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Smooth graph signal interpolation for big data

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arXiv 2018年

作者： Heimowitz, Ayelet Eldar, Yonina C. Program in Applied and Computational Mathematics Princeton University PrincetonNJ United States Faculty of Math and Computer Science Weizmann Institute of Science Rehovot Israel

In this paper we present the Markov variation, a smoothness measure which offers a probabilistic interpretation of graph signal smoothness. This measure is then used to develop an optimization framework for graph signal interpolation. Our approach is based on diffusion embedding vectors and the connection between diffusion maps and signal processing on graphs. As diffusion embedding vectors may be expensive to compute for large graphs, we present a computationally efficient method, based on the Nyström extension, for interpolation of signals over a graph. We demonstrate our approach on the MNIST dataset and a dataset of daily average temperatures around the US. We show that our method outperforms state of the art graph signal interpolation methods on both datasets, and that our computationally efficient reconstruction achieves slightly reduced accuracy with a large speedup. Copyright © 2018, The Authors. All rights reserved.

关键词： computational efficiency

An efficient bidiagonalization algorithm for combined CPU-accelerator environments

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An efficient bidiagonalization algorithm for combined CPU-ac...

IASTED International Conference on Parallel and Distributed Computing and Networks, PDCN 2009

作者： Yamamoto, Yusaku Fukaya, Takeshi Uneyama, Takashi Nakamura, Yoshimasa Dept. of Computational Science and Engineering Graduate School of Engineering Nagoya University Nagoya 464-8603 Japan Division of Multidisciplinary Chemistry Institute for Chemical Research Kyoto University Kyoto 611-0011 Japan Dept. of Applied Mathematics and Physics Graduate School of Informatics Kyoto University/JST SORST Kyoto 606-8501 Japan

ISBN: (纸本)9780889867840

In computing the singular values of a square matrix, transformation of the input matrix to bidiagonal form accounts for most of the computation time. In this paper, we consider speeding up this process using a combination of CPU and floating-point accelerator. As an algorithm for bidiagonalization, we can use the conventional Householder's method or Bischof's two-phase algorithm, which can use the level-3 BLAS efficiently. We can also choose to store the whole matrix in the CPU memory or in the on-board memory of the accelerator. So there are four possible strategies. We investigate the advantages and disadvantages of each strategy and construct an analytical performance model for each of them. Using the models, we predict the performance of bidiagonalzation on the CSX600 accelerator and show that it is the best to achieve high performance to use Bischof's algorithm with the matrix stored in the on-board memory. This conclusion should hold for many other accelerators with similar performance characteristics.

关键词： Linear transformations

T-Cal: an optimal test for the calibration of predictive models

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The Journal of Machine Learning Research

The Journal of Machine Learning Research 2023年第1期24卷 15959-16030页

作者： Donghwan Lee Xinmeng Huang Hamed Hassani Edgar Dobriban Graduate Group in Applied Mathematics and Computational Science University of Pennsylvania Philadelphia PA Department of Electrical and Systems Engineering University of Pennsylvania Philadelphia PA Department of Statistics and Data Science University of Pennsylvania Philadelphia PA

The prediction accuracy of machine learning methods is steadily increasing, but the calibration of their uncertainty predictions poses a significant challenge. Numerous works focus on obtaining well-calibrated predictive models, but less is known about reliably assessing model calibration. This limits our ability to know when algorithms for improving calibration have a real effect, and when their improvements are merely artifacts due to random noise in finite datasets. In this work, we consider detecting mis-calibration of predictive models using a finite validation dataset as a hypothesis testing problem. The null hypothesis is that the predictive model is calibrated, while the alternative hypothesis is that the deviation from calibration is sufficiently *** find that detecting mis-calibration is only possible when the conditional probabilities of the classes are sufficiently smooth functions of the predictions. When the conditional class probabilities are Hölder continuous, we propose T-Cal, a minimax optimal test for calibration based on a debiased plug-in estimator of the ℓ2-Expected Calibration Error (ECE). We further propose adaptive T-Cal, a version that is adaptive to unknown smoothness. We verify our theoretical findings with a broad range of experiments, including with several popular deep neural net architectures and several standard post-hoc calibration methods. T-Cal is a practical general-purpose tool, which--combined with classical tests for discrete-valued predictors--can be used to test the calibration of virtually any probabilistic classification method. T-Cal is available at https://***/dh7401/T-Cal.

关键词： uncertainty quantification calibration nonparametric statistics hypothesis testing minimax optimality

Understanding the Influence of Digraphs on Decentralized Optimization: Effective Metrics, Lower Bound, and Optimal Algorithm

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arXiv 2023年

作者： Liang, Liyuan Huang, Xinmeng Xin, Ran Yuan, Kun School of Mathematics Science Peking University China Graduate Group in Applied Mathematics and Computational Science University of Pennsylvania United States ByteDance Applied Machine Learning China Center for Machine Learning Research Peking University China

This paper investigates the influence of directed networks on decentralized stochastic non-convex optimization associated with column-stochastic mixing matrices. Surprisingly, we find that the canonical spectral gap, a widely used metric in undirected networks, is insufficient to characterize the impact of directed topology on decentralized algorithms. To overcome this limitation, we introduce a novel metric termed equilibrium skewness. This metric, together with the spectral gap, accurately and comprehensively captures the influence of column-stochastic mixing matrices on decentralized stochastic algorithms. With these two metrics, we clarify, for the first time, how the directed network topology influences the performance of prevalent algorithms such as Push-Sum and Push-DIGing. Furthermore, we establish the first lower bound of the convergence rate for decentralized stochastic non-convex algorithms over directed networks. Since existing algorithms cannot match our lower bound, we further propose the MG-Push-DIGing algorithm, which integrates Push-DIGing with a multi-round gossip technique. MG-Push-DIGing attains our lower bound up to logarithmic factors, demonstrating its near-optimal performance and the tightness of the lower bound. Numerical experiments verify our theoretical results. Copyright © 2023, The Authors. All rights reserved.

关键词： Topology

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

MONGE-AMPÈRE FLOW FOR GENERATIVE MODELING

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arXiv 2018年

作者： Zhang, Linfeng Weinan, E. Wang, Lei Program in Applied and Computational Mathematics Princeton University United States Department of Mathematics and Program in Applied and Computational Mathematics Princeton University United States Center for Data Science Peking University Beijing Institute of Big Data Research Beijing100871 China Institute of Physics Chinese Academy of Sciences Beijing100190 China

We present a deep generative model, named Monge-Ampère flow, which builds on continuous-time gradient flow arising from the Monge-Ampère equation in optimal transport theory. The generative map from the latent space to the data space follows a dynamical system, where a learnable potential function guides a compressible fluid to flow towards the target density distribution. Training of the model amounts to solving an optimal control problem. The Monge-Ampère flow has tractable likelihoods and supports efficient sampling and inference. One can easily impose symmetry constraints in the generative model by designing suitable scalar potential functions. We apply the approach to unsupervised density estimation of the MNIST dataset and variational calculation of the two-dimensional Ising model at the critical point. This approach brings insights and techniques from Monge-Ampère equation, optimal transport, and fluid dynamics into reversible flow-based generative models. Copyright © 2018, The Authors. All rights reserved.

关键词： Statistical mechanics

HARADA’S CONJECTURE II FOR THE FINITE GENERAL LINEAR GROUPS AND UNITARY GROUPS

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arXiv 2023年

作者： Masahiro, Sugimoto Doctoral Program in Mathematics Degree Programs in Pure and Applied Sciences Graduate School of Science and Technology University of Tsukuba Japan

K. Harada conjectured for any finite group G, the product of sizes of all conjugacy classes is divisible by the product of degrees of all irreducible characters. We study this conjecture when G is the general linear group over a finite field. We show the conjecture holds if the order of the field is sufficiently *** Codes 20C15 Copyright © 2023, The Authors. All rights reserved.

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Enhanced energy coupling for indirect-drive fast-ignition fusion targets (vol 17, pg 435, 2020)

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NATURE PHYSICS 2020年第7期16卷 815-815页

作者： Zhang, F. Cai, H. B. Zhou, W. M. Dai, Z. S. Shan, L. Q. Xu, H. Chen, J. B. Ge, F. J. Tang, Q. Zhang, W. S. Wei, L. Liu, D. X. Gu, J. F. Du, H. B. Bi, B. Wu, S. Z. Li, J. Lu, F. Zhang, H. Zhang, B. He, M. Q. Yu, M. H. Yang, Z. H. Wang, W. W. Zhang, H. S. Cui, B. Yang, L. Wu, J. F. Qi, W. Cao, L. H. Li, Z. Liu, H. J. Yang, Y. M. Ren, G. L. Tian, C. Yuan, Z. Q. Zheng, W. D. Cao, L. F. Zhou, C. T. Zou, S. Y. Gu, Y. Q. Du, K. Ding, Y. K. Zhang, B. H. Zhu, S. P. Zhang, W. Y. He, X. T. Science and Technology on Plasma Physics Laboratory Laser Fusion Research Center CAEP Mianyang China (GRID:grid.249079.1) (ISNI:0000 0004 0369 4132) Institute of Applied Physics and Computational Mathematics Beijing China (GRID:grid.418809.c) (ISNI:0000 0000 9563 2481) HEDPS Center for Applied Physics and Technology Peking University Beijing China (GRID:grid.11135.37) (ISNI:0000 0001 2256 9319) Institute of Applied Physics and Computational Mathematics Beijing China (GRID:grid.418809.c) (ISNI:0000 0000 9563 2481) National University of Defense Technology College of Computing Science Changsha China (GRID:grid.412110.7) (ISNI:0000 0000 9548 2110) Shenzhen Technology University Center for Advanced Material Diagnostic Technology Shenzhen China (GRID:grid.499351.3) (ISNI:0000 0004 6353 6136) Science and Technology on Plasma Physics Laboratory Laser Fusion Research Center CAEP Mianyang China (GRID:grid.249079.1) (ISNI:0000 0004 0369 4132) Institute of Applied Physics and Computational Mathematics Beijing China (GRID:grid.418809.c) (ISNI:0000 0000 9563 2481) China Academy of Engineering Physics Graduate School Beijing China (GRID:grid.249079.1) (ISNI:0000 0004 0369 4132)

An amendment to this paper has been published and can be accessed via a link at the top of the paper.

关键词： Energy science and technology Laser-produced plasmas

Artificial neural network approach for turbulence models: A local framework

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Physical Review Fluids 2021年第8期6卷 084612-084612页

作者： Chenyue Xie Xiangming Xiong Jianchun Wang Program in Applied and Computational Mathematics Princeton University Princeton New Jersey 08544 USA Department of Mechanics and Aerospace Engineering Southern University of Science and Technology Shenzhen 518055 People's Republic of China

A local artificial neural network (LANN) framework is developed for turbulence modeling. The Reynolds-averaged Navier-Stokes (RANS) unclosed terms are reconstructed by the artificial neural network based on the local coordinate system which is orthogonal to the curved walls. We verify the proposed model in the flows over periodic hills. The correlation coefficients of the RANS unclosed terms predicted by the LANN model can be made larger than 0.96 in an a priori analysis, and the relative error of the unclosed terms can be made smaller than 18%. In an a posteriori analysis, detailed comparisons are made on the results of RANS simulations using the LANN, global artificial neural network (GANN), Spalart-Allmaras (SA), and shear stress transport (SST) k−ω models. It is shown that the LANN model performs better than the GANN, SA, and SST k−ω models in the prediction of the average velocity, wall-shear stress, and average pressure, which gives the results that are essentially indistinguishable from the direct numerical simulation data. The LANN model trained at low Reynolds number, Re=2800, can be directly applied to the cases of high Reynolds numbers, Re=5600, 10 595, 19 000, and 37 000, with accurate predictions. Furthermore, the LANN model is verified for flows over periodic hills with varying slopes. These results suggest that the LANN framework has a great potential to be applied to complex turbulent flows with curved walls.

关键词： Compressible boundary layers Turbulence Artificial neural networks