Persistent Homology Method Research Articles

Homogenization of word relationships in schizophrenia: Topological analysis of cortical semantic representations.

Patients with schizophrenia typically exhibit symptoms of disorganized thought and display concreteness and over-inclusion in verbal reports, depending on the level of abstraction. While concreteness and over-inclusion may appear contradictory, the underlying psychopathology that explains these symptoms remains unclear. In the current study, we used functional magnetic resonance imaging with an encoding modeling approach to examine how concepts of various words, represented as brain activity, are anomalously connected at different levels of abstraction in patients with schizophrenia. Fourteen individuals diagnosed with schizophrenia and 17 healthy controls underwent functional magnetic resonance imaging to measure brain activity representing concepts of various words. We used a persistent homology (PH) method to analyze the topological structures of word representations in schizophrenia patients, healthy controls, and random data, across different levels of abstraction by varying dissimilarity scales in the representation space. The results revealed that patients with schizophrenia exhibited more homogeneous word relationships across different levels of abstraction compared with healthy controls. Additionally, topological structures exhibited a shift toward a random network structure in patients with schizophrenia compared with controls. The PH method successfully distinguished semantic representations of patients with schizophrenia from those of controls. The current results provide an explanation for the mechanisms underlying the deficits in abstraction ability observed in schizophrenia. The isotopic connection of individual concepts reflects both the reduction of contextual connections at a semantically fine-grained scale and the absence of clear boundaries between related concepts at a coarse scale, which lead to concreteness and over-inclusion, respectively.

Investigating rough single-fracture permeabilities with persistent homology

Abstract. The permeability of rock fractures is a crucial parameter for flow processes in the subsurface. In the last few decades, different methods were developed to investigate on permeability in fractures, such as flow-through experiments, numerical flow simulations, or empirical equations. In recent years, the topological method of persistent homology was also used to estimate the permeability of fracture networks and porous rocks but not for rough single fractures yet. Hence, we apply persistent homology analysis on a decimetre-scale, rough sandstone bedding joint. To investigate the influence of roughness, three different data sets are created to perform the analysis: (1) 200 µm, (2) 100 µm, and (3) 50 µm resolutions. All estimated permeabilities were then compared to values derived by experimental air permeameter measurements and numerical flow simulation. The results reveal that persistent homology analysis is able to estimate the permeability of a single fracture, even if it tends to slightly overestimate permeabilities compared to conventional methods. Previous studies using porous media showed the same overestimation trend. Furthermore, the expenditure of time for persistent homology analysis, as well as air permeameter measurements and numerical flow simulation, was compared, which showed that persistent homology analysis can be also an acceptable alternative method.

Topological data analysis expands the genotype to phenotype map for 3D maize root system architecture.

A central goal of biology is to understand how genetic variation produces phenotypic variation, which has been described as a genotype to phenotype (G to P) map. The plant form is continuously shaped by intrinsic developmental and extrinsic environmental inputs, and therefore plant phenomes are highly multivariate and require comprehensive approaches to fully quantify. Yet a common assumption in plant phenotyping efforts is that a few pre-selected measurements can adequately describe the relevant phenome space. Our poor understanding of the genetic basis of root system architecture is at least partially a result of this incongruence. Root systems are complex 3D structures that are most often studied as 2D representations measured with relatively simple univariate traits. In prior work, we showed that persistent homology, a topological data analysis method that does not pre-suppose the salient features of the data, could expand the phenotypic trait space and identify new G to P relations from a commonly used 2D root phenotyping platform. Here we extend the work to entire 3D root system architectures of maize seedlings from a mapping population that was designed to understand the genetic basis of maize-nitrogen relations. Using a panel of 84 univariate traits, persistent homology methods developed for 3D branching, and multivariate vectors of the collective trait space, we found that each method captures distinct information about root system variation as evidenced by the majority of non-overlapping QTL, and hence that root phenotypic trait space is not easily exhausted. The work offers a data-driven method for assessing 3D root structure and highlights the importance of non-canonical phenotypes for more accurate representations of the G to P map.

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.

Formation of features based on computational topology methods

The use of traditional methods of algebraic topology to obtain information about the shape of an object is associated with the problem of forming a small amount of information, namely, Betti numbers and Euler characteristics. The central tool for topological data analysis is the persistent homology method, which summarizes the geometric and topological information in the data using persistent diagrams and barcodes. Based on persistent homology methods, topological data can be analyzed to obtain information about the shape of an object. The construction of persistent barcodes and persistent diagrams in computational topology does not allow one to construct a Hilbert space with a scalar product. The possibility of applying the methods of topological data analysis is based on mapping persistent diagrams into a Hilbert space; one of the ways of such mapping is a method of constructing a persistence landscape. It has an advantage of being reversible, so it does not lose any information and has persistence properties. The paper considers mathematical models and functions for representing persistence landscape objects based on the persistent homology method. Methods for converting persistent barcodes and persistent diagrams into persistence landscape functions are considered. Associated with persistence landscape functions is a persistence landscape kernel that forms a mapping into a Hilbert space with a dot product. A formula is proposed for determining a distance between the persistence landscapes, which allows the distance between images of objects to be found. The persistence landscape functions map persistent diagrams into a Hilbert space. Examples of determining the distance between images based on the construction of persistence landscape functions for these images are given. Representations of topological characteristics in various models of computational topology are considered. Results for one-parameter persistence modules are extended onto multi-parameter persistence modules.

Modeling shear-induced solid-liquid transition of granular materials using persistent homology

This paper investigates the transition between the solid and liquid phases of sheared granular materials from the perspective of the contact network. Tools from persistent homology are employed to quantify the dynamics of contact network during the solid-liquid transition from a global perspective, and two important topological invariants, i.e., components and loops, are mainly investigated from discrete numerical simulations. The highly heterogeneous composition of the contact network is revealed, and a rationale partition threshold for distinguishing between strong and weak contact subnetworks can be determined through the emergence and death of these topological invariants. During the shearing process, we recognize mechanical precursors forecasting the occurrence of solid-liquid transition when the assembly is still stable. Furthermore, we provide the panorama of the solid-liquid transition from the evolution of contact network and its homology groups. Finally, this study suggests that the persistent homology method is capable of quantitatively bridging the microscopic dynamics with macroscopic responses through the contact network, which paves an efficient way to further include the evolution of the contact network in the constitutive modeling of granular materials.

Neighborhood Path Complex for the Quantitative Analysis of the Structure and Stability of Carboranes

Thanks to the tremendous progress in data, computing power and algorithms, AI-based material mining and design have gained much attention. However, building high-performance AI models requires efficient material structure representation. In this work, we propose a structural characterization method based on the neighborhood path complex for the first time. Specifically, we use persistent neighborhood path homology to obtain the structural features by introducing a filtration. This approach preserves more elemental information, as well as the corresponding physicochemical information, through the directed edges of the neighborhood digraph. To validate our model, we perform cross-validation with the carborane structures. The Pearson coefficient for stability prediction is as high as 0.903, which is a 15.5% improvement compared to the traditional persistent homology method. In addition, we constructed a prediction model based on the neighborhood path complex, and the Pearson coefficients for the prediction of carboranes’ HOMO, LUMO, and HOMO–LUMO gaps were 0.915, 0.946, and 0.941, respectively. The results show that our proposed method can effectively extract structural information and achieve accurate material property prediction.

Dynamic ECG signal quality evaluation based on persistent homology and GoogLeNet method.

Cardiovascular disease is a serious health problem. Continuous Electrocardiograph (ECG) monitoring plays a vital role in the early detection of cardiovascular disease. As the Internet of Things technology continues to mature, wearable ECG signal monitors have been widely used. However, dynamic ECG signals are extremely susceptible to contamination. Therefore, it is necessary to evaluate the quality of wearable dynamic ECG signals. The topological data analysis method (TDA) with persistent homology, which can effectively capture the topological information of high-dimensional data space, has been widely studied. In this study, a brand-new quality assessment method of wearable dynamic ECG signals was proposed based on the TDA with persistent homology method. The point cloud of an ECG signal was constructed, and then the complex sequence was generated and displayed as a persistent barcode. Finally, GoogLeNet based on the transfer learning model with a 10-fold cross-validation method was used to train the classification model. A total of 12-leads ECGs Dataset and single-lead ECGs Dataset, established based on the 2011 PhysioNet/CinC challenge dataset, were both used to verify the performance of this method. In the study, 773 "acceptable" and 225 "unacceptable" signals were used as 12-leads ECGs Dataset. We relabeled 12,000 ECG signals in the challenge dataset, and treated them as single-lead ECGs Dataset after empty lead detection and balance datasets. Compared with the traditional ECG signal quality assessment method mainly based on waveform characteristics and time-frequency characteristics, the performance of the quality assessment method proposed. In this study, the classification performance of the proposed method are fairly great, mAcc = 98.04%, F1 = 98.40%, Se = 97.15%, Sp = 98.93% for 12-leads ECGs Dataset and mAcc = 98.55%, F1 = 98.62%, Se = 98.37%, Sp = 98.85% for single-lead ECGs Dataset.

Topological data analysis for geographical information science using persistent homology

Topological data analysis (TDA) is an emerging field of research, which considers the application of topology to data analysis. Recently, these methods have been successfully applied to research problems in the field of geographical information science (GIS) and there is much potential for future applications. In this article, we provide an introduction to the fundamentals of TDA for GIS researchers and practitioners and highlight specific benefits that TDA methods provide relative to some conventional methods. We focus on the method of persistent homology, which is the most commonly used TDA method. We describe how persistent homology can be applied to data types commonly encountered in the GIScience domain, namely sets of points, networks and sequences of images. We also describe the application of persistent homology to two specific GIS problems, which are the point pattern analysis of UK city pubs and the analysis of UK rainfall radar imagery. In each case we stress the specific benefits of TDA methods that include, for example, generating an output signature in a form that can be subject to subsequent analyses; identification of void regions in point patterns; and providing a relatively simple method to track objects in spatio-temporal images.

Topological data analysis of protein structure and inter/intra-molecular interaction changes attributable to amino acid mutations

The presence of some amino acid mutations in the amino acid sequence that determines a protein’s structure can significantly affect that 3D structure and its biological function. However, the effects upon structural and functional changes differ for each displaced amino acid, and it is very difficult to predict these changes in advance. Although computer simulations are very effective at predicting conformational changes, they struggle to determine whether the amino acid mutation of interest induces sufficient conformational changes, unless the researcher is a specialist in molecular structure calculations. Therefore, we created a framework that efficiently utilizes molecular dynamics and persistent homology methods to identify amino acid mutations that induce structural changes. We show that this framework can be used not only to predict conformational changes produced by amino acid mutations but also to extract groups of mutations that significantly alter similar molecular interactions, by capturing the resultant protein–protein interaction changes.

Distances and isomorphism between networks: stability and convergence of network invariants

We develop the theoretical foundations of a generalized Gromov-Hausdorff distance between functions on networks that has recently been applied to various subfields of topological data analysis and optimal transport. These functional representations of networks, or networks for short, specialize in the finite setting to (possibly asymmetric) adjacency matrices and derived representations such as distance or kernel matrices. Existing literature utilizing these constructions cannot, however, benefit from continuous formulations because the continuum limits of finite networks under this distance are not well-understood. For example, while there are currently numerous persistent homology methods on finite networks, it is unclear if these methods produce well-defined persistence diagrams in the infinite setting. We resolve this situation by introducing the collection of compact networks that arises by taking continuum limits of finite networks and developing sampling results showing that this collection admits well-defined persistence diagrams. Compared to metric spaces, the isomorphism class of the generalized Gromov-Hausdorff distance over networks is rather complex, and contains representatives having different cardinalities and different topologies. We provide an exact characterization of a suitable notion of isomorphism for compact networks as well as alternative, stronger characterizations under additional topological regularity assumptions. Toward data applications, we describe a unified framework for developing quantitatively stable network invariants, provide basic examples, and cast existing results on the stability of persistent homology methods in this extended framework. To illustrate our theoretical results, we introduce a model of directed circles with finite reversibility and characterize their Dowker persistence diagrams.

Formation of Machine Learning Features Based on the Construction of Tropical Functions

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.

Dynamic topology analysis for spatial patterns of multifocal lesions on MRI.

Quantitatively analysing the spatial patterns of multifocal lesions on clinical MRI is an important step towards a better understanding of the disease and for precision medicine, which is yet to be properly explored by feature engineering and deep learning methods. Network science addresses this issue by explicitly modeling the inter-lesion topology. However, the construction of the informative graph with optimal edge sparsity and quantification of community graph structures are the current challenges in network science. In this paper, we address these challenges with a novel Dynamic Topology Analysis framework on the basis of persistent homology, aiming to investigate the predictive values of global geometry and local clusters of multifocal lesions. Firstly, Dynamic Hierarchical Network is proposed to construct informative global and community-level topology over multi-scale networks from sparse to dense. Multi-scale global topology is constructed with a nested sequence of Rips complexes, from which a new K-simplex Filtration is designed to generate a higher-level topological abstraction for community identification based on the connectivity of k-simplices in the Rips Complex. Secondly, to quantify multi-scale community structures, we design a new Decomposed Community Persistence algorithm to track the dynamic evolution of communities, and then summarise the evolutionary communities incorporated with a customisable descriptor. The quantified community features are encapsulated with global geometric invariants for topological pattern analysis. The proposed framework was evaluated on both diagnostic differentiation and prognostic prediction for multiple sclerosis that is a typical multifocal disease, and achieved ROC_AUC 0.875 and 0.767, respectively, outperforming seven state-of-the-art persistent homology methods and the reported performance of six feature engineering and deep learning methods.

Topological Analysis of Syntactic Structures

We use the persistent homology method of topological data analysis and dimensional analysis techniques to study data of syntactic structures of world languages. We analyze relations between syntactic parameters in terms of dimensionality, of hierarchical clustering structures, and of non-trivial loops. We show there are relations that hold across language families and additional relations that are family-specific. We then analyze the trees describing the merging structure of persistent connected components for languages in different language families and we show that they partly correlate to historical phylogenetic trees but with significant differences. We also show the existence of interesting non-trivial persistent first homology groups in various language families. We give examples where explicit generators for the persistent first homology can be identified, some of which appear to correspond to homoplasy phenomena, while others may have an explanation in terms of historical linguistics, corresponding to known cases of syntactic borrowing across different language subfamilies.

Топологический анализ невыровненных аудио- и текстовых данных

We have performed preliminary work on topological analysis of audio and text data for unsupervised speech processing. The work assumes that phoneme frequencies and contextual relationships are similar in the acoustic and text domains for the same language. Accordingly, this allowed the creation of a mapping between these spaces that considers their geometric structure. As a first step, generative methods based on variational autoencoders were chosen to map audio and text data into two latent vector spaces. In the next stage, persistent homology methods are used to analyze the topological structure of two spaces. Although the results obtained support the idea of the similarity of the two spaces, further research is needed to correctly map acoustic and text spaces, as well as to evaluate the real effect of including topological information in the autoencoder training process.

Avalanche dynamics in sheared athermal particle packings occurs via localized bursts predicted by unstable linear response.

Under applied shear strain, granular and amorphous materials deform via particle rearrangements, which can be small and localized or organized into system-spanning avalanches. While the statistical properties of avalanches under quasi-static shear are well-studied, the dynamics during avalanches is not. In numerical simulations of sheared soft spheres, we find that avalanches can be decomposed into bursts of localized deformations, which we identify using an extension of persistent homology methods. We also study the linear response of unstable systems during an avalanche, demonstrating that eigenvalue dynamics are highly complex during such events, and that the most unstable eigenvector is a poor predictor of avalanche dynamics. Instead, we modify existing tools that identify localized excitations in stable systems, and apply them to these unstable systems with non-positive definite Hessians, quantifying the evolution of such excitations during avalanches. We find that bursts of localized deformations in the avalanche almost always occur at localized excitations identified using the linear spectrum. These new tools will provide an improved framework for validating and extending mesoscale elastoplastic models that are commonly used to explain avalanche statistics in glasses and granular matter.

Geodesic-based distance reveals nonlinear topological features in neural activity from mouse visual cortex.

An increasingly popular approach to the analysis of neural data is to treat activity patterns as being constrained to and sampled from a manifold, which can be characterized by its topology. The persistent homology method identifies the type and number of holes in the manifold, thereby yielding functional information about the coding and dynamic properties of the underlying neural network. In this work, we give examples of highly nonlinear manifolds in which the persistent homology algorithm fails when it uses the Euclidean distance because it does not always yield a good approximation to the true distance distribution of a point cloud sampled from a manifold. To deal with this issue, we instead estimate the geodesic distance which is a better approximation of the true distance distribution and can therefore be used to successfully identify highly nonlinear features with persistent homology. To document the utility of the method, we utilize a toy model comprised of a circular manifold, built from orthogonal sinusoidal coordinate functions and show how the choice of metric determines the performance of the persistent homology algorithm. Furthermore, we explore the robustness of the method across different manifold properties, like the number of samples, curvature and amount of added noise. We point out strategies for interpreting its results as well as some possible pitfalls of its application. Subsequently, we apply this analysis to neural data coming from the Visual Coding-Neuropixels dataset recorded at the Allen Institute in mouse visual cortex in response to stimulation with drifting gratings. We find that different manifolds with a non-trivial topology can be seen across regions and stimulus properties. Finally, we interpret how these changes in manifold topology along with stimulus parameters and cortical region inform how the brain performs visual computation.

MFCIS: an automatic leaf-based identification pipeline for plant cultivars using deep learning and persistent homology

Recognizing plant cultivars reliably and efficiently can benefit plant breeders in terms of property rights protection and innovation of germplasm resources. Although leaf image-based methods have been widely adopted in plant species identification, they seldom have been applied in cultivar identification due to the high similarity of leaves among cultivars. Here, we propose an automatic leaf image-based cultivar identification pipeline called MFCIS (Multi-feature Combined Cultivar Identification System), which combines multiple leaf morphological features collected by persistent homology and a convolutional neural network (CNN). Persistent homology, a multiscale and robust method, was employed to extract the topological signatures of leaf shape, texture, and venation details. A CNN-based algorithm, the Xception network, was fine-tuned for extracting high-level leaf image features. For fruit species, we benchmarked the MFCIS pipeline on a sweet cherry (Prunus avium L.) leaf dataset with >5000 leaf images from 88 varieties or unreleased selections and achieved a mean accuracy of 83.52%. For annual crop species, we applied the MFCIS pipeline to a soybean (Glycine max L. Merr.) leaf dataset with 5000 leaf images of 100 cultivars or elite breeding lines collected at five growth periods. The identification models for each growth period were trained independently, and their results were combined using a score-level fusion strategy. The classification accuracy after score-level fusion was 91.4%, which is much higher than the accuracy when utilizing each growth period independently or mixing all growth periods. To facilitate the adoption of the proposed pipelines, we constructed a user-friendly web service, which is freely available at http://www.mfcis.online.

Persistent homology in two-dimensional atomic networks.

The topology of two-dimensional network materials is investigated by persistent homology analysis. The constraint of two dimensions allows for a direct comparison of key persistent homology metrics (persistence diagrams, cycles, and Betti numbers) with more traditional metrics such as the ring-size distributions. Two different types of networks are employed in which the topology is manipulated systematically. In the first, comparatively rigid networks are generated for a triangle-raft model, which are representative of materials such as silica bilayers. In the second, more flexible networks are generated using a bond-switching algorithm, which are representative of materials such as graphene. Bands are identified in the persistence diagrams by reference to the length scales associated with distorted polygons. The triangle-raft models with the largest ordering allow specific bands Bn (n = 1, 2, 3, …) to be allocated to configurations of atoms separated by n bonds. The persistence diagrams for the more disordered network models also display bands albeit less pronounced. The persistent homology method thereby provides information on n-body correlations that is not accessible from structure factors or radial distribution functions. An analysis of the persistent cycles gives the primitive ring statistics, provided the level of disorder is not too large. The method also gives information on the regularity of rings that is unavailable from a ring-statistics analysis. The utility of the persistent homology method is demonstrated by its application to experimentally-obtained configurations of silica bilayers and graphene.

A persistent homology method with modified filtration to characterize the phase trajectory of a turbulent wake flow

The topological features of recurrent phase trajectories of a turbulent wake are studied using a modified persistent homology method. In the general persistent homology computation, the input data are considered as isolated points in a high-dimensional space. Networks with various spatial resolutions are constructed based on these points. When the resolution is low, many edges among neighboring points are created as they satisfy the distance threshold. However, most of these edges do not reflect new topology other than the phase trajectory itself. Therefore, our modified method discards the duplicated edges in the network. Only the phase trajectory and the essential topological connections, which have a local minimum distance in the network, are used to represent the topological structure of a phase trajectory. The homology of the recurrent loop reflects the topology complexity of a trajectory in the phase space, and the first Betti number can be used to classify the trajectories according to the number of self-crossings, which characterizes the trajectory complexity. A significant number of trajectories have only one or a few self-crossings. There are also complex trajectories that contain more than 100 self-crossings. The topological distribution classified using the first Betti number follows a power law.