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Regularized matrix data clustering and its application to image analysis

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arXiv 2018年

作者： Gao, Xu Shen, Weining Ombao, Hernando Department of Statistics University of California IrvineCA United States Program on Applied Mathematics & Computational Science King Abdullah University of Science and Technology Saudi Arabia

In this paper, we propose a regularized mixture probabilistic model to cluster matrix data and apply it to brain signals. The approach is able to capture the sparsity (low rank, small/zero values) of the original signals by introducing regularization terms into the likelihood function. Through a modified EM algorithm, our method achieves the optimal solution with low computational cost. Theoretical results are also provided to establish the consistency of the proposed estimators. Simulations show the advantages of the proposed method over other existing methods. We also apply the approach to two real datasets from different experiments. Promising results imply that the proposed method successfully characterizes signals with different patterns while yielding insightful scientific interpretation. Copyright © 2018, The Authors. All rights reserved.

关键词： Matrix algebra

Bayesian Parameter Inference for Partially Observed Stochastic Volterra Equations

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arXiv 2023年

作者： Jasra, Ajay Ruzayqat, Hamza Wu, Amin Applied Mathematics & Computational Science Program Computer Electrical and Mathematical Sciences and Engineering Division King Abdullah University of Science and Technology Thuwal23955 Saudi Arabia

In this article we consider Bayesian parameter inference for a type of partially observed stochastic Volterra equation (SVE). SVEs are found in many areas such as physics and mathematical finance. In the latter field they can be used to represent long memory in unobserved volatility processes. In many cases of practical interest, SVEs must be time-discretized and then parameter inference is based upon the posterior associated to this time-discretized process. Based upon recent studies on time-discretization of SVEs (e.g. [24]) we use Euler-Maruyama methods for the afore-mentioned discretization. We then show how multilevel Markov chain Monte Carlo (MCMC) methods [19] can be applied in this context. In the examples we study, we give a proof that shows that the cost to achieve a 4 mean square error (MSE) of O(ϵ2), ϵ > 0, is O(ϵ− 4/2H+1 ), where H is the Hurst parameter. If one uses a single level MCMC method then the cost is O(ϵ−2(2H+3)/ 2H+1 ) to achieve the same MSE. We illustrate these results in the context of state-space and stochastic volatility models, with the latter applied to real *** Codes 62F15, 62M20, 60J10, 60J22, 65C40 © 2023, CC BY.

关键词： Mean square error

Diffusion spreadability as a probe of the microstructure of complex media across length scales

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arXiv 2021年

作者： Torquato, Salvatore Department of Chemistry Princeton Institute for the Science and Technology of Materials Program in Applied and Computational Mathematics Princeton University PrincetonNJ08544 United States

Understanding time-dependent diffusion processes in multiphase media is of great importance in physics, chemistry, materials science, petroleum engineering and biology. Consider the time-dependent problem of mass transfer of a solute between two phases and assume that the solute is initially distributed in one phase (phase 2) and absent from the other (phase 1). We desire the fraction of total solute present in phase 1 as a function of time, S(t), which we call the spreadability, since it is a measure of the spreadability of diffusion information as a function of time. We derive exact direct-space formulas for S(t) in any Euclidean space dimension d in terms of the autocovariance function as well as corresponding Fourier representations of S(t) in terms of the spectral density, which are especially useful when scattering information is available experimentally or theoretically. These are singular results because they are rare examples of mass transport problems where exact solutions are possible. We derive closed-form general formulas for the short- and long-time behaviors of the spreadability in terms of crucial small- and large-scale microstructural information, respectively. The long-time behavior of S(t) enables one to distinguish the entire spectrum of microstructures that span from hyperuniform to nonhyperuniform media. For hyperuniform media, disordered or not, we show that the "excess" spreadability, S(∞) − S(t), decays to its long-time behavior exponentially faster than that of any nonhyperuniform two-phase medium, the "slowest" being antihyperuniform media. The stealthy hyperuniform class is characterized by an excess spreadability with the fastest decay rate among all translationally invariant microstructures. We obtain exact results for S(t) for a variety of specific ordered and disordered model microstructures across dimensions that span from hyperuniform to antihyperuniform media. Moreover, we establish a remarkable connection between the spreadability

关键词： Microstructure

DeePCG: Constructing coarse-grained models via deep neural networks

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arXiv 2018年

作者： Weinan, E. Zhang, Linfeng Han, Jiequn Wang, Han Car, Roberto Program in Applied and Computational Mathematics Princeton University PrincetonNJ08544 United States Institute of Applied Physics and Computational Mathematics Fenghao East Road 2 Beijing100094 China Caep Software Center for High Performance Numerical Simulation Huayuan Road 6 Beijing100088 China Program in Applied and Computational Mathematics Princeton Institute for Science and Technology of Materials Princeton University PrincetonNJ08544 United States Department of Mathematics and Program in Applied and Computational Mathematics Princeton University PrincetonNJ08544 United States Beijing Institute of Big Data Research Beijing100871 China

We introduce a general framework for constructing coarse-grained potential models without ad hoc approximations such as limiting the potential to two- A nd/or three-body contributions. The scheme, called Deep Coarse-Grained Potential (abbreviated DeePCG), exploits a carefully crafted neural network to construct a many-body coarse-grained potential. The network is trained with full atomistic data in a way that preserves the natural symmetries of the system. The resulting model is very accurate and can be used to sample the configurations of the coarse-grained variables in a much faster way than with the original atomistic model. As an application we consider liquid water and use the oxygen coordinates as the coarse-grained variables, starting from a full atomistic simulation of this system at the ab initio molecular dynamics level. We find that the two-body, three-body, and higher-order oxygen correlation functions produced by the coarse-grained and full atomistic models agree very well with each other, illustrating the effectiveness of the DeePCG model on a rather challenging task. Copyright © 2018, The Authors. All rights reserved.

关键词： Deep neural networks

IMPROVED EFFICIENCY OF MULTILEVEL MONTE CARLO FOR STOCHASTIC PDE THROUGH STRONG PAIRWISE COUPLING

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arXiv 2021年

作者： Chada, Neil K. Hoel, Håkon Jasra, Ajay Zouraris, Georgios E. Applied Mathematics and Computational Science Program King Abdullah University of Science and Technology Thuwal23955-6900 Saudi Arabia Mathematics for Uncertainty Quantification RWTH Aachen University Aachen Germany Department of Mathematics and Applied Mathematics Voutes Campus Crete HeraklionGR-700 13 Greece

Multilevel Monte Carlo (MLMC) has become an important methodology in applied mathematics for reducing the computational cost of weak approximations. For many problems, it is well-known that strong pairwise coupling of numerical solutions in the multilevel hierarchy is needed to obtain efficiency gains. In this work, we show that strong pairwise coupling indeed is also important when MLMC is applied to stochastic partial differential equations (SPDE) of reaction-diffusion type, as it can improve the rate of convergence and thus improve tractability. For the MLMC method with strong pairwise coupling that was developed and studied numerically on filtering problems in [Chernov et al., Numer. Math., 147 (2021), 71-125], we prove that the rate of computational efficiency is higher than for existing methods. We also provide numerical comparisons with alternative coupling ideas on linear and nonlinear SPDE to illustrate the importance of this *** Codes 65C05, 65N35, 60H35, 65C30 © 2021, CC BY.

关键词： Monte Carlo methods

Diffusion curvature for estimating local curvature in high dimensional data 22

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Diffusion curvature for estimating local curvature in high d...

Proceedings of the 36th International Conference on Neural Information Processing Systems

作者： Dhananjay Bhaskar Kincaid MacDonald Oluwadamilola Fasina Dawson Thomas Bastian Rieck Ian Adelstein Smita Krishnaswamy Department of Genetics Yale University Department of Mathematics Yale University Applied Mathematics Program Yale University Department of Mathematics Department of Physics Yale University Institute of AI for Health Helmholtz Pioneer Campus Helmholtz Munich Department of Computer Science Department of Genetics Applied Mathematics Program Program for Computational Biology and Bioinformatics Yale University

ISBN: (纸本)9781713871088

We introduce a new intrinsic measure of local curvature on point-cloud data called diffusion curvature. Our measure uses the framework of diffusion maps, including the data diffusion operator, to structure point cloud data and define local curvature based on the laziness of a random walk starting at a point or region of the data. We show that this laziness directly relates to volume comparison results from Riemannian geometry. We then extend this scalar curvature notion to an entire quadratic form using neural network estimations based on the diffusion map of point-cloud data. We show applications of both estimations on toy data, single-cell data and on estimating local Hessian matrices of neural network loss landscapes.

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Unbiased Estimation using Underdamped Langevin Dynamics

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arXiv 2022年

作者： Ruzayqat, Hamza Chada, Neil K. Jasra, Ajay Applied Mathematics and Computational Science Program Computer Electrical and Mathematical Sciences and Engineering Division King Abdullah University of Science and Technology Thuwal23955-6900 Saudi Arabia

In this work we consider the unbiased estimation of expectations w.r.t. probability measures that have non-negative Lebesgue density, and which are known point-wise up-to a normalizing constant. We focus upon developing an unbiased method via the underdamped Langevin dynamics, which has proven to be popular of late due to applications in statistics and machine learning. Specifically in continuous-time, the dynamics can be constructed so that as the time goes to infinity they admit the probability of interest as a stationary measure. In many cases, time-discretized versions of the underdamped Langevin dynamics are used in practice which are run only with a fixed number of iterations. We develop a novel scheme based upon doubly randomized estimation as in [17, 19], which requires access only to time-discretized versions of the dynamics. The proposed scheme aims to remove the dicretization bias and the bias resulting from running the dynamics for a finite number of iterations. We prove, under standard assumptions, that our estimator is of finite variance and either has finite expected cost, or has finite cost with a high probability. To illustrate our theoretical findings we provide numerical experiments which verify our theory, which include challenging examples from Bayesian statistics and statistical *** Codes 60J22, 65C05, 65C40, 82C31, 62G08, 35Q56 © 2022, CC BY.

关键词： Dynamics

Unbiased Parameter Estimation for Partially Observed Diffusions

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arXiv 2023年

作者： Awadelkarim, Elsiddig Jasra, Ajay Ruzayqat, Hamza Applied Mathematics and Computational Science Program Computer Electrical and Mathematical Sciences and Engineering Division King Abdullah University of Science and Technology Thuwal23955-6900 Saudi Arabia

In this article we consider the estimation of static parameters for partially observed diffusion process with discrete-time observations over a fixed time interval. In particular, we assume that one must time-discretize the partially observed diffusion process and work with the model with bias and consider maximizing the resulting log-likelihood. Using a novel double randomization scheme, based upon Markovian stochastic approximation we develop a new method to unbiasedly estimate the static parameters, that is, to obtain the maximum likelihood estimator with no time discretization bias. Under assumptions we prove that our estimator is unbiased and investigate the method in several numerical examples, showing that it can empirically out-perform existing unbiased *** Codes 60J22, 62M05, 65C40, 62M20 © 2023, CC BY.

关键词： Parameter estimation

Quantifying behavioral data sets of criminal activity

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Quantifying behavioral data sets of criminal activity

2010 AAAI Spring Symposium

作者： Toole, Jameson Eagle, Nathan Plotkin, Joshua Department of Physics University of Michigan Ann Arbor MI 48109 United States Santa Fe Institute 1399 Hyde Park Rd Santa Fe NM 87501 United States Department of Biology and Program in Applied Mathematics and Computational Science University of Pennsylvania Philadelphia PA 19104 United States

ISBN: (纸本)9781577354550

With the increased availability of rich behavioral data sets, we present a novel combination of tools to analyze to analyze this information. Using criminal offense records as an example, we employ cross-correlation measures, eigenvalue spectrum analysis, and results from random matrix theory to identify spatiotemporal patterns. Finally, with multivariate autoregressive models, we demonstrate a possible source of structure within the data. instructions. © 2010, Association for the Advancement of Artificial Intelligence.

关键词： Spectrum analysis