Computing Minimal Presentations and Bigraded Betti Numbers of 2-Parameter Persistent Homology

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

Topological methods are very rarely used in structural health monitoring (SHM), or indeed in structural dynamics generally, especially when considering the structure and topology of observed data. Topological methods can provide a way of proposing new metrics and methods of scrutinising data, that otherwise may be overlooked. In this work, a method of quantifying the shape of data, via a topic called topological data analysis will be introduced. The main tool within topological data analysis is persistent homology. Persistent homology is a method of quantifying the shape of data over a range of length scales. The required background and a method of computing persistent homology is briefly introduced here. Ideas from topological data analysis are applied to a Z24 Bridge case study, to scrutinise different data partitions, classified by the conditions at which the data were collected. A metric, from topological data analysis, is used to compare between the partitions. The results presented demonstrate that the presence of damage alters the manifold shape more significantly than the effects present from temperature.

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.

Topological Data Analysis is a set of mathematical methods for capturing the structure of data forms. TDA has been used to cope with noisy signals and time series. TDA is used to discover the key characteristics of data, which are shapes. One of the basic techniques for TDA is persistent homology which is used to do a study of time series and signal processing. Human activity analysis is a well-researched issue in the field of vision, with large research on the topic. The produced time- series data is typically high-dimensional, intrinsic, non-linear, and non-stationary, leading to pattern changes. The data acquired by a person’s numerous sensors, as well as their placement and orientation, often produces noisy and complicated data, resulting in poor model performance. The implementation method must be data-driven, independent of sensor location and orientation, and employ a coordinate and parameter-free strategy to identify the variable time series data. We need technology that can represent a low-dimensional manifold compactly, identify emergent behaviors, and globalize the system. The persistent homology having property being co- ordinate free. Sensor placement has little effect on time series factorization using persistent homology. This paper gives an overview of topological approaches as well as the problems of human activity analysis, as well as recommendations for the future.

BackgroundAttention-deficit/hyperactivity disorder (ADHD) is currently diagnosed by a diagnostic interview, mainly based on subjective reports from parents or teachers. It is necessary to develop methods that rely on objectively measureable neurobiological data to assess brain-behavior relationship in patients with ADHD. We investigated the application of a topological data analysis tool, Mapper, to analyze the brain functional connectivity data from ADHD patients.MethodsTo quantify the disease severity using the neuroimaging data, the decomposition of individual functional networks into normal and disease components by the healthy state model (HSM) was performed, and the magnitude of the disease component (MDC) was computed. Topological data analysis using Mapper was performed to distinguish children with ADHD (n = 196) from typically developing controls (TDC) (n = 214).ResultsIn the topological data analysis, the partial clustering results of patients with ADHD and normal subjects were shown in a chain-like graph. In the correlation analysis, the MDC showed a significant increase with lower intelligence scores in TDC. We also found that the rates of comorbidity in ADHD significantly increased when the deviation of the functional connectivity from HSM was large. In addition, a significant correlation between ADHD symptom severity and MDC was found in part of the dataset.ConclusionsThe application of HSM and topological data analysis methods in assessing the brain functional connectivity seem to be promising tools to quantify ADHD symptom severity and to reveal the hidden relationship between clinical phenotypic variables and brain connectivity.

Single-cell technologies have revolutionized our ability to interrogate biological systems at unprecedented resolution, revealing complex cellular heterogeneity and dynamic processes that underlie development, disease, and immune responses. However, the high dimensionality and nonlinear structure of single-cell data present substantial analytical challenges. Topological data analysis offers a powerful mathematical framework for capturing the intrinsic shape of data, providing novel insights that complement and extend traditional statistical and machine learning methods. By leveraging tools such as persistent homology and the Mapper algorithm, TDA enables the detection of subtle, multiscale patterns – including rare cell populations, transitional states, and branching trajectories – that are often obscured by conventional approaches. In this review, we explore the theoretical foundations of topological data analysis and examine its emerging applications across single-cell transcriptomics, proteomics, and spatial biology. We highlight how this approach can unveil previously unrecognized biological phenomena, from alternative differentiation paths to complex tissue architectures, and discuss the growing ecosystem of computational tools that support its use. As single-cell datasets become increasingly large and multimodal, topological data analysis stands out as a uniquely robust and interpretable approach, with the potential to deepen our understanding of cellular identity and function in health and disease. TDA is specially suited for fields such as systems immunology since it can capture the complex, nonlinear structures inherent in high-dimensional immune data helping to identify distinct immune cell states, differentiation pathways, and dynamic responses to infection or therapy. This topological perspective complements traditional statistical approaches, providing a robust, scale-invariant framework for uncovering hidden organization within the immune system’s complexity.

Topological data analysis is a family of recent mathematical techniques seeking to understand the ‘shape’ of data, and has been used to understand the structure of the descriptor space produced from a standard chemical informatics software from the point of view of solubility. We have used the mapper algorithm, a TDA method that creates low-dimensional representations of data, to create a network visualization of the solubility space. While descriptors with clear chemical implications are prominent features in this space, reflecting their importance to the chemical properties, an unexpected and interesting correlation between chlorine content and rings and their implication for solubility prediction is revealed. A parallel representation of the chemical space was generated using persistent homology applied to molecular graphs. Links between this chemical space and the descriptor space were shown to be in agreement with chemical heuristics. The use of persistent homology on molecular graphs, extended by the use of norms on the associated persistence landscapes allow the conversion of discrete shape descriptors to continuous ones, and a perspective of the application of these descriptors to quantitative structure property relations is presented.

The microstructure of porous electrodes of solid oxide cells (SOCs) significantly impacts their electrochemical performance. Therefore, various attempts have been made to analyze their complex porous microstructure in three dimensions. The obtained microstructures have been quantitatively evaluated using structural metrics, such as volume fraction, surface area density, and tortuosity factor of the constituent phases, as well as double- and triple-phase boundaries. These structural characteristics are useful to correlate the electrode performance with their microstructure, thereby determining optimal electrode design. However, these intuitive metrics often fails to explain the electrode performance. In fact, structural changes during long term operation of SOCs are subtle so that the performance degradation of the electrodes cannot be fully detected by these intuitive characteristics. Therefore, it is necessary to find a way to extract hidden structural metrics that characterize the electrode microstructure and to clarify their correlation with electrode performance.One of the promising candidates for such structural metrics is topological information. Topology is a structural property that is invariant to successive deformation operations, such as connected structures and the number of holes in the structures. Topology-based structural analysis has many applications in material science, where macroscopic properties of materials are correlated with topological information about their internal structures (e.g., crystal structures). For example, Wang et al. [1] performed a topological analysis on the crystal structure of double-phase steel to characterize its structure and to predict its macroscopic properties, such as stress-strain curves. Application of topological data analysis is also found in the analysis of SOFC electrodes; Pawlowski et al. [2] attempted to capture the structural changes in SOFC anodes during long-term operation from a topological perspective.In addition, topological analysis is regarded as a tool for dimensional reduction of complex structural information. Therefore, the information extracted from the topological analysis can be used to evaluate conventional metrics of porous microstructures, such as surface area and triple-phase boundary. If these metrics can be accurately evaluated from a limited number of information from the topological analysis, the numerical cost is expected to be significantly reduced.Therefore, this study investigates the applicability of the topological analysis to the electrode microstructure analysis. Fig. 1 shows the schematic diagram of the structural analysis in this study. First, persistent homology analysis [3] is employed to extract the topological information of the electrodes, where the birth and death of topological features are detected during the filtration process within the structure datasets to produce the persistent diagram (PD). The persistent diagram is then discretized and concatenated to obtain the persistent image (PI). Subsequently, principal component analysis (PCA) is conducted to further reduce the dimensionality of the structural information extracted in the persistent image. The obtained values of the principal components are correlated with the conventional structural metrics of the porous electrodes, such as volume fraction, surface area density, and triple-phase boundary density.Moreover, neural networks (NNs) consisting of fully-connected layers are constructed to quantify the structural metrics from the principal components. The constructed neural networks are trained using the real electrode microstructure datasets obtained using the focused ion beam and scanning electron microstructure [4]. To validate the developed neural networks, they are compared with the convolutional neural network (CNN), which directly quantifies the microstructural metrics from the three-dimensional structures, in terms of quantification accuracy, the number of training datasets, and required training time.The values obtained from the topological analysis followed by the principal component analysis are found to contain essential structural information in the porous electrodes. This is implied by the fact that the principal component values obtained from the real structure datasets are significantly different from those from the artificial sphere-packing structures, even though the conventional structural metrics, i.e., volume fraction and surface density, are identical between the real and artificial structures. In addition, the principal component values have sensitivity to the conventional statistical metrics, because the structures with different solid compositions form clusters in different locations in a principal component space. These suggest that the topological information will be useful not only to detect subtle structural changes in the porous electrodes undetectable in conventional structural metrics, but also to reduce the dimensionality of the complex porous electrodes without losing the information quality.Since the information extracted using the persistent homology and principal component analysis possess essential information about the electrode structures, conventional microstructural metrics are accurately quantified by the constructed neural network. It should be emphasized that the number of parameters in the neural network that need to be adjusted during the training process is significantly reduced compared with the convolutional neural network. As a result, the number of required datasets for the training of the network is significantly reduced. These clearly indicate the effectiveness of the dimensional reduction in the quantification of the electrode microstructures.

Topological data analysis and its main method, persistent homology, provide a toolkit for computing topological information of high-dimensional and noisy data sets. Kernels for one-parameter persistent homology have been established to connect persistent homology with machine learning techniques with applicability on shape analysis, recognition and classification. We contribute a kernel construction for multi-parameter persistence by integrating a one-parameter kernel weighted along straight lines. We prove that our kernel is stable and efficiently computable, which establishes a theoretical connection between topological data analysis and machine learning for multivariate data analysis.

Persistent homology is a fundamental tool in topological data analysis used for the most diverse applications. Information captured by persistent homology is commonly visualized using scatter plots representations. Despite being widely adopted, such a visualization technique limits user understanding and is prone to misinterpretation. This article proposes a new approach for the efficient computation of persistence cycles, a geometric representation of the features captured by persistent homology. We illustrate the importance of rendering persistence cycles when analyzing scalar fields, and we discuss the advantages that our approach provides compared to other techniques in topology-based visualization. We provide an efficient implementation of our approach based on discrete Morse theory, as a new module for the Topology Toolkit. We show that our implementation has comparable performance with respect to state-of-the-art toolboxes while providing a better framework for visually analyzing persistent homology information.

Over the last two decades, topological data analysis (TDA) has emerged as a very powerful data analytic approach that can deal with various data modalities of varying complexities. One of the most commonly used tools in TDA is persistent homology (PH), which can extract topological properties from data at various scales. The aim of this article is to introduce TDA concepts to a statistical audience and provide an approach to analyzing multivariate time series data. The application's focus will be on multivariate brain signals and brain connectivity networks. Finally, this paper concludes with an overview of some open problems and potential application of TDA to modeling directionality in a brain network, as well as the casting of TDA in the context of mixed effect models to capture variations in the topological properties of data collected from multiple subjects.

The Euler characteristic is an invariant of a topological space that in a precise sense captures its canonical notion of size, akin to the cardinality of a set. The Euler characteristic is closely related to the homology of a space, as it can be expressed as the alternating sum of its Betti numbers, whenever the sum is well-defined. Thus, one says that homology categorifies the Euler characteristic. In his work on the generalisation of cardinality-like invariants, Leinster introduced the magnitude of a metric space, a real number that gives the number of of the space. Recently, Leinster and Shulman introduced a homology theory for metric spaces, called magnitude homology, which categorifies the magnitude of a space. When studying a metric space, one is often only interested in the metric space up to a rescaling of the distance of the points by a non-negative real number. The magnitude function describes how the effective number of points changes as one scales the distance, and it is completely encoded in the Euler characteristic of magnitude homology. When studying a finite metric space in topological data analysis using persistent homology, one approximates the space through a nested sequence of simplicial complexes so as to recover topological information about the space by studying the homology of this sequence. Here we relate magnitude homology and persistent homology as two different ways of computing homology of filtered simplicial sets.

The natural pseudo-distance \(d_G\) associated with a group G of self-homeomorphisms of a topological space X is a pseudo-metric developed to compare real-valued functions defined on X, when the equivalence between functions is expressed by the group G. In this paper, we illustrate \(d_G\), its role in topological data analysis, its main properties and its link with persistent homology.

This chapter describes a new approach for studying the stability of stochastic delay equations by investigating their time series using topological data analysis (TDA). The approach is illustrated utilizing two stochastic delay equations. The first model equation is the stochastic version of Hayes equation—a scalar autonomous delay equation—where the noise is an additive term. The second model equation is the stochastic version of Mathieu’s equation—a time-periodic delay equation. In the latter, noise is added via a multiplicative term in the time-periodic coefficient. The time series is generated using Euler–Maruyama method and a corresponding point cloud is obtained using the Takens’ embedding. The point cloud is then analyzed using a tool from TDA known as persistent homology. The results of this study show that the described approach can be used for analyzing datasets of delay dynamical systems that are described using constant as well as time-periodic coefficients. The presented approach can be used for signals generated from both numerical simulation and experiments. It can be used as a tool to study the stability of stochastic delay equations for which there are currently a limited number of analysis tools.

In this paper, we explore how to use topological tools to compare dimension reduction methods. We first make a brief overview of some of the methods often used in dimension reduction such as isometric feature mapping, Laplacian Eigenmaps, fast independent component analysis, kernel ridge regression, and t-distributed stochastic neighbor embedding. We then give a brief overview of some of the topological notions used in topological data analysis, such as barcodes, persistent homology, and Wasserstein distance. Theoretically, when these methods are applied on a data set, they can be interpreted differently. From EEG data embedded into a manifold of high dimension, we discuss these methods and we compare them across persistent homologies of dimensions 0, 1, and 2, that is, across connected components, tunnels and holes, shells around voids, or cavities. We find that from three dimension clouds of points, it is not clear how distinct from each other the methods are, but Wasserstein and Bottleneck distances, topological tests of hypothesis, and various methods show that the methods qualitatively and significantly differ across homologies. We can infer from this analysis that topological persistent homologies do change dramatically at seizure, a finding already obtained in previous analyses. This suggests that looking at changes in homology landscapes could be a predictor of seizure.