\tanh$ show geometric properties where excursion set boundary volumes can converge to zero, remain constant, or grow exponentially, influenced by a spectral parameter computable from the activation function and network depth.

2. Classification into Fractal and Kac-Rice Classes:

The authors introduce the Covariance Regularity Index (CRI) to characterize the regularity of a network's covariance function. They categorize neural networks into two classes based on this index:

• Fractal Class: Only networks with activations yielding a CRI smaller than one belong here, leading to fractal behaviors in boundary geometries.
• Kac-Rice Class: Networks with CRI greater than one exhibit classical behavior, enabling the use of classical Kac-Rice formulas to derive expected boundary volumes.

3. Theoretical and Practical Implications:

The findings imply that the geometric features inherent in neural networks could inform the design of more efficient architectures, where certain classes might be preferable depending on the desired behavior. For instance, networks with fractal boundaries may be better suited for diverse or complex pattern recognition tasks due to higher complexity representation capabilities.

Implications and Future Directions

Theoretical Implications:

This paper deepens the mathematical understanding of neural networks, reinforcing the characterization of DNNs using geometric and probabilistic tools. The classification based on CRI offers a structured way to predict network behavior from its spectral properties, potentially guiding hyperparameter selection in network design.

Practical Implications:

Understanding the fractal or regular nature of a network's excursion set geometries can assist in applications where model interpretability or complexity management is critical. For example, networks prone to a fractal class behavior could be utilized in applications requiring detailed feature extraction or dynamic adaptation to input changes.

Speculation on AI Future Developments:

Looking forward, the integration of geometric analysis with deep learning could foster the development of networks with tailored complexity and stability properties, pushing the boundaries of applications in machine learning from dynamic environments to complex datasets requiring intricate pattern recognition.

This paper contributes a crucial step in marrying the geometric and probabilistic perspective with the practical application of deep neural networks, providing a robust foundation for future investigation into the multi-faceted complexities of AI models.