Topological Data Analysis with Bregman Divergences

<p>Over the last few years, the amount of large and complex data in the public domain has increased enormously and new challenges arose in the representation, analysis and visualization of such data. Considering the number of space missions that provided and will provide remote sensing data, there is still the need of a system that can be dispatched in several remote repositories and being accessible from a single client of commodity hardware.</p><p>To tackle this challenge, at the DLR Institute for Software Technology we have defined a dual backend frontend system, enabling the interactive analysis and visualization of large-scale remote sensing data. The basis for all visualization and interaction approaches is CosmoScout VR, a visualization tool developed internally at DLR, and publicly available on Github, that allows the visualization of complex planetary data and large simulation data in real-time. The dual component of this system is based on an MPI framework, called Viracocha, that enables the analysis of large data remotely, and allows the efficient network usage about sending compact and partial results for interactive visualization in CosmoScout as soon as they are computed.</p><p>A node-based interface is defined within the visualization tool, and this lets a domain expert to easily define customized pipelines for processing and visualizing the remote data. Each “node” of this interface is either linked with a feature extraction module, defined in Viracocha, or to a rendering module defined directly in CosmoScout. Being this interface completely customizable by a user, multiple pipelines can be defined over the same dataset to enhance even more the visualization feedback for analysis purposes.</p><p>Being an ongoing project, on top of these tools, as a novel strategy in EO data processing and visualization, we plan to define and implement strategies based on Topological Data Analysis (TDA). TDA is an emerging set of technique for processing the data considering its topological features. These include both the geometric information associated to a point, as well all the non-geometric scalar values, like temperature and pressure, to name a few, that can be captured during a monitoring mission. One of the major theories behind TDA is Discrete Morse Theory, that, given a scalar value, is used to define a gradient on such function, extract the critical points, identify the region-of-influence of each critical point, and so on. This strategy is parameter free and enables a domain scientist to process large datasets without a prior knowledge of it.</p><p>An interesting research question, that it will be investigated during this project is the correlation of changes of critical points at different time steps, and the identification of deformation (or changes) across time in the original dataset.</p>

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.

In this paper we develop an alternative topological data analysis (TDA) approach for studying graph representations of time series of dynamical systems. Specifically, we show how persistent homology, a tool from TDA, can be used to yield a compressed, multi-scale representation of the graph that can distinguish between dynamic states such as periodic and chaotic behavior. We show the approach for two graph constructions obtained from the time series. In the first approach the time series is embedded into a point cloud which is then used to construct an undirected k-nearest-neighbor graph. The second construct relies on the recently developed ordinal partition framework. In either case, a pairwise distance matrix is then calculated using the shortest path between the graph's nodes, and this matrix is utilized to define a filtration of a simplicial complex that enables tracking the changes in homology classes over the course of the filtration. These changes are summarized in a persistence diagram's two-dimensional summary of changes in the topological features. We then extract existing as well as new geometric and entropy point summaries from the persistence diagram and compare to other commonly used network characteristics. Our results show that persistence-based point summaries yield a clearer distinction of the dynamic behavior and are more robust to noise than existing graph-based scores, especially when combined with ordinal graphs.

Detecting structural changes in time-series data is crucial in many applications. However, the changes in the data may appear as global structural changes that cannot be detected by conventional methods. In recent years, Topological Data Analysis (TDA) has been used to detect such global structural changes. In TDA, information on connected components or holes of data is encoded into a two-dimensional plot called a persistence diagram (PD), which can be used to detect global changes in time-series. However, only a few studies on TDA conducted change detection assuming probabilistic structure on PDs. In this paper, we introduce probability structures into PD, with which we conduct change detection on the basis of the minimum description length principle. We propose the following two methods: (1) A parametric method: We employ the Gaussian mixture model for PD modeling and then detect global changes by tracking the changes in the optimal number of mixture components. (2) A non-parametric method: We employ kernel densities for PD modeling and then detect changes by tracking changes in their global complexity. These methods not only improve the detection accuracy of global structural changes but also provide the explainability of global changes. We showcase the effectiveness of the proposed methods using synthetic data and real-world financial time-series data.

Although there is no doubt that multi-parameter persistent homology is a useful tool to analyze multi-variate data, efficient ways to compute these modules are still lacking in the available topological data analysis toolboxes. Other issues, such as interpretation and visualization of the output, remain difficult to solve. Software visualizing multi-parameter persistence diagrams is currently only available for 2-dimensional persistence modules. One of the simplest invariants for a multi-parameter persistence module is its rank invariant, defined as the function that counts the number of linearly independent homology classes that live in the filtration through a given pair of values of the multi-parameter. We propose a step towards interpretation and visualization of the rank invariant for persistence modules for any given number of parameters. We show how discrete Morse theory may be used to compute the rank invariant, proving that it is completely determined by its values at points whose coordinates are critical with respect to a discrete Morse gradient vector field. These critical points partition the set of all lines of positive slope in the parameter space into equivalence classes such that the rank invariant along lines in the same class are also equivalent. We show that we can deduce the persistence diagrams for all lines in a given class from the persistence diagram of a representative line in that class.

Topological data analysis is an approach to study the shape of a data set by means of topology. Its main object of study is the persistence diagram, which represents the topological features of the data set at different spatial resolutions. Multiple data sets can be compared by the similarity of their diagrams to understand their behaviors relative to each other. The bottleneck and Wasserstein distances are often used as a tool to indicate the similarity. In this paper, we introduce the cosine similarity as a new indicator for the similarity between persistence diagrams and investigate its properties. Furthermore, it leads to the new notion of orthogonality between persistence diagrams. It turns out that the orthogonality refers to perfect dissimilarity between persistence diagrams under the cosine similarity. Through data demonstration, the cosine similarity is shown to be more accurate than the standard distances to measure the similarity between persistence diagrams.

Diagnosing neuropsychiatric systemic lupus erythematosus (NPSLE) and differentiating it from systemic lupus erythematosus (SLE) without neuropsychiatric manifestations remains a substantial clinical challenge due to the absence of specific biomarkers. Topological data analysis (TDA) is a novel computational technique that enables the visualization, exploration, and analysis of complex data structures. This study aimed to identify distinct neuroimaging biomarkers in patients with NPSLE (NPSLE group) and differentiate them from patients with SLE without neuropsychiatric symptoms (non-NPSLE group) by employing TDA. We conducted a retrospective cohort study involving 30 patients with NPSLE and 30 without neuropsychiatric symptoms between 2005 and 2020. TDA was utilized to extract topological features, specifically connected components and holes, from fluid-attenuated inversion recovery (FLAIR) sequences obtained via brain magnetic resonance imaging (MRI). Summary statistics, including critical point count, persistence lifetime, centroid coordinates, perimeter, area, and filamentarity, were derived from persistence diagrams. Multiple logistic regression analyses, adjusted for age, cerebrovascular comorbidities, and 50% hemolytic unit of complement levels, demonstrated a significant association between NPSLE and the perimeter of the holes (odds ratio [OR]: 1.67, 95% confidence interval [CI]: 1.07-2.63, p = 0.025) and the area of the holes (OR: 4.42, 95% CI: 1.35-19.6, p = 0.026) of the identified topological features. Additionally, both areas under the receiver operating characteristic curve (AUC) exceeded 0.8, indicating good diagnostic accuracy. This study identified novel neuroimaging biomarkers for the diagnosis of NPSLE. The application of TDA to brain MRI features in patients with SLE proved to be a valuable diagnostic tool, particularly through the analysis of persistence diagrams.

Continuation of point clouds via persistence diagrams

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.

Metal–organic frameworks (MOFs), self-assembled porous materials synthesized from metal ions and organic ligands, are promising candidates for the direct capture of CO2 from the atmosphere. In this work, we developed a regression model to predict the optimal component of the MOF that governs the amount of CO2 adsorption per volume based on experimentally observed adsorption and structure data combined with MOF adsorption sites. The structural descriptors were generated by topological data analysis with persistence diagrams, an advanced mathematical method for quantifying the rings and cavities within the MOF. This enables us to analyze direct effects and significance of the geometric structure of the MOF on the efficiency of CO2 adsorption in a novel way. The proposed approach is proved to be highly correlated with experimental data and thus offers an effective screening tool for MOFs with optimized structures.

The Vietoris–Rips filtration is a versatile tool in topological data analysis. It is a sequence of simplicial complexes built on a metric space to add topological structure to an otherwise disconnected set of points. It is widely used because it encodes useful information about the topology of the underlying metric space. This information is of-ten extracted from its so-called persistence diagram. Unfortunately, this filtration is often too large to construct in full. We show how to construct an O(n)-size filtered simplicial complex on an n-point metric space such that its persistence diagram is a good approximation to that of the Vietoris–Rips filtration. This new filtration can be constructed in O(n log n) time. The constant factors in both the size and the run-ning time depend only on the doubling dimension of the metric space and the desired tightness of the approximation. For the first time, this makes it computationally tractable to approximate the persistence di-agram of the Vietoris–Rips filtration across all scales for large data sets. We describe two different sparse filtrations. The first is a zigzag filtration that removes points as the scale increases. The second is a (non-zigzag) filtration that yields the same persistence diagram. Both methods are based on a hierarchical net-tree and yield the same guar-antees.

This application paper presents a comprehensive experimental eval-uation of the suitability of Topological Data Analysis (TDA) for the quantitative comparison of turbulent flows. Specifically, our study documents the usage of the persistence diagram of the max-ima of flow enstrophy (an established vorticity indicator), for the topological representation of 180 ensemble members, generated by a coarse sampling of the parameter space of five numerical solvers. We document five main hypotheses reported by domain experts, describing their expectations regarding the variability of the flows generated by the distinct solver configurations. We contribute three evaluation protocols to assess the validation of the above hypothe-ses by two comparison measures: (i) a standard distance used in scientific imaging (the <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$L_{2}$</tex> norm) and (ii) an established topological distance between persistence diagrams (the <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$L_{2}$</tex> -Wasserstein metric). Extensive experiments on the input ensemble demonstrate the supe-riority of the topological distance (ii) to report as close to each other flows which are expected to be similar by domain experts, due to the configuration of their vortices. Overall, the insights reported by our study bring an experimental evidence of the suitability of TDA for representing and comparing turbulent flows, thereby providing to the fluid dynamics community confidence for its usage in future work. Also, our flow data and evaluation protocols provide to the TDA community an application-approved benchmark for the evaluation and design of further topological distances.

We propose to use Topological Data Analysis (TDA) to characterize the morphological development of brain organoids. We combine TDA with clustering strategies to characterize the morphology of three developmental stages of segmented brain organoid images. We calculate a linear regression of the H1 feature diagrams as well as entropy, dispersion, and average persistence of H0 and H1 features separately for each developmental stage. We also compute pairwise average Wasserstein distances between features, and within and inter clusters for each developmental stage. To explore all feature vectors from each group, we calculate K-L divergence on the t-SNE reduction of the TDA results.Early stages are characterized by a persistence diagram with a low slope and intercept, a high entropy within features, and a high Wasserstein distance between clusters near and far from the origin. The opposite is true for later stages. K-L divergences are particularly high between mid-stages and early or late stages, and punctually between some clusters. Results highlight specific morphological patterns at 14 days, corresponding to neuroepithelial formations.KeywordsBrain organoidCharacterisationTDA-clustering

Topological data analysis (TDA) provides a set of data analysis tools for extracting embedded topological structures from complex high-dimensional datasets. In recent years, TDA has been a rapidly growing field which has found success in a wide range of applications, including signal processing, neuroscience and network analysis. In these applications, the online detection of changes is of crucial importance, but this can be highly challenging since such changes often occur in low-dimensional embeddings within high-dimensional data streams. We thus propose a new method, called PERsistence diagram-based ChangE-PoinT detection (PERCEPT), which leverages the learned topological structure from TDA to sequentially detect changes. PERCEPT follows two key steps: it first learns the embedded topology as a point cloud via persistence diagrams, then applies a nonparametric monitoring approach for detecting changes in the resulting point cloud distributions. This yields a nonparametric, topology-aware framework which can efficiently detect online geometric changes. We investigate the effectiveness of PERCEPT over existing methods in a suite of numerical experiments where the data streams have an embedded topological structure. We then demonstrate the usefulness of PERCEPT in two applications on solar flare monitoring and human gesture detection.

Topological data analysis (TDA) is a mathematically well-founded set of methods to derive robust information about the structure and topology of data. It has been applied successfully in several biological contexts. Derived primarily from algebraic topology, TDA rigorously identifies persistent features in complex data, making it well-suited to better understand the key features of three-dimensional chromosome structure. Chromosome structure has a significant influence in many diverse genomic processes and has recently been shown to relate to cellular differentiation. While there exist many methods to study specific substructures of chromosomes, we are still missing a global view of all geometric features of chromosomes. By applying TDA to the study of chromosome structure through differentiation across three cell lines, we provide insight into principles of chromosome folding and looping. We identify persistent connected components and one-dimensional topological features of chromosomes and characterize them across cell types and stages of differentiation. Availability: Scripts to reproduce the results from this study can be found at https://github.com/Kingsford-Group/hictda