Abstract The Mapper Algorithm is a powerful tool for representing the topology of a dataset’s structure as a similarity graph for the purposes of exploratory analysis. Despite Mapper’s ability to simplify complex high-dimensional data representations, interpreting the structure of the output graph remains a challenge. The conventional method of interpreting the Mapper graph by coloring nodes by features values is infeasible for high-dimensional data due to time limitations and the potential for subjectivity-related oversights. We present a novel method to enhance the interpretability of the Mapper algorithm. Specifically, we propose adapting eXplainable Artificial Intelligence techniques to determine feature importance, offering both local and global interpretations. Our approach can be used to assist domain experts in understanding functional differences across Mapper graphs, enabling them to draw meaningful conclusions from the graph’s structure. To validate our approach, we conducted experiments on five real-world medical datasets and the MNIST handwritten digit dataset. Our evaluation methods consist of a combination of visualization, classification tasks, and alignment of interpretations to existing literature. The results demonstrate our method’s effectiveness in providing a means to interpret Mapper graphs by highlighting the roles of specific features in the graph—such as pixel regions in MNIST and genes in TCGA datasets.

Topological methods have recently gained traction as powerful tools for extracting insights from high-dimensional data, forming the foundation of an approach known as Topological Data Analysis (TDA). Among the key developments in TDA is the Mapper algorithm, which constructs graph-based representations of complex datasets, capturing their topological structure at a user-defined resolution. The Mapper algorithm has shown promise across various applications, particularly in biomedical data analysis. However, its application requires careful selection of several parameters, especially the clustering algorithm and its settings. Without prior knowledge and a deep understanding of the data, these choices are non-trivial and can be a major barrier for researchers aiming to leverage Mapper effectively. In this work, we introduce enhancements to the Mapper algorithm to address this challenge. Specifically, we investigate the integration of ensemble learning (EL) techniques into Mapper’s graph construction to eliminate the need for arbitrary parameter selection. Additionally, we propose a data-driven criterion for selecting the clustering method best suited to the Mapper algorithm. Our experimental results demonstrate that the proposed approach enables the construction of Mapper graphs that accurately capture the underlying structure of the input data, all without manual parameter tuning.

Abstract In topological data analysis (TDA), persistence diagrams (PDs) have been a successful tool. To compare them, Wasserstein and bottleneck distances are commonly used. We address the shortcomings of these metrics and show a way to investigate them in a systematic way by introducing bottleneck profiles. This leads to a notion of discrete Prokhorov metrics for PDs as a generalization of the bottleneck distance. These metrics satisfy a stability result and can be used to bound Wasserstein metrics from above and from below. We provide algorithms to compute the newly introduced quantities and end with an discussion about experiments.

The Mapper algorithm is a technique from TDA used to create low-dimensional graph-based representations of high-dimensional data, proven effective in numerous exploratory data analysis tasks. The Mapper algorithm’s output depends on several user-chosen parameters, and selecting their values is a non-trivial choice, significantly narrowing its potential application in real-world scenarios. Research attempting to assist in selection of the parameters has been very limited to date. This paper is the first one to address the selection of Mapper’s three parameters simultaneously. The proposed idea incorporates the concept of Ensemble Learning into the Mapper algorithm. Using several datasets with known labels, we show that our method outperforms two baselines in recovering the dataset structure.

Norms of Persistent Homology, introduced in topological data analysis, are indicators of system instability, analogous to the changing predictability that is captured in financial market uncertainty indexes. Differentiating observed and unobserved measures of uncertainty, this paper demonstrates a contemporaneous link between persistence norms and uncertainty of all known types. Extending to a dynamic modelling framework provides evidence that unobserved uncertainty Granger causes persistence norms. Further, study of dynamic correlations between persistence norms, volatility and unobserved uncertainty shows that correlations between unobserved uncertainty and persistence norms increase ahead of market crashes. Evidence from the dynamic analysis of the persistence norm–uncertainty relationship provided herein supports the early conclusion that persistence norms provide a signal of impending market crashes. Persistence norms therefore have potential as a further tool in asset pricing, capturing signals from financial time series beyond volatility.

Neuronal morphology is a fundamental factor influencing information processing within neurons and networks. Dendritic morphology in particular can widely vary among cell classes, brain regions, and animal species. Thus, accurate quantitative descriptions allowing classification of large sets of neurons is essential for their structural and functional characterization. Current robust and unbiased computational methods that characterize groups of neurons are scarce. In this work, we introduce a novel technique to study dendritic morphology, complementing and advancing many of the existing techniques. Our approach is to conceptualize the notion of a Sholl descriptor and associate, for each morphological feature, and to each neuron, a function of the radial distance from the soma, taking values in a metric space. Functional distances give rise to pseudo-metrics on sets of neurons which are then used to perform the two distinct tasks of clustering and classification. To illustrate the use of Sholl descriptors, four datasets were retrieved from the large public repository https://neuromorpho.org/ comprising neuronal reconstructions from different species and brain regions. Sholl descriptors were subsequently computed, and standard clustering methods enhanced with detection and metric learning algorithms were then used to objectively cluster and classify each dataset. Importantly, our descriptors outperformed conventional morphometric techniques (L-Measure metrics) in several of the tested datasets. Therefore, we offer a novel and effective approach to the analysis of diverse neuronal cell types, and provide a toolkit for researchers to cluster and classify neurons.

Corporate failure resonates widely, leaving practitioners searching for understanding of default risk. Managers seek to steer away from trouble, credit providers to avoid risky loans and investors to mitigate losses. Applying Topological Data Analysis tools, this paper explores whether failing firms from the United States organise neatly along the five predictors of default proposed by the Z-score models. Each firm is represented as a point in a five-dimensional point cloud, each dimension being one of the five predictors. Visualising that cloud using Ball Mapper reveals failing firms are not always located in similar regions of the point cloud, that is they are not concentrated in an easily split out area of the space. As new modelling approaches vie to better predict firm failure, often using black boxes to deliver potentially over-fitting models, a timely reminder is sounded on the importance of evidencing the identification process. Value is added to the understanding of where in the parameter space failure occurs, and how firms might act to move away from financial distress. Further, lenders may find opportunity amongst subsets of firms that are traditionally considered to be in danger of bankruptcy, but which the Ball Mapper plots developed herein clarify actually sit in characteristic spaces where failure has not occurred.

Persistent homology (PH) is a rigorous mathematical theory that provides a robust descriptor of data in the form of persistence diagrams (PDs). PDs exhibit, however, complex structure and are difficult to integrate in today's machine learning workflows. This paper introduces persistence bag-of-words: a novel and stable vectorized representation of PDs that enables the seamless integration with machine learning. Comprehensive experiments show that the new representation achieves state-of-the-art performance and beyond in much less time than alternative approaches.

A topology preserving skeleton is a synthetic representation of an object that retains its topology and many of its significant morphological properties. The process of obtaining the skeleton, referred to as skeletonization or thinning, is a very active research area. It plays a central role in reducing the amount of information to be processed during image analysis and visualization, computer-aided diagnosis, or by pattern recognition algorithms. This paper introduces a novel topology preserving thinning algorithm, which removes simple cells-a generalization of simple points-of a given cell complex. The test for simple cells is based on acyclicity tables automatically produced in advance with homology computations. Using acyclicity tables render the implementation of thinning algorithms straightforward. Moreover, the fact that tables are automatically filled for all possible configurations allows to rigorously prove the generality of the algorithm and to obtain fool-proof implementations. The novel approach enables, for the first time, according to our knowledge, to thin a general unstructured simplicial complex. Acyclicity tables for cubical and simplicial complexes and an open source implementation of the thinning algorithm are provided as an additional material to allow their immediate use in the vast number of applications arising in medical imaging and beyond.

In this paper we present a new algorithm for computing the homology of regular CW-complexes. This algorithm is based on the coreduction algorithm due to Mrozek and Batko and consists essentially of a geometric preprocessing algorithm for the standard chain complex generated by a CW-complex. By employing the concept of S-complexes the original chain complex can—in all known practical cases—be reduced to a significantly smaller S-complex with isomorphic homology, which can then be computed using standard methods. Furthermore, we demonstrate that in the context of non-uniform cubical grids this method significantly improves currently available algorithms based on uniform cubical grids.