TopoInVis Connect: Topology meets Artificial Intelligencebookmark_border

The contract data visualization tools make with their users is that a chart is a faithful and accurate visual representation of the numbers it is made from. Motivated by wanting to make tools that are more flexible while still rigorously structure preserving, we propose a methodology for fully specifying arbitrary data to visualization mappings in a manner that translates to pseudocode. We propose that fiber bundles provide a uniform interface for describing a variety of underlying data - tables, images, networks, etc. - in a manner that independently encodes the mathematical structure of the continuity and the fields of the dataset. Modeling the data structures that store the datasets as sheaves provides a framework for specifying visualization methods that are designed to work regardless of how the dataset is stored - whether the data is on disk, distributed, or on demand. Specifying the visualization library components as natural transforms of sheaves means that the constraints that the component must satisfy to be structure preserving can be specified as the set of morphisms on the data and graphic sheaves, including the structure on the topology and fields of the data. Using category theory to formally express how visual elements are constructed means we can translate those expectations into pseudocode, which can then be used to build tools that enforce the expectation that a visualization tool is faithfully translating between numbers and charts.

We introduce the Continuous Edit Distance (CED), an elastic and geodesic metric for time-varying persistence diagrams (TVPDs). Being elastic, the CED goes beyond existing strategies for comparing temporal sequences of persistence diagrams based on Wasserstein distance integration (e.g., [61]), which require a perfect temporal alignment of the input sequences and which are therefore sensitive to temporal shifts or stretches. Being geodesic, in contrast to standard elastic metrics for time series, the CED brings the perspective of being amenable to the computation of geodesics and barycenters of TVPDs for future work. We describe how input temporal sequences of persistence diagrams can be converted into a TVPD and how to compute our metric via dynamic programming, yielding interpretable alignments between the input sequences. Empirically, the CED is robust to additive perturbations (both in time and birth-death values), recovers temporal shifts, supports temporal pattern search, and achieves MDS-based clustering performance comparable to or better than standard elastic distances on real datasets. A C++ implementation of our algorithms is available at https://github.com/sebastien-tchitchek/ContinuousEditDistance.

Trait-induced merge trees (TIMTs) provide a robust topology-based method for selecting and browsing feature level sets in multivariate data by analyzing the distance field induced by a user-specified trait in attribute space. Existing TIMT computations typically sample this distance field at mesh vertices and assume piecewise-linear interpolation, although the Euclidean distance-to-trait function is generally not piecewise linear on the original mesh. As a result, the resulting merge tree may miss zero-valued features and may perturb the locations and values of minima and merge events. We study the exact computation of TIMTs for piecewise-linear bivariate fields, focusing first on point traits. We show that the restricted sublevel sets inside each tetrahedron are convex and therefore have trivial local merge-tree structure, implying that global topological changes arise only through gluing across simplex boundaries. Based on this observation, we construct a weighted graph whose merge tree is isomorphic to the exact merge tree of the induced distance field. We further relate TIMTs to Jacobi sets, showing how nonzero edge events of the TIMT are localized by the singular structure of the underlying bivariate map. We establish a theoretical upper bound on the error of the vertex-sampled linear interpolation, expressed in terms of the maximum length of projected mesh edges in the range. We discuss extensions to line, line-segment, and finite point-set traits, and implement the method robustly using CGAL and VTK, demonstrating results on both synthetic and real-world datasets.

Exploring chemical reaction latent spaces is essential for understanding the organization of complex reaction landscapes. Previous work has shown that transformer-based models can generate high-quality latent representations, known as reaction fingerprints, that enable the visualization of reaction spaces as tree-structured atlases. However, interpreting these reaction atlases remains largely manual and time-consuming, limiting their scalability in modern cheminformatics workflows. Although users can readily identify branches, neighborhoods, and transitions, understanding their underlying chemical significance still requires examining numerous individual reactions. We present a framework that combines interactive reaction-atlas visualization with large language model (LLM)-based explanations. Building on reaction atlases, our approach introduces two complementary LLM-driven components: a Neighborhood Explainer, which summarizes and compares local regions of the atlas, and a Path Explainer, which analyzes transitions of reactions with similar latent space representations along paths defined by user-selected endpoints. The framework supports both label-free and label-aware settings, operating either directly on reaction SMILES (Simplified Molecular Input Line Entry System) representations or by incorporating reaction class labels. We demonstrate the framework on the Schneider 50K dataset, a curated benchmark comprising approximately 50,000 atom-mapped chemical reactions, and show that explanation-guided exploration enables users to interpret local neighborhoods, compare related branches, analyze reaction-space transitions, and identify locally inconsistent regions within chemical reaction latent spaces.

Time-varying bivariate fields arise in many scientific applications, where the relationship between two scalar quantities evolves over time. While topological methods such as merge trees provide an effective framework for identifying and tracking features in univariate data, analogous approaches for bivariate fields remain comparatively underexplored. Reeb spaces extend topological analysis to multivariate data by representing fiber connectivity through a collection of interconnected sheets, making these sheets natural candidates for describing bivariate structures. However, establishing temporal correspondences between sheets is challenging due to the structural complexity of Reeb spaces, sensitivity to noise, and the difficulty of defining meaningful similarity measures across timesteps. We present a framework for tracking Reeb space sheets in time-varying bivariate fields. The method establishes correspondences between sheets in consecutive timesteps using complementary similarity measures defined in the spatial domain and the range space. We evaluate the method on a synthetic torus dataset and two time-varying molecular electronic structure datasets. The results show that Reeb space sheet tracking reveals persistent structures and highlights interesting intervals of temporal change. Overall, the results demonstrate that Reeb space sheets can serve as trackable topological structures and provide a foundation for the visual analysis of time-varying bivariate data.

Topological Data Analysis (TDA) provides an effective framework for visualizing the underlying shape of high-dimensional data, revealing structural insights often obscured by traditional methods. In particular, the Mapper algorithm generates graph-based representations that capture global connectivity and local similarity within high-dimensional point clouds. While time-series data is frequently clustered without explicit temporal timestamps to prevent biasing similarity metrics, the resulting topological structures of TC wind field data can inherently preserve the temporal sequence. This phenomenon is particularly evident in the Mapper graphs of tropical cyclone (TC) wind fields. We present a novel visualization tool designed to recover and project this latent temporal information onto the topological map. Applying this tool to a longitudinal study of tropical cyclones from 2012–2025 reveals that specific wind field states are present among TCs with high intensities.

Understanding how spatial patterns evolve over time is a complex task that often arises in the analysis of public health data. In this work, we investigate the use of vineyards from topological data analysis (TDA) in this setting by applying them to time series data related to the overdose epidemic in the state of Ohio. We begin by proposing statistical tests that can be used in order to evaluate whether vineyards are a reasonable technique to study a spatiotemporal dataset. We then apply these tests to the data of drug overdose deaths in Ohio and, finding the data suitable, perform a subsequent analysis using vineyards to visualize the evolution of local maxima of death rates throughout the Ohio overdose epidemic. We conclude by developing statistical methods to quantify the significance and uncertainty of vineyard features and by exploring how vineyard-derived summaries can be used for forecasting.