A kernel for multi-parameter persistent homology

Abstract: This book presents a rigorous, interdisciplinary investigation into the convergence of Artificial Intelligence (AI) and Topological Data Analysis (TDA) as a transformative framework for modeling and interpreting high-dimensional data structures. It addresses a fundamental challenge in modern data science: traditional statistical and machine learning techniques often struggle to preserve the global geometric and topological properties of complex datasets. By leveraging tools from algebraic topology—such as persistent homology, simplicial complexes, and Betti numbers—TDA enables the extraction of robust, multi-scale topological features from noisy, sparse, and nonlinear data. The book introduces a comprehensive framework in which topological descriptors are integrated into AI pipelines through persistence diagrams, barcodes, and vectorized representations. Methodologies include differentiable TDA layers, topological regularization in deep learning, manifold learning via Mapper and Reeb graphs, and Bayesian inference with topological priors. Applications span across domains including neuroscience, genomics, medical imaging, finance, and computer vision. Empirical results and case studies demonstrate how topology-aware AI models enhance robustness, reduce overfitting, and provide semantically meaningful representations of data. The book concludes by identifying open challenges—such as the scalability and differentiability of topological operations—and outlines a roadmap for future developments in topology-native machine learning. Through this synthesis, the work establishes TDA not only as a diagnostic tool but as a foundational principle for next-generation AI systems in high-dimensional data environments. Keywords Topological Data Analysis, Artificial Intelligence, Persistent Homology, High-Dimensional Data, Simplicial Complexes, Betti Numbers, Manifold Learning, Mapper Algorithm, Reeb Graphs, Dimensionality Reduction, Topological Priors, Differentiable TDA, Topological Regularization, Federated Learning, Bayesian Inference, Explainable AI, Algebraic Topology, Complex Systems, Geometric Machine Learning, Shape-Aware AI

Understanding streamflow data can be important climatic indicators for environmental risk problems such as flooding. Recently, topological data analysis (TDA) gave a new insight in data analysis. The main idea in TDA is to used results based on topology to develop tools for studying qualitative features or shape-like structure of data. Persistent homology (PH) is one of the tools in TDA that focuses on aspects of topological features in data that persists across multiple scales. So the question here is, can PH detect flood based on streamflow data. Therefore, the first attempt of streamflow analysis using PH was conducted at Guillemard Bridge Station, Kelantan River, Malaysia. Analysis for streamflow data during dry period, wet period and flood events were perform using TDA approach. The analysis result shows that PH can detect the pattern of topological features in streamflow data. The analysis suggests that the presence of short-lived topological features indicates dry period while long-lived topological features for wet period. Based on the streamflow data of flood events, PH consistently captured long-lived topological features of the data.Understanding streamflow data can be important climatic indicators for environmental risk problems such as flooding. Recently, topological data analysis (TDA) gave a new insight in data analysis. The main idea in TDA is to used results based on topology to develop tools for studying qualitative features or shape-like structure of data. Persistent homology (PH) is one of the tools in TDA that focuses on aspects of topological features in data that persists across multiple scales. So the question here is, can PH detect flood based on streamflow data. Therefore, the first attempt of streamflow analysis using PH was conducted at Guillemard Bridge Station, Kelantan River, Malaysia. Analysis for streamflow data during dry period, wet period and flood events were perform using TDA approach. The analysis result shows that PH can detect the pattern of topological features in streamflow data. The analysis suggests that the presence of short-lived topological features indicates dry period while long-lived topologica...

Topological data analysis (TDA) is a rapidly evolving field in applied mathematics and data science that leverages tools from topology to uncover robust, shape-driven, and explainable insights in complex datasets. The main workhorse is persistent homology, a technique rooted in algebraic topology. Paired with topological deep learning (TDL) or topological machine learning, persistent homology has achieved tremendous success in a wide variety of applications in science, engineering, medicine, and industry. However, persistent homology has many limitations due to its high-level abstraction, insensitivity to non-topological changes, and restriction to point cloud data. This paper presents a comprehensive review of TDA and TDL beyond persistent homology. It analyzes how persistent topological Laplacians and Dirac operators provide spectral representations to capture both topological invariants and homotopic evolution. Other formulations are presented in terms of sheaf theory, Mayer topology, and interaction topology. For data on differentiable manifolds, techniques rooted in differential topology, such as persistent de Rham cohomology, persistent Hodge Laplacian, and Hodge decomposition, are reviewed. For one-dimensional (1D) curves embedded in 3-space, approaches from geometric topology are discussed, including multiscale Gauss-link integrals, persistent Jones polynomials, and persistent Khovanov homology. This paper further discusses the appropriate selection of topological tools for different input data, such as point clouds, sequential data, data on manifolds, curves embedded in 3-space, and data with additional non-geometric information. A review is also given of various topological representations, software packages, and machine learning vectorizations. Finally, this review ends with concluding remarks.

Abstract: Neurodegenerative diseases, such as Alzheimer's and Parkinson's, are characterized by progressive neuronal deterioration, leading to cognitive and motor impairments. Early detection is crucial for effective intervention and management. This study proposes a novel framework utilizing Topological Data Analysis (TDA), particularly persistent homology, to identify early biomarkers from neuroimaging and physiological data. This study explores the application of TDA techniques—specifically persistent homology and topological invariants—for the early detection of neurodegenerative diseases such as Alzheimer's. By analyzing functional brain connectivity, gait dynamics, and molecular structures associated with disease progression, the research demonstrates that TDA can capture subtle, yet critical, topological signatures indicative of early neurodegenerative changes. Layer-wise topological complexity in neural networks and persistent homology in clinical datasets reveal distinct structural degradation patterns. The integration of TDA with machine learning models enhances diagnostic accuracy, interpretability, and generalization, offering a promising direction for non-invasive, data-driven diagnostics in neurology.By capturing the intrinsic topological features of brain networks, the framework aims to distinguish between healthy and early-stage neurodegenerative conditions, offering a robust tool for early diagnosis.Topological Data Analysis (TDA) offers a robust mathematical framework for extracting shape-based features from high-dimensional and noisy biomedical data. Keywords: Topological Data Analysis, Persistent Homology, Neurodegenerative Disease, Alzheimer's, Brain Connectivity, Early Diagnosis, Betti Numbers, Gait Analysis, Functional Networks, Machine Learning, Biomedical Data

Persistent homology is a powerful notion rooted in topological data analysis which allows for retrieving the essential topological features of an object. The attention on persistent homology is constantly growing in a large number of application domains, such as biology and chemistry, astrophysics, automatic classification of images, sensor and social network analysis. Thus, an increasing number of researchers is now approaching to persistent homology as a tool to be used in their research activity. At the same time, the literature lacks of tools for introducing beginners to this topic, especially if they do not have a strong mathematical background in algebraic topology. We propose here two complementary tools which meet this requirement. The first one is a web-based user-guide equipped with interactive examples to facilitate the comprehension of the notions at the basis of persistent homology. The second one is an interactive tool, with a specific focus on shape analysis, developed for studying persistence pairs by visualizing them directly on the input complex.

In recent years, persistent homology (PH) and topological data analysis (TDA) have gained increasing attention in the fields of shape recognition, image analysis, data analysis, machine learning, computer vision, computational biology, brain functional networks, financial networks, haze detection, etc. In this article, we will focus on stock markets and demonstrate how TDA can be useful in this regard. We first explain signatures that can be detected using TDA, for three toy models of topological changes. We then showed how to go beyond network concepts like nodes (0-simplex) and links (1-simplex), and the standard minimal spanning tree or planar maximally filtered graph picture of the cross correlations in stock markets, to work with faces (2-simplex) or any k-dim simplex in TDA. By scanning through a full range of correlation thresholds in a procedure called filtration, we were able to examine robust topological features (i.e. less susceptible to random noise) in higher dimensions. To demonstrate the advantages of TDA, we collected time-series data from the Straits Times Index and Taiwan Capitalization Weighted Stock Index (TAIEX), and then computed barcodes, persistence diagrams, persistent entropy, the bottleneck distance, Betti numbers, and Euler characteristic. We found that during the periods of market crashes, the homology groups become less persistent as we vary the characteristic correlation. For both markets, we found consistent signatures associated with market crashes in the Betti numbers, Euler characteristics, and persistent entropy, in agreement with our theoretical expectations.

One of the main methods of computational topology and topological data analysis is persistent homology, which combines geometric and topological information about an object using persistent diagrams and barcodes. The persistent homology method from computational topology provides a balance between reducing the data dimension and characterizing the internal structure of an object. Combining machine learning and persistent homology is hampered by topological representations of data, distance metrics, and representation of data objects. The paper considers mathematical models and functions for representing persistent landscape objects based on the persistent homology method. The persistent landscape functions allow you to map persistent diagrams to Hilbert space. The representations of topological functions in various machine learning models are considered. An example of finding the distance between images based on the construction of persistent landscape functions is given. Based on the algebra of polynomials in the barcode space, which are used as coordinates, the distances in the barcode space are determined by comparing intervals from one barcode to another and calculating penalties. For these purposes, tropical functions are used that take into account the basic structure of the barcode space. Methods for constructing rational tropical functions are considered. An example of finding the distance between images based on the construction of tropical functions is given. To increase the variety of parameters (machine learning features), filtering of object scanning by rows from left to right and scanning by columns from bottom to top are built. This adds spatial information to topological information. The method of constructing persistent landscapes is compatible with the approach of constructing tropical rational functions when obtaining persistent homologies.

Topological Data Analysis (TDA) and its mainstay computational device, persistent homology (PH), has established a strong track record of providing researchers across the data-driven sciences with new insights and methodologies by characterizing low-dimensional geometric structures in high-dimensional data. When combined with machine learning (ML) methods, PH is valued as a discriminating-feature extraction tool. This work highlights many of the recent successes at the intersection of TDA and ML, introduces some of the foundational mathematics underpinning TDA, and summarizes the efforts to strengthen the bridge between TDA and ML. Thus, this document is a launching point for experimentalists and theoreticians to consider what can be learned from the shape of their data.

Topology at the undergraduate level is often a purely theoretical mathematics course, introducing concepts from point-set topology or possibly algebraic or geometric topology. However, the last two decades have seen an explosion of growth in applied topology and topological data analysis, which are topics that can be presented in an accessible way to undergraduate students and can encourage exciting projects. For the past several years, the Topology course at Macalester College has included content from point-set and algebraic topology, as well as applied topology, culminating in a project chosen by the students. In the course, students work through a topology scavenger hunt as an activity to introduce the ideas and software behind some of the primary tools in topological data analysis, namely, persistent homology and mapper. This scavenger hunt includes a variety of point clouds of varying dimensions, such as an annulus in 2D, a bouquet of loops in 3D, a sphere in 4D, and a torus in 400D. The students' goal is to analyze each point cloud with a variety of software. This activity can fit nicely into a course where students have been introduced to some of the fundamentals of point-set topology such as connectedness, continuity, compactness, etc. of arbitrary topologies as well as tools from algebraic topology such as simplicial complexes and simplicial homology, which is accessible through the lens of linear algebra. The activity takes approximately a week of class time to provide a brief introduction to persistent homology and mapper, as well as some software resources to perform these computations, and then a week outside of class time for students to work on the scavenger hunt. After completing this activity, students are able to extend the ideas learned in the scavenger hunt to an open-ended capstone project. Examples of past projects include: using persistence to explore the relationship between country development and geography, to analyze congressional voting patterns, and to classify genres of a large corpus of texts by combining with tools from natural language processing and machine learning.

Persistent homology (PH) is a powerful burgeoning technique from Topological data analysis (TDA) that leverages machinery drawn from algebraic topology. PH records the appearance and disappearance of essential topological features of an object that persist across various scales or resolutions, and it is immune to noise. PH is independent of parameters, dimension and coordinates. In recent years, PH is catching an eye of the machine learning community due to the challenges offered by nature of data available today. But due to unapproachable introductory literature on PH, beginners who have the mathematical aptitude but no background in algebraic topology find it difficult to understand the concepts and techniques involved. On the other side, researchers are working effortlessly to bring together machine learning and TDA, as both combined can do wonders. This paper is an attempt to introduce the theory of PH step by step for the beginner and illustrate the concepts involved by using toy data sets. The main purpose of this research work is to introduce extensions of PH that helps researchers to apply tools from statistics and machine learning.

Abstract: This chapter explores the integration of advanced AI techniques with topological data analysis (TDA) to decode complex high-dimensional data structures. It examines how AI-driven TDA can identify and analyze the geometric and topological features of datasets, offering deeper insights into patterns, clusters, and anomalies that traditional methods might overlook. By leveraging machine learning algorithms such as neural networks, graph neural networks, and autoencoders, AI enhances the ability to process and interpret high-dimensional data from various fields, including biology, finance, and network analysis. The chapter also discusses the application of persistent homology and multi-scale analysis in AI-driven TDA, emphasizing the future potential of these technologies in fields such as genomics, neuroscience, and sensor networks. Challenges related to computational complexity, scalability, and interpretability are addressed, as well as emerging trends in the field. Keywords: advanced AI, topological data analysis, TDA, high-dimensional data, machine learning, neural networks, graph neural networks, autoencoders, persistent homology, multi-scale analysis, data patterns, data clusters, anomaly detection, genomics, neuroscience, sensor networks, computational complexity.

Topological data analysis is an emerging concept of data analysis for characterizing shapes. A state-of-the-art tool in topological data analysis is persistent homology, which is expected to summarize quantified topological and geometric features. Although persistent homology is useful for revealing the topological and geometric information, it is difficult to interpret the parameters of persistent homology themselves and difficult to directly relate the parameters to physical properties. In this study, we focus on connectivity and apertures of flow channels detected from persistent homology analysis. We propose a method to estimate permeability in fracture networks from parameters of persistent homology. Synthetic 3D fracture network patterns and their direct flow simulations are used for the validation. The results suggest that the persistent homology can estimate fluid flow in fracture network based on the image data. This method can easily derive the flow phenomena based on the information of the structure.

This paper introduces a novel categorical framework that unifies classical algebraic topology with modern topological data analysis through the lens of category theory. We develop the theory of persistence categories as a natural generalization of persistence modules, establishing functorial relationships between classical topological invariants and their persistent counterparts. Our approach reveals deep connections between sheaf cohomology, spectral sequences, and multi-parameter persistence, providing a rigorous mathematical foundation for understanding the stability and structure of topological features in data. We prove that persistent homology can be viewed as a particular instance of a more general categorical construction that encompasses both classical and computational topology. Furthermore, we establish new stability theorems for categorical persistence and demonstrate how classical results in algebraic topology can be lifted to the persistent setting through appropriate functorial constructions. We present practical applications in data science, computational biology, and machine learning, demonstrating the effectiveness of our theoretical framework through concrete implementations and computational experiments.

fMRI is the preeminent method for collecting signals from the human brain in vivo, for using these signals in the service of functional discovery, and relating these discoveries to anatomical structure. Numerous computational and mathematical techniques have been deployed to extract information from the fMRI signal. Yet, the application of Topological Data Analyses (TDA) remain limited to certain sub-areas such as connectomics (that is, with summarized versions of fMRI data). While connectomics is a natural and important area of application of TDA, applications of TDA in the service of extracting structure from the (non-summarized) fMRI data itself are heretofore nonexistent. “Structure” within fMRI data is determined by dynamic fluctuations in spatially distributed signals over time, and TDA is well positioned to help researchers better characterize mass dynamics of the signal by rigorously capturing shape within it. To accurately motivate this idea, we a) survey an established method in TDA (“persistent homology”) to reveal and describe how complex structures can be extracted from data sets generally, and b) describe how persistent homology can be applied specifically to fMRI data. We provide explanations for some of the mathematical underpinnings of TDA (with expository figures), building ideas in the following sequence: a) fMRI researchers can and should use TDA to extract structure from their data; b) this extraction serves an important role in the endeavor of functional discovery, and c) TDA approaches can complement other established approaches toward fMRI analyses (for which we provide examples). We also provide detailed applications of TDA to fMRI data collected using established paradigms, and offer our software pipeline for readers interested in emulating our methods. This working overview is both an inter-disciplinary synthesis of ideas (to draw researchers in TDA and fMRI toward each other) and a detailed description of methods that can motivate collaborative research.

Machine learning, and especially deep learning, is rapidly gaining acceptance and clinical usage in a wide range of image analysis applications and is regarded as providing high performance in detecting anatomical structures and identification and classification of patterns of disease in medical images. However, there are many roadblocks to the widespread implementation of machine learning in clinical image analysis, including differences in data capture leading to different measurements, high dimensionality of imaging and other medical data, and the black-box nature of machine learning, with a lack of insight into relevant features. Techniques such as radiomics have been used in traditional machine learning approaches to model the mathematical relationships between adjacent pixels in an image and provide an explainable framework for clinicians and researchers. Newer paradigms, such as topological data analysis (TDA), have recently been adopted to design and develop innovative image analysis schemes that go beyond the abilities of pixel-to-pixel comparisons. TDA can automatically construct filtrations of topological shapes of image texture through a technique known as persistent homology (PH); these features can then be fed into machine learning models that provide explainable outputs and can distinguish different image classes in a computationally more efficient way, when compared to other currently used methods. The aim of this review is to introduce PH and its variants and to review TDA’s recent successes in medical imaging studies.