🍩 Database of Original & Non-Theoretical Uses of Topology

(found 18 matches in 0.002737s)
  1. Gene Expression Data Classification Using Topology and Machine Learning Models (2022)

    Tamal K. Dey, Sayan Mandal, Soham Mukherjee
    Abstract Interpretation of high-throughput gene expression data continues to require mathematical tools in data analysis that recognizes the shape of the data in high dimensions. Topological data analysis (TDA) has recently been successful in extracting robust features in several applications dealing with high dimensional constructs. In this work, we utilize some recent developments in TDA to curate gene expression data. Our work differs from the predecessors in two aspects: (1) Traditional TDA pipelines use topological signatures called barcodes to enhance feature vectors which are used for classification. In contrast, this work involves curating relevant features to obtain somewhat better representatives with the help of TDA. This representatives of the entire data facilitates better comprehension of the phenotype labels. (2) Most of the earlier works employ barcodes obtained using topological summaries as fingerprints for the data. Even though they are stable signatures, there exists no direct mapping between the data and said barcodes.
  2. Some Applications of TDA on Financial Markets (2022)

    Miguel Angel Ruiz-Ortiz, José Carlos Gómez-Larrañaga, Jesús Rodríguez-Viorato
    Abstract The Topological Data Analysis (TDA) has had many applications. However, financial markets has been studied slightly through TDA. Here we present a quick review of some recent applications of TDA on financial markets and propose a new turbulence index based on persistent homology -- the fundamental tool for TDA -- that seems to capture critical transitions on financial data, based on our experiment with SP500 data before 2020 stock market crash in February 20, 2020, due to the COVID-19 pandemic. We review applications in the early detection of turbulence periods in financial markets and how TDA can help to get new insights while investing and obtain superior risk-adjusted returns compared with investing strategies using classical turbulence indices as VIX and the Chow's index based on the Mahalanobis distance. Furthermore, we include an introduction to persistent homology so the reader could be able to understand this paper without knowing TDA.
  3. Persistent Homology in Cosmic Shear - II. A Tomographic Analysis of DES-Y1 (2022)

    Sven Heydenreich, Benjamin Brück, Pierre Burger, Joachim Harnois-Déraps, Sandra Unruh, Tiago Castro, Klaus Dolag, Nicolas Martinet
    Abstract We demonstrate how to use persistent homology for cosmological parameter inference in a tomographic cosmic shear survey. We obtain the first cosmological parameter constraints from persistent homology by applying our method to the first-year data of the Dark Energy Survey. To obtain these constraints, we analyse the topological structure of the matter distribution by extracting persistence diagrams from signal-to-noise maps of aperture masses. This presents a natural extension to the widely used peak count statistics. Extracting the persistence diagrams from the cosmo-SLICS, a suite of \textlessi\textgreaterN\textlessi/\textgreater-body simulations with variable cosmological parameters, we interpolate the signal using Gaussian processes and marginalise over the most relevant systematic effects, including intrinsic alignments and baryonic effects. For the structure growth parameter, we find , which is in full agreement with other late-time probes. We also constrain the intrinsic alignment parameter to \textlessi\textgreaterA\textlessi/\textgreater = 1.54 ± 0.52, which constitutes a detection of the intrinsic alignment effect at almost 3\textlessi\textgreaterσ\textlessi/\textgreater.
  4. Exploring the Geometry and Topology of Neural Network Loss Landscapes (2022)

    Stefan Horoi, Jessie Huang, Bastian Rieck, Guillaume Lajoie, Guy Wolf, Smita Krishnaswamy
    Abstract Recent work has established clear links between the generalization performance of trained neural networks and the geometry of their loss landscape near the local minima to which they converge. This suggests that qualitative and quantitative examination of the loss landscape geometry could yield insights about neural network generalization performance during training. To this end, researchers have proposed visualizing the loss landscape through the use of simple dimensionality reduction techniques. However, such visualization methods have been limited by their linear nature and only capture features in one or two dimensions, thus restricting sampling of the loss landscape to lines or planes. Here, we expand and improve upon these in three ways. First, we present a novel “jump and retrain” procedure for sampling relevant portions of the loss landscape. We show that the resulting sampled data holds more meaningful information about the network’s ability to generalize. Next, we show that non-linear dimensionality reduction of the jump and retrain trajectories via PHATE, a trajectory and manifold-preserving method, allows us to visualize differences between networks that are generalizing well vs poorly. Finally, we combine PHATE trajectories with a computational homology characterization to quantify trajectory differences.
  5. Toroidal Topology of Population Activity in Grid Cells (2022)

    Richard J. Gardner, Erik Hermansen, Marius Pachitariu, Yoram Burak, Nils A. Baas, Benjamin A. Dunn, May-Britt Moser, Edvard I. Moser
    Abstract The medial entorhinal cortex is part of a neural system for mapping the position of an individual within a physical environment1. Grid cells, a key component of this system, fire in a characteristic hexagonal pattern of locations2, and are organized in modules3 that collectively form a population code for the animal’s allocentric position1. The invariance of the correlation structure of this population code across environments4,5 and behavioural states6,7, independent of specific sensory inputs, has pointed to intrinsic, recurrently connected continuous attractor networks (CANs) as a possible substrate of the grid pattern1,8–11. However, whether grid cell networks show continuous attractor dynamics, and how they interface with inputs from the environment, has remained unclear owing to the small samples of cells obtained so far. Here, using simultaneous recordings from many hundreds of grid cells and subsequent topological data analysis, we show that the joint activity of grid cells from an individual module resides on a toroidal manifold, as expected in a two-dimensional CAN. Positions on the torus correspond to positions of the moving animal in the environment. Individual cells are preferentially active at singular positions on the torus. Their positions are maintained between environments and from wakefulness to sleep, as predicted by CAN models for grid cells but not by alternative feedforward models12. This demonstration of network dynamics on a toroidal manifold provides a population-level visualization of CAN dynamics in grid cells.
  6. Determining Clinically Relevant Features in Cytometry Data Using Persistent Homology (2022)

    Soham Mukherjee, Darren Wethington, Tamal K. Dey, Jayajit Das
    Abstract Cytometry experiments yield high-dimensional point cloud data that is difficult to interpret manually. Boolean gating techniques coupled with comparisons of relative abundances of cellular subsets is the current standard for cytometry data analysis. However, this approach is unable to capture more subtle topological features hidden in data, especially if those features are further masked by data transforms or significant batch effects or donor-to-donor variations in clinical data. We present that persistent homology, a mathematical structure that summarizes the topological features, can distinguish different sources of data, such as from groups of healthy donors or patients, effectively. Analysis of publicly available cytometry data describing non-naïve CD8+ T cells in COVID-19 patients and healthy controls shows that systematic structural differences exist between single cell protein expressions in COVID-19 patients and healthy controls. We identify proteins of interest by a decision-tree based classifier, sample points randomly and compute persistence diagrams from these sampled points. The resulting persistence diagrams identify regions in cytometry datasets of varying density and identify protruded structures such as ‘elbows’. We compute Wasserstein distances between these persistence diagrams for random pairs of healthy controls and COVID-19 patients and find that systematic structural differences exist between COVID-19 patients and healthy controls in the expression data for T-bet, Eomes, and Ki-67. Further analysis shows that expression of T-bet and Eomes are significantly downregulated in COVID-19 patient non-naïve CD8+ T cells compared to healthy controls. This counter-intuitive finding may indicate that canonical effector CD8+ T cells are less prevalent in COVID-19 patients than healthy controls. This method is applicable to any cytometry dataset for discovering novel insights through topological data analysis which may be difficult to ascertain otherwise with a standard gating strategy or existing bioinformatic tools.

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  7. Topological Data Analysis: Concepts, Computation, and Applications in Chemical Engineering (2021)

    Alexander D. Smith, Paweł Dłotko, Victor M. Zavala
    Abstract A primary hypothesis that drives scientific and engineering studies is that data has structure. The dominant paradigms for describing such structure are statistics (e.g., moments, correlation functions) and signal processing (e.g., convolutional neural nets, Fourier series). Topological Data Analysis (TDA) is a field of mathematics that analyzes data from a fundamentally different perspective. TDA represents datasets as geometric objects and provides dimensionality reduction techniques that project such objects onto low-dimensional descriptors. The key properties of these descriptors (also known as topological features) are that they provide multiscale information and that they are stable under perturbations (e.g., noise, translation, and rotation). In this work, we review the key mathematical concepts and methods of TDA and present different applications in chemical engineering.
  8. Persistent Homology in Cosmic Shear: Constraining Parameters With Topological Data Analysis (2021)

    Sven Heydenreich, Benjamin Brück, Joachim Harnois-Déraps
    Abstract In recent years, cosmic shear has emerged as a powerful tool for studying the statistical distribution of matter in our Universe. Apart from the standard two-point correlation functions, several alternative methods such as peak count statistics offer competitive results. Here we show that persistent homology, a tool from topological data analysis, can extract more cosmological information than previous methods from the same data set. For this, we use persistent Betti numbers to efficiently summarise the full topological structure of weak lensing aperture mass maps. This method can be seen as an extension of the peak count statistics, in which we additionally capture information about the environment surrounding the maxima. We first demonstrate the performance in a mock analysis of the KiDS+VIKING-450 data: We extract the Betti functions from a suite of \textlessi\textgreaterN\textlessi/\textgreater-body simulations and use these to train a Gaussian process emulator that provides rapid model predictions; we next run a Markov chain Monte Carlo analysis on independent mock data to infer the cosmological parameters and their uncertainties. When comparing our results, we recover the input cosmology and achieve a constraining power on that is 3% tighter than that on peak count statistics. Performing the same analysis on 100 deg\textlesssup\textgreater2\textlesssup/\textgreater of \textlessi\textgreaterEuclid\textlessi/\textgreater-like simulations, we are able to improve the constraints on \textlessi\textgreaterS\textlessi/\textgreater\textlesssub\textgreater8\textlesssub/\textgreater and Ω\textlesssub\textgreaterm\textlesssub/\textgreater by 19% and 12%, respectively, while breaking some of the degeneracy between \textlessi\textgreaterS\textlessi/\textgreater\textlesssub\textgreater8\textlesssub/\textgreater and the dark energy equation of state. To our knowledge, the methods presented here are the most powerful topological tools for constraining cosmological parameters with lensing data.
  9. Towards a Philological Metric Through a Topological Data Analysis Approach (2020)

    Eduardo Paluzo-Hidalgo, Rocio Gonzalez-Diaz, Miguel A. Gutiérrez-Naranjo
    Abstract The canon of the baroque Spanish literature has been thoroughly studied with philological techniques. The major representatives of the poetry of this epoch are Francisco de Quevedo and Luis de Góngora y Argote. They are commonly classified by the literary experts in two different streams: Quevedo belongs to the Conceptismo and G\ńgora to the Culteranismo. Besides, traditionally, even if Quevedo is considered the most representative of the Conceptismo, Lope de Vega is also considered to be, at least, closely related to this literary trend. In this paper, we use Topological Data Analysis techniques to provide a first approach to a metric distance between the literary style of these poets. As a consequence, we reach results that are under the literary experts' criteria, locating the literary style of Lope de Vega, closer to the one of Quevedo than to the one of G\'ǵora.
  10. 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.
  11. Quantifying Genetic Innovation: Mathematical Foundations for the Topological Study of Reticulate Evolution (2020)

    Michael Lesnick, Raúl Rabadán, Daniel I. S. Rosenbloom
    Abstract A topological approach to the study of genetic recombination, based on persistent homology, was introduced by Chan, Carlsson, and Rabadán in 2013. This associates a sequence of signatures called barcodes to genomic data sampled from an evolutionary history. In this paper, we develop theoretical foundations for this approach. First, we present a novel formulation of the underlying inference problem. Specifically, we introduce and study the novelty profile, a simple, stable statistic of an evolutionary history which not only counts recombination events but also quantifies how recombination creates genetic diversity. We propose that the (hitherto implicit) goal of the topological approach to recombination is the estimation of novelty profiles. We then study the problem of obtaining a lower bound on the novelty profile using barcodes. We focus on a low-recombination regime, where the evolutionary history can be described by a directed acyclic graph called a galled tree, which differs from a tree only by isolated topological defects. We show that in this regime, under a complete sampling assumption, the \$1\textasciicircum\mathrm\st\\$ barcode yields a lower bound on the novelty profile, and hence on the number of recombination events. For \$i\textgreater1\$, the \$i\textasciicircum\\mathrm\th\\\$ barcode is empty. In addition, we use a stability principle to strengthen these results to ones which hold for any subsample of an arbitrary evolutionary history. To establish these results, we describe the topology of the Vietoris--Rips filtrations arising from evolutionary histories indexed by galled trees. As a step towards a probabilistic theory, we also show that for a random history indexed by a fixed galled tree and satisfying biologically reasonable conditions, the intervals of the \$1\textasciicircum\\mathrm\st\\\$ barcode are independent random variables. Using simulations, we explore the sensitivity of these intervals to recombination.
  12. Revisiting Abnormalities in Brain Network Architecture Underlying Autism Using Topology-Inspired Statistical Inference (2018)

    Sourabh Palande, Vipin Jose, Brandon Zielinski, Jeffrey Anderson, P. Thomas Fletcher, Bei Wang
    Abstract A large body of evidence relates autism with abnormal structural and functional brain connectivity. Structural covariance magnetic resonance imaging (scMRI) is a technique that maps brain regions with covarying gray matter densities across subjects. It provides a way to probe the anatomical structure underlying intrinsic connectivity networks (ICNs) through analysis of gray matter signal covariance. In this article, we apply topological data analysis in conjunction with scMRI to explore network-specific differences in the gray matter structure in subjects with autism versus age-, gender-, and IQ-matched controls. Specifically, we investigate topological differences in gray matter structure captured by structural correlation graphs derived from three ICNs strongly implicated in autism, namely the salience network, default mode network, and executive control network. By combining topological data analysis with statistical inference, our results provide evidence of statistically significant network-specific structural abnormalities in autism.
  13. Persistent Homology of Time-Dependent Functional Networks Constructed From Coupled Time Series (2017)

    Bernadette J. Stolz, Heather A. Harrington, Mason A. Porter
    Abstract We use topological data analysis to study “functional networks” that we construct from time-series data from both experimental and synthetic sources. We use persistent homology with a weight rank clique filtration to gain insights into these functional networks, and we use persistence landscapes to interpret our results. Our first example uses time-series output from networks of coupled Kuramoto oscillators. Our second example consists of biological data in the form of functional magnetic resonance imaging data that were acquired from human subjects during a simple motor-learning task in which subjects were monitored for three days during a five-day period. With these examples, we demonstrate that (1) using persistent homology to study functional networks provides fascinating insights into their properties and (2) the position of the features in a filtration can sometimes play a more vital role than persistence in the interpretation of topological features, even though conventionally the latter is used to distinguish between signal and noise. We find that persistent homology can detect differences in synchronization patterns in our data sets over time, giving insight both on changes in community structure in the networks and on increased synchronization between brain regions that form loops in a functional network during motor learning. For the motor-learning data, persistence landscapes also reveal that on average the majority of changes in the network loops take place on the second of the three days of the learning process.
  14. Inference of Ancestral Recombination Graphs Through Topological Data Analysis (2016)

    Pablo G. Cámara, Arnold J. Levine, Raúl Rabadán
    Abstract The recent explosion of genomic data has underscored the need for interpretable and comprehensive analyses that can capture complex phylogenetic relationships within and across species. Recombination, reassortment and horizontal gene transfer constitute examples of pervasive biological phenomena that cannot be captured by tree-like representations. Starting from hundreds of genomes, we are interested in the reconstruction of potential evolutionary histories leading to the observed data. Ancestral recombination graphs represent potential histories that explicitly accommodate recombination and mutation events across orthologous genomes. However, they are computationally costly to reconstruct, usually being infeasible for more than few tens of genomes. Recently, Topological Data Analysis (TDA) methods have been proposed as robust and scalable methods that can capture the genetic scale and frequency of recombination. We build upon previous TDA developments for detecting and quantifying recombination, and present a novel framework that can be applied to hundreds of genomes and can be interpreted in terms of minimal histories of mutation and recombination events, quantifying the scales and identifying the genomic locations of recombinations. We implement this framework in a software package, called TARGet, and apply it to several examples, including small migration between different populations, human recombination, and horizontal evolution in finches inhabiting the Galápagos Islands., Evolution occurs through different mechanisms, including point mutations, gene duplication, horizontal gene transfer, and recombinations. Some of these mechanisms cannot be captured by tree graphs. We present a framework, based on the mathematical tools of computational topology, that can explicitly accommodate both recombination and mutation events across the evolutionary history of a sample of genomic sequences. This approach generates a new type of summary graph and algebraic structures that provide quantitative information on the evolutionary scale and frequency of recombination events. The accompanying software, TARGet, is applied to several examples, including migration between sexually-reproducing populations, human recombination, and recombination in Darwin’s finches.
  15. Multidimensional Persistence in Biomolecular Data (2015)

    Kelin Xia, Guo-Wei Wei
    Abstract Persistent homology has emerged as a popular technique for the topological simplification of big data, including biomolecular data. Multidimensional persistence bears considerable promise to bridge the gap between geometry and topology. However, its practical and robust construction has been a challenge. We introduce two families of multidimensional persistence, namely pseudo-multidimensional persistence and multiscale multidimensional persistence. The former is generated via the repeated applications of persistent homology filtration to high dimensional data, such as results from molecular dynamics or partial differential equations. The latter is constructed via isotropic and anisotropic scales that create new simiplicial complexes and associated topological spaces. The utility, robustness and efficiency of the proposed topological methods are demonstrated via protein folding, protein flexibility analysis, the topological denoising of cryo-electron microscopy data, and the scale dependence of nano particles. Topological transition between partial folded and unfolded proteins has been observed in multidimensional persistence. The separation between noise topological signatures and molecular topological fingerprints is achieved by the Laplace-Beltrami flow. The multiscale multidimensional persistent homology reveals relative local features in Betti-0 invariants and the relatively global characteristics of Betti-1 and Betti-2 invariants.
  16. Multiresolution Persistent Homology for Excessively Large Biomolecular Datasets (2015)

    Kelin Xia, Zhixiong Zhao, Guo-Wei Wei
    Abstract Although persistent homology has emerged as a promising tool for the topological simplification of complex data, it is computationally intractable for large datasets. We introduce multiresolution persistent homology to handle excessively large datasets. We match the resolution with the scale of interest so as to represent large scale datasets with appropriate resolution. We utilize flexibility-rigidity index to access the topological connectivity of the data set and define a rigidity density for the filtration analysis. By appropriately tuning the resolution of the rigidity density, we are able to focus the topological lens on the scale of interest. The proposed multiresolution topological analysis is validated by a hexagonal fractal image which has three distinct scales. We further demonstrate the proposed method for extracting topological fingerprints from DNA molecules. In particular, the topological persistence of a virus capsid with 273 780 atoms is successfully analyzed which would otherwise be inaccessible to the normal point cloud method and unreliable by using coarse-grained multiscale persistent homology. The proposed method has also been successfully applied to the protein domain classification, which is the first time that persistent homology is used for practical protein domain analysis, to our knowledge. The proposed multiresolution topological method has potential applications in arbitrary data sets, such as social networks, biological networks, and graphs.
  17. 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.