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Cryo-EM reconstruction of continuous heterogeneity by Laplacian spectral volumes

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arXiv 2019年

作者： Moscovich, Amit Halevi, Amit Andén, Joakim Singer, Amit Program in Applied & Computational Mathematics Princeton University Princeton NJ United States Center for Computational Mathematics Flatiron Institute New YorkNY United States Department of Mathematics Princeton University Princeton NJ United States

Single-particle electron cryomicroscopy is an essential tool for high-resolution 3D reconstruction of proteins and other biological macromolecules. An important challenge in cryo-EM is the reconstruction of non-rigid molecules with parts that move and deform. Traditional reconstruction methods fail in these cases, resulting in smeared reconstructions of the moving parts. This poses a major obstacle for structural biologists, who need high-resolution reconstructions of entire macromolecules, moving parts included. To address this challenge, we present a new method for the reconstruction of macromolecules exhibiting continuous heterogeneity. The proposed method uses projection images from multiple viewing directions to construct a graph Laplacian through which the manifold of three-dimensional conformations is analyzed. The 3D molecular structures are then expanded in a basis of Laplacian eigenvectors, using a novel generalized tomographic reconstruction algorithm to compute the expansion coefficients. These coefficients, which we name spectral volumes, provide a high-resolution visualization of the molecular dynamics. We provide a theoretical analysis and evaluate the method empirically on several simulated data *** Codes 62P10, 65R32, 94A08 Copyright © 2019, The Authors. All rights reserved.

关键词： Macromolecules

Beyond trees: classification with sparse pairwise dependencies

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The Journal of Machine Learning Research

The Journal of Machine Learning Research 2020年第1期21卷 7728-7760页

作者： Yaniv Tenzer Amit Moscovich Mary Frances Dorn Boaz Nadler Clifford Spiegelman Department of Computer Science and Applied Mathematics Weizmann Institute of Science Rehovot Israel Program in Applied and Computational Mathematics Princeton University Princeton NJ Los Alamos National Laboratory Los Alamos NM Department of Statistics Texas A&M University College Station TX

Several classification methods assume that the underlying distributions follow tree-structured graphical models. Indeed, trees capture statistical dependencies between pairs of variables, which may be crucial to attaining low classification errors. In this setting, the optimal classifier is linear in the log-transformed univariate and bivariate densities that correspond to the tree edges. In practice, observed data may not be well approximated by trees. Yet, motivated by the importance of pairwise dependencies for accurate classification, here we propose to approximate the optimal decision boundary by a sparse linear combination of the univariate and bivariate log-transformed densities. Our proposed approach is semi-parametric in nature: we non-parametrically estimate the univariate and bivariate densities, remove pairs of variables that are nearly independent using the Hilbert-Schmidt independence criterion, and finally construct a linear SVM using the retained log-transformed densities. We demonstrate on synthetic and real data sets, that our classifier, named SLB (sparse log-bivariate density), is competitive with other popular classification methods.

关键词： binary classification graphical model Bayesian network sparsity semi-parametric

Algorithms for Solving High Dimensional PDEs: From Nonlinear Monte Carlo to Machine Learning

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arXiv 2020年

作者： Weinan, E. Han, Jiequn Jentzen, Arnulf Department of Mathematics Princeton University United States Program in Applied and Computational Mathematics Princeton University United States Faculty of Mathematics and Computer Science University of Münster Germany

In recent years, tremendous progress has been made on numerical algorithms for solving partial differential equations (PDEs) in a very high dimension, using ideas from either nonlinear (multilevel) Monte Carlo or deep learning. They are potentially free of the curse of dimensionality for many different applications and have been proven to be so in the case of some nonlinear Monte Carlo methods for nonlinear parabolic PDEs. In this paper, we review these numerical and theoretical advances. In addition to algorithms based on stochastic reformulations of the original problem, such as the multilevel Picard iteration and the Deep BSDE method, we also discuss algorithms based on the more traditional Ritz, Galerkin, and least square formulations. We hope to demonstrate to the reader that studying PDEs as well as control and variational problems in very high dimensions might very well be among the most promising new directions in mathematics and scientific computing in the near *** Codes 65C05, 65K10, 65M75, 90C06 Copyright © 2020, The Authors. All rights reserved.

关键词： Monte Carlo methods

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

Possible origin of the discrepancy in Peierls stresses of fcc metals: First-principles simulations of dislocation mobility in aluminum

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Physical Review B 2013年第6期88卷 064106-064106页

作者： Ilgyou Shin Emily A. Carter Department of Mechanical and Aerospace Engineering Program in Applied and Computational Mathematics and Andlinger Center for Energy and the Environment Princeton University Princeton New Jersey 08544-5263 USA

Dislocation motion governs the strength and ductility of metals, and the Peierls stress (σp) quantifies dislocation mobility. σp measurements carry substantial uncertainty in face-centered cubic (fcc) metals, and σp values can differ by up to two orders of magnitude. We perform first-principles simulations based on orbital-free density functional theory (OFDFT) to calculate the most accurate currently possible σp for the motion of 12〈110〉111 dislocations in fcc Al. We predict the σps of screw and edge dislocations (dissociated in their equilibrium state) to be 1.9×10−4G and 4.9×10−5G, respectively (G is the shear modulus). These values fall within the range of measurements from mechanical deformation tests (10−4–10−5G). OFDFT also finds a new metastable structure for a screw dislocation not seen in earlier simulations, in which a dislocation core on the glide plane does not dissociate into partials. The corresponding σp for this undissociated dislocation is predicted to be 1.1×10−2G, which agrees with typical Bordoni peak measurements (10−2–10−3G). The calculated σps for dissociated and undissociated screw dislocations differ by two orders of magnitude. The presence of undissociated, as well as dissociated, screw dislocations may resolve the decades-long mystery in fcc metals regarding the two orders of magnitude discrepancy in σp measurements.

关键词： ALUMINUM DUCTILITY STRENGTH of materials STRAINS & stresses (Mechanics) DISLOCATIONS in metals FACE centered cubic structure

Generalization error of GAN from the discriminator's perspective

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arXiv 2021年

作者： Yang, Hongkang Weinan, E. Program in Applied and Computational Mathematics Princeton University United States Department of Mathematics Princeton University United States Beijing Institute of Big Data Research China

The generative adversarial network (GAN) is a well-known model for learning high-dimensional distributions, but the mechanism for its generalization ability is not understood. In particular, GAN is vulnerable to the memorization phenomenon, the eventual convergence to the empirical distribution. We consider a simplified GAN model with the generator replaced by a density, and analyze how the discriminator contributes to generalization. We show that with early stopping, the generalization error measured by Wasserstein metric escapes from the curse of dimensionality, despite that in the long term, memorization is inevitable. In addition, we present a hardness of learning result for *** Codes 68T07, 62G07, 60-08 Copyright © 2021, The Authors. All rights reserved.

关键词： Probability distributions

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

Complexity measures for neural networks with general activation functions using path-based norms

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arXiv 2020年

作者： Li, Zhong Ma, Chao Wu, Lei School of Mathematical Sciences Peking University China Department of Mathematics Stanford University United States Program in Applied and Computational Mathematics Princeton University United States

A simple approach is proposed to obtain complexity controls for neural networks with general activation functions. The approach is motivated by approximating the general activation functions with one-dimensional ReLU networks, which reduces the problem to the complexity controls of ReLU networks. Specifically, we consider two-layer networks and deep residual networks, for which path-based norms are derived to control complexities. We also provide preliminary analyses of the function spaces induced by these norms and a priori estimates of the corresponding regularized estimators. Copyright © 2020, The Authors. All rights reserved.

关键词： Chemical activation

Towards optimization of a human-inspired heuristic for solving explore-exploit problems

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Towards optimization of a human-inspired heuristic for solvi...

IEEE Annual Conference on Decision and Control

作者： Paul Reverdy Robert C. Wilson Philip Holmes Naomi E. Leonard Department of Mechanical and Aerospace Engineering Princeton University Department of Psychology and Neuroscience Institute Princeton University Program in Applied and Computational Mathematics Princeton University

ISBN: (纸本)9781467320658

Motivated by models of human decision making, we consider a heuristic solution for explore-exploit problems. In a numerical example we show that, with appropriate parameter values, the algorithm performs well. However, the parameters of the algorithm trade off exploration against exploitation in a complicated way so that finding the optimal parameter values is not obvious. We show that the optimal parameter values can be analytically computed in some cases and prove that suboptimal parameter tunings can provide robustness to modeling error. The analytic results suggest a feedback control law for dynamically optimizing parameters.

关键词： optimal parameter Heuristics numerical example control law Solve modeling error Exploration