1,721,043 research outputs found
An application of topological data analysis to gravitational wave detection problem
No full textIn this talk, we will discuss a topological data analysis based classification method for detecting gravitational wave signals. The method is combined with machine learning algorithm as the topological features of gravitational waves are trained for classification. We will discuss the usability of signal topology for the detection problem and related issues including stability of the method and its remedy as an important issue when applied to real applications.2
Exact multi-parameter persistent homology of time-series data: Fast and variable topological inferences
Get PDFWe propose a novel exact multi-parameter persistent homology method for analyzing time-series data utilizing the Liouville torus. In the field of topological data analysis (TDA), the conventional approach to analyzing time-series data often involves sliding window embedding. From the perspective of Takens\u27 embedding theorem, we justify the analysis of the Liouville torus in TDA and discuss the similarities and differences between the Liouville torus and sliding window embedding approaches. We develop a multi-parameter filtration method based on Fourier decomposition and provide an exact formula of persistent homology with its one-parameter reduction of the multi-parameter filtration. The conventional TDA of time-series data via sliding window is known to be computationally expensive, but the proposed method yields the exact barcode formula with the symmetry of the Liouville torus promptly, which significantly reduces computational complexity while demonstrating comparable or superior performance compared to the existing TDA methods. Furthermore, the proposed method provides a way of obtaining various topological inferences by exploring different filtration rays within the multi-parameter filtration space, utilizing the nearly real-time computational capabilities of the proposed method. The advantages of the proposed method significantly improve the efficiency and flexibility of TDA when handling extensive time-series data within machine learning workflows.37 pages, Sixth Edition. Further rationalize our method. And we have added an experiment using 1-nearest neighborhood classification and Fourier coefficient classification to compare our method. There is no mathematical chang
Weighted Isolation and Random Cut Forest Algorithms for Anomaly Detection
No full textRandom cut forest (RCF) algorithms have been developed for anomaly detection,
particularly in time series data. The RCF algorithm is an improved version of
the isolation forest (IF) algorithm. Unlike the IF algorithm, the RCF algorithm
can determine whether real-time input contains an anomaly by inserting the
input into the constructed tree network. Various RCF algorithms, including
Robust RCF (RRCF), have been developed, where the cutting procedure is
adaptively chosen probabilistically. The RRCF algorithm demonstrates better
performance than the IF algorithm, as dimension cuts are decided based on the
geometric range of the data, whereas the IF algorithm randomly chooses
dimension cuts. However, the overall data structure is not considered in both
IF and RRCF, given that split values are chosen randomly. In this paper, we
propose new IF and RCF algorithms, referred to as the weighted IF (WIF) and
weighted RCF (WRCF) algorithms, respectively. Their split values are determined
by considering the density of the given data. To introduce the WIF and WRCF, we
first present a new geometric measure, a density measure, which is crucial for
constructing the WIF and WRCF. We provide various mathematical properties of
the density measure, accompanied by theorems that support and validate our
claims through numerical examples.Comment: 45 pages, 28 figure
Numerical Approximations of Solutions for Reaction-Diffusion Equations with Machine Learning – Homogeneous Neumann Boundary Conditions and Long Time Integrations
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The Interconnectivity Vector: A Finite-Dimensional Vector Representation of Persistent Homology
No full textPersistent Homology (PH) is a useful tool to study the underlying structure of a data set. Persistence Diagrams (PDs), which are 2D multisets of points, are a concise summary of the information found by studying the PH of a data set. However, PDs are difficult to incorporate into a typical machine learning workflow. To that end, two main methods for representing PDs have been developed: kernel methods and vectorization methods. In this talk we will discuss a new finite-dimensional vector, called the interconnectivity vector, representation of a PD adapted from bag-of-words (BoW). This new representation is constructed to demonstrate the connections between the homological features of a data set. This initial definition of the interconnectivity vector proves to be unstable, but we introduce a stabilized version of the vector and prove its stability with respect to small perturbations in the inputs. We evaluate both versions of the presented vectorization on several data sets and show their high discriminative power.1
A note on high-precision approximation of asymptotically decaying solution and orthogonal decomposition
No full textIn some physical applications, the decaying rate of asymptotically decaying solution is more important than the solution magnitude itself in understanding the physical system such as the late-time behavior of decaying fields in black hole space-time. In Khanna (J Sci Comput 56(2):366-380, 2013), it was emphasized that high-precision arithmetic and high-order methods are required to capture numerically the correct decaying rate of the late-time radiative tails of black-hole system in order to prevent roundoff errors from inducing a wrong power-law decay rate in the numerical approximation. In this paper, we explain how roundoff errors induce a wrong decay mode in the numerical approximation using simple linear differential equations. Then we describe the orthogonal decomposition method as a possible technique to remove wrong decaying modes induced by roundoff errors in the numerical approximation.11Nsciescopu
Topological data analysis of vascular disease: I A Theoretical framework
No full textVascular disease is a leading cause of death world wide and therefore the treatment thereof is critical. Understanding and classifying the types and levels of stenosis can lead to more accurate and better treatment of vascular disease. In this paper, we propose a new methodology using topological data analysis, which can serve as a supplementary way of diagnosis to currently existing methods. We show that we may use persistent homology as a tool to measure stenosis levels for various types of stenotic vessels. We first propose the critical failure value, which is an application of the 1-dimensional homology to stenotic vessels as a generalization of the percent stenosis. We then propose the spherical projection method, which is meant to allow for future classification of different types and levels of stenosis. We use the 2-dimensional homology of the spherical projection and showed that it can be used as a new index of vascular characterization. The main interest of this paper is to focus on the theoretical development of the framework for the proposed method using a simple set of vascular data.11Nscopu
The exact formula of the optimal penalty parameter value of the spectral penalty method for differential equations
No full textSpectral penalty methods were originally introduced to deal with the stability of the spectral solution coupled with the boundary conditions for differential equations [Funaro 1986, Funaro and Gottlieb 1988]. Later the penalty method was used for spectral methods to be implemented in irregular domains in multiple dimensions. It has been also shown that there are close relations between discontinuous Galerkin methods and spectral penalty methods in multi-domain and element setting. In addition to stability, the penalty method provides a better accuracy because of its asymptotic behavior in the neighborhood of boundaries. The optimal value of the penalty parameter for accuracy has not been studied thoroughly for the exact form. In this short note, we consider a simple differential equation and study the optimal value of the penalty parameter by minimizing the error in maximum norm. We focus on the optimization for the case of Chebyshev spectral collocation method. We provide its exact form and verify it numerically. (C) 2020 Elsevier Inc.All rights reserved.11Nsciescopu
Iterative spectral mollification and conjugation for successive edge detection
No full textFourier concentration method is a well-established method for finding edges given a set of truncated Fourier data. However, it is hard to distinguish weak jumps from the Gibbs oscillations with the method unless the concentration is highly refined locally. In this note, we show that it is possible to find all edges whether strong or weak by using the method iteratively. The iterative method first finds strong edges followed by an iterative procedure in which previously found edges are smoothed by local adaptive mollifier. The iterative method is advantageous when the ratio of the largest to smallest jump magnitudes is significantly large.11Yscopu
Adaptive Gaussian radial basis function methods for initial value problems: Construction and comparison with adaptive multiquadric radial basis function methods
No full textAdaptive radial basis function (RBF) methods have been developed recently in Gu and Jung (2020) based on the multiquadric (MQ) RBFs for solving initial value problems (IVPs). The proposed adaptive RBF methods utilize the free parameter in order to adaptively enhance the local convergence of the numerical solution. Methods pertaining to the polynomial interpolation yield only fixed rate of convergence regardless of the solution smoothness while the proposed methods use the smoothness of the solution, given in derivatives of the solution, to control the rate of convergence. In this paper, for the completion of the development of the adaptive RBF methods, we develop various adaptive Gaussian RBF methods for solving IVPs by modifying the classical solvers such as the Euler's method, midpoint method, Adams-Bashforth method and Adams-Moulton method by replacing the polynomial basis with the Gaussian RBFs. For each development, we compare the performance with the adaptive MQ-RBF methods and explain when and why the adaptive Gaussian methods are better or not than the MQ-RBF ones. We provide the collection of modifications with the MQ and Gaussian RBFs. We also provide the stability regions for the adaptive Gaussian methods. Numerical results confirm that the adaptivity enhances accuracy and convergence and also show the differences and similarities between MQ and Gaussian RBFs in their performance - we found that the adaptive MQ-RBF method has larger stability region than the Gaussian RBF method. Both MQ and Gaussian RBF methods yield the desired order of convergence while the superiority of one method to the other depends on the method and the problem considered. (C) 2020 Elsevier B.V. All rights reserved.11Nsciescopu
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