Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

From Classical to Topological Neural Networks Under Uncertainty

Sarah Harkins Dayton, Layal Bou Hamdan, Ioannis D. Schizas, David L. Boothe, Vasileios Maroulas

TL;DR

The chapter tackles the challenge of robust, uncertain decision-making in military AI by integrating neural networks, topological data analysis, and Bayesian methods across vision, time series, and graphs. It advances a spectrum of topology-informed models, including persistence-diagram based clustering and Bayesian processing, topological CNNs, simplicial and cell complex networks, and sheaf-theoretic approaches, all designed to quantify and leverage uncertainty. Key contributions span practical architectures like TCNNs, BTCNNs, BSNNs, and ToFU, along with principled statistical frameworks for persistence diagrams (KDE, PPPs, Bayes factors) that enable rigorous inference. The proposed methods improve interpretability, robustness to noise and transformations, and calibrated uncertainty—crucial attributes for mission-critical systems in contested environments. Collectively, the work demonstrates how topology and probabilistic reasoning can be embedded into deep learning to enhance generalization, reliability, and human–machine collaboration in defense applications.

Abstract

This chapter explores neural networks, topological data analysis, and topological deep learning techniques, alongside statistical Bayesian methods, for processing images, time series, and graphs to maximize the potential of artificial intelligence in the military domain. Throughout the chapter, we highlight practical applications spanning image, video, audio, and time-series recognition, fraud detection, and link prediction for graphical data, illustrating how topology-aware and uncertainty-aware models can enhance robustness, interpretability, and generalization.

From Classical to Topological Neural Networks Under Uncertainty

TL;DR

The chapter tackles the challenge of robust, uncertain decision-making in military AI by integrating neural networks, topological data analysis, and Bayesian methods across vision, time series, and graphs. It advances a spectrum of topology-informed models, including persistence-diagram based clustering and Bayesian processing, topological CNNs, simplicial and cell complex networks, and sheaf-theoretic approaches, all designed to quantify and leverage uncertainty. Key contributions span practical architectures like TCNNs, BTCNNs, BSNNs, and ToFU, along with principled statistical frameworks for persistence diagrams (KDE, PPPs, Bayes factors) that enable rigorous inference. The proposed methods improve interpretability, robustness to noise and transformations, and calibrated uncertainty—crucial attributes for mission-critical systems in contested environments. Collectively, the work demonstrates how topology and probabilistic reasoning can be embedded into deep learning to enhance generalization, reliability, and human–machine collaboration in defense applications.

Abstract

This chapter explores neural networks, topological data analysis, and topological deep learning techniques, alongside statistical Bayesian methods, for processing images, time series, and graphs to maximize the potential of artificial intelligence in the military domain. Throughout the chapter, we highlight practical applications spanning image, video, audio, and time-series recognition, fraud detection, and link prediction for graphical data, illustrating how topology-aware and uncertainty-aware models can enhance robustness, interpretability, and generalization.
Paper Structure (27 sections, 33 equations, 12 figures)

This paper contains 27 sections, 33 equations, 12 figures.

Table of Contents

  1. Introduction
  2. Foundational Artificial Neural Networks
  3. Convolutional Neural Networks
  4. Recurrent Neural Networks
  5. Transformers
  6. Graph Neural Networks
  7. Graph Convolutional Networks
  8. GraphSAGE and GraphSAINT
  9. Graph Attention Networks
  10. Bayesian Neural Networks
  11. Topological Methods
  12. Learning Frameworks in Persistence Diagram Space
  13. Clustering in Persistence Diagram Space
  14. Statistical Modeling of Random Persistence Diagrams
  15. Bayesian Modeling of Random Persistence Diagrams
  16. ...and 12 more sections

Figures (12)

  • Figure 1: General architecture of an artificial neural network.
  • Figure 2: A convolution operation that is applied in a convolutional layer.
  • Figure 3: GAT methodology. (a) For a pair of neighboring nodes $i$ and $j$, node features are first linearly transformed ($\mathbf{W}\,{\mathbf{h}}_{i}$, $\mathbf{W}\,{\mathbf{h}}_{j}$), concatenated, passed through a LeakyReLU, and normalized with a softmax to yield attention coefficients ${\boldsymbol{\alpha}}_{ij}$. (b) Each node aggregates its neighbors’ features (including a self-loop) using the learned weights ${\boldsymbol{\alpha}}_{ij}$. Outputs from multiple heads are combined to produce the updated representation ${\mathbf{h}}^{\,\prime}$.
  • Figure 4: Simplicial complex
  • Figure 5: Persistent homology of a circular point cloud.
  • ...and 7 more figures