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Hypergraphs capture multi-way relationships in data, and they have consequently seen a number of applications in higher-order network analysis, computer vision, geometry processing, and machine learning. In this paper, we develop theoretical foundations for studying the space of hypergraphs using ingredients from optimal transport. By enriching a hypergraph with probability measures on its nodes and hyperedges, as well as relational information capturing local and global structures, we obtain a general and robust framework for studying the collection of all hypergraphs. First, we introduce a hypergraph distance based on the co-optimal transport framework of Redko et al. and study its theoretical properties. Second, we formalize common methods for transforming a hypergraph into a graph as maps between the space of hypergraphs and the space of graphs, and study their functorial properties and Lipschitz bounds. Finally, we demonstrate the versatility of our Hypergraph Co-Optimal Transport (HyperCOT) framework through various examples. 

Deep models are known to be vulnerable to data adversarial attacks, and many adversarial training techniques have been developed to improve their adversarial robustness. While data adversaries attack model predictions through modifying data, little is known about their impact on the neuron activations produced by the model, which play a crucial role in determining the model’s predictions and interpretability. In this work, we aim to develop a topological understanding of adversarial training to enhance its interpretability. We analyze the topological structure—in particular, mapper graphs—of neuron activations of data samples produced by deep adversarial training. Each node of a mapper graph represents a cluster of activations, and two nodes are connected by an edge if their corresponding clusters have a nonempty intersection. We provide an interactive visualization tool that demonstrates the utility of our topological framework in exploring the activation space. We found that stronger attacks make the data samples more indistinguishable in the neuron activation space that leads to a lower accuracy. Our tool also provides a natural way to identify the vulnerable data samples that may be useful in improving model robustness. 

Deep models are known to be vulnerable to data adversarial attacks, and many adversarial training techniques have been developed to improve their adversarial robustness. While data adversaries attack model predictions through modifying data, little is known about their impact on the neuron activations produced by the model, which play a crucial role in determining the model’s predictions and interpretability. In this work, we aim to develop a topological understanding of adversarial training to enhance its interpretability. We analyze the topological structure—in particular, mapper graphs—of neuron activations of data samples produced by deep adversarial training. Each node of a mapper graph represents a cluster of activations, and two nodes are connected by an edge if their corresponding clusters have a nonempty intersection. We provide an interactive visualization tool that demonstrates the utility of our topological framework in exploring the activation space. We found that stronger attacks make the data samples more indistinguishable in the neuron activation space that leads to a lower accuracy. Our tool also provides a natural way to identify the vulnerable data samples that may be useful in improving model robustness. 

Topological data analysis (TDA) is a branch of computational mathematics, bridging algebraic topology and data science, that provides compact, noise-robust representations of complex structures. Deep neural networks (DNNs) learn millions of parameters associated with a series of transformations defined by the model architecture resulting in high-dimensional, difficult to interpret internal representations of input data. As DNNs become more ubiquitous across multiple sectors of our society, there is increasing recognition that mathematical methods are needed to aid analysts, researchers, and practitioners in understanding and interpreting how these models' internal representations relate to the final classification. In this paper we apply cutting edge techniques from TDA with the goal of gaining insight towards interpretability of convolutional neural networks used for image classification. We use two common TDA approaches to explore several methods for modeling hidden layer activations as high-dimensional point clouds, and provide experimental evidence that these point clouds capture valuable structural information about the model's process. First, we demonstrate that a distance metric based on persistent homology can be used to quantify meaningful differences between layers and discuss these distances in the broader context of existing representational similarity metrics for neural network interpretability. Second, we show that a mapper graph can provide semantic insight as to how these models organize hierarchical class knowledge at each layer. These observations demonstrate that TDA is a useful tool to help deep learning practitioners unlock the hidden structures of their models. 

Abstract Metabolism is intertwined with various cellular processes, including controlling cell fate, influencing tumorigenesis, participating in stress responses and more. Metabolism is a complex, interdependent network, and local perturbations can have indirect effects that are pervasive across the metabolic network. Current analytical and technical limitations have long created a bottleneck in metabolic data interpretation. To address these shortcomings, we developed Metaboverse, a user-friendly tool to facilitate data exploration and hypothesis generation. Here we introduce algorithms that leverage the metabolic network to extract complex reaction patterns from data. To minimize the impact of missing measurements within the network, we introduce methods that enable pattern recognition across multiple reactions. Using Metaboverse, we identify a previously undescribed metabolite signature that correlated with survival outcomes in early stage lung adenocarcinoma patients. Using a yeast model, we identify metabolic responses suggesting an adaptive role of citrate homeostasis during mitochondrial dysfunction facilitated by the citrate transporter, Ctp1. We demonstrate that Metaboverse augments the user’s ability to extract meaningful patterns from multi-omics datasets to develop actionable hypotheses. 

Abstract We propose a novel method for the computation of Jacobi sets in 2D domains. The Jacobi set is a topological descriptor based on Morse theory that captures gradient alignments among multiple scalar fields, which is useful for multi-field visualization. Previous Jacobi set computations use piecewise linear approximations on triangulations that result in discretization artifacts like zig-zag patterns. In this paper, we utilize a local bilinear method to obtain a more precise approximation of Jacobi sets by preserving the topology and improving the geometry. Consequently, zig-zag patterns on edges are avoided, resulting in a smoother Jacobi set representation. Our experiments show a better convergence with increasing resolution compared to the piecewise linear method. We utilize this advantage with an efficient local subdivision scheme. Finally, our approach is evaluated qualitatively and quantitatively in comparison with previous methods for different mesh resolutions and across a number of synthetic and real-world examples.