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arXiv 2024年

作者： Landa, Boris Kluger, Yuval Ma, Rong Department of Electrical Engineering Yale University United States Program in Applied Mathematics Yale University United States Department of Biostatistics Harvard University United States Interdepartmental Program in Computational Biology and Bioinformatics Yale University United States Department of Pathology Yale University School of Medicine United States

Embedding high-dimensional data into a low-dimensional space is an indispensable component of data analysis. In numerous applications, it is necessary to align and jointly embed multiple datasets from different studies or experimental conditions. Such datasets may share underlying structures of interest but exhibit individual distortions, resulting in misaligned embeddings using traditional techniques. In this work, we propose Entropic Optimal Transport (EOT) eigenmaps, a principled approach for aligning and jointly embedding a pair of datasets with theoretical guarantees. Our approach leverages the leading singular vectors of the EOT plan matrix between two datasets to extract their shared underlying structure and align the datasets accordingly in a common embedding space. We interpret our approach as an inter-data variant of the classical Laplacian eigenmaps and diffusion maps embeddings, showing that it enjoys many favorable analogous properties. We then analyze a data-generative model where two observed high-dimensional datasets share latent variables on a common low-dimensional manifold, but each dataset is subject to data-specific translation, scaling, nuisance structures, and noise. We show that in a high-dimensional asymptotic regime, the EOT plan recovers the shared manifold structure by approximating a kernel function evaluated at the locations of the latent variables. Subsequently, we provide a geometric interpretation of our embedding by relating it to the eigenfunctions of population-level operators encoding the density and geometry of the shared manifold. Finally, we showcase the performance of our approach for data integration and embedding through simulations and analyses of real-world biological data, demonstrating its advantages over alternative methods in challenging scenarios. Copyright © 2024, The Authors. All rights reserved.

关键词： Embeddings

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Scalable Cluster-Consistency Statistics for Robust Multi-Object Matching

Scalable Cluster-Consistency Statistics for Robust Multi-Obj...

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International Conference on 3D Imaging, Modeling, Processing, Visualization and Transmission (3DIMPVT)

作者： Yunpeng Shi Shaohan Li Tyler Maunu Gilad Lerman Program in Applied and Computational Mathematics Princeton University School of Mathematics University of Minnesota Brandeis University

ISBN: (纸本)9781665426893

We develop new statistics for robustly filtering corrupted keypoint matches in the structure from motion pipeline. The statistics are based on consistency constraints that arise within the clustered structure of the graph of keypoint matches. The statistics are designed to give smaller values to corrupted matches and than uncorrupted matches. These new statistics are combined with an iterative reweighting scheme to filter keypoints, which can then be fed into any standard structure from motion pipeline. This filtering method can be efficiently implemented and scaled to massive datasets as it only requires sparse matrix multiplication. We demonstrate the efficacy of this method on synthetic and real structure from motion datasets and show that it achieves state-of-the-art accuracy and speed in these tasks.

关键词： Matched filters Structure from motion Three-dimensional displays Filtering Image edge detection Pipelines FCC

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Deformations of acid-mediated invasive tumors in a model with Allee effect

arXiv

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arXiv 2024年

作者： Carter, Paul Doelman, Arjen van Heijster, Peter Levy, Daniel Maini, Philip Okey, Erin Yeung, Paige Department of Mathematics University of California Irvine United States Mathematical Institute Leiden University Leiden Netherlands Mathematical and Statistical Methods—Biometris Wageningen University & Research Wageningen Netherlands Program in Applied and Computational Mathematics Princeton University Princeton United States Mathematical Institute University of Oxford Oxford United Kingdom Division of Applied Mathematics Brown University ProvidenceRI United States Massachusetts Institute of Technology Cambridge United States

We consider a Gatenby–Gawlinski-type model of invasive tumors in the presence of an Allee effect. We describe the construction of bistable one-dimensional traveling fronts using singular perturbation techniques in different parameter regimes corresponding to tumor interfaces with, or without, acellular gap. By extending the front as a planar interface, we perform a stability analysis to long wavelength perturbations transverse to the direction of front propagation and derive a simple stability criterion for the front in two spatial dimensions. In particular we find that in general the presence of the acellular gap indicates transversal instability of the associated planar front, which can lead to complex interfacial dynamics such as the development of finger-like protrusions and/or different invasion speeds. © 2024, CC BY.

关键词： Perturbation techniques

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INTERACTION OF LIQUID CRYSTALS WITH A RIGID BODY

arXiv

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arXiv 2023年

作者： Binz, Tim Brandt, Felix Hieber, Matthias Roy, Arnab Princeton University Program in Applied & Computational Mathematics Fine Hall Washington Road PrincetonNJ08544 United States Technische Universität Darmstadt Schloßgartenstraße 7 Darmstadt64289 Germany

This article investigates the interaction of nematic liquid crystals modeled by a simplified Ericksen-Leslie model with a rigid body. It is shown that this problem is locally strongly well-posed, and that it also admits a unique, global strong solution for initial data close to constant equilibria. The proof of the global strong solution relies on a new splitting method for the director in a mean value zero and average *** Codes 35Q35, 76A15, 74F10, 76D03, 35K59 © 2023, CC BY.

关键词： Nematic liquid crystals

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A Neural Network-Based Approach to Normality Testing for Dependent Data

arXiv

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arXiv 2023年

作者： Kim, Minwoo Genton, Marc G. Huser, Raphaël Castruccio, Stefano Department of Statistics Pusan National University Korea Republic of Statistics program King Abdullah University of Science and Technology Saudi Arabia Department of Applied and Computational Mathematics and Statistics University of Notre Dame United States

There is a wide availability of methods for testing normality under the assumption of independent and identically distributed data. When data are dependent in space and/or time, however, assessing and testing the marginal behavior is considerably more challenging, as the marginal behavior is impacted by the degree of dependence. We propose a new approach to assess normality for dependent data by non-linearly incorporating existing statistics from normality tests as well as sample moments such as skewness and kurtosis through a neural network. We calibrate (deep) neural networks by simulated normal and non-normal data with a wide range of dependence structures and we determine the probability of rejecting the null hypothesis. We compare several approaches for normality tests and demonstrate the superiority of our method in terms of statistical power through an extensive simulation study. A real world application to global temperature data further demonstrates how the degree of spatio-temporal aggregation affects the marginal normality in the data. © 2023, CC BY.

关键词： Higher order statistics

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MULTI-TARGET DETECTION WITH ROTATIONS

arXiv

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arXiv 2021年

作者： Bendory, Tamir Lan, Ti-Yen Marshall, Nicholas F. Rukshin, Iris Singer, Amit School of Electrical Engineering Tel Aviv University Tel Aviv Israel Program in Applied and Computational Mathematics Princeton University Princeton NJ USA Department of Mathematics Oregon State University Corvallis OR USA Program in Applied and Computational Mathematics Princeton University Princeton NJ USA Program in Applied and Computational Mathematics and the Department of Mathematics Princeton University Princeton NJ USA

We consider the multi-target detection problem of estimating a two-dimensional target image from a large noisy measurement image that contains many randomly rotated and translated copies of the target image. Motivated by single-particle cryo-electron microscopy, we focus on the low signal-to-noise regime, where it is difficult to estimate the locations and orientations of the target images in the measurement. Our approach uses autocorrelation analysis to estimate rotationally and translationally invariant features of the target image. We demonstrate that, regardless of the level of noise, our technique can be used to recover the target image when the measurement is sufficiently large. Copyright © 2021, The Authors. All rights reserved.

关键词： Signal to noise ratio

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Unbiased Approximations for Stationary Distributions of McKean-Vlasov SDEs

arXiv

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arXiv 2024年

作者： Awadelkarim, Elsiddig 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 Department of Mathematics City University of Hong Kong Hong Kong School of Data Science The Chinese University of Hong Kong Shenzhen Shenzhen China

We consider the development of unbiased estimators, to approximate the stationary distribution of Mckean-Vlasov stochastic differential equations (MVSDEs). These are an important class of processes, which frequently appear in applications such as mathematical finance, biology and opinion dynamics. Typically the stationary distribution is unknown and indeed one cannot simulate such processes exactly. As a result one commonly requires a time-discretization scheme which results in a discretization bias and a bias from not being able to simulate the associated stationary distribution. To overcome this bias, we present a new unbiased estimator taking motivation from the literature on unbiased Monte Carlo. We prove the unbiasedness of our estimator, under assumptions. In order to prove this we require developing ergodicity results of various discrete time processes, through an appropriate discretization scheme, towards the invariant measure. Numerous numerical experiments are provided, on a range of MVSDEs, to demonstrate the effectiveness of our unbiased estimator. Such examples include the Currie-Weiss model, a 3D neuroscience model and a parameter estimation problem. © 2024, CC BY-NC-SA.

关键词： Stochastic systems

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Viscoelastic fluid flow in a slowly varying planar contraction: the role of finite extensibility on the pressure drop

arXiv

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arXiv 2024年

作者： Mahapatra, Bimalendu Ruangkriengsin, Tachin Stone, Howard A. Boyko, Evgeniy Faculty of Mechanical Engineering Technion – Israel Institute of Technology Haifa3200003 Israel Program in Applied and Computational Mathematics Princeton University PrincetonNJ08544 United States Department of Mechanical and Aerospace Engineering Princeton University PrincetonNJ08544 United States

We analyze the steady viscoelastic fluid flow in slowly varying contracting channels of arbitrary shape and present a theory based on the lubrication approximation for calculating the flow rate–pressure drop relation at low and high Deborah (De) numbers. Unlike most prior theoretical studies leveraging the Oldroyd-B model, we describe the fluid viscoelasticity using a FENE-CR model and examine how the polymer chains’ finite extensibility impacts the pressure drop. We employ the low-Deborah-number lubrication analysis to provide analytical expressions for the pressure drop up to O(De4). We further consider the ultra-dilute limit and exploit a one-way coupling between the parabolic velocity and elastic stresses to calculate the pressure drop of the FENE-CR fluid for arbitrary values of the Deborah number. Such an approach allows us to elucidate elastic stress contributions governing the pressure drop variations and the effect of finite extensibility for all De. We validate our theoretical predictions with two-dimensional numerical simulations and find excellent agreement. We show that, at low Deborah numbers, the pressure drop of the FENE-CR fluid monotonically decreases with De, similar to the previous results for the Oldroyd-B and FENE-P fluids. However, at high Deborah numbers, in contrast to a linear decrease for the Oldroyd-B fluid, the pressure drop of the FENE-CR fluid exhibits a non-monotonic variation due to finite extensibility, first decreasing and then increasing with de. Nevertheless, even at sufficiently high Deborah numbers, the pressure drop of the FENE-CR fluid in the ultra-dilute and lubrication limits is lower than the corresponding Newtonian pressure drop. © 2024, CC BY.

关键词： Non Newtonian flow

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English-Chinese Translation using Neural Machine Models

English-Chinese Translation using Neural Machine Models

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Telecommunications, Optics and Computer Science (TOCS), IEEE Conference on

作者： Qi Liu Pengfei Ou Applied and Computational Mathematical Sciences Program University of Washington Seattle the United States Department of Mathematics University of Illinois at Urbana Champaign Urbana the United States

ISBN: (纸本)9781665470544

During the last few years, neural machine translation (NMT) as a notable branch of machine translation has been increasing its popularity both in research and in practice. In particular, neural machine translation between English and Chinese, which are two of the most widely used languages all over the world, is receiving more and more attention. In this review, we discuss current mainstream models for neural machine translation, including recurrent neural network (RNN), convolution neural network (CNN), and self-attention network (SAN) or transformer. The mechanisms of these models are illustrated. Moreover, examples of studies and applications of them are analyzed. In addition, comparisons on the performance of different models are implemented.

关键词： Recurrent neural networks Training data Transformers Optics Tokenization Data models Telecommunications

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Parameter Estimation for Partially Observed McKean-Vlasov Diffusions

arXiv

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arXiv 2024年

作者： Jasra, Ajay Maama, Mohamed Tempone, Raul School of Data Science The Chinese University of Hong Kong CN Shenzhen China 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 Chair of Mathematics for Uncertainty Quantification RWTH Aachen University Aachen52062 Germany

In this article we consider likelihood-based estimation of static parameters for a class of partially observed McKean-Vlasov (POMV) diffusion process with discrete-time observations over a fixed time interval. In particular, using the framework of [5] we develop a new randomized multilevel Monte Carlo method for estimating the parameters, based upon Markovian stochastic approximation methodology. New Markov chain Monte Carlo algorithms for the POMV model are introduced facilitating the application of [5]. We prove, under assumptions, that the expectation of our estimator is biased, but with expected small and controllable bias. Our approach is implemented on several examples. Copyright © 2024, The Authors. All rights reserved.

关键词： Stochastic systems

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