Persistent topology for natural data analysis - A survey

Natural data offer a hard challenge to data analysis. One set of tools is being developed by several teams to face this difficult task: Persistent topology. After a brief introduction to this theory, some applications to the analysis and classification of cells, lesions, music pieces, gait, oil and gas reservoirs, cyclones, galaxies, bones, brain connections, languages, handwritten and gestured letters are shown

Variable sets over an algebra of lifetimes: a contribution of lattice theory to the study of computational topology

A topos theoretic generalisation of the category of sets allows for modelling spaces which vary according to time intervals. Persistent homology, or more generally, persistence is a central tool in topological data analysis, which examines the structure of data through topology. The basic techniques have been extended in several different directions, permuting the encoding of topological features by so called barcodes or equivalently persistence diagrams. The set of points of all such diagrams determines a complete Heyting algebra that can explain aspects of the relations between persistent bars through the algebraic properties of its underlying lattice structure. In this paper, we investigate the topos of sheaves over such algebra, as well as discuss its construction and potential for a generalised simplicial homology over it. In particular we are interested in establishing a topos theoretic unifying theory for the various flavours of persistent homology that have emerged so far, providing a global perspective over the algebraic foundations of applied and computational topology.Comment: 20 pages, 12 figures, AAA88 Conference proceedings at Demonstratio Mathematica. The new version has restructured arguments, clearer intuition is provided, and several typos correcte

Mathematics at the eve of a historic transition in biology

A century ago physicists and mathematicians worked in tandem and established quantum mechanism. Indeed, algebras, partial differential equations, group theory, and functional analysis underpin the foundation of quantum mechanism. Currently, biology is undergoing a historic transition from qualitative, phenomenological and descriptive to quantitative, analytical and predictive. Mathematics, again, becomes a driving force behind this new transition in biology.Comment: 5 pages, 2 figure

Designing a topological algorithm for 3D activity recognition

Voxel carving is a non-invasive and low-cost technique that is used for the reconstruction of a 3D volume from images captured from a set of cameras placed around the object of interest. In this paper we propose a method to topologically analyze a video sequence of 3D reconstructions representing a tennis player performing different forehand and backhand strokes with the aim of providing an approach that could be useful in other sport activities

Canonical tensor model through data analysis -- Dimensions, topologies, and geometries --

The canonical tensor model (CTM) is a tensor model in Hamilton formalism and is studied as a model for gravity in both classical and quantum frameworks. Its dynamical variables are a canonical conjugate pair of real symmetric three-index tensors, and a question in this model was how to extract spacetime pictures from the tensors. We give such an extraction procedure by using two techniques widely known in data analysis. One is the tensor-rank (or CP, etc.) decomposition, which is a certain generalization of the singular value decomposition of a matrix and decomposes a tensor into a number of vectors. By regarding the vectors as points forming a space, topological properties can be extracted by using the other data analysis technique called persistent homology, and geometries by virtual diffusion processes over points. Thus, time evolutions of the tensors in the CTM can be interpreted as topological and geometric evolutions of spaces. We have performed some initial investigations of the classical equation of motion of the CTM in terms of these techniques for a homogeneous fuzzy circle and homogeneous two- and three-dimensional fuzzy spheres as spaces, and have obtained agreement with the general relativistic system obtained previously in a formal continuum limit of the CTM. It is also demonstrated by some concrete examples that the procedure is general for any dimensions and topologies, showing the generality of the CTM.Comment: 44 pages, 16 figures, minor correction

Analyzing Single Cell RNA Sequencing with Topological Nonnegative Matrix Factorization

Single-cell RNA sequencing (scRNA-seq) is a relatively new technology that has stimulated enormous interest in statistics, data science, and computational biology due to the high dimensionality, complexity, and large scale associated with scRNA-seq data. Nonnegative matrix factorization (NMF) offers a unique approach due to its meta-gene interpretation of resulting low-dimensional components. However, NMF approaches suffer from the lack of multiscale analysis. This work introduces two persistent Laplacian regularized NMF methods, namely, topological NMF (TNMF) and robust topological NMF (rTNMF). By employing a total of 12 datasets, we demonstrate that the proposed TNMF and rTNMF significantly outperform all other NMF-based methods. We have also utilized TNMF and rTNMF for the visualization of popular Uniform Manifold Approximation and Projection (UMAP) and t-distributed stochastic neighbor embedding (t-SNE)

Ventricular Fibrillation and Tachycardia Detection Using Features Derived from Topological Data Analysis

A rapid and accurate detection of ventricular arrhythmias is essential to take appropriate therapeutic actions when cardiac arrhythmias occur. Furthermore, the accurate discrimination between arrhythmias is also important, provided that the required shocking therapy would not be the same. In this work, the main novelty is the use of the mathematical method known as Topological Data Analysis (TDA) to generate new types of features which can contribute to the improvement of the detection and classification performance of cardiac arrhythmias such as Ventricular Fibrillation (VF) and Ventricular Tachycardia (VT). The electrocardiographic (ECG) signals used for this evaluation were obtained from the standard MIT-BIH and AHA databases. Two input data to the classify are evaluated: TDA features, and Persistence Diagram Image (PDI). Using the reduced TDA-obtained features, a high average accuracy near 99% was observed when discriminating four types of rhythms (98.68% to VF; 99.05% to VT; 98.76% to normal sinus; and 99.09% to Other rhythms) with specificity values higher than 97.16% in all cases. In addition, a higher accuracy of 99.51% was obtained when discriminating between shockable (VT/VF) and non-shockable rhythms (99.03% sensitivity and 99.67% specificity). These results show that the use of TDA-derived geometric features, combined in this case this the k-Nearest Neighbor (kNN) classifier, raises the classification performance above results in previous works. Considering that these results have been achieved without preselection of ECG episodes, it can be concluded that these features may be successfully introduced in Automated External Defibrillation (AED) and Implantable Cardioverter Defibrillation (ICD) therapie

Learning Persistent Community Structures in Dynamic Networks via Topological Data Analysis

Dynamic community detection methods often lack effective mechanisms to ensure temporal consistency, hindering the analysis of network evolution. In this paper, we propose a novel deep graph clustering framework with temporal consistency regularization on inter-community structures, inspired by the concept of minimal network topological changes within short intervals. Specifically, to address the representation collapse problem, we first introduce MFC, a matrix factorization-based deep graph clustering algorithm that preserves node embedding. Based on static clustering results, we construct probabilistic community networks and compute their persistence homology, a robust topological measure, to assess structural similarity between them. Moreover, a novel neural network regularization TopoReg is introduced to ensure the preservation of topological similarity between inter-community structures over time intervals. Our approach enhances temporal consistency and clustering accuracy on real-world datasets with both fixed and varying numbers of communities. It is also a pioneer application of TDA in temporally persistent community detection, offering an insightful contribution to field of network analysis. Code and data are available at the public git repository: https://github.com/kundtx/MFC_TopoRegComment: AAAI 202

A Fuzzy Approach for Topological Data Analysis

Geometry and topology are becoming more powerful and dominant in data analysis because of their outstanding characteristics. It has emerged recently as a promising research area, known as Topological Data Analysis (TDA), for modern computer science. In recent years, the Mapper algorithm, an outstanding TDA representative, is increasingly completed with a stabilized theoretical foundation and practical applications and diverse, intuitive, user-friendly implementations. From a theoretical perspective, the Mapper algorithm is still a fuzzy clustering algorithm, with a visualization capability to extract the shape summary of data. However, its outcomes are still very sensitive to the parameter choice, including resolution and function. Therefore, there is a need to reduce the dependence on its parameters significantly. This idea is exciting and can be solved thanks to the outstanding characteristics of fuzzy clustering. The Mapper clustering ability is getting more potent by the support from well-known techniques. Therefore, this combination is expected to usefully and powerfully solve some problems encountered in many fields. The main research goal of this thesis is to approach TDA by fuzzy theory to create the interrelationships between them in terms of clustering. Explicitly speaking, the Mapper algorithm represents TDA, and the Fuzzy $C$-Means (FCM) algorithm represents fuzzy theory. They are combined to promote their advantages and overcome their disadvantages. On the one hand, the FCM algorithm helps the Mapper algorithm simplify the choice of parameters to obtain the most informative presentation and is even more efficient in data clustering. On the other hand, the FCM algorithm is equipped with the outstanding features of the Mapper algorithm in simplifying and visualizing data with qualitative analysis. This thesis focuses on conquering and achieving the following aims: (1) Summarizing the theoretical foundations and practical applications of the Mapper algorithm in the flow of literature with improved versions and various implementations. (2) Optimizing the cover choice of the Mapper algorithm in the direction of dividing the filter range automatically into irregular intervals with a random overlapping percentage by using the FCM algorithm. (3) Constructing a novel method for mining data that can exhibit the same clustering ability as the FCM algorithm and reveal some meaningful relationships by visualizing the global shape of data supplied by the Mapper algorithm.Geometrie a topologie se stávají silnějšími a dominantnějšími v analýze dat díky svým vynikajícím vlastnostem. Nedávno se objevila jako slibná výzkumná oblast, známá jako topologická analýza dat (TDA), pro moderní informatiku. V posledních letech je algoritmus Mapper, vynikající představitel TDA, stále více doplněn o stabilizovaný teoretický základ a praktické aplikace a rozmanité, intuitivní a uživatelsky přívětivé implementace. Z teoretického hlediska je algoritmus Mapper stále fuzzy shlukovací algoritmus se schopností vizualizace extrahovat souhrn tvaru dat. Jeho výsledky jsou však stále velmi citlivé na volbu parametrů, včetně rozlišení a funkce. Proto je potřeba výrazně snížit závislost na jeho parametrech. Tato myšlenka je vzrušující a lze ji vyřešit díky vynikajícím vlastnostem fuzzy shlukování. Schopnost shlukování Mapperu je stále silnější díky podpoře známých technik. Proto se očekává, že tato kombinace užitečně a účinně vyřeší některé problémy, se kterými se setkáváme v mnoha oblastech. Hlavním výzkumným cílem této práce je přiblížit TDA pomocí fuzzy teorie a vytvořit mezi nimi vzájemné vztahy z hlediska shlukování. Explicitně řečeno, algoritmus Mapper představuje TDA a algoritmus Fuzzy $C$-Means (FCM) představuje fuzzy teorii. Jsou kombinovány, aby podpořily své výhody a překonaly své nevýhody. Na jedné straně algoritmus FCM pomáhá algoritmu Mapper zjednodušit výběr parametrů pro získání nejinformativnější prezentace a je ještě efektivnější při shlukování dat. Na druhé straně je algoritmus FCM vybaven vynikajícími vlastnostmi algoritmu Mapper pro zjednodušení a vizualizaci dat pomocí kvalitativní analýzy. Tato práce se zaměřuje na dobývání a dosažení následujících cílů: (1) Shrnutí teoretických základů a praktických aplikací Mapperova algoritmu v toku literatury s vylepšenými verzemi a různými implementacemi. (2) Optimalizace volby pokrytí algoritmu Mapper ve směru automatického rozdělení rozsahu filtru do nepravidelných intervalů s náhodně se překrývajícím procentem pomocí algoritmu FCM. (3) Vytvoření nové metody pro těžbu dat, která může vykazovat stejnou schopnost shlukování jako algoritmus FCM a odhalit některé smysluplné vztahy vizualizací globálního tvaru dat poskytovaných algoritmem Mapper.460 - Katedra informatikyvyhově