Noise-Robust Topology Estimation of 2D Image Data via Neural Networks and Persistent Homology

Persistent Homology (PH) and Artificial Neural Networks (ANNs) offer contrasting approaches to inferring topological structure from data. In this study, we examine the noise robustness of a supervised neural network trained to predict Betti numbers in 2D binary images. We compare an ANN approach against a PH pipeline based on cubical complexes and the Signed Euclidean Distance Transform (SEDT), which is a widely adopted strategy for noise-robust topological analysis. Using one synthetic and two real-world datasets, we show that ANNs can outperform this PH approach under noise, likely due to their capacity to learn contextual and geometric priors from training data. Though still emerging, the use of ANNs for topology estimation offers a compelling alternative to PH under structural noise.

Time Series Analysis of Spiking Neural Systems via Transfer Entropy and Directed Persistent Homology

We present a topological framework for analysing neural time series that integrates Transfer Entropy (TE) with directed Persistent Homology (PH) to characterize information flow in spiking neural systems. TE quantifies directional influence between neurons, producing weighted, directed graphs that reflect dynamic interactions. These graphs are then analyzed using PH, enabling assessment of topological complexity across multiple structural scales and dimensions. We apply this TE+PH pipeline to synthetic spiking networks trained on logic gate tasks, image-classification networks exposed to structured and perturbed inputs, and mouse cortical recordings annotated with behavioral events. Across all settings, the resulting topological signatures reveal distinctions in task complexity, stimulus structure, and behavioral regime. Higher-dimensional features become more prominent in complex or noisy conditions, reflecting interaction patterns that extend beyond pairwise connectivity. Our findings offer a principled approach to mapping directed information flow onto global organizational patterns in both artificial and biological neural systems. The framework is generalizable and interpretable, making it well suited for neural systems with time-resolved and binary spiking data.

Diffusion in Zero-Shot Learning for Environmental Audio

Zero-shot learning enables models to generalize to unseen classes by leveraging semantic information, bridging the gap between training and testing sets with non-overlapping classes. While much research has focused on zero-shot learning in computer vision, the application of these methods to environmental audio remains underexplored, with poor performance in existing studies. Generative methods, which have demonstrated success in computer vision, are notably absent from environmental audio zero-shot learning, where classification-based approaches dominate. To address this gap, this work investigates generative methods for zero-shot learning in environmental audio. Two successful generative models from computer vision are adapted: a cross-aligned and distribution-aligned variational autoencoder (CADA-VAE) and a leveraging invariant side generative adversarial network (LisGAN). Additionally, a novel diffusion model conditioned on class auxiliary data is introduced. The diffusion model generates synthetic data for unseen classes, which is combined with seen-class data to train a classifier. Experiments are conducted on two environmental audio datasets, ESC-50 and FSC22. Results show that the diffusion model significantly outperforms all baseline methods, achieving more than 25% higher accuracy on the ESC-50 test partition. This work establishes the diffusion model as a promising generative approach for zero-shot learning and introduces the first benchmark of generative methods for environmental audio zero-shot learning, providing a foundation for future research in the field. Code is provided at https://github.com/ysims/ZeroDiffusion for the novel ZeroDiffusion method.

Synthetic Data Generation and Deep Learning for the Topological Analysis of 3D Data

This research uses deep learning to estimate the topology of manifolds represented by sparse, unordered point cloud scenes in 3D. A new labelled dataset was synthesised to train neural networks and evaluate their ability to estimate the genus of these manifolds. This data used random homeomorphic deformations to provoke the learning of visual topological features. We demonstrate that deep learning models could extract these features and discuss some advantages over existing topological data analysis tools that are based on persistent homology. Semantic segmentation was used to provide additional geometric information in conjunction with topological labels. Common point cloud multi-layer perceptron and transformer networks were both used to compare the viability of these methods. The experimental results of this pilot study support the hypothesis that, with the aid of sophisticated synthetic data generation, neural networks can perform segmentation-based topological data analysis. While our study focused on simulated data, the accuracy achieved suggests a potential for future applications using real data.

Topology Estimation of Simulated 4D Image Data by Combining Downscaling and Convolutional Neural Networks

Four-dimensional image-type data can quickly become prohibitively large, and it may not be feasible to directly apply methods, such as persistent homology or convolutional neural networks, to determine the topological characteristics of these data because they can encounter complexity issues. This study aims to determine the Betti numbers of large four-dimensional image-type data. The experiments use synthetic data, and demonstrate that it is possible to circumvent these issues by applying downscaling methods to the data prior to training a convolutional neural network, even when persistent homology software indicates that downscaling can significantly alter the homology of the training data. When provided with downscaled test data, the neural network can estimate the Betti numbers of the original samples with reasonable accuracy.