Vietoris Rips Research Articles

Vietoris–Rips complexes of platonic solids

Vietoris–Rips complexes of platonic solids

An approach to extract topological information from Intuitionistic Fuzzy Sets and their application in obtaining a Natural Hierarchical Clustering Algorithm

Topological data analysis (TDA) is a powerful mathematical framework that extracts valuable insights about the shape and structure of complex datasets by identifying and analyzing underlying topological features, including connected components, holes, voids, and higher-dimensional counterparts. TDA achieves this by constructing a simplicial complex from the data, comprising simple shapes like points, lines, triangles, and higher-dimensional counterparts. The Vietoris–Rips complex (VRC), a widely used method, constructs simplicial complexes by measuring the distances between data points in a metric space. While the link between Intuitionistic Fuzzy Sets (IFSs) and TDA is clear through intuitionistic fuzzy distance measures, the literature has not explored the application of TDA for extracting topological features from uncertain and vague datasets using IFSs. In this paper, we propose a novel approach that leverages Intuitionistic Fuzzy Distance Measures (IFDMs) to generate the Intuitionistic Fuzzy Vietoris–Rips Complex (IFVRC) of IFSs. Specifically, we investigate the persistence of invariant relations of topological features between IFVRCs constructed using different distances, including Atanassov’s Euclidean distance (l2IFS), Boran and Akay’s distance (D), and Liu’s distance measure (dL), over artificially generated IFSs. Furthermore, we demonstrate that the generation of IFVRC can be viewed as a Natural Hierarchical Clustering Algorithm (NHCA) for IFSs. Finally, we apply the proposed IFVRC as a Natural Hierarchical Clustering Algorithm to cluster intuitionistic fuzzy datasets related to Car and Building materials. Moreover, we incorporate a relatively large dataset, the GTZAN dataset from the Kaggle datasets repository, to classify music genres based on their topological information.

Persistent homology based Bottleneck distance in hypergraph products

In this paper, we extended the technique of measuring similarity between topological spaces using bottle neck distance between persistence diagrams to hypergraph networks. Finding a relationship between the bottleneck distance of the Cartesian product of topological spaces and the bottleneck distance of individual spaces, we are trying to ease the comparative study of the Cartesian product of topological spaces. The Cartesian product and the strong product of weighted hypergraphs are defined, and the relationship between the bottleneck distance between hypergraph products and the bottleneck distance between individual hypergraphs is determined. For this, clique complex filtration and the Vietoris–Rips filtration in unweighted and weighted hypergraphs are defined and used.

Curvature Sets Over Persistence Diagrams

We study a family of invariants of compact metric spaces that combines the Curvature Sets defined by Gromov in the 1980 s with Vietoris–Rips Persistent Homology. For given integers k≥0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$k\\ge 0$$\\end{document} and n≥1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$n\\ge 1$$\\end{document} we consider the dimension k Vietoris–Rips persistence diagrams of all subsets of a given metric space with cardinality at most n. We call these invariants persistence sets and denote them as Dn,kVR\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\ extbf{D}}_{n,k}^{\ extrm{VR}}$$\\end{document}. We first point out that this family encompasses the usual Vietoris–Rips diagrams. We then establish that (1) for certain range of values of the parameters n and k, computing these invariants is significantly more efficient than computing the usual Vietoris–Rips persistence diagrams, (2) these invariants have very good discriminating power and, in many cases, capture information that is imperceptible through standard Vietoris–Rips persistence diagrams, and (3) they enjoy stability properties analogous to those of the usual Vietoris–Rips persistence diagrams. We precisely characterize some of them in the case of spheres and surfaces with constant curvature using a generalization of Ptolemy’s inequality. We also identify a rich family of metric graphs for which D4,1VR\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\ extbf{D}}_{4,1}^{\ extrm{VR}}$$\\end{document} fully recovers their homotopy type by studying split-metric decompositions. Along the way we prove some useful properties of Vietoris–Rips persistence diagrams using Mayer–Vietoris sequences. These yield a geometric algorithm for computing the Vietoris–Rips persistence diagram of a space X with cardinality 2k+2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$2k+2$$\\end{document} with quadratic time complexity as opposed to the much higher cost incurred by the usual algebraic algorithms relying on matrix reduction.

Dory: Computation of persistence diagrams up to dimension two for Vietoris–Rips filtrations of large data sets

Persistent homology (PH) is an approach to topological data analysis (TDA) that computes multi-scale topologically invariant properties of high-dimensional data that are robust to noise. While PH has revealed useful patterns across various applications, computational requirements have limited applications to small data sets of a few thousand points. We present Dory, an efficient and scalable algorithm that can compute the persistent homology of sparse Vietoris–Rips complexes on larger data sets, up to and including dimension two and over the field Z2. As an application, we compute the PH of the human genome at high resolution as revealed by a genome-wide Hi-C data set containing approximately three million points. Extant algorithms were unable to process it, whereas Dory processed it within five minutes, using less than five GB of memory. Results show that the topology of the human genome changes significantly upon treatment with auxin, a molecule that degrades cohesin, corroborating the hypothesis that cohesin plays a crucial role in loop formation in DNA.

Enhancing the Vietoris–Rips simplicial complex for topological data analysis: applications in cancer gene expression datasets

AbstractThe aim of this study is to enhance the extraction of informative features from complex data through the application of topological data analysis (TDA) using novel topological overlapping measures. Topological data analysis has emerged as a promising methodology for extracting meaningful insights from complex datasets. Existing approaches in TDA often involve extrapolating data points using distance correlation measures, which subsequently constrain downstream predictive tasks. Our objective is to improve the construction of the Vietoris–Rips simplicial complex by introducing topological overlapping measures. These measures take into account the interplay of direct connection strengths and shared neighbours, leading to the identification of persistent topological features. We propose the utilisation of topological overlapping measures to optimise the construction of the Vietoris–Rips simplicial complex, offering a more refined representation of complex data structures. The application of topological overlapping measures results in the identification of plentiful persistent topological features. This enhancement contributes to an improvement of up to 20% in cancer phenotype prediction across different cancer types. Our study demonstrates the effectiveness of utilising topological overlapping measures in optimising the construction of the Vietoris–Rips simplicial complex. The identified persistent topological features significantly enhance the predictive accuracy of cancer phenotypes. This novel approach has the potential to advance the field of topological data analysis and improve our understanding of complex data structures, particularly in the context of cancer research and predictive modelling. Further exploration and application of these measures may yield valuable insights in various domains dealing with intricate datasets.

Vietoris–Rips persistent homology, injective metric spaces, and the filling radius

Vietoris–Rips persistent homology, injective metric spaces, and the filling radius

Persistence-based clustering with outlier-removing filtration

This article describes a non-parametric clustering algorithm with an outlier removal step. Our method is based on tools from topological data analysis: we define a new filtration on metric spaces which is a variant of the Vietoris–Rips filtration that adds information about the points' nearest neighbor to the persistence diagram. We prove a stability theorem for this filtration, and evaluate our method on point cloud and graph datasets, showing that it can compete with state-of-the-art methods while being non-parametric.

Lower Bounds on the Homology of Vietoris–Rips Complexes of Hypercube Graphs

We provide novel lower bounds on the Betti numbers of Vietoris–Rips complexes of hypercube graphs of all dimensions and at all scales. In more detail, let Qn\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$Q_n$$\\end{document} be the vertex set of 2n\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$2^n$$\\end{document} vertices in the n-dimensional hypercube graph, equipped with the shortest path metric. Let VR(Qn;r)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ extrm{VR}(Q_n;r)$$\\end{document} be its Vietoris–Rips complex at scale parameter r≥0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$r \\ge 0$$\\end{document}, which has Qn\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$Q_n$$\\end{document} as its vertex set, and all subsets of diameter at most r as its simplices. For integers r<r′\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$r<r'$$\\end{document} the inclusion VR(Qn;r)↪VR(Qn;r′)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ extrm{VR}(Q_n;r)\\hookrightarrow \ extrm{VR}(Q_n;r')$$\\end{document} is nullhomotopic, meaning no persistent homology bars have length longer than one, and we therefore focus attention on the individual spaces VR(Qn;r)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ extrm{VR}(Q_n;r)$$\\end{document}. We provide lower bounds on the ranks of homology groups of VR(Qn;r)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ extrm{VR}(Q_n;r)$$\\end{document}. For example, using cross-polytopal generators, we prove that the rank of H2r-1(VR(Qn;r))\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$H_{2^r-1}(\ extrm{VR}(Q_n;r))$$\\end{document} is at least 2n-(r+1)nr+1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$2^{n-(r+1)}\\left( {\\begin{array}{c}n\\\\ r+1\\end{array}}\\right) $$\\end{document}. We also prove a version of homology propagation: if q≥1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$q\\ge 1$$\\end{document} and if p is the smallest integer for which rankHq(VR(Qp;r))≠0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ extrm{rank}H_q(\ extrm{VR}(Q_p;r))\ e 0$$\\end{document}, then rankHq(VR(Qn;r))≥∑i=pn2i-pi-1p-1·rankHq(VR(Qp;r))\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ extrm{rank}H_q(\ extrm{VR}(Q_n;r)) \\ge \\sum _{i=p}^n 2^{i-p} \\left( {\\begin{array}{c}i-1\\\\ p-1\\end{array}}\\right) \\cdot \ extrm{rank}H_q(\ extrm{VR}(Q_p;r))$$\\end{document} for all n≥p\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$n \\ge p$$\\end{document}. When r≤3\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$r\\le 3$$\\end{document}, this result and variants thereof provide tight lower bounds on the rank of Hq(VR(Qn;r))\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$H_q(\ extrm{VR}(Q_n;r))$$\\end{document} for all n, and for each r≥4\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$r \\ge 4$$\\end{document} we produce novel lower bounds on the ranks of homology groups. Furthermore, we show that for each r≥2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$r\\ge 2$$\\end{document}, the homology groups of VR(Qn;r)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ extrm{VR}(Q_n;r)$$\\end{document} for n≥2r+1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$n \\ge 2r+1$$\\end{document} contain propagated homology not induced by the initial cross-polytopal generators.

On Vietoris–Rips complexes of finite metric spaces with scale 2

On Vietoris–Rips complexes of finite metric spaces with scale 2

Automatic epileptic seizure detection based on persistent homology.

Epilepsy is a prevalent brain disease, which is quite difficult-to-treat or cure. This study developed a novel automatic seizure detection method based on the persistent homology method. In this study, a Vietoris-Rips (VR) complex filtration model was constructed based on the EEG data. And the persistent homology method was applied to calculate the VR complex filtration barcodes to describe the topological changes of EEG recordings. Afterward, the barcodes as the topological characteristics of EEG signals were fed into the GoogLeNet for classification. The persistent homology is applicable for multi-channel EEG data analysis, where the global topological information is calculated and the features are extracted by considering the multi-channel EEG data as a whole, without the multiple calculations or the post-stitching. Three databases were used to evaluate the proposed approach and the results showed that the approach had high performances in the epilepsy detection. The results obtained from the CHB-MIT Database recordings revealed that the proposed approach can achieve a segment-based averaged accuracy, sensitivity and specificity values of 97.05%, 96.71% and 97.38%, and achieve an event-based averaged sensitivity value of 100% with 1.22s average detection latency. In addition, on the Siena Scalp Database, the proposed method yields averaged accuracy, sensitivity and specificity values of 96.42%, 95.23% and 97.6%. Multiple tasks of the Bonn Database also showed achieved accuracy of 99.55%, 98.63%, 98.28% and 97.68%, respectively. The experimental results on these three EEG databases illustrate the efficiency and robustness of our approach for automatic detection of epileptic seizure.

Topological analysis of functional connectivity in Parkinson's disease.

Parkinson's disease (PD) is a clinically heterogeneous disorder, which mainly affects patients' motor and non-motor function. Functional connectivity was preliminary explored and studied through resting state functional magnetic resonance imaging (rsfMRI). Through the topological analysis of 54 PD scans and 31 age-matched normal controls (NC) in the Neurocon dataset, leveraging on rsfMRI data, the brain functional connection and the Vietoris-Rips (VR) complex were constructed. The barcodes of the complex were calculated to reflect the changes of functional connectivity neural circuits (FCNC) in brain network. The 0-dimensional Betti number β0 means the number of connected branches in VR complex. The average number of connected branches in PD group was greater than that in NC group when the threshold δ ≤ 0.7. Two-sample Mann-Whitney U test and false discovery rate (FDR) correction were used for statistical analysis to investigate the FCNC changes between PD and NC groups. In PD group, under threshold of 0.7, the number of FCNC involved was significantly differences and these brain regions include the Cuneus_R, Lingual_R, Fusiform_R and Heschl_R. There are also significant differences in brain regions in the Frontal_Inf_Orb_R and Pallidum_R, when the threshold increased to 0.8 and 0.9 (p < 0.05). In addition, when the length of FCNC was medium, there was a significant statistical difference between the PD group and the NC group in the Neurocon dataset and the Parkinson's Progression Markers Initiative (PPMI) dataset. Topological analysis based on rsfMRI data may provide comprehensive information about the changes of FCNC and may provide an alternative for clinical differential diagnosis.

On Vietoris–Rips Complexes (with Scale 3) of Hypercube Graphs

On Vietoris–Rips Complexes (with Scale 3) of Hypercube Graphs

Vietoris–Rips metric thickenings of the circle

Vietoris–Rips metric thickenings of the circle

Vietoris–Rips complexes of metric spaces near a metric graph

Vietoris–Rips complexes of metric spaces near a metric graph

Topological data analysis of noise: Uniform unimodal distributions

Random point clouds determine random simplicial complexes which can be characterized via homology. In this work we consider Vietoris–Rips complexes built on top of random point clouds in the D-dimensional Euclidian space in two setups: (i) a homogeneous Poisson point process (HPPP) and (ii) N i.i.d. points sampled from a uniform elliptical distribution. We derive expected topological summaries for such complexes in terms of the average Betti numbers (ABN) and of the newly introduced average integral persistence landscape (AIPL). The latter summary is derived from the average persistence landscape (APL) and preserves its 1-norm. In the HPPP model, we derive general analytical expressions for the density of ABN and AIPL and their 1-norms. For finite samples, we establish the functional form of decline of the 1-norm of the APL with the decrease of the sample size and/or increase of the cloud dimensionality. Furthermore, we derive analytical estimates of boundary effects in a disk, an ellipse, a D-dimensional ball, and 3-dimensional spheroids. We develop asymptotic models accounting for degeneration of a highly eccentric ellipse or a prolate spheroid to a line-segment and of an oblate spheroid to a disk, which describe the decline or growth of the 1-norm of the APL upon degeneration of an elliptic or spheroidal point cloud to a one- or two-dimensional cloud, respectively. Our results can serve as a “null model” for selection of the hyperparameters of the method D and N, as well as for the interpretation of observations in a broad range of real-world problems.

Vietoris thickenings and complexes have isomorphic homotopy groups

We study the relationship between metric thickenings and simplicial complexes associated to coverings of metric spaces. Let \({\mathcal {U}}\) be a cover of a separable metric space X by open sets with a uniform diameter bound. The Vietoris complex \({\mathcal {V}}({\mathcal {U}})\) contains all simplices with vertex set contained in some \(U \in {\mathcal {U}}\), and the Vietoris metric thickening \({\mathcal {V}}^\textrm{m}({\mathcal {U}})\) is the space of probability measures with support in some \(U \in {\mathcal {U}}\), equipped with an optimal transport metric. We show that \({\mathcal {V}}^\textrm{m}({\mathcal {U}})\) and \({\mathcal {V}}({\mathcal {U}})\) have isomorphic homotopy groups in all dimensions. In particular, by choosing the cover \({\mathcal {U}}\) appropriately, we get isomorphisms between the homotopy groups of Vietoris–Rips metric thickenings and simplicial complexes \(\pi _k(\textrm{VR}^\textrm{m}(X;r))\cong \pi _k(\textrm{VR}(X;r))\) for all integers \(k\ge 0\), where both spaces are defined using the convention “diameter \(< r\)” (instead of \(\le r\)). Similarly, we get isomorphisms between the homotopy groups of Čech metric thickenings and simplicial complexes \(\pi _k(\check{\mathrm {{C}}}^\textrm{m}(X;r))\cong \pi _k(\check{\mathrm {{C}}}(X;r))\) for all integers \(k\ge 0\), where both spaces are defined using open balls (instead of closed balls).

Change point detection in multi-agent systems based on higher-order features

Change point detection (CPD) for multi-agent systems helps one to evaluate the state and better control the system. Multivariate CPD methods solve the d × T time series well; however, the multi-agent systems often produce the N × d × T dimensional data, where d is the dimension of multivariate observations, T is the total observation time, and N is the number of agents. In this paper, we propose two valid approaches based on higher-order features, namely, the Betti number feature extraction and the Persistence feature extraction, to compress the d-dimensional features into one dimension so that general CPD methods can be applied to higher-dimensional data. First, a topological structure based on the Vietoris-Rips complex is constructed on each time-slice snapshot. Then, the Betti number and persistence of the topological structures are obtained to separately constitute two feature matrices for change point estimates. Higher-order features primarily describe the data distribution on each snapshot and are, therefore, independent of the node correspondence cross snapshots, which gives our methods unique advantages in processing missing data. Experiments in multi-agent systems demonstrate the significant performance of our methods. We believe that our methods not only provide a new tool for dimensionality reduction and missing data in multi-agent systems but also have the potential to be applied to a wider range of fields, such as complex networks.

Persistent Homology for Breast Tumor Classification Using Mammogram Scans

An important tool in the field of topological data analysis is persistent homology (PH), which is used to encode abstract representations of the homology of data at different resolutions in the form of persistence barcode (PB). Normally, one will obtain one PB from a digital image when using a sublevel-set filtration method. In this work, we built more than one PB representation of a single image based on a landmark selection method, known as local binary patterns (LBP), which encode different types of local texture from a digital image. Starting from the top-left corner of any 3-by-3 patch selected from an input image, the LBP process starts by subtracting the central pixel value from its eight neighboring pixel values. Then, each cell is assigned with 1 if the subtraction outcome is positive, and 0 otherwise, to obtain an 8-bit binary representation. This process will identify a set of landmark pixels to represent 0-simplices and use Vietoris–Rips filtration to obtain its corresponding PB. Using LBP, we can construct up to 56 PBs from a single image if we restrict to only using the binary codes that have two circular transitions between 1 and 0. The information within these 56 PBs contain detailed local and global topological and geometrical information, which can be used to design effective machine learning models. We used four different PB vectorizations, namely, persistence landscapes, persistence images, Betti curves (barcode binning), and PB statistics. We tested the effectiveness of the proposed landmark-based PH on two publicly available breast abnormality detection datasets using mammogram scans. The sensitivity and specificity of the landmark-based PH obtained was over 90% and 85%, respectively, in both datasets for the detection of abnormal breast scans. Finally, the experimental results provide new insights on using different PB vectorizations with sublevel set filtrations and landmark-based Vietoris–Rips filtration from digital mammogram scans.

Method for persistent topological features extraction of schizophrenia patients' electroencephalography signal based on persistent homology.

With the development of network science and graph theory, brain network research has unique advantages in explaining those mental diseases, the neural mechanism of which is unclear. Additionally, it can provide a new perspective in revealing the pathophysiological mechanism of brain diseases from the system level. The selection of threshold plays an important role in brain networks construction. There are no generally accepted criteria for determining the proper threshold. Therefore, based on the topological data analysis of persistent homology theory, this study developed a multi-scale brain network modeling analysis method, which enables us to quantify various persistent topological features at different scales in a coherent manner. In this method, the Vietoris–Rips filtering algorithm is used to extract dynamic persistent topological features by gradually increasing the threshold in the range of full-scale distances. Subsequently, the persistent topological features are visualized using barcodes and persistence diagrams. Finally, the stability of persistent topological features is analyzed by calculating the Bottleneck distances and Wasserstein distances between the persistence diagrams. Experimental results show that compared with the existing methods, this method can extract the topological features of brain networks more accurately and improves the accuracy of diagnostic and classification. This work not only lays a foundation for exploring the higher-order topology of brain functional networks in schizophrenia patients, but also enhances the modeling ability of complex brain systems to better understand, analyze, and predict their dynamic behaviors.