arXiv subjects

Elizabeth Munch

Publications and source records attributed to Elizabeth Munch.

At least 19 recordsLinked to original sources

arXiv2026

Canopies: A Generalization of Vines and Vineyards for Parameterized Persistence

In this paper, we provide a new construction for studying parameterized persistence, called a canopy. We give two versions of this construction: the A-canopy, retaining all information about points on the diagonal of the persistence diagram; and the D-canopy, encoding the information of the "standard" persistence diagram. We do this by making a simple but major modification in the persistence bundle representation information: namely, rather than tracking a point in the persistence diagram, we instead track some choice of pairs of simplices that created said point. This viewpoint is a combinatorial version of tracking the chain complex information rather than just the output of persistence. We show how to construct the canopies from any filtered filtration function, proving, using the algebraic structure of filtered chain complexes, that different choices of pairs result in homeomorphic structures. Finally, we showcase the power of our approach by using canopies to define vines even in the presence of points with multiplicity; to discuss monodromy; and to obtain some immediate results linking non-trivial monodromy in the persistent homology transform with the existence of non-Hausdorff points in the canopy.

cs.CG↗

arXiv2026

Topological summaries of fingerprint ridge patterns carry identity information

Fingerprints are the most widely deployed biometric. Verifying whether two impressions come from the same finger typically relies on minutiae, small landmarks such as skin ridge endings and bifurcations. These landmarks are extracted through a multi-stage pipeline of image enhancement, skeletonization, minutiae detection, and alignment. We investigate an alternative: using topological data analysis to represent the full pattern of skin ridges and valleys directly, bypassing minutiae detection and the downstream matching pipeline. We apply persistent homology, a topological tool that tracks how loops in the ridge pattern form and fill in across spatial scales, producing multi-scale summaries of ridge geometry. We develop and compare a range of verification methods on a standard benchmark dataset, FVC2000 DB1. Even the simplest topological summaries, with no trained parameters, substantially outperform geometry-only baselines. A trained method achieves an AUC of 0.91, while an optimal-transport method excels at the strictest false-accept thresholds, suggesting they capture different aspects of the ridge pattern. Fusing these two approaches yields the best performance at every low false-accept threshold we examine. Our results establish that these topological summaries capture substantial fingerprint identity information, far more effective for verification than raw pixel-level geometry. Because the entire pipeline is openly specified, it offers a transparent complement to minutiae-based systems, and we provide a modular framework for constructing, evaluating, and combining topological verification methods.

cs.CV↗

arXiv2026

Bounding the interleaving distance on concrete categories using a loss function

The interleaving distance is arguably the most widely used metric in topological data analysis (TDA) due to its applicability to a wide array of inputs of interest, such as (multiparameter) persistence modules, Reeb graphs, merge trees, and zigzag modules. However, computation of the interleaving distance in the vast majority of this settings is known to be NP-hard, limiting its use in practical settings. Inspired by the work of Chambers et al. on the interleaving distance for mapper graphs, we solve a more general problem bounding the interleaving distance between generalized persistence modules on concrete categories via a loss function. This loss function measures how far an assignment, which can be thought of as an interleaving that might not commute, is from defining a true interleaving. We give settings for which the loss can be computed in polynomial time, including for certain assumptions on $k$-parameter persistence modules.

math.AT↗

arXiv2025

The Walk-Length Filtration for Persistent Homology on Weighted Directed Graphs

Directed graphs arise in many applications where computing persistent homology helps to encode the shape and structure of the input information. However, there are only a few ways to turn the directed graph information into an undirected simplicial complex filtration required by the standard persistent homology framework. In this paper, we present a new filtration constructed from a directed graph, called the walk-length filtration. This filtration mirrors the behavior of small walks visiting certain collections of vertices in the directed graph. We show that, while the persistence is not stable under the usual $L_\infty$-style network distance, a generalized $L_1$-style distance is, indeed, stable. We further provide an algorithm for its computation, and investigate the behavior of this filtration in examples, including cycle networks and synthetic hippocampal networks with a focus on comparison to the often used Dowker filtration.

cs.CG↗

arXiv2025

On the Stability of the Euler Characteristic Transform for a Perturbed Embedding

The Euler Characteristic Transform (ECT) is a robust method for shape classification. It takes an embedded shape and, for each direction, computes a piecewise constant function representing the Euler Characteristic of the shape's sublevel sets, which are defined by the height function in that direction. It has applications in TDA inverse problems, such as shape reconstruction, and is also employed with machine learning methodologies. In this paper, we define a distance between the ECTs of two distinct geometric embeddings of the same abstract simplicial complex and provide an upper bound for this distance. The Super Lifted Euler Characteristic Transform (SELECT), a related construction, extends the ECT to scalar fields defined on shapes. We establish a similar distance bound for SELECT, specifically when applied to fields defined on embedded simplicial complexes.

cs.CG↗

arXiv2025

The Kinetic Hourglass Data Structure for Computing the Bottleneck Distance of Dynamic Data

The kinetic data structure (KDS) framework is a powerful tool for maintaining various geometric configurations of continuously moving objects. In this work, we introduce the kinetic hourglass, a novel KDS implementation designed to compute the bottleneck distance for geometric matching problems. We detail the events and updates required for handling general graphs, accompanied by a complexity analysis. Furthermore, we demonstrate the utility of the kinetic hourglass by applying it to compute the bottleneck distance between two persistent homology transforms (PHTs) derived from shapes in $\mathbb{R}^2$, which are topological summaries obtained by computing persistent homology from every direction in $\mathbb{S}^1$.

cs.DS↗

arXiv2025

Towards an Optimal Bound for the Interleaving Distance on Mapper Graphs

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.

cs.CG↗

arXiv2024

A Distance for Geometric Graphs via the Labeled Merge Tree Interleaving Distance

Geometric graphs appear in many real-world data sets, such as road networks, sensor networks, and molecules. We investigate the notion of distance between embedded graphs and present a metric to measure the distance between two geometric graphs via merge trees. In order to preserve as much useful information as possible from the original data, we introduce a way of rotating the sublevel set to obtain the merge trees via the idea of the directional transform. We represent the merge trees using a surjective multi-labeling scheme and then compute the distance between two representative matrices. We show some theoretically desirable qualities and present two methods of computation: approximation via sampling and exact distance using a kinetic data structure, both in polynomial time. We illustrate its utility by implementing it on two data sets.

cs.DS↗

arXiv2023

An Invitation to the Euler Characteristic Transform

The Euler characteristic transform (ECT) is a simple to define yet powerful representation of shape. The idea is to encode an embedded shape using sub-level sets of a a function defined based on a given direction, and then returning the Euler characteristics of these sublevel sets. Because the ECT has been shown to be injective on the space of embedded simplicial complexes, it has been used for applications spanning a range of disciplines, including plant morphology and protein structural analysis. In this survey article, we present a comprehensive overview of the Euler characteristic transform, highlighting the main idea on a simple leaf example, and surveying its its key concepts, theoretical foundations, and available applications.

cs.CG↗

arXiv2023

Bounding the Interleaving Distance for Mapper Graphs with a Loss Function

Data consisting of a graph with a function mapping into $\mathbb{R}^d$ arise in many data applications, encompassing structures such as Reeb graphs, geometric graphs, and knot embeddings. As such, the ability to compare and cluster such objects is required in a data analysis pipeline, leading to a need for distances between them. In this work, we study the interleaving distance on discretization of these objects, called mapper graphs when $d=1$, where functor representations of the data can be compared by finding pairs of natural transformations between them. However, in many cases, computation of the interleaving distance is NP-hard. For this reason, we take inspiration from recent work by Robinson to find quality measures for families of maps that do not rise to the level of a natural transformation, called assignments. We then endow the functor images with the extra structure of a metric space and define a loss function which measures how far an assignment is from making the required diagrams of an interleaving commute. Finally we show that the computation of the loss function is polynomial with a given assignment. We believe this idea is both powerful and translatable, with the potential to provide approximations and bounds on interleavings in a broad array of contexts.

cs.CG↗

arXiv2023

Using topological data analysis to compare inter-subject variability across resting state functional MRI brain representations

In neuroimaging, extensive post-processing of resting-state functional MRI (rfMRI) data is necessary for its application and investigation in relation to brain-behavior associations. Such post-processing is used to derive brain representations, lower dimensional feature sets used for brain-behavior association studies. A brain representation involves a choice of dimension reduction (a parcellation into regions or networks) and a choice of feature type, such as spatial topography, connectivity matrix, amplitude. However, widespread variability in rfMRI brain representations has hindered both reproducibility and knowledge accumulation across the field. Brain representation choice effects measurements of inter-subject variability, which muddies the comparison and integration of findings. We leveraged persistent homology on the subject-space topologies induced by 34 different brain representations to enable direct comparison of brain representations in the context of individual differences. Our findings reveal the importance of considering feature type when comparing results derived from different brain representations, suggesting best practices for assessing the replicability and generalizability of brain-behavior research in rfMRI data.

cs.CG↗

arXiv2023

NervePool: A Simplicial Pooling Layer

For deep learning problems on graph-structured data, pooling layers are important for down sampling, reducing computational cost, and to minimize overfitting. We define a pooling layer, nervePool, for data structured as simplicial complexes, which are generalizations of graphs that include higher-dimensional simplices beyond vertices and edges; this structure allows for greater flexibility in modeling higher-order relationships. The proposed simplicial coarsening scheme is built upon partitions of vertices, which allow us to generate hierarchical representations of simplicial complexes, collapsing information in a learned fashion. NervePool builds on the learned vertex cluster assignments and extends to coarsening of higher dimensional simplices in a deterministic fashion. While in practice the pooling operations are computed via a series of matrix operations, the topological motivation is a set-theoretic construction based on unions of stars of simplices and the nerve complex.

cs.CG↗

arXiv2023

A Topological Framework for Identifying Phenomenological Bifurcations in Stochastic Dynamical Systems

Changes in the parameters of dynamical systems can cause the state of the system to shift between different qualitative regimes. These shifts, known as bifurcations, are critical to study as they can indicate when the system is about to undergo harmful changes in its behavior. In stochastic dynamical systems, there is particular interest in P-type (phenomenological) bifurcations, which can include transitions from a mono-stable state to multi-stable states, the appearance of stochastic limit cycles, and other features in the probability density function (PDF) of the system's state. Current practices are limited to systems with small state spaces, cannot detect all possible behaviours of the PDFs, and mandate human intervention for visually identifying the change in the PDF. In contrast, this study presents a new approach based on Topological Data Analysis (TDA) that uses superlevel persistence to mathematically quantify P-type bifurcations in stochastic systems through a "homological bifurcation plot'' -- which shows the changing ranks of 0th and 1st homology groups. Using these plots, we demonstrate the successful detection of P-bifurcations on the stochastic Duffing, Raleigh-Vander Pol and Quintic Oscillators given their analytical PDFs, and elaborate on how to generate an estimated homological bifurcation plot given a kernel density estimate (KDE) of these systems by employing a tool for finding topological consistency between PDFs and KDEs.

math.DS↗

arXiv2023

Robust Zero-crossings Detection in Noisy Signals using Topological Signal Processing

We explore a novel application of zero-dimensional persistent homology from Topological Data Analysis (TDA) for bracketing zero-crossings of both one-dimensional continuous functions, and uniformly sampled time series. We present an algorithm and show its robustness in the presence of noise for a range of sampling frequencies. In comparison to state-of-the-art software-based methods for finding zeros of a time series, our method generally converges faster, provides higher accuracy, and is capable of finding all the roots in a given interval instead of converging only to one of them. We also present and compare options for automatically setting the persistence threshold parameter that influences the accurate bracketing of the roots.

cs.CG↗

arXiv2022

Comparing Embedded Graphs Using Average Branching Distance

Graphs drawn in the plane are ubiquitous, arising from data sets through a variety of methods ranging from GIS analysis to image classification to shape analysis. A fundamental problem in this type of data is comparison: given a set of such graphs, can we rank how similar they are, in such a way that we capture their geometric "shape" in the plane? In this paper we explore a method to compare two such embedded graphs, via a simplified combinatorial representation called a tail-less merge tree which encodes the structure based on a fixed direction. First, we examine the properties of a distance designed to compare merge trees called the branching distance, and show that the distance as defined in previous work fails to satisfy some of the requirements of a metric. We incorporate this into a new distance function called average branching distance to compare graphs by looking at the branching distance for merge trees defined over many directions. Despite the theoretical issues, we show that the definition is still quite useful in practice by using our open-source code to cluster data sets of embedded graphs.

cs.CG↗

arXiv2022

Detecting bifurcations in dynamical systems with CROCKER plots

Existing tools for bifurcation detection from signals of dynamical systems typically are either limited to a special class of systems, or they require carefully chosen input parameters, and significant expertise to interpret the results. Therefore, we describe an alternative method based on persistent homology -- a tool from Topological Data Analysis (TDA) -- that utilizes Betti numbers and CROCKER plots. Betti numbers are topological invariants of topological spaces, while the CROCKER plot is a coarsened but easy to visualize data representation of a one-parameter varying family of persistence barcodes. The specific bifurcations we investigate are transitions from periodic to chaotic behavior or vice versa in a one-parameter family of differential equations. We validate our methods using numerical experiments on ten dynamical systems and contrast the results with existing tools that use the maximum Lyapunov exponent. We further prove the relationship between the Wasserstein distance to the empty diagram and the norm of the Betti vector, which shows that an even more simplified version of the information has the potential to provide insight into the bifurcation parameter. The results show that our approach reveals more information about the shape of the periodic attractor than standard tools, and it has more favorable computational time in comparison to the Rosenstein algorithm for computing the Lyapunov exponent from time series.

math.DS↗

arXiv2022

Temporal Network Analysis Using Zigzag Persistence

This work presents a framework for studying temporal networks using zigzag persistence, a tool from the field of Topological Data Analysis (TDA). The resulting approach is general and applicable to a wide variety of time-varying graphs. For example, these graphs may correspond to a system modeled as a network with edges whose weights are functions of time, or they may represent a time series of a complex dynamical system. We use simplicial complexes to represent snapshots of the temporal networks that can then be analyzed using zigzag persistence. We show two applications of our method to dynamic networks: an analysis of commuting trends on multiple temporal scales, e.g., daily and weekly, in the Great Britain transportation network, and the detection of periodic/chaotic transitions due to intermittency in dynamical systems represented by temporal ordinal partition networks. Our findings show that the resulting zero- and one-dimensional zigzag persistence diagrams can detect changes in the networks' shapes that are missed by traditional connectivity and centrality graph statistics.

cs.CG↗

arXiv2022

Persistent Homology of Coarse Grained State Space Networks

This work is dedicated to the topological analysis of complex transitional networks for dynamic state detection. Transitional networks are formed from time series data and they leverage graph theory tools to reveal information about the underlying dynamic system. However, traditional tools can fail to summarize the complex topology present in such graphs. In this work, we leverage persistent homology from topological data analysis to study the structure of these networks. We contrast dynamic state detection from time series using a coarse-grained state-space network (CGSSN) and topological data analysis (TDA) to two state of the art approaches: ordinal partition networks (OPNs) combined with TDA and the standard application of persistent homology to the time-delay embedding of the signal. We show that the CGSSN captures rich information about the dynamic state of the underlying dynamical system as evidenced by a significant improvement in dynamic state detection and noise robustness in comparison to OPNs. We also show that because the computational time of CGSSN is not linearly dependent on the signal's length, it is more computationally efficient than applying TDA to the time-delay embedding of the time series.

stat.ML↗