🍩 Database of Original & Non-Theoretical Uses of Topology

(found 25 matches in 0.003878s)

  2. Topological Singularity Detection at Multiple Scales (2023)

    Julius Von Rohrscheidt, Bastian Rieck
    Abstract The manifold hypothesis, which assumes that data lies on or close to an unknown manifold of low intrinsic dimension, is a staple of modern machine learning research. However, recent work has shown that real-world data exhibits distinct non-manifold structures, i.e. singularities, that can lead to erroneous findings. Detecting such singularities is therefore crucial as a precursor to interpolation and inference tasks. We address this issue by developing a topological framework that (i) quantifies the local intrinsic dimension, and (ii) yields a Euclidicity score for assessing the ’manifoldness’ of a point along multiple scales. Our approach identifies singularities of complex spaces, while also capturing singular structures and local geometric complexity in image data.

  3. Relational Persistent Homology for Multispecies Data With Application to the Tumor Microenvironment (2023)

    Bernadette J. Stolz, Jagdeep Dhesi, Joshua A. Bull, Heather A. Harrington, Helen M. Byrne, Iris H. R. Yoon
    Abstract Topological data analysis (TDA) is an active field of mathematics for quantifying shape in complex data. Standard methods in TDA such as persistent homology (PH) are typically focused on the analysis of data consisting of a single entity (e.g., cells or molecular species). However, state-of-the-art data collection techniques now generate exquisitely detailed multispecies data, prompting a need for methods that can examine and quantify the relations among them. Such heterogeneous data types arise in many contexts, ranging from biomedical imaging, geospatial analysis, to species ecology. Here, we propose two methods for encoding spatial relations among different data types that are based on Dowker complexes and Witness complexes. We apply the methods to synthetic multispecies data of a tumor microenvironment and analyze topological features that capture relations between different cell types, e.g., blood vessels, macrophages, tumor cells, and necrotic cells. We demonstrate that relational topological features can extract biological insight, including the dominant immune cell phenotype (an important predictor of patient prognosis) and the parameter regimes of a data-generating model. The methods provide a quantitative perspective on the relational analysis of multispecies spatial data, overcome the limits of traditional PH, and are readily computable.

  4. Time-Inhomogeneous Diffusion Geometry and Topology (2022)

    Guillaume Huguet, Alexander Tong, Bastian Rieck, Jessie Huang, Manik Kuchroo, Matthew Hirn, Guy Wolf, Smita Krishnaswamy
    Abstract Diffusion condensation is a dynamic process that yields a sequence of multiscale data representations that aim to encode meaningful abstractions. It has proven effective for manifold learning, denoising, clustering, and visualization of high-dimensional data. Diffusion condensation is constructed as a time-inhomogeneous process where each step first computes and then applies a diffusion operator to the data. We theoretically analyze the convergence and evolution of this process from geometric, spectral, and topological perspectives. From a geometric perspective, we obtain convergence bounds based on the smallest transition probability and the radius of the data, whereas from a spectral perspective, our bounds are based on the eigenspectrum of the diffusion kernel. Our spectral results are of particular interest since most of the literature on data diffusion is focused on homogeneous processes. From a topological perspective, we show diffusion condensation generalizes centroid-based hierarchical clustering. We use this perspective to obtain a bound based on the number of data points, independent of their location. To understand the evolution of the data geometry beyond convergence, we use topological data analysis. We show that the condensation process itself defines an intrinsic diffusion homology. We use this intrinsic topology as well as an ambient topology to study how the data changes over diffusion time. We demonstrate both homologies in well-understood toy examples. Our work gives theoretical insights into the convergence of diffusion condensation, and shows that it provides a link between topological and geometric data analysis.

  6. Persistent Homology Based Graph Convolution Network for Fine-Grained 3D Shape Segmentation (2021)

    Chi-Chong Wong, Chi-Man Vong
    Abstract Fine-grained 3D segmentation is an important task in 3D object understanding, especially in applications such as intelligent manufacturing or parts analysis for 3D objects. However, many challenges involved in such problem are yet to be solved, such as i) interpreting the complex structures located in different regions for 3D objects; ii) capturing fine-grained structures with sufficient topology correctness. Current deep learning and graph machine learning methods fail to tackle such challenges and thus provide inferior performance in fine-grained 3D analysis. In this work, methods in topological data analysis are incorporated with geometric deep learning model for the task of fine-grained segmentation for 3D objects. We propose a novel neural network model called Persistent Homology based Graph Convolution Network (PHGCN), which i) integrates persistent homology into graph convolution network to capture multi-scale structural information that can accurately represent complex structures for 3D objects; ii) applies a novel Persistence Diagram Loss (ℒPD) that provides sufficient topology correctness for segmentation over the fine-grained structures. Extensive experiments on fine-grained 3D segmentation validate the effectiveness of the proposed PHGCN model and show significant improvements over current state-of-the-art methods.

  7. Data-Driven and Automatic Surface Texture Analysis Using Persistent Homology (2021)

    Melih C. Yesilli, Firas A. Khasawneh
    Abstract Surface roughness plays an important role in analyzing engineering surfaces. It quantifies the surface topography and can be used to determine whether the resulting surface finish is acceptable or not. Nevertheless, while several existing tools and standards are available for computing surface roughness, these methods rely heavily on user input thus slowing down the analysis and increasing manufacturing costs. Therefore, fast and automatic determination of the roughness level is essential to avoid costs resulting from surfaces with unacceptable finish, and user-intensive analysis. In this study, we propose a Topological Data Analysis (TDA) based approach to classify the roughness level of synthetic surfaces using both their areal images and profiles. We utilize persistent homology from TDA to generate persistence diagrams that encapsulate information on the shape of the surface. We then obtain feature matrices for each surface or profile using Carlsson coordinates, persistence images, and template functions. We compare our results to two widely used methods in the literature: Fast Fourier Transform (FFT) and Gaussian filtering. The results show that our approach yields mean accuracies as high as 97%. We also show that, in contrast to existing surface analysis tools, our TDA-based approach is fully automatable and provides adaptive feature extraction.

  8. Interpretable Phase Detection and Classification With Persistent Homology (2020)

    Alex Cole, Gregory J. Loges, Gary Shiu
    Abstract We apply persistent homology to the task of discovering and characterizing phase transitions, using lattice spin models from statistical physics for working examples. Persistence images provide a useful representation of the homological data for conducting statistical tasks. To identify the phase transitions, a simple logistic regression on these images is sufficient for the models we consider, and interpretable order parameters are then read from the weights of the regression. Magnetization, frustration and vortex-antivortex structure are identified as relevant features for characterizing phase transitions.

  9. Contagion Dynamics for Manifold Learning (2020)

    Barbara I. Mahler
    Abstract Contagion maps exploit activation times in threshold contagions to assign vectors in high-dimensional Euclidean space to the nodes of a network. A point cloud that is the image of a contagion map reflects both the structure underlying the network and the spreading behaviour of the contagion on it. Intuitively, such a point cloud exhibits features of the network's underlying structure if the contagion spreads along that structure, an observation which suggests contagion maps as a viable manifold-learning technique. We test contagion maps as a manifold-learning tool on a number of different real-world and synthetic data sets, and we compare their performance to that of Isomap, one of the most well-known manifold-learning algorithms. We find that, under certain conditions, contagion maps are able to reliably detect underlying manifold structure in noisy data, while Isomap fails due to noise-induced error. This consolidates contagion maps as a technique for manifold learning.

  10. Finding Universal Structures in Quantum Many-Body Dynamics via Persistent Homology (2020)

    Daniel Spitz, Jürgen Berges, Markus K. Oberthaler, Anna Wienhard
    Abstract Inspired by topological data analysis techniques, we introduce persistent homology observables and apply them in a geometric analysis of the dynamics of quantum field theories. As a prototype application, we consider simulated data of a two-dimensional Bose gas far from equilibrium. We discover a continuous spectrum of dynamical scaling exponents, which provides a refined classification of nonequilibrium universal phenomena. A possible explanation of the underlying processes is provided in terms of mixing wave turbulence and vortex kinetics components in point clouds. We find that the persistent homology scaling exponents are inherently linked to the geometry of the system, as the derivation of a packing relation reveals. The approach opens new ways of analyzing quantum many-body dynamics in terms of robust topological structures beyond standard field theoretic techniques.

  11. Geometric Anomaly Detection in Data (2020)

    Bernadette J. Stolz, Jared Tanner, Heather A. Harrington, Vidit Nanda
    Abstract The quest for low-dimensional models which approximate high-dimensional data is pervasive across the physical, natural, and social sciences. The dominant paradigm underlying most standard modeling techniques assumes that the data are concentrated near a single unknown manifold of relatively small intrinsic dimension. Here, we present a systematic framework for detecting interfaces and related anomalies in data which may fail to satisfy the manifold hypothesis. By computing the local topology of small regions around each data point, we are able to partition a given dataset into disjoint classes, each of which can be individually approximated by a single manifold. Since these manifolds may have different intrinsic dimensions, local topology discovers singular regions in data even when none of the points have been sampled precisely from the singularities. We showcase this method by identifying the intersection of two surfaces in the 24-dimensional space of cyclo-octane conformations and by locating all of the self-intersections of a Henneberg minimal surface immersed in 3-dimensional space. Due to the local nature of the topological computations, the algorithmic burden of performing such data stratification is readily distributable across several processors.

  12. The Accumulated Persistence Function, a New Useful Functional Summary Statistic for Topological Data Analysis, With a View to Brain Artery Trees and Spatial Point Process Applications (2019)

    C.A.N. Biscio, J. Møller
    Abstract We start with a simple introduction to topological data analysis where the most popular tool is called a persistence diagram. Briefly, a persistence diagram is a multiset of points in the plane describing the persistence of topological features of a compact set when a scale parameter varies. Since statistical methods are difficult to apply directly on persistence diagrams, various alternative functional summary statistics have been suggested, but either they do not contain the full information of the persistence diagram or they are two-dimensional functions. We suggest a new functional summary statistic that is one-dimensional and hence easier to handle, and which under mild conditions contains the full information of the persistence diagram. Its usefulness is illustrated in statistical settings concerned with point clouds and brain artery trees. The supplementary materials include additional methods and examples, technical details, and the R code used for all examples. © 2019, © 2019 American Statistical Association, Institute of Mathematical Statistics, and Interface Foundation of North America.

  14. Topological Data Analysis as a Morphometric Method: Using Persistent Homology to Demarcate a Leaf Morphospace (2018)

    Mao Li, Hong An, Ruthie Angelovici, Clement Bagaza, Albert Batushansky, Lynn Clark, Viktoriya Coneva, Michael J. Donoghue, Erika Edwards, Diego Fajardo, Hui Fang, Margaret H. Frank, Timothy Gallaher, Sarah Gebken, Theresa Hill, Shelley Jansky, Baljinder Kaur, Phillip C. Klahs, Laura L. Klein, Vasu Kuraparthy, Jason Londo, Zoë Migicovsky, Allison Miller, Rebekah Mohn, Sean Myles, Wagner C. Otoni, J. C. Pires, Edmond Rieffer, Sam Schmerler, Elizabeth Spriggs, Christopher N. Topp, Allen Van Deynze, Kuang Zhang, Linglong Zhu, Braden M. Zink, Daniel H. Chitwood
    Abstract Current morphometric methods that comprehensively measure shape cannot compare the disparate leaf shapes found in seed plants and are sensitive to processing artifacts. We explore the use of persistent homology, a topological method applied as a filtration across simplicial complexes (or more simply, a method to measure topological features of spaces across different spatial resolutions), to overcome these limitations. The described method isolates subsets of shape features and measures the spatial relationship of neighboring pixel densities in a shape. We apply the method to the analysis of 182,707 leaves, both published and unpublished, representing 141 plant families collected from 75 sites throughout the world. By measuring leaves from throughout the seed plants using persistent homology, a defined morphospace comparing all leaves is demarcated. Clear differences in shape between major phylogenetic groups are detected and estimates of leaf shape diversity within plant families are made. The approach predicts plant family above chance. The application of a persistent homology method, using topological features, to measure leaf shape allows for a unified morphometric framework to measure plant form, including shapes, textures, patterns, and branching architectures.

  16. Shape Terra: Mechanical Feature Recognition Based on a Persistent Heat Signature (2017)

    Ramy Harik, Yang Shi, Stephen Baek
    Abstract This paper presents a novel approach to recognizing mechanical features through a multiscale persistent heat signature similarity identification technique. First, heat signature is computed using a modified Laplacian in the application of the heat kernel. Regularly, matrices tend to include an indicator to the manifold curvature (the cotangent in our case), but we add a mesh uniformity factor to overcome mesh proportionality and skewness. Second, once heat retention values are computed, we apply persistent homology to extract significant subsets of the global mesh at different time intervals. Subsets are computed based on similarity of heat retention levels and/or retention values. Third, we present a multiscale persistence identification approach where we scan the part at different persistence levels to detect the presence of a feature. Once features are recognized and their geometrical descriptors identified, the next stage in future work will be feature matching.

  17. Raw Material Flow Optimization as a Capacitated Vehicle Routing Problem: A Visual Benchmarking Approach for Sustainable Manufacturing (2017)

    Michele Dassisti, Yasamin Eslami, Matin Mohaghegh
    Abstract Optimisation problem concerning material flows, to increase the efficiency while reducing relative resource consumption is one of the most pressing problems today. The focus point of this study is to propose a new visual benchmarking approach to select the best material-flow path from the depot to the production lines, referring to the well-known Capacitated Vehicle Routing Problem (CVRP). An example industrial case study is considered to this aim. Two different solution techniques were adopted (namely Mixed Integer Linear Programming and the Ant Colony Optimization) in searching optimal solutions to the CVRP. The visual benchmarking proposed, based on the persistent homology approach, allowed to support the comparison of the optimal solutions based on the entropy of the output in different scenarios. Finally, based on the non-standard measurements of Crossing Length Percentage (CLP), the visual benchmarking procedure makes it possible to find the most practical and applicable solution to CVRP by considering the visual attractiveness and the quality of the routes.

  18. Topological Data Analysis of Financial Time Series: Landscapes of Crashes (2017)

    Marian Gidea, Yuri Katz
    Abstract We explore the evolution of daily returns of four major US stock market indices during the technology crash of 2000, and the financial crisis of 2007-2009. Our methodology is based on topological data analysis (TDA). We use persistence homology to detect and quantify topological patterns that appear in multidimensional time series. Using a sliding window, we extract time-dependent point cloud data sets, to which we associate a topological space. We detect transient loops that appear in this space, and we measure their persistence. This is encoded in real-valued functions referred to as a 'persistence landscapes'. We quantify the temporal changes in persistence landscapes via their \$L\textasciicircump\$-norms. We test this procedure on multidimensional time series generated by various non-linear and non-equilibrium models. We find that, in the vicinity of financial meltdowns, the \$L\textasciicircump\$-norms exhibit strong growth prior to the primary peak, which ascends during a crash. Remarkably, the average spectral density at low frequencies of the time series of \$L\textasciicircump\$-norms of the persistence landscapes demonstrates a strong rising trend for 250 trading days prior to either dotcom crash on 03/10/2000, or to the Lehman bankruptcy on 09/15/2008. Our study suggests that TDA provides a new type of econometric analysis, which goes beyond the standard statistical measures. The method can be used to detect early warning signals of imminent market crashes. We believe that this approach can be used beyond the analysis of financial time series presented here.

  19. Analysis of Kolmogorov Flow and Rayleigh–Bénard Convection Using Persistent Homology (2016)

    Miroslav Kramár, Rachel Levanger, Jeffrey Tithof, Balachandra Suri, Mu Xu, Mark Paul, Michael F. Schatz, Konstantin Mischaikow
    Abstract We use persistent homology to build a quantitative understanding of large complex systems that are driven far-from-equilibrium. In particular, we analyze image time series of flow field patterns from numerical simulations of two important problems in fluid dynamics: Kolmogorov flow and Rayleigh–Bénard convection. For each image we compute a persistence diagram to yield a reduced description of the flow field; by applying different metrics to the space of persistence diagrams, we relate characteristic features in persistence diagrams to the geometry of the corresponding flow patterns. We also examine the dynamics of the flow patterns by a second application of persistent homology to the time series of persistence diagrams. We demonstrate that persistent homology provides an effective method both for quotienting out symmetries in families of solutions and for identifying multiscale recurrent dynamics. Our approach is quite general and it is anticipated to be applicable to a broad range of open problems exhibiting complex spatio-temporal behavior.

  20. Using Persistent Homology and Dynamical Distances to Analyze Protein Binding (2016)

    Violeta Kovacev-Nikolic, Peter Bubenik, Dragan Nikolić, Giseon Heo
    Abstract Persistent homology captures the evolution of topological features of a model as a parameter changes. The most commonly used summary statistics of persistent homology are the barcode and the persistence diagram. Another summary statistic, the persistence landscape, was recently introduced by Bubenik. It is a functional summary, so it is easy to calculate sample means and variances, and it is straightforward to construct various test statistics. Implementing a permutation test we detect conformational changes between closed and open forms of the maltose-binding protein, a large biomolecule consisting of 370 amino acid residues. Furthermore, persistence landscapes can be applied to machine learning methods. A hyperplane from a support vector machine shows the clear separation between the closed and open proteins conformations. Moreover, because our approach captures dynamical properties of the protein our results may help in identifying residues susceptible to ligand binding; we show that the majority of active site residues and allosteric pathway residues are located in the vicinity of the most persistent loop in the corresponding filtered Vietoris-Rips complex. This finding was not observed in the classical anisotropic network model.

  21. Statistical Topological Data Analysis - A Kernel Perspective (2015)

    Roland Kwitt, Stefan Huber, Marc Niethammer, Weili Lin, Ulrich Bauer
    Abstract We consider the problem of statistical computations with persistence diagrams, a summary representation of topological features in data. These diagrams encode persistent homology, a widely used invariant in topological data analysis. While several avenues towards a statistical treatment of the diagrams have been explored recently, we follow an alternative route that is motivated by the success of methods based on the embedding of probability measures into reproducing kernel Hilbert spaces. In fact, a positive definite kernel on persistence diagrams has recently been proposed, connecting persistent homology to popular kernel-based learning techniques such as support vector machines. However, important properties of that kernel enabling a principled use in the context of probability measure embeddings remain to be explored. Our contribution is to close this gap by proving universality of a variant of the original kernel, and to demonstrate its effective use in two-sample hypothesis testing on synthetic as well as real-world data.

  22. CD8 T-Cell Reactivity to Islet Antigens Is Unique to Type 1 While CD4 T-Cell Reactivity Exists in Both Type 1 and Type 2 Diabetes (2014)

    Ghanashyam Sarikonda, Jeremy Pettus, Sonal Phatak, Sowbarnika Sachithanantham, Jacqueline F. Miller, Johnna D. Wesley, Eithon Cadag, Ji Chae, Lakshmi Ganesan, Ronna Mallios, Steve Edelman, Bjoern Peters, Matthias von Herrath
    Abstract Previous cross-sectional analyses demonstrated that CD8+ and CD4+ T-cell reactivity to islet-specific antigens was more prevalent in T1D subjects than in healthy donors (HD). Here, we examined T1D-associated epitope-specific CD4+ T-cell cytokine production and autoreactive CD8+ T-cell frequency on a monthly basis for one year in 10 HD, 33 subjects with T1D, and 15 subjects with T2D. Autoreactive CD4+ T-cells from both T1D and T2D subjects produced more IFN-γ when stimulated than cells from HD. In contrast, higher frequencies of islet antigen-specific CD8+ T-cells were detected only in T1D. These observations support the hypothesis that general beta-cell stress drives autoreactive CD4+ T-cell activity while islet over-expression of MHC class I commonly seen in T1D mediates amplification of CD8+ T-cells and more rapid beta-cell loss. In conclusion, CD4+ T-cell autoreactivity appears to be present in both T1D and T2D while autoreactive CD8+ T-cells are unique to T1D. Thus, autoreactive CD8+ cells may serve as a more T1D-specific biomarker.

  23. A Topology-Based Object Representation for Clasping, Latching and Hooking (2013)

    J. A. Stork, F. T. Pokorny, D. Kragic
    Abstract We present a loop-based topological object representation for objects with holes. The representation is used to model object parts suitable for grasping, e.g. handles, and it incorporates local volume information about these. Furthermore, we present a grasp synthesis framework that utilizes this representation for synthesizing caging grasps that are robust under measurement noise. The approach is complementary to a local contact-based force-closure analysis as it depends on global topological features of the object. We perform an extensive evaluation with four robotic hands on synthetic data. Additionally, we provide real world experiments using a Kinect sensor on two robotic platforms: a Schunk dexterous hand attached to a Kuka robot arm as well as a Nao humanoid robot. In the case of the Nao platform, we provide initial experiments showing that our approach can be used to plan whole arm hooking as well as caging grasps involving only one hand.

  24. Alpha, Betti and the Megaparsec Universe: On the Topology of the Cosmic Web (2011)

    Rien Van De Weygaert, Gert Vegter, Herbert Edelsbrunner, Bernard J. T. Jones, Pratyush Pranav, Changbom Park, Wojciech A. Hellwing, Bob Eldering, Nico Kruithof, E. G. P. Bos, Johan Hidding, Job Feldbrugge, Eline Ten Have, Matti Van Engelen, Manuel Caroli, Monique Teillaud
    Abstract We study the topology of the Megaparsec Cosmic Web in terms of the scale-dependent Betti numbers, which formalize the topological information content of...

  25. A Barcode Shape Descriptor for Curve Point Cloud Data (2004)

    Anne Collins, Afra Zomorodian, Gunnar Carlsson, Leonidas J. Guibas
    Abstract In this paper, we present a complete computational pipeline for extracting a compact shape descriptor for curve point cloud data (PCD). Our shape descriptor, called a barcode, is based on a blend of techniques from differential geometry and algebraic topology. We also provide a metric over the space of barcodes, enabling fast comparison of PCDs for shape recognition and clustering. To demonstrate the feasibility of our approach, we implement our pipeline and provide experimental evidence in shape classification and parametrization.