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.

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.

MOTIVATION: Topological methods have recently emerged as a reliable and interpretable framework for extracting information from high-dimensional data, leading to the creation of a branch of applied mathematics called Topological Data Analysis (TDA). Since then, TDA has been progressively adopted in biomedical research. Biological data collection can result in enormous datasets, comprising thousands of features and spanning diverse datatypes. This presents a barrier to initial data analysis as the fundamental structure of the dataset becomes hidden, obstructing the discovery of important features and patterns. TDA provides a solution to obtain the underlying shape of datasets over continuous resolutions, corresponding to key topological features independent of noise. TDA has the potential to support future developments in healthcare as biomedical datasets rise in complexity and dimensionality. Previous applications extend across the fields of neuroscience, oncology, immunology and medical image analysis. TDA has been used to reveal hidden subgroups of cancer patients, construct organizational maps of brain activity and classify abnormal patterns in medical images. The utility of TDA is broad and to understand where current achievements lie, we have evaluated the present state of TDA in cancer data analysis. RESULTS: This article aims to provide an overview of TDA in Cancer Research. A brief introduction to the main concepts of TDA is provided to ensure that the article is accessible to readers who are not familiar with this field. Following this, a focussed literature review on the field is presented, discussing how TDA has been applied across heterogeneous datatypes for cancer research.