Bei Wang

Emilie Purvine, Davis Brown, Brett Jefferson et al.

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 into the 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 we 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 into 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.

Brenda Praggastis, Davis Brown, Carlos Ortiz Marrero et al.

Deep neural networks used for image classification often use convolutional filters to extract distinguishing features before passing them to a linear classifier. Most interpretability literature focuses on providing semantic meaning to convolutional filters to explain a model's reasoning process and confirm its use of relevant information from the input domain. Fully connected layers can be studied by decomposing their weight matrices using a singular value decomposition, in effect studying the correlations between the rows in each matrix to discover the dynamics of the map. In this work we define a singular value decomposition for the weight tensor of a convolutional layer, which provides an analogous understanding of the correlations between filters, exposing the dynamics of the convolutional map. We validate our definition using recent results in random matrix theory. By applying the decomposition across the linear layers of an image classification network we suggest a framework against which interpretability methods might be applied using hypergraphs to model class separation. Rather than looking to the activations to explain the network, we use the singular vectors with the greatest corresponding singular values for each linear layer to identify those features most important to the network. We illustrate our approach with examples and introduce the DeepDataProfiler library, the analysis tool used for this study.

Erin Wolf Chambers, Ishika Ghosh, Elizabeth Munch et al.

Mapper graphs are widely used tools in topological data analysis and visualization. They can be understood as discrete approximations of Reeb graphs, providing insight into the shape and connectivity of complex data. Given a high-dimensional point cloud together with a real-valued function defined on it, a mapper graph summarizes the induced topological structure: each node represents a local neighborhood, and edges connect nodes whose corresponding neighborhoods overlap. Our focus is the interleaving distance for mapper graphs, arising as a discretized analogue of the interleaving distance for Reeb graphs-a quantity known to be NP-hard to compute. This distance measures how similar two mapper graphs are by quantifying how much they must be ``stretched'' to be made comparable. Recent work introduced a loss function that gives an upper bound on this distance. The loss evaluates how far a given collection of maps, called an assignment, is from being a true interleaving. Importantly, it is computationally tractable, offering a practical way to bound the distance, however the quality of the bound is dependent on the choice of assignment. In this paper, we develop the first framework for bounding the interleaving distance on mapper graphs. We present the bound in two ways: first, by formulating an integer linear program (ILP) that determines whether an $n$-interleaving exists for a given $n$; and second, by constructing an ILP that identifies an assignment with minimal loss for that $n$. We also evaluate the method on small examples where the interleaving distance is known, and on benchmark and simulated datasets, demonstrating the utility of the approach for classification tasks based on mapper graphs.

Archit Rathore, Sunipa Dev, Jeff M. Phillips et al.

Word vector embeddings have been shown to contain and amplify biases in data they are extracted from. Consequently, many techniques have been proposed to identify, mitigate, and attenuate these biases in word representations. In this paper, we utilize interactive visualization to increase the interpretability and accessibility of a collection of state-of-the-art debiasing techniques. To aid this, we present Visualization of Embedding Representations for deBiasing system ("VERB"), an open-source web-based visualization tool that helps the users gain a technical understanding and visual intuition of the inner workings of debiasing techniques, with a focus on their geometric properties. In particular, VERB offers easy-to-follow use cases in exploring the effects of these debiasing techniques on the geometry of high-dimensional word vectors. To help understand how various debiasing techniques change the underlying geometry, VERB decomposes each technique into interpretable sequences of primitive transformations and highlights their effect on the word vectors using dimensionality reduction and interactive visual exploration. VERB is designed to target natural language processing (NLP) practitioners who are designing decision-making systems on top of word embeddings, and also researchers working with fairness and ethics of machine learning systems in NLP. It can also serve as a visual medium for education, which helps an NLP novice to understand and mitigate biases in word embeddings.

Xinyuan Yan, Rita Sevastjanova, Sinie van der Ben et al.

Large language models (LLMs) produce high-dimensional embeddings that capture rich semantic and syntactic relationships between words, sentences, and concepts. Investigating the topological structures of LLM embedding spaces via mapper graphs enables us to understand their underlying structures. Specifically, a mapper graph summarizes the topological structure of the embedding space, where each node represents a topological neighborhood (containing a cluster of embeddings), and an edge connects two nodes if their corresponding neighborhoods overlap. However, manually exploring these embedding spaces to uncover encoded linguistic properties requires considerable human effort. To address this challenge, we introduce a framework for semi-automatic annotation of these embedding properties. To organize the exploration process, we first define a taxonomy of explorable elements within a mapper graph such as nodes, edges, paths, components, and trajectories. The annotation of these elements is executed through two types of customizable LLM-based agents that employ perturbation techniques for scalable and automated analysis. These agents help to explore and explain the characteristics of mapper elements and verify the robustness of the generated explanations. We instantiate the framework within a visual analytics workspace and demonstrate its effectiveness through case studies. In particular, we replicate findings from prior research on BERT's embedding properties across various layers of its architecture and provide further observations into the linguistic properties of topological neighborhoods.

Md Rahat-uz- Zaman, Bei Wang, Paul Rosen

Deep neural networks (DNNs) achieve state-of-the-art performance in many vision tasks, yet understanding their internal behavior remains challenging, particularly how different layers and activation channels contribute to class separability. We introduce ChannelExplorer, an interactive visual analytics tool for analyzing image-based outputs across model layers, emphasizing data-driven insights over architecture analysis for exploring class separability. ChannelExplorer summarizes activations across layers and visualizes them using three primary coordinated views: a Scatterplot View to reveal inter- and intra-class confusion, a Jaccard Similarity View to quantify activation overlap, and a Heatmap View to inspect activation channel patterns. Our technique supports diverse model architectures, including CNNs, GANs, ResNet and Stable Diffusion models. We demonstrate the capabilities of ChannelExplorer through four use-case scenarios: (1) generating class hierarchy in ImageNet, (2) finding mislabeled images, (3) identifying activation channel contributions, and(4) locating latent states' position in Stable Diffusion model. Finally, we evaluate the tool with expert users.

Lin Yan, Talha Bin Masood, Farhan Rasheed et al.

Merge trees, a type of topological descriptor, serve to identify and summarize the topological characteristics associated with scalar fields. They present a great potential for the analysis and visualization of time-varying data. First, they give compressed and topology-preserving representations of data instances. Second, their comparisons provide a basis for studying the relations among data instances, such as their distributions, clusters, outliers, and periodicities. A number of comparative measures have been developed for merge trees. However, these measures are often computationally expensive since they implicitly consider all possible correspondences between critical points of the merge trees. In this paper, we perform geometry-aware comparisons of merge trees using labeled interleaving distances. The main idea is to decouple the computation of a comparative measure into two steps: a labeling step that generates a correspondence between the critical points of two merge trees, and a comparison step that computes distances between a pair of labeled merge trees by encoding them as matrices. We show that our approach is general, computationally efficient, and practically useful. Our general framework makes it possible to integrate geometric information of the data domain in the labeling process. At the same time, it reduces the computational complexity since not all possible correspondences have to be considered. We demonstrate via experiments that such geometry-aware merge tree comparisons help to detect transitions, clusters, and periodicities of time-varying datasets, as well as to diagnose and highlight the topological changes between adjacent data instances.

Youjia Zhou, Methun Kamruzzaman, Patrick Schnable et al.

High-throughput technologies to collect field data have made observations possible at scale in several branches of life sciences. The data collected can range from the molecular level (genotypes) to physiological (phenotypic traits) and environmental observations (e.g., weather, soil conditions). These vast swathes of data, collectively referred to as phenomics data, represent a treasure trove of key scientific knowledge on the dynamics of the underlying biological system. However, extracting information and insights from these complex datasets remains a significant challenge owing to their multidimensionality and lack of prior knowledge about their complex structure. In this paper, we present Pheno-Mapper, an interactive toolbox for the exploratory analysis and visualization of large-scale phenomics data. Our approach uses the mapper framework to perform a topological analysis of the data, and subsequently render visual representations with built-in data analysis and machine learning capabilities. We demonstrate the utility of this new tool on real-world plant (e.g., maize) phenomics datasets. In comparison to existing approaches, the main advantage of Pheno-Mapper is that it provides rich, interactive capabilities in the exploratory analysis of phenomics data, and it integrates visual analytics with data analysis and machine learning in an easily extensible way. In particular, Pheno-Mapper allows the interactive selection of subpopulations guided by a topological summary of the data and applies data mining and machine learning to these selected subpopulations for in-depth exploration.

Lin Yan, Talha Bin Masood, Raghavendra Sridharamurthy et al.

In topological data analysis and visualization, topological descriptors such as persistence diagrams, merge trees, contour trees, Reeb graphs, and Morse-Smale complexes play an essential role in capturing the shape of scalar field data. We present a state-of-the-art report on scalar field comparison using topological descriptors. We provide a taxonomy of existing approaches based on visualization tasks associated with three categories of data: single fields, time-varying fields, and ensembles. These tasks include symmetry detection, periodicity detection, key event/feature detection, feature tracking, clustering, and structure statistics. Our main contributions include the formulation of a set of desirable mathematical and computational properties of comparative measures, and the classification of visualization tasks and applications that are enabled by these measures.

Fangfei Lan, Michael Young, Lauren Anderson et al.

We present a state-of-the-art report on visualization in astrophysics. We survey representative papers from both astrophysics and visualization and provide a taxonomy of existing approaches based on data analysis tasks. The approaches are classified based on five categories: data wrangling, data exploration, feature identification, object reconstruction, as well as education and outreach. Our unique contribution is to combine the diverse viewpoints from both astronomers and visualization experts to identify challenges and opportunities for visualization in astrophysics. The main goal is to provide a reference point to bring modern data analysis and visualization techniques to the rich datasets in astrophysics.

Youjia Zhou, Nathaniel Saul, Ilkin Safarli et al.

The mapper construction is a powerful tool from topological data analysis that is designed for the analysis and visualization of multivariate data. In this paper, we investigate a method for stitching a pair of univariate mappers together into a bivariate mapper, and study topological notions of information gains, referred to as topological gains, during such a process. We further provide implementations that visualize such topological gains for mapper graphs.

Youjia Zhou, Archit Rathore, Emilie Purvine et al.

We study hypergraph visualization via its topological simplification. We explore both vertex simplification and hyperedge simplification of hypergraphs using tools from topological data analysis. In particular, we transform a hypergraph to its graph representations known as the line graph and clique expansion. A topological simplification of such a graph representation induces a simplification of the hypergraph. In simplifying a hypergraph, we allow vertices to be combined if they belong to almost the same set of hyperedges, and hyperedges to be merged if they share almost the same set of vertices. Our proposed approaches are general, mathematically justifiable, and they put vertex simplification and hyperedge simplification in a unifying framework.

Mingzhe Li, Sourabh Palande, Lin Yan et al.

Merge trees are a type of topological descriptors that record the connectivity among the sublevel sets of scalar fields. They are among the most widely used topological tools in visualization. In this paper, we are interested in sketching a set of merge trees. That is, given a large set T of merge trees, we would like to find a much smaller basis set S such that each tree in T can be approximately reconstructed from a linear combination of merge trees in S. A set of high-dimensional vectors can be sketched via matrix sketching techniques such as principal component analysis and column subset selection. However, up until now, topological descriptors such as merge trees have not been known to be sketchable. We develop a framework for sketching a set of merge trees that combines the Gromov-Wasserstein probabilistic matching with techniques from matrix sketching. We demonstrate the applications of our framework in sketching merge trees that arise from time-varying scientific simulations. Specifically, our framework obtains a much smaller representation of a large set of merge trees for downstream analysis and visualization. It is shown to be useful in identifying good representatives and outliers with respect to a chosen basis. Finally, our work shows a promising direction of utilizing randomized linear algebra within scientific visualization.

Youjia Zhou, Nithin Chalapathi, Archit Rathore et al.

The mapper algorithm is a popular tool from topological data analysis for extracting topological summaries of high-dimensional datasets. In this paper, we present Mapper Interactive, a web-based framework for the interactive analysis and visualization of high-dimensional point cloud data. It implements the mapper algorithm in an interactive, scalable, and easily extendable way, thus supporting practical data analysis. In particular, its command-line API can compute mapper graphs for 1 million points of 256 dimensions in about 3 minutes (4 times faster than the vanilla implementation). Its visual interface allows on-the-fly computation and manipulation of the mapper graph based on user-specified parameters and supports the addition of new analysis modules with a few lines of code. Mapper Interactive makes the mapper algorithm accessible to nonspecialists and accelerates topological analytics workflows.

Bei Wang, Jianping An

This paper addresses the importance of full-image supervision for monocular depth estimation. We propose a semi-supervised architecture, which combines both unsupervised framework of using image consistency and supervised framework of dense depth completion. The latter provides full-image depth as supervision for the former. Ego-motion from navigation system is also embedded into the unsupervised framework as output supervision of an inner temporal transform network, making monocular depth estimation better. In the evaluation, we show that our proposed model outperforms other approaches on depth estimation.

Archit Rathore, Nithin Chalapathi, Sourabh Palande et al.

Deep neural networks such as GoogLeNet, ResNet, and BERT have achieved impressive performance in tasks such as image and text classification. To understand how such performance is achieved, we probe a trained deep neural network by studying neuron activations, i.e., combinations of neuron firings, at various layers of the network in response to a particular input. With a large number of inputs, we aim to obtain a global view of what neurons detect by studying their activations. In particular, we develop visualizations that show the shape of the activation space, the organizational principle behind neuron activations, and the relationships of these activations within a layer. Applying tools from topological data analysis, we present TopoAct, a visual exploration system to study topological summaries of activation vectors. We present exploration scenarios using TopoAct that provide valuable insights into learned representations of neural networks. We expect TopoAct to give a topological perspective that enriches the current toolbox of neural network analysis, and to provide a basis for network architecture diagnosis and data anomaly detection.

Ingrid Hotz, Roxana Bujack, Christoph Garth et al.

Mathematical concepts and tools have shaped the field of visualization in fundamental ways and played a key role in the development of a large variety of visualization techniques. In this chapter, we sample the visualization literature to provide a taxonomy of the usage of mathematics in visualization, and to identify a fundamental set of mathematics that should be taught to students as part of an introduction to contemporary visualization research. Within the scope of this chapter, we are unable to provide a full review of all mathematical foundations of visualization; rather, we identify a number of concepts that are useful in visualization, explain their significance, and provide references for further reading.

Lin Yan, Yusu Wang, Elizabeth Munch et al.

Physical phenomena in science and engineering are frequently modeled using scalar fields. In scalar field topology, graph-based topological descriptors such as merge trees, contour trees, and Reeb graphs are commonly used to characterize topological changes in the (sub)level sets of scalar fields. One of the biggest challenges and opportunities to advance topology-based visualization is to understand and incorporate uncertainty into such topological descriptors to effectively reason about their underlying data. In this paper, we study a structural average of a set of labeled merge trees and use it to encode uncertainty in data. Specifically, we compute a 1-center tree that minimizes its maximum distance to any other tree in the set under a well-defined metric called the interleaving distance. We provide heuristic strategies that compute structural averages of merge trees whose labels do not fully agree. We further provide an interactive visualization system that resembles a numerical calculator that takes as input a set of merge trees and outputs a tree as their structural average. We also highlight structural similarities between the input and the average and incorporate uncertainty information for visual exploration. We develop a novel measure of uncertainty, referred to as consistency, via a metric-space view of the input trees. Finally, we demonstrate an application of our framework through merge trees that arise from ensembles of scalar fields. Our work is the first to employ interleaving distances and consistency to study a global, mathematically rigorous, structural average of merge trees in the context of uncertainty visualization.

Bei Wang, Jianping An, Jiayan Cao

Object detection in point cloud data is one of the key components in computer vision systems, especially for autonomous driving applications. In this work, we present Voxel-FPN, a novel one-stage 3D object detector that utilizes raw data from LIDAR sensors only. The core framework consists of an encoder network and a corresponding decoder followed by a region proposal network. Encoder extracts multi-scale voxel information in a bottom-up manner while decoder fuses multiple feature maps from various scales in a top-down way. Extensive experiments show that the proposed method has better performance on extracting features from point data and demonstrates its superiority over some baselines on the challenging KITTI-3D benchmark, obtaining good performance on both speed and accuracy in real-world scenarios.

René Corbet, Ulderico Fugacci, Michael Kerber et al.

Topological data analysis and its main method, persistent homology, provide a toolkit for computing topological information of high-dimensional and noisy data sets. Kernels for one-parameter persistent homology have been established to connect persistent homology with machine learning techniques. We contribute a kernel construction for multi-parameter persistence by integrating a one-parameter kernel weighted along straight lines. We prove that our kernel is stable and efficiently computable, which establishes a theoretical connection between topological data analysis and machine learning for multivariate data analysis.

Paul Rosen, Mustafa Hajij, Bei Wang

Node-link diagrams are a popular method for representing graphs that capture relationships between individuals, businesses, proteins, and telecommunication endpoints. However, node-link diagrams may fail to convey insights regarding graph structures, even for moderately sized data of a few hundred nodes, due to visual clutter. We propose to apply the mapper construction -- a popular tool in topological data analysis -- to graph visualization, which provides a strong theoretical basis for summarizing the data while preserving their core structures. We develop a variation of the mapper construction targeting weighted, undirected graphs, called {\mog}, which generates homology-preserving skeletons of graphs. We further show how the adjustment of a single parameter enables multi-scale skeletonization of the input graph. We provide a software tool that enables interactive explorations of such skeletons and demonstrate the effectiveness of our method for synthetic and real-world data.

Tim Sodergren, Jessica Hair, Jeff M. Phillips et al.

We present an interactive visualization system for exploring the coverage in sensor networks with uncertain sensor locations. We consider a simple case of uncertainty where the location of each sensor is confined to a discrete number of points sampled uniformly at random from a region with a fixed radius. Employing techniques from topological data analysis, we model and visualize network coverage by quantifying the uncertainty defined on its simplicial complex representations. We demonstrate the capabilities and effectiveness of our tool via the exploration of randomly distributed sensor networks.

Paul Rosen, Anil Seth, Betsy Mills et al.

The current generation of radio and millimeter telescopes, particularly the Atacama Large Millimeter Array (ALMA), offers enormous advances in observing capabilities. While these advances represent an unprecedented opportunity to facilitate scientific understanding, the increased complexity in the spatial and spectral structure of these ALMA data cubes lead to challenges in their interpretation. In this paper, we perform a feasibility study for applying topological data analysis and visualization techniques never before tested by the ALMA community. Through techniques based on contour trees, we seek to improve upon existing analysis and visualization workflows of ALMA data cubes, in terms of accuracy and speed in feature extraction. We review our application development process in building effective analysis and visualization capabilities for the astrophysicists. We also summarize effective design practices by identifying domain-specific needs of simplicity, integrability, and reproducibility, in order to best target and service the large astrophysics community.