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High-dimensional data fusion via joint manifold learning

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High-dimensional data fusion via joint manifold learning

2010 AAAI Fall Symposium

作者： Davenport, Mark A. Hegde, Chinmay Duarte, Marco F. Baraniuk, Richard G. Department of Statistics Stanford University Stanford CA 94305 United States Department of Electrical and Computer Engineering Rice University Houston TX 77005 United States Program in Applied and Computational Mathematics Princeton University Princeton NJ 08544 United States

ISBN: (纸本)9781577354888

The emergence of low-cost sensing architectures for diverse modalities has made it possible to deploy sensor networks that acquire large amounts of very high-dimensional data. To cope with such a data deluge, manifold models are often developed that provide a powerful theoretical and algorithmic framework for capturing the intrinsic structure of data governed by a low-dimensional set of ***, these models do not typically take into account dependencies among multiple sensors. We thus propose a new joint manifold framework for data ensembles that exploits such dependencies. We show that joint manifold structure can lead to improved performance for manifold learning. Additionally, we leverage recent results concerning random projections of manifolds to formulate a universal, network-scalable dimensionality reduction scheme that efficiently fuses the data from all sensors. Copyright © 2010, Association for the Advancement of Artificial Intelligence. All rights reserved.

关键词： Sensor networks

On the Hamiltonian Hopf bifurcations in the 3D Hénon-Heiles family

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Journal of Dynamics and Differential Equations 2002年第3期14卷 675-695页

作者： Hanßmann, Heinz Van Der Meer, Jan-Cees Program for Applied and Computational Mathematics Princeton University Princeton NJ United States Institut für Reine und Angewandte Mathematik RWTH Aachen Aachen Germany Faculteit Wiskunde en Informatica Technische Universiteit Eindhoven 5600 MB Eindhoven PB513 Netherlands

An axially symmetric perturbed isotropic harmonic oscillator undergoes several bifurcations as the parameter λ adjusting the relative strength of the two terms in the cubic potential is varied. We show that three of these bifurcations are Hamiltonian Hopf bifurcations. To this end we analyse an appropriately chosen normal form. It turns out that the linear behaviour is not that of a typical Hamiltonian Hopf bifurcation as the eigen-values completely vanish at the bifurcation. However, the nonlinear structure is that of a Hamiltonian Hopf bifurcation. The result is obtained by formulating geometric criteria involving the normalized Hamiltonian and the reduced phase space. © 2002 Plenum Publishing Corporation.

关键词： Bifurcation Hamiltonian Hopf bifurcation Hamiltonian system Hénon-Heiles family Normal form Reduction Relative equilibria Transversality conditions

Scalable Cluster-Consistency Statistics for Robust Multi-Object Matching

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arXiv 2022年

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

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. Copyright © 2022, The Authors. All rights reserved.

关键词： Statistics

Deterministic guarantees for Burer–Monteiro factorizations of smooth semidefinite programs

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arXiv 2018年

作者： Boumal, Nicolas Voroninski, Vladislav Bandeira, Afonso S. Mathematics Department Program in Applied and Computational Mathematics Princeton University Department of Mathematics Center for Data Science Courant Institute of Mathematical Sciences New York University

We consider semidefinite programs (SDPs) with equality constraints. The variable to be optimized is a positive semidefinite matrix X of size n. Following the Burer–Monteiro approach, we optimize a factor Y of size n× p instead, such that X = YY. This ensures positive semidefiniteness at no cost and can reduce the dimension of the problem if p is small, but results in a non-convex optimization problem with a quadratic cost function and quadratic equality constraints in Y. In this paper, we show that if the set of constraints on Y regularly defines a smooth manifold, then, despite non-convexity, first- and second-order necessary optimality conditions are also sufficient, provided p is large enough. For smaller values of p, we show a similar result holds for almost all (linear) cost functions. Under those conditions, a global optimum Y maps to a global optimum X = YY of the SDP. We deduce old and new consequences for SDP relaxations of the generalized eigenvector problem, the trust-region subproblem and quadratic optimization over several spheres, as well as for the Max-Cut and Orthogonal-Cut SDPs which are common relaxations in stochastic block modeling and synchronization of rotations. Copyright © 2018, The Authors. All rights reserved.

关键词： Stochastic systems

Adaptive coupling of a deep neural network potential to a classical force field

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arXiv 2018年

作者： Zhang, Linfeng Wang, Han Weinan, E. Program in Applied and Computational Mathematics Princeton University PrincetonNJ08544 United States Laboratory of Computational Physics Institute of Applied Physics and Computational Mathematics Huayuan Road 6 Beijing100088 China Department of Mathematics and Program in Applied and Computational Mathematics Princeton University PrincetonNJ08544 United States Center for Data Science and Beijing International Center for Mathematical Research Peking University China Beijing Institute of Big Data Research Beijing100871 China

An adaptive modeling method (AMM) that couples a deep neural network potential and a classical force field is introduced to address the accuracy-efficiency dilemma faced by the molecular simulation community. The AMM simulated system is decomposed into three types of regions. The first type captures the important phenomena in the system and requires high accuracy, for which we use the Deep Potential Molecular Dynamics (DeePMD) model in this work. The DeePMD model is trained to accurately reproduce the statistical properties of the ab initio molecular dynamics. The second type does not require high accuracy and a classical force field is used to describe it in an efficient way. The third type is used for a smooth transition between the first and the second types of regions. By using a force interpolation scheme and imposing a thermodynamics force in the transition region, we make the DeePMD region embedded in the AMM simulated system as if it were embedded in a system that is fully described by the accurate potential. A representative example of the liquid water system is used to show the feasibility and promise of this method. Copyright © 2018, The Authors. All rights reserved.

关键词： Deep neural networks

Interatomic potentials for hydrogen in α–iron based on density functional theory

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Physical Review B 2009年第17期79卷 174101-174101页

作者： Ashwin Ramasubramaniam Mitsuhiro Itakura Emily A. Carter Program in Applied and Computational Mathematics Princeton University Princeton NJ 08544 United States Center for Computational Science and E-systems Japan Atomic Energy Agency Taito-ku Tokyo 110-0015 6-9-3 Higashiueno Japan Department of Mechanical and Aerospace Engineering Program in Applied and Computational Mathematics Princeton University Princeton NJ 08544-5263 United States

We present two interatomic potentials for hydrogen in α–iron based on the embedded atom method potentials for iron developed by Mendelev et al. [Philos. Mag. 83, 3977 (2003)] and Ackland et al. [J. Phys.: Condens. Matter 16, S2629 (2004)]. Since these latter potentials are unique among existing iron potentials in their ability to produce the same core structure for screw dislocations as density functional theory (DFT) calculations, our interatomic potentials for hydrogen in iron also inherit this important feature. We use an extensive database of energies and atomic configurations from DFT calculations to fit the cross interaction of hydrogen with iron. Detailed tests on the dissolution and diffusion of hydrogen in bulk α–iron, as well as the binding of H to vacancies, free surfaces, and dislocations, indicate that our potentials are in excellent overall agreement with DFT calculations.

关键词： density functional theory diffusion dissolving hydrogen iron potential energy functions screw dislocations vacancies (crystal)

Chaotic behavior of interacting elliptical instability modes

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The Physics of Fluids

The Physics of Fluids 1988年第1期31卷 6-8页

作者： Carl S. Hellberg Steven A. Orszag Department of Physics Princeton University Princeton New Jersey 08544 Program in Applied and Computational Mathematics Princeton University Princeton New Jersey 08544

A three‐mode projection of the Navier–Stokes equations for nonlinear perturbations to an elliptical vortex is studied numerically. It is found that, as the Reynolds number increases, the perturbations undergo a sequence of period doublings leading to chaos according to the Feigenbaum scenario [J. Statis. Phys. 19, 25 (1978); Phys. Lett. 74 A, 375 (1979)].

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Domain Formation in the Plasma Membrane: Roles of Nonequilibrium Lipid Transport and Membrane Proteins

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Physical Review Letters 2008年第17期100卷 178102-178102页

作者： Jun Fan Maria Sammalkorpi Mikko Haataja Department of Mechanical and Aerospace Engineering Princeton Institute for the Science and Technology of Materials (PRISM) and Program in Applied and Computational Mathematics (PACM) Princeton University Princeton New Jersey 08544 USA

Compositional lipid domains (“lipid rafts”) in plasma membranes are believed to be important components of many cellular processes. The mechanisms by which cells regulate the sizes and lifetimes of these spatially extended domains are poorly understood at the moment. Here we show that the competition between phase separation in an immiscible lipid system and active cellular lipid transport processes naturally leads to the formation of such domains. Furthermore, we demonstrate that local interactions with immobile membrane proteins can spatially localize the rafts and lead to further clustering.

关键词： Lipids

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