🍩 Database of Original & Non-Theoretical Uses of Topology

(found 8 matches in 0.002491s)

  1. A Simplified Algorithm for Identifying Abnormal Changes in Dynamic Networks (2022)

    Bouchaib Azamir, Driss Bennis, Bertrand Michel
    Abstract Topological data analysis has recently been applied to the study of dynamic networks. In this context, an algorithm was introduced and helps, among other things, to detect early warning signals of abnormal changes in the dynamic network under study. However, the complexity of this algorithm increases significantly once the database studied grows. In this paper, we propose a simplification of the algorithm without affecting its performance. We give various applications and simulations of the new algorithm on some weighted networks. The obtained results show clearly the efficiency of the introduced approach. Moreover, in some cases, the proposed algorithm makes it possible to highlight local information and sometimes early warning signals of local abnormal changes.

  2. Topological Detection of Trojaned Neural Networks (2021)

    Songzhu Zheng, Yikai Zhang, Hubert Wagner, Mayank Goswami, Chao Chen
    Abstract Deep neural networks are known to have security issues. One particular threat is the Trojan attack. It occurs when the attackers stealthily manipulate the model’s behavior through Trojaned training samples, which can later be exploited. Guided by basic neuroscientific principles, we discover subtle – yet critical – structural deviation characterizing Trojaned models. In our analysis we use topological tools. They allow us to model high-order dependencies in the networks, robustly compare different networks, and localize structural abnormalities. One interesting observation is that Trojaned models develop short-cuts from shallow to deep layers. Inspired by these observations, we devise a strategy for robust detection of Trojaned models. Compared to standard baselines it displays better performance on multiple benchmarks.

  3. A Probabilistic Topological Approach to Feature Identification Using a Stochastic Robotic Swarm (2018)

    Ragesh K. Ramachandran, Sean Wilson, Spring Berman
    Abstract This paper presents a novel automated approach to quantifying the topological features of an unknown environment using a swarm of robots with local sensing and limited or no access to global position information. The robots randomly explore the environment and record a time series of their estimated position and the covariance matrix associated with this estimate. After the robots’ deployment, a point cloud indicating the free space of the environment is extracted from their aggregated data. Tools from topological data analysis, in particular the concept of persistent homology, are applied to a subset of the point cloud to construct barcode diagrams, which are used to determine the numbers of different types of features in the domain. We demonstrate that our approach can correctly identify the number of topological features in simulations with zero to four features and in multi-robot experiments with one to three features.

  4. The Extended Persistent Homology Transform of Manifolds With Boundary (2022)

    Katharine Turner, Vanessa Robins, James Morgan
    Abstract The Extended Persistent Homology Transform (XPHT) is a topological transform which takes as input a shape embedded in Euclidean space, and to each unit vector assigns the extended persistence module of the height function over that shape with respect to that direction. We can define a distance between two shapes by integrating over the sphere the distance between their respective extended persistence modules. By using extended persistence we get finite distances between shapes even when they have different Betti numbers. We use Morse theory to show that the extended persistence of a height function over a manifold with boundary can be deduced from the extended persistence for that height function restricted to the boundary, alongside labels on the critical points as positive or negative critical. We study the application of the XPHT to binary images; outlining an algorithm for efficient calculation of the XPHT exploiting relationships between the PHT of the boundary curves to the extended persistence of the foreground.

  5. A Novel Multi-Task Machine Learning Classifier for Rare Disease Patterning Using Cardiac Strain Imaging Data (2024)

    Nanda K. Siva, Yashbir Singh, Quincy A. Hathaway, Partho P. Sengupta, Naveena Yanamala
    Abstract To provide accurate predictions, current machine learning-based solutions require large, manually labeled training datasets. We implement persistent homology (PH), a topological tool for studying the pattern of data, to analyze echocardiography-based strain data and differentiate between rare diseases like constrictive pericarditis (CP) and restrictive cardiomyopathy (RCM). Patient population (retrospectively registered) included those presenting with heart failure due to CP (n = 51), RCM (n = 47), and patients without heart failure symptoms (n = 53). Longitudinal, radial, and circumferential strains/strain rates for left ventricular segments were processed into topological feature vectors using Machine learning PH workflow. In differentiating CP and RCM, the PH workflow model had a ROC AUC of 0.94 (Sensitivity = 92%, Specificity = 81%), compared with the GLS model AUC of 0.69 (Sensitivity = 65%, Specificity = 66%). In differentiating between all three conditions, the PH workflow model had an AUC of 0.83 (Sensitivity = 68%, Specificity = 84%), compared with the GLS model AUC of 0.68 (Sensitivity = 52% and Specificity = 76%). By employing persistent homology to differentiate the “pattern” of cardiac deformations, our machine-learning approach provides reasonable accuracy when evaluating small datasets and aids in understanding and visualizing patterns of cardiac imaging data in clinically challenging disease states.

  6. Topological Data Analysis in Medical Imaging: Current State of the Art (2023)

    Yashbir Singh, Colleen M. Farrelly, Quincy A. Hathaway, Tim Leiner, Jaidip Jagtap, Gunnar E. Carlsson, Bradley J. Erickson
    Abstract Machine learning, and especially deep learning, is rapidly gaining acceptance and clinical usage in a wide range of image analysis applications and is regarded as providing high performance in detecting anatomical structures and identification and classification of patterns of disease in medical images. However, there are many roadblocks to the widespread implementation of machine learning in clinical image analysis, including differences in data capture leading to different measurements, high dimensionality of imaging and other medical data, and the black-box nature of machine learning, with a lack of insight into relevant features. Techniques such as radiomics have been used in traditional machine learning approaches to model the mathematical relationships between adjacent pixels in an image and provide an explainable framework for clinicians and researchers. Newer paradigms, such as topological data analysis (TDA), have recently been adopted to design and develop innovative image analysis schemes that go beyond the abilities of pixel-to-pixel comparisons. TDA can automatically construct filtrations of topological shapes of image texture through a technique known as persistent homology (PH); these features can then be fed into machine learning models that provide explainable outputs and can distinguish different image classes in a computationally more efficient way, when compared to other currently used methods. The aim of this review is to introduce PH and its variants and to review TDA’s recent successes in medical imaging studies.

  7. A Topological Approach to Selecting Models of Biological Experiments (2019)

    M. Ulmer, Lori Ziegelmeier, Chad M. Topaz
    Abstract We use topological data analysis as a tool to analyze the fit of mathematical models to experimental data. This study is built on data obtained from motion tracking groups of aphids in [Nilsen et al., PLOS One, 2013] and two random walk models that were proposed to describe the data. One model incorporates social interactions between the insects via a functional dependence on an aphid’s distance to its nearest neighbor. The second model is a control model that ignores this dependence. We compare data from each model to data from experiment by performing statistical tests based on three different sets of measures. First, we use time series of order parameters commonly used in collective motion studies. These order parameters measure the overall polarization and angular momentum of the group, and do not rely on a priori knowledge of the models that produced the data. Second, we use order parameter time series that do rely on a priori knowledge, namely average distance to nearest neighbor and percentage of aphids moving. Third, we use computational persistent homology to calculate topological signatures of the data. Analysis of the a priori order parameters indicates that the interactive model better describes the experimental data than the control model does. The topological approach performs as well as these a priori order parameters and better than the other order parameters, suggesting the utility of the topological approach in the absence of specific knowledge of mechanisms underlying the data.

  8. Algebraic Topology-Based Machine Learning Using MRI Predicts Outcomes in Primary Sclerosing Cholangitis (2022)

    Yashbir Singh, William A. Jons, John E. Eaton, Mette Vesterhus, Tom Karlsen, Ida Bjoerk, Andreas Abildgaard, Kristin Kaasen Jorgensen, Folseraas Trine, Derek Little, Aliya F. Gulamhusein, Kosta Petrovic, Anne Negard, Gian Marco Conte, Joseph D. Sobek, Jaidip Jagtap, Sudhakar K. Venkatesh, Gregory J. Gores, Nicholas F. LaRusso, Konstantinos N. Lazaridis, Bradley J. Erickson
    Abstract Background: Primary sclerosing cholangitis (PSC) is a chronic cholestatic liver disease that can lead to cirrhosis and hepatic decompensation. However, predicting future outcomes in patients with PSC is challenging. Our aim was to extract magnetic resonance imaging (MRI) features that predict the development of hepatic decompensation by applying algebraic topology-based machine learning (ML). Methods: We conducted a retrospective multicenter study among adults with large duct PSC who underwent MRI. A topological data analysis-inspired nonlinear framework was used to predict the risk of hepatic decompensation, which was motivated by algebraic topology theory-based ML. The topological representations (persistence images) were employed as input for classifcation to predict who developed early hepatic decompensation within one year after their baseline MRI. Results: We reviewed 590 patients; 298 were excluded due to poor image quality or inadequate liver coverage, leaving 292 potentially eligible subjects, of which 169 subjects were included in the study. We trained our model using contrast-enhanced delayed phase T1-weighted images on a single center derivation cohort consisting of 54 patients (hepatic decompensation, n = 21; no hepatic decompensation, n = 33) and a multicenter independent validation cohort of 115 individuals (hepatic decompensation, n = 31; no hepatic decompensation, n = 84). When our model was applied in the independent validation cohort, it remained predictive of early hepatic decompensation (area under the receiver operating characteristic curve = 0.84). Conclusions: Algebraic topology-based ML is a methodological approach that can predict outcomes in patients with PSC and has the potential for application in other chronic liver diseases