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arXiv 2023年

作者： Lin, Buyu Ge, Changhao Liu, Jun S. Department of Statistics Harvard University United States Graduate Group in Applied Mathematics and Computational Science University of Pennsylvania United States

We introduce a class of generic spike-and-slab priors for high-dimensional linear regression with grouped variables and present a Coordinate-ascent Variational Inference (CAVI) algorithm for obtaining an optimal variational Bayes approximation. Using parameter expansion for a specific, yet comprehensive, family of slab distributions, we obtain a further gain in computational efficiency. The method can be easily extended to fitting additive models. Theoretically, we present general conditions on the generic spike-and-slab priors that enable us to derive the contraction rates for both the true posterior and the VB posterior for linear regression and additive models, of which some previous theoretical results can be viewed as special cases. Our simulation studies and real data application demonstrate that the proposed method is superior to existing methods in both variable selection and parameter estimation. Our algorithm is implemented in the R package GVSSB. Copyright © 2023, The Authors. All rights reserved.

关键词： computational efficiency

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Lower Bounds and Nearly Optimal Algorithms in Distributed Learning with Communication Compression

arXiv

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arXiv 2022年

作者： Huang, Xinmeng Chen, Yiming Yin, Wotao Yuan, Kun Graduate Group in Applied Mathematics and Computational Science Univ. of Pennsylvania United States DAMO Academy Alibaba Group China

Recent advances in distributed optimization and learning have shown that communication compression is one of the most effective means of reducing communication. While there have been many results on convergence rates under communication compression, a theoretical lower bound is still missing. Analyses of algorithms with communication compression have attributed convergence to two abstract properties: the unbiased property or the contractive property. They can be applied with either unidirectional compression (only messages from workers to server are compressed) or bidirectional compression. In this paper, we consider distributed stochastic algorithms for minimizing smooth and non-convex objective functions under communication compression. We establish a convergence lower bound for algorithms whether using unbiased or contractive compressors in unidirection or bidirection. To close the gap between the lower bound and the existing upper bounds, we further propose an algorithm, NEOLITHIC, which almost reaches our lower bound (up to logarithm factors) under mild conditions. Our results also show that using contractive bidirectional compression can yield iterative methods that converge as fast as those using unbiased unidirectional compression. The experimental results validate our findings. Copyright © 2022, The Authors. All rights reserved.

关键词： Iterative methods

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The knowledge gradient for sequential decision making with stochastic binary feedbacks 33

The knowledge gradient for sequential decision making with s...

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33rd International Conference on Machine Learning, ICML 2016

作者： Wang, Yingfei Wang, Chu Powell, Warren Department of Computer Science Princeton University PrincetonNJ08540 United States Program in Applied and Computational Mathematics Princeton University PrincetonNJ08544 United States Department of Operations Research and Financial Engineering Princeton University PrincetonNJ08544 United States

ISBN: (纸本)9781510829008

We consider the problem of sequentially making decisions that are rewarded by "successes" and "failures" which can be predicted through an unknown relationship that depends on a partially controllable vector of attributes for each instance. The learner takes an active role in selecting samples from the instance pool. The goal is to maximize the probability of success, either after the offline training phase or minimizing regret in online learning. Our problem is motivated by real-world applications where observations are time consuming and/or expensive. With the adaptation of an online Bayesian linear classifier, we develop a knowledge-gradient type policy to guide the experiment by maximizing the expected value of information of labeling each alternative, in order to reduce the number of expensive physical experiments. We provide a finite-time analysis of the estimated error and demonstrate the performance of the proposed algorithm on both synthetic problems and benchmark UCI datasets.

关键词： Benchmarking

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Beyond trees: classification with sparse pairwise dependencies

The Journal of Machine Learning Research

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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

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Physics-informed machine learning with smoothed particle hydrodynamics: Hierarchy of reduced Lagrangian models of turbulence

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Physical Review Fluids 2023年第5期8卷 054602-054602页

作者： Michael Woodward Yifeng Tian Criston Hyett Chris Fryer Mikhail Stepanov Daniel Livescu Michael Chertkov Graduate Interdisciplinary Program in Applied Mathematics University of Arizona Tucson Arizona 85721 USA Department of Mathematics University of Arizona Tucson Arizona 85721 USA Computer Computational and Statistical Sciences Division LANL Los Alamos New Mexico 87544 USA

Building efficient, accurate, and generalizable reduced-order models of developed turbulence remains a major challenge. This manuscript approaches this problem by developing a hierarchy of parameterized reduced Lagrangian models for turbulent flows, and it investigates the effects of enforcing physical structure through smoothed particle hydrodynamics (SPH) versus relying on neural networks (NNs) as universal function approximators. Starting from NN parametrizations of a Lagrangian acceleration operator, this hierarchy of models gradually incorporates a weakly compressible and parameterized SPH framework, which enforces physical symmetries, such as Galilean, rotational, and translational invariances. Within this hierarchy, two new parameterized smoothing kernels are developed to increase the flexibility of the learn-able SPH simulators. For each model we experiment with different loss functions which are minimized using gradient based optimization, where efficient computations of gradients are obtained by using automatic differentiation and sensitivity analysis. Each model within the hierarchy is trained on two data sets associated with weakly compressible homogeneous isotropic turbulence: (1) a validation set using weakly compressible SPH; and (2) a high-fidelity set from direct numerical simulations. Numerical evidence shows that encoding more SPH structure improves generalizability to different turbulent Mach numbers and time shifts, and that including the novel parameterized smoothing kernels improves the accuracy of SPH at the resolved scales.

关键词： Sensitivity analysis

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NOMAD: Nonlinear Manifold Decoders for Operator Learning

arXiv

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arXiv 2022年

作者： Seidman, Jacob H. Kissas, Georgios Perdikaris, Paris Pappas, George J. Graduate Program in Applied Mathematics and Computational Science University of Pennsylvania United States Department of Mechanical Engineering and Applied Mechanics University of Pennsylvania United States Department of Electrical and Systems Engineering University of Pennsylvania United States

Supervised learning in function spaces is an emerging area of machine learning research with applications to the prediction of complex physical systems such as fluid flows, solid mechanics, and climate modeling. By directly learning maps (operators) between infinite dimensional function spaces, these models are able to learn discretization invariant representations of target functions. A common approach is to represent such target functions as linear combinations of basis elements learned from data. However, there are simple scenarios where, even though the target functions form a low dimensional submanifold, a very large number of basis elements is needed for an accurate linear representation. Here we present NOMAD, a novel operator learning framework with a nonlinear decoder map capable of learning finite dimensional representations of nonlinear submanifolds in function spaces. We show this method is able to accurately learn low dimensional representations of solution manifolds to partial differential equations while outperforming linear models of larger size. Additionally, we compare to state-of-the-art operator learning methods on a complex fluid dynamics benchmark and achieve competitive performance with a significantly smaller model size and training cost. © 2022, CC BY-NC-SA.

关键词： Benchmarking

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Machine learning based non-Newtonian fluid model with molecular fidelity

arXiv

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arXiv 2020年

作者： Lei, Huan Wu, Lei Weinan, E. Department of Computational Mathematics Science & Engineering Department of Statistics & Probability Michigan State University MI48824 United States Department of Mathematics and Program in Applied and Computational Mathematics Princeton University NJ08544 United States

We introduce a machine-learning-based framework for constructing continuum non-Newtonian fluid dynamics model directly from a micro-scale description. Polymer solution is used as an example to demonstrate the essential ideas. To faithfully retain molecular fidelity, we establish a micro-macro correspondence via a set of encoders for the micro-scale polymer configurations and their macro-scale counterparts, a set of nonlinear conformation tensors. The dynamics of these conformation tensors can be derived from the micro-scale model and the relevant terms can be parametrized using machine learning. The final model, named the deep non-Newtonian model (DeePN2), takes the form of conventional non-Newtonian fluid dynamics models, with a new form of the objective tensor derivative. Numerical results demonstrate the accuracy of DeePN2. Copyright © 2020, The Authors. All rights reserved.

关键词： Viscous flow

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Parallel Multicore CSB Format and Its Sparse Matrix Vector Multiplication

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Advances in Linear Algebra & Matrix Theory 2014年第1期4卷 1-8页

作者： Bing Yang Shuo Gu Tong-Xiang Gu Cong Zheng Xing-Ping Liu Graduate School of Chinese Academy of Engineering Physics Beijing China Laboratory of Computational Physics Institute of Applied Physics and Computational Mathematics BeijingChina School of Electronic Engineering University of Electronic Science and Technology of China Chengdu China

Sparse Matrix Vector Multiplication (SpMV) is one of the most basic problems in scientific and engineering computations. It is the basic operation in many realms, such as solving linear systems or eigenvalue problems. Nowadays, more than 90 percent of the world’s highest performance parallel computers in the top 500 use multicore architecture. So it is important practically to design the efficient methods of computing SpMV on multicore parallel computers. Usually, algorithms based on compressed sparse row (CSR) format suffer from a number of nonzero elements on each row so hardly as to use the multicore structure efficiently. Compressed Sparse Block (CSB) format is an effective storage format which can compute SpMV efficiently in a multicore computer. This paper presents a parallel multicore CSB format and SpMV based on it. We carried out numerical experiments on a parallel multicore computer. The results show that our parallel multicore CSB format and SpMV algorithm can reach high speedup, and they are highly scalable for banded matrices.

关键词： SpMV Multicore Parallel Computers Parallel Multicore CSB Format

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Hyperuniformity of quasicrystals

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Physical Review B 2017年第5期95卷 054119-054119页

作者： Erdal C. Oğuz Joshua E. S. Socolar Paul J. Steinhardt Salvatore Torquato Department of Chemistry Department of Physics Princeton Institute for the Science and Technology of Materials and Program in Applied and Computational Mathematics Princeton University Princeton New Jersey 08540 USA

Hyperuniform systems, which include crystals, quasicrystals, and special disordered systems, have attracted considerable recent attention, but rigorous analyses of the hyperuniformity of quasicrystals have been lacking because the support of the spectral intensity is dense and discontinuous. We employ the integrated spectral intensity Z(k) to quantitatively characterize the hyperuniformity of quasicrystalline point sets generated by projection methods. The scaling of Z(k) as k tends to zero is computed for one-dimensional quasicrystals and shown to be consistent with independent calculations of the variance, σ2(R), in the number of points contained in an interval of length 2R. We find that one-dimensional quasicrystals produced by projection from a two-dimensional lattice onto a line of slope 1/τ fall into distinct classes determined by the width of the projection window. For a countable dense set of widths, Z(k)∼k4; for all others, Z(k)∼k2. This distinction suggests that measures of hyperuniformity define new classes of quasicrystals in higher dimensions as well.

关键词： Microstructure Structural properties

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Towards Trajectory Planning for a 6-Degree-of-Freedom Robot Manipulator Considering the Orientation of the End-effector Using Computer Algebra 10

Towards Trajectory Planning for a 6-Degree-of-Freedom Robot ...

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10th International Symposium on Symbolic Computation in Software science - Work in Progress Workshop, SCSS 2024 WiP

作者： Okazaki, Takumu Terui, Akira Mikawa, Masahiko Master's Program in Mathematics Graduate School of Science and Technology University of Tsukuba Tsukuba305-8571 Japan Institute of Pure and Applied Sciences University of Tsukuba Tsukuba305-8571 Japan Institute of Library Information and Media Science University of Tsukuba Tsukuba305-8550 Japan

We consider the trajectory planning of a 6-Degree-of-Freedom (DOF) robot manipulator using computer algebra, with controlling the orientation of the end-effector. As a first step towards the objective, we present a solution to the inverse kinematics problem of the manipulator such that the orientation of the end-effector remains constant using computer algebra. © 2024 CEUR-WS. All rights reserved.

关键词： End effectors

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