arXiv subjects

Robin Belton

Publications and source records attributed to Robin Belton.

8 recordsLinked to original sources

arXiv2026

A Directed Cubical Framework for k-critical Bipersistence Modules

In topological data analysis, multiparameter persistence modules (MPMs) are used to analyze data with multiple parameters. Summarizing topological information for MPMs is a non-trivial task; one useful summary of bipersistence modules is the fibered barcode \cite{lesnick2015interactive,lesnick2025fast}. In this paper, we use a directed topological construction of the parameter space of a bifiltration to extend the fibered barcode. Specifically, given a $k$-critical bifiltration $F$, we build an associated directed Euclidean cubical complex $C$ whose directed paths each correspond to a one-parameter subfiltration of $F$. The path space of $C$, $\di{C}$, is used to define the \emph{extended fibered barcode}, which contains all possible monotonically increasing paths in $C$. We show that there can be paths in $\di{C}$ which generate filtrations not obtainable along a line with positive slope, and that all lines with positive slope have a path in $\di{C}$ that generate an equivalent barcode. We introduce two types of equivalence for these restricted modules: combinatorial equivalence of diagrams, and simplicial equivalence of the filtrations generating the modules. We show that removing certain cubes from $C$ yields a subcomplex on which dihomotopic paths induce simplicially equivalent filtrations, and hence combinatorially equivalent diagrams.

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arXiv2026

The Inverse Mapper Problem for Two-Dimensional Data

The Mapper algorithm is a popular graphical tool for data visualization and exploration in topological data analysis (TDA). It takes as input a data set $X$ and returns a graph that aims to capture the underlying topological features of these data. The Mapper algorithm has several parameters that need to be specified by the user: a $\textit{filter function}$ $f:X\to Y$ where $X$ is often a point cloud data set in $\mathbb{R}^n$ and $Y$ is the $\textit{reference space}$ that is often a lower dimensional Euclidean space, a $\textit{cover}$ of $f(X)$, and a $\textit{clustering algorithm}$ that is applied to the pre-image of each element in the cover. In this paper, we consider the following question: What graphs can occur as the output of the Mapper algorithm? It is known that when $Y=\mathbb{R}^3$, any graph can be a Mapper graph. In this work, we focus on a more practical case where the filter function is $f:\mathbb{R}^2\to\mathbb{R}$. We consider two different scenarios. In the first scenario, we fix the parameters of the Mapper algorithm, but vary the data. This way we characterize exactly the graphs that can occur for several choices of parameters. In the second scenario, we fix the data set but vary the parameters. This can be seen as a way to understand how much the output can differ depending on the choices of parameters.In both settings, we find close connections to classical topics in graph drawing.

cs.CG↗

arXiv2025

Discrete Level Set Persistence for Finite Discrete Functions

We study sublevel set and superlevel set persistent homology on discrete functions through the perspective of finite ordered sets of both linearly ordered and cyclically ordered domains. Finite ordered sets also serve as the codomain of our functions making all arguments finite and discrete. We prove duality of filtrations of sublevel sets and superlevel sets that undergirths a range of duality results of sublevel set persistent homology without the need to invoke complications of continuous functions or classical Morse theory. We show that Morse-like behavior can be achieved for flat extrema without assuming genericity. Additionally, we show that with inversion of order, one can compute sublevel set persistence from superlevel set persistence, and vice versa via a duality result that does not require the boundary to be treated as a special case. Furthermore, we discuss aspects of barcode construction rules, surgery of circular and linearly ordered sets, as well as surgery on auxiliary structures such as box snakes, which segment the ordered set by extrema and monotones.

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arXiv2025

G-Mapper: Learning a Cover in the Mapper Construction

The Mapper algorithm is a visualization technique in topological data analysis (TDA) that outputs a graph reflecting the structure of a given dataset. However, the Mapper algorithm requires tuning several parameters in order to generate a ``nice" Mapper graph. This paper focuses on selecting the cover parameter. We present an algorithm that optimizes the cover of a Mapper graph by splitting a cover repeatedly according to a statistical test for normality. Our algorithm is based on G-means clustering which searches for the optimal number of clusters in $k$-means by iteratively applying the Anderson-Darling test. Our splitting procedure employs a Gaussian mixture model to carefully choose the cover according to the distribution of the given data. Experiments for synthetic and real-world datasets demonstrate that our algorithm generates covers so that the Mapper graphs retain the essence of the datasets, while also running significantly faster than a previous iterative method.

cs.LG↗

arXiv2024

Any Graph is a Mapper Graph

The Mapper algorithm is a popular tool for visualization and data exploration in topological data analysis. We investigate an inverse problem for the Mapper algorithm: Given a dataset $X$ and a graph $G$, does there exist a set of Mapper parameters such that the output Mapper graph of $X$ is isomorphic to $G$? We provide constructions that affirmatively answer this question. Our results demonstrate that it is possible to engineer Mapper parameters to generate a desired graph.

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arXiv2022

Extremal Event Graphs: A (Stable) Tool for Analyzing Noisy Time Series Data

Local maxima and minima, or extremal events, in experimental time series can be used as a coarse summary to characterize data. However, the discrete sampling in recording experimental measurements suggests uncertainty on the true timing of extrema during the experiment. This in turn gives uncertainty in the timing order of extrema within the time series. Motivated by applications in genomic time series and biological network analysis, we construct a weighted directed acyclic graph (DAG) called an extremal event DAG using techniques from persistent homology that is robust to measurement noise. Furthermore, we define a distance between extremal event DAGs based on the edit distance between strings. We prove several properties including local stability for the extremal event DAG distance with respect to pairwise $L_{\infty}$ distances between functions in the time series data. Lastly, we provide algorithms, publicly free software, and implementations on extremal event DAG construction and comparison.

cs.DS↗

arXiv2022

Combinatorial Conditions for Directed Collapsing

The purpose of this article is to study directed collapsibility of directed Euclidean cubical complexes. One application of this is in the nontrivial task of verifying the execution of concurrent programs. The classical definition of collapsibility involves certain conditions on a pair of cubes of the complex. The direction of the space can be taken into account by requiring that the past links of vertices remain homotopy equivalent after collapsing. We call this type of collapse a link-preserving directed collapse. In this paper, we give combinatorially equivalent conditions for preserving the topology of the links, allowing for the implementation of an algorithm for collapsing a directed Euclidean cubical complex. Furthermore, we give conditions for when link-preserving directed collapses preserve the contractability and connectedness of directed path spaces, as well as examples when link-preserving directed collapses do not preserve the number of connected components of the path space between the minimum and a given vertex.

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arXiv2019

Towards Directed Collapsibility

In the directed setting, the spaces of directed paths between fixed initial and terminal points are the defining feature for distinguishing different directed spaces. The simplest case is when the space of directed paths is homotopy equivalent to that of a single path; we call this the trivial space of directed paths. Directed spaces that are topologically trivial may have non-trivial spaces of directed paths, which means that information is lost when the direction of these topological spaces is ignored. We define a notion of directed collapsibility in the setting of a directed Euclidean cubical complex using the spaces of directed paths of the underlying directed topological space relative to an initial or a final vertex. In addition, we give sufficient conditions for a directed Euclidean cubical complex to have a contractible or a connected space of directed paths from a fixed initial vertex. We also give sufficient conditions for the path space between two vertices in a Euclidean cubical complex to be disconnected. Our results have applications to speeding up the verification process of concurrent programming and to understanding partial executions in concurrent programs.

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