Abstract:
Abstract: Analyzing and effectively visualizing brain data remains a fundamental challenge of modern research. Neural circuits contain intricate dendritic trees (10-6 m), functional networks that span the whole brain, and an astronomical number of synaptic connections (1014 ). Neural activity varies at the level of individual spikes (10-3 s) and over a lifetime of development and experience. Moreover, much research seeks to correlate neural activity with: (1) structures of the external world (being perceived by the brain); (2) behaviors (being directed by the brain); or (3) reports of mental states (e.g., experiences, imaginations, intentions, etc.; each being created by the brain). As description of each of these domains is a formidable challenge in its own right, the study of how brains coordinate perception, experiences, and actions in real-world contexts remains a monumental task. Recent advances in Topological Data Visualization (TDV) provide new tools to explore old data and there is great promise in application of TDV to the vast trove of extant neuroscience data. In this talk, I will give a brief introduction to common brain science datasets for applied topologists, highlight previous examples of topological analysis of such, and overview some curated datasets here provided for collaborative exploration throughout the workshop.
Abstract:
Abstract: Mapper graphs are a widely used tool in topological data analysis and visualization. They can be viewed as discrete approximations of Reeb graphs, offering insight into the shape and connectivity of complex data. Given a high-dimensional point cloud X equipped with a function f: X → R, a mapper graph provides a summary of the topological structure of X induced by f, where each node represents a local neighborhood, and edges connect nodes whose corresponding neighborhoods overlap. Our focus is the interleaving distance for mapper graphs, which arises from a discretization of the interleaving distance for Reeb graphs. Notably, computing both distances is NP-hard in general. The interleaving distance for mapper graphs quantifies their similarity by measuring the extent to which they must be "stretched" to become comparable. Recent work introduced a loss function that provides an upper bound on the interleaving distance for mapper graphs. More specifically, this loss function evaluates how far a given assignment is from being a true interleaving and is computationally tractable, offering a practical way to estimate the distance.
In this talk, I will present a categorical formulation of mapper graphs and introduce the first computational framework for estimating their interleaving distance via an associated loss function. Since the quality of this bound depends on the chosen assignment between graphs, we formulate the problem as an integer linear program to find an optimal assignment. I will demonstrate that, on small examples where the true interleaving distance can be computed by hand, our optimized bound matches the exact value. Finally, I will show results from an experiment on an image benchmarking dataset, where we compute pairwise mapper loss values and use them for image classification, illustrating the practical potential of this approach.
Abstract: The Mapper algorithm is a fundamental tool in exploratory topological data analysis for identifying connectivity and topological clustering in data. Derived from the nerve construction, Mapper graphs can contain additional information about clustering density when considering the higher-dimensional skeleta. To observe two-dimensional features, and capture one-dimensional topology, we construct **2-Mapper**. A common issue in using Mapper algorithms is parameter choice. We develop tools to choose 2-Mapper parameters that reflect persistent Betti-1 information. Computationally, we study how cover choice affects 2-Mapper and analyze this through a computational Multiscale Mapper algorithm. We test our constructions on three-dimensional shape data, including the Klein bottle.
Software demo: 2-Mapper and Multiscale 2-Mapper is an algorithm to compute a filtration of mapper complexes across a tower of cover choices. In this talk we will discuss the software implementation available to compute Multiscale Mapper using Python. Our software is available on GitHub (GitHub - hfr1tz3/TwoMapper: 2-Mapper is a higher dimensional generalization of the mapper graph.)
Abstract: This talk presents a topological framework for analyzing complex, high-dimensional datasets by leveraging the combined power of simplicial complexes, simplicial maps, and filtration techniques. At the core of the discussion is the Mapper algorithm, a tool from Topological Data Analysis (TDA) that constructs a graph-based summary of data by translating geometric and topological relationships into a combinatorial structure.
We begin by introducing the foundational idea of representing data through simplicial complexes, where points and their relationships are encoded using vertices, edges, and higher-dimensional simplices. We then examine how simplicial maps allow us to relate different complexes and preserve their topological structure during transformations.
The notion of filtration is introduced to study how these topological summaries evolve as we vary parameters such as bin size or overlap in the cover. We demonstrate how changes in these parameters reveal persistent features and structural transitions within the data, offering insights that are robust to noise and not easily captured by classical techniques.
Abstract: Topological Data Analysis (TDA) provides a powerful framework for extracting structure from complex data by studying its shape. This talk presents recent work on visualizing maps between high-dimensional spaces to detect correlations between datasets, alongside new adaptations of TDA to settings where representative sampling is impossible. This includes the integration of TDA with machine learning methodologies, particularly in contexts where traditional sampling is impractical, to analyze infinite datasets effectively. Time permitting, we will also talk about relations defined on three or more sets, including a generalization of the Dowker's theorem and provide applications to knot theory and comparative cancer genomics.
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Abstract: Topological data analysis (TDA) has emerged as a powerful framework for capturing the intrinsic structure of data. By summarizing the connectivity and critical structures within scalar fields, topological descriptors such as merge trees and Morse complexes provide a robust way to detect, track, and compare meaningful features across time and conditions. These tools are particularly valuable in scientific domains, ranging from neuroscience to climate science, where data is complex, noisy, and dynamic.
In this talk, I present a unified view of recent advances that integrate topological descriptors with optimal transport theory to enable robust and scalable analysis across multiple settings. I begin by introducing a sketching framework for merge trees, where dimensionality reduction is achieved through matrix sketching and Gromov-Wasserstein-based embeddings, allowing us to extract representative structural "modes" from large ensembles of simulations or repeated measurements. Next, I present GWMT, a probabilistic framework for feature tracking in time-varying scalar fields, which models merge trees as measure networks and applies partial optimal transport to establish soft correspondences across time. This allows us to track features that appear, disappear, split, or merge, and to quantify uncertainty in the tracking process. I demonstrate this framework through a real-world application to cloud system tracking using satellite data, highlighting its ability to capture both physical events and complex spatial-temporal dynamics. Finally, I explore the use of optimal transport distances to compare Morse complexes, enabling efficient and structurally meaningful alignment of gradient-based topological features. This approach supports tasks such as feature correspondence and classification, and offers a new perspective on comparing topological structures across scientific datasets.
Abstract: Perceptual spaces are mental workspaces in which the elements of a sensory or cognitive domain correspond to points, and distances between these points correspond to perceptual dissimilarity. Perceptual spaces thus characterize how sensory stimuli are organized and how they can support operations such as grouping and generalization. Since it is a fundamental tenet that similar percepts should reflect similar patterns of neural activity, perceptual spaces also provide clues to neural mechanisms. The classic example is Maxwell’s linkage between the observation that color is a three-parameter perceptual domain, and that color vision is based on three classes of photoreceptors with three distinct spectral sensitivities.
I will present an overview of plausible geometries of perceptual spaces and then consider several strategies for inferring the geometry of a perceptual space from psychophysical experiments. One focus will be the contrast between spaces of low-level visual features, and spaces with high semantic content. A second focus will be a space of visual textures. This is a 10-dimensional space in which the local geometry can be rigorously characterized, and it provides an opportunity to examine how the geometry of the space is transformed by task. A third focus will be on graphical models, and some open questions that arise when trying to bridge the gap between graphical models and more standard geometric ones.
Support: NIH EY07977, NSF 2014217.
Abstract:
Abstract: Cycle representatives of persistent homology classes can be used to provide descriptions of topological features in data. However, the non-uniqueness of these representatives creates ambiguity and can lead to many different interpretations of the same set of classes. One approach to solving this problem is to optimize the choice of representative against some measure that is meaningful in the context of the data. In this work, we provide a study of the effectiveness and computational cost of several ℓ1-minimization optimization procedures for constructing homological cycle bases for persistent homology with rational coefficients in dimension one, including uniform-weighted and length-weighted edge-loss algorithms as well as uniform-weighted and area-weighted triangle-loss algorithms. We conduct these optimizations via standard linear programming methods, applying general-purpose solvers to optimize over column bases of simplicial boundary matrices.
Our key findings are:
(i) optimization is effective in reducing the size of cycle representatives, though the extent of the
reduction varies according to the dimension and distribution of the underlying data, (ii)
the computational cost of optimizing a basis of cycle representatives exceeds the cost of computing
such a basis, in most datasets we consider, (iii) the choice of linear solvers matters a lot to the
computation time of optimizing cycles, (iv) the computation time of solving an integer program is not
significantly longer than the computation time of solving a linear program for most of the cycle
representatives, using the Gurobi linear solver, (v) strikingly, whether requiring integer solutions
or not, we almost always obtain a solution with the same cost and almost all solutions found have
entries in {-1, 0, 1} and therefore, are also solutions to a restricted
ℓ0
optimization problem, and (vi) we obtain qualitatively different results for generators in
Erdős–Rényi
random clique complexes than in real-world and synthetic point cloud data.
Abstract: Topological descriptors such as contour trees are widely utilized in scientific data analysis and visualization, with applications from materials science to climate simulations. It is desirable to preserve topological descriptors when data compression is part of the scientific workflow for these applications. However, classic error-bounded lossy compressors for volumetric data do not guarantee the preservation of topological descriptors, despite imposing strict pointwise error bounds. In this work, we introduce a general framework for augmenting any lossy compressor to preserve the topology of the data during compression. Specifically, our framework quantifies the adjustments (to the decompressed data) needed to preserve the contour tree and then employs a custom variable-precision encoding scheme to store these adjustments. We demonstrate the utility of our framework in augmenting classic compressors (such as SZ3, TTHRESH, and ZFP) and deep learning-based compressors (such as Neurcomp) with topological guarantees. This is based on a joint work with Nathaniel Gorski, Xin Liang, Hanqi Guo, and Lin Yan. DOI: 10.1109/TVCG.2025.3567054.
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Abstract: In this talk, I will present recent collaborative work on inverse mapper problems. We will look at questions of the form: if I am given a graph G, can I create an open cover, and a filter function whose mapper is G? What if we only want our mapper graph to contain at least n triangles? What about loops, or spheres? Answers to these questions highly depend on whether we are working with clustering algorithms (for data), or path-connected components. We will begin with some interesting examples of Mappers that exhibit wild behavior (even under reasonable clustering algorithms) and then we will look at general techniques that we can use to answer questions of this form in the setting of path-connected components.
Abstract: Topological data analysis (TDA) has been used to visualize, summarize, and understand complex data in science and engineering, ranging from climate and neuroscience to cosmology. However, data's ever-increasing complexity and size pose grand challenges to traditional methodologies and necessitate TDA to understand essential features, sensitivities, and uncertainties from science simulations, experiments, and observations.
This talk covers three topics addressing these challenges by enriching methodologies and tools of topology-based visualization for scientific data exploration. First, I will present a merge-tree-based comparative measure using labeled interleaving distances for scalar fields. Such merge tree comparison helps detect transitions, clusters, and periodicities for time-varying datasets; the metric makes it possible to derive the structural average of labeled merge trees for uncertainty visualization. Second, I will illustrate using merge trees to quantify the structural stability of vector field features to perturbations. Specifically, this framework can enhance feature tracking, selection, and comparison in climate reanalysis data for tropical cyclone (TC) tracking. Third, I will present my work on developing advanced data reduction techniques and software that preserve topological features in data for in situ and post hoc analysis and visualization at extreme scales.
Abstract: Antibodies are small molecules in your body which can bind to specific target molecules, which makes them an attractive object of study in many fields. Synthesizing them, however, is difficult and expensive due to their protein-based makeup. Aptamers are a potential alternative; they are small strands of RNA or DNA which can fold themselves up and bind to a specific target molecule, much like an antibody. Unlike antibodies, RNA/DNA sequences are easy to replicate and synthesize, on the orders of billions at a time, using PCR or PCR-like processes. The harder problem is finding which aptamers we want to replicate; we use the Mapper algorithm from topological data analysis to investigate SELEX, the traditional process used for this problem, as well as the aptamers themselves to visualize families of aptamers and their interactions with regards to how well they bind to specific target molecules.
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