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Tropical geometry of deep neural networks

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arXiv 2018年

作者： Zhang, Liwen Naitzat, Gregory Lim, Lek-Heng Department of Computer Science University of Chicago ChicagoIL United States Department of Statistics University of Chicago ChicagoIL United States Computational and Applied Mathematics Initiative University of Chicago ChicagoIL United States

We establish, for the first time, connections between feedforward neural networks with ReLU activation and tropical geometry — we show that the family of such neural networks is equivalent to the family of tropical rational maps. Among other things, we deduce that feedforward ReLU neural networks with one hidden layer can be characterized by zonotopes, which serve as building blocks for deeper networks;we relate decision boundaries of such neural networks to tropical hypersurfaces, a major object of study in tropical geometry;and we prove that linear regions of such neural networks correspond to vertices of polytopes associated with tropical rational functions. An insight from our tropical formulation is that a deeper network is exponentially more expressive than a shallow network.14T05, 62M45, 68T01 Copyright © 2018, The Authors. All rights reserved.

关键词： Tropics

Fundamental solutions and gegenbauer expansions of helmholtz operators in riemannian spaces of constant curvature

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arXiv 2018年

作者： Cohl, Howard S. Dang, Thinh H. Dunster, T.M. Applied and Computational Mathematics Division National Institute of Standards and Technology Mission ViejoCA92694 United States Department of Computer Science George Washington University WashingtonDC20052 United States Department of Mathematics and Statistics San Diego State University San DiegoCA92182 United States

. We perform global and local analysis of oscillatory and damped spherically symmetric fundamental solutions for Helmholtz operators 2 in d-dimensional, R-radius hyperbolic Hd R and hyperspherical Sd R geometry, which represent Riemannian manifolds with positive constant and negative constant sectional curvature respectively. In particular, we compute closed-form expressions for fundamental solutions of 2 on Hd R, +2 on Sd R, and present two candidate fundamental solutions for 2 on Sd R. Flat-space limits, with their corresponding asymptotic representations, are used to restrict proportionality constants for these fundamental solutions. In order to accomplish this, we summarize and derive new large degree asymptotics for associated Legendre and Ferrers functions of the first and second kind. Furthermore, we prove that our fundamental solutions on the hyperboloid are unique due to their decay at infinity. To derive Gegenbauer polynomial expansions of our fundamental solutions for Helmholtz operators on hyperspheres and hyperboloids, we derive a collection of infinite series addition theorems for Ferrers and associated Legendre functions which are generalizations and extensions of the addition theorem for Gegenbauer polynomials. Using these addition theorems, in geodesic polar coordinates for dimensions greater than or equal to three, we compute Gegenbauer polynomial expansions for these fundamental solutions, and azimuthal Fourier expansions in *** Codes 31C12, 32Q45, 33C05, 33C45, 35A08, 35J05, 42A16 Copyright © 2018, The Authors. All rights reserved.

关键词： Geometry

Weighted graphlets and deep neural networks for protein structure classification

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arXiv 2019年

作者： Guo, Hongyu Newaz, Khalique Emrich, Scott Milenković, Tijana Li, Jun Department of Applied and Computational Mathematics and Statistics University of Notre Dame Notre DameIN46556 United States Department of Computer Science and Engineering University of Notre Dame Notre DameIN46556 United States Eck Institute for Global Health University of Notre Dame Notre DameIN46556 United States Interdisciplinary Center for Network Science & Applications University of Notre Dame Notre DameIN46556 United States Department of Electrical Engineering & Computer Science University of Tennessee KnoxvilleTN37996 United States

As proteins with similar structures often have similar functions, analysis of protein structures can help predict protein functions and is thus important. We consider the problem of protein structure classification, which computationally classifies the structures of proteins into pre-defined groups. We develop a weighted network that depicts the protein structures, and more importantly, we propose the first graphlet-based measure that applies to weighted networks. Further, we develop a deep neural network (DNN) composed of both convolutional and recurrent layers to use this measure for classification. Put together, our approach shows dramatic improvements in performance over existing graphlet-based approaches on 36 real datasets. Even comparing with the state-of-the-art approach, it almost halves the classification error. In addition to protein structure networks, our weighted-graphlet measure and DNN classifier can potentially be applied to classification of other weighted networks in computational biology as well as in other domains. Copyright © 2019, The Authors. All rights reserved.

关键词： Deep neural networks

Singular value decomposition of complexes

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arXiv 2018年

作者： Brake, Danielle A. Hauenstein, Jonathan D. Schreyer, Frank-Olaf Sommese, Andrew J. Stillman, Michael E. Department of Mathematics University of Wisconsin - Eau Claire Eau ClaireWI54702 United States Department of Applied and Computational Mathematics and Statistics University of Notre Dame Notre DameIN46556 United States Mathematik und Informatik Universität des Saarlandes Campus E2 4 SaarbrückenD-66123 Germany Department of Mathematics Cornell University IthacaNY14853 United States

Singular value decompositions of matrices are widely used in numerical linear algebra with many applications. In this paper, we extend the notion of singular value decompositions to finite complexes of real vector spaces. We provide two methods to compute them and present several applications. Copyright © 2018, The Authors. All rights reserved.

关键词： Singular value decomposition

Nonlinear concentration patterns and bands in autochemotactic suspensions

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Physical Review E 2018年第5期98卷 052411-052411页

作者： Enkeleida Lushi Raymond E. Goldstein Michael J. Shelley Department of Mathematics New Jersey Institute of Technology Newark New Jersey 07102 USA Department of Applied Mathematics and Theoretical Physics University of Cambridge Cambridge CB3 0WA United Kingdom Center for Computational Biology Flatiron Institute Simons Foundation New York New York 10010 USA Courant Institute of Mathematical Sciences New York University New York New York 10012 USA

In suspensions of microorganisms, pattern formation can arise from the interplay of chemotaxis and the fluid flows collectively generated by the organisms themselves. Here we investigate the resulting pattern formation in square and elongated domains in the context of two distinct models of locomotion in which the chemoattractant dynamics is fully coupled to the fluid flows and swimmer motion. Analyses for both models reveal an aggregative instability due to chemotaxis, independent of swimmer shape and type, and a hydrodynamic instability for “pusher” swimmers. We discuss the similarities and differences between the models. Simulations reveal a critical length scale of the swimmer aggregates and this feature can be utilized to stabilize swimmer concentration patterns into quasi-one-dimensional bands by varying the domain size. These concentration bands transition to traveling pulses under an external chemoattractant gradient, as observed in experiments with chemotactic bacteria.

关键词： Cell aggregation Chemotaxis Collective behavior Fluid-particle interactions Locomotion Microfluidics Self-organization Swimming

Anomalous stopping of laser-accelerated intense proton beam in dense ionized matter

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arXiv 2020年

作者： Ren, Jieru Deng, Zhigang Qi, Wei Chen, Benzheng Ma, Bubo Wang, Xing Yin, Shuai Feng, Jianhua Liu, Wei Hoffmann, Dieter H.H. Wang, Shaoyi Fan, Quanping Cui, Bo He, Shukai Cao, Zhurong Zhao, Zongqing Cao, Leifeng Gu, Yuqiu Zhu, Shaoping Cheng, Rui Zhou, Xianming Xiao, Guoqing Zhao, Hongwei Zhang, Yihang Zhang, Zhe Li, Yutong Wu, Dong Zhou, Weimin Zhao, Yongtao MOE Key Laboratory for Nonequilibrium Synthesis and Modulation of Condensed Matter School of Science Xi'an Jiaotong University Xi'an710049 China Science and Technology on Plasma Physics Laboratory Laser Fusion Research Center China Academy of Engineering Physics Mianyang621900 China Institute for Fusion Theory and Simulation Department of Physics Zhejiang University Hangzhou310058 China Xi'an Technological University Xi'an710021 China Institute of Applied Physics and Computational Mathematics Beijing100094 China Graduate School China Academy of Engineering Physics Beijing100088 China Institute of Modern Physics Chinese Academy of Sciences Lanzhou710049 China Xianyang Normal University Xianyang712000 China Institute of Physics Chinese Academy of Sciences Beijing100190 China

Ultrahigh-intensity lasers (1018-1022W/cm2) have opened up new perspectives in many fields of research and application [1-5]. By irradiating a thin foil, an ultrahigh accelerating field (1012 V/m) can be formed and multi-MeV ions with unprecedentedly high intensity (1010A/cm2) in short time scale (∼ps) are produced [6-14]. Such beams provide new options in radiography [15], high-yield neutron sources [16], high-energy-density-matter generation [17], and ion fast ignition [18, 19]. An accurate understanding of the nonlinear behavior of beam transport in matter is crucial for all these applications. We report here the first experimental evidence of anomalous stopping of a laser-generated high-current proton beam in well-characterized dense ionized matter. The observed stopping power is one order of magnitude higher than single-particle slowing-down theory predictions. We attribute this phenomenon to collective effects where the intense beam drives an decelerating electric field approaching 1GV/m in the dense ionized matter. This finding will have considerable impact on the future path to inertial fusion energy. Copyright © 2020, The Authors. All rights reserved.

关键词： Neutron sources

ELSI - An open infrastructure for electronic structure solvers

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arXiv 2019年

作者： Yu, Victor Wen-Zhe Campos, Carmen Dawson, William García, Alberto Havu, Ville Hourahine, Ben Huhn, William P. Jacquelin, Mathias Jia, Weile Keçeli, Murat Laasner, Raul Li, Yingzhou Lin, Lin Lu, Jianfeng Moussa, Jonathan Roman, Jose E. Vázquez-Mayagoitia, Álvaro Yang, Chao Blum, Volker Department of Mechanical Engineering and Materials Science Duke University DurhamNC27708 United States Departament de Sistemes Informàtics i Computació Universitat Politècnica de València València Spain RIKEN Center for Computational Science Kobe650-0047 Japan BellaterraE-08193 Spain Department of Applied Physics Aalto University AaltoFI-00076 Finland SUPA University of Strathclyde GlasgowG4 0NG United Kingdom Computational Research Division Lawrence Berkeley National Laboratory BerkeleyCA94720 United States Department of Mathematics University of California BerkeleyCA94720 United States Computational Science Division Argonne National Laboratory ArgonneIL60439 United States Department of Mathematics Duke University DurhamNC27708 United States Department of Physics Duke University DurhamNC27708 United States Department of Chemistry Duke University DurhamNC27708 United States Molecular Sciences Software Institute BlacksburgVA24060 United States

Routine applications of electronic structure theory to molecules and periodic systems need to compute the electron density from given Hamiltonian and, in case of non-orthogonal basis sets, overlap matrices. System sizes can range from few to thousands or, in some examples, millions of atoms. Different discretization schemes (basis sets) and different system geometries (finite non-periodic vs. infinite periodic boundary conditions) yield matrices with different structures. The ELectronic Structure Infrastructure (ELSI) project provides an open-source software interface to facilitate the implementation and optimal use of high-performance solver libraries covering cubic scaling eigensolvers, linear scaling density-matrix-based algorithms, and other reduced scaling methods in between. In this paper, we present recent improvements and developments inside ELSI, mainly covering (1) new solvers connected to the interface, (2) matrix layout and communication adapted for parallel calculations of periodic and/or spin-polarized systems, (3) routines for density matrix extrapolation in geometry optimization and molecular dynamics calculations, and (4) general utilities such as parallel matrix I/O and JSON output. The ELSI interface has been integrated into four electronic structure code projects (DFTB+, DGDFT, FHI-aims, SIESTA), allowing us to rigorously benchmark the performance of the solvers on an equal footing. Based on results of a systematic set of large-scale benchmarks performed with Kohn-Sham density-functional theory and density-functional tight-binding theory, we identify factors that strongly affect the efficiency of the solvers, and propose a decision layer that assists with the solver selection process. Finally, we describe a reverse communication interface encoding matrix-free iterative solver strategies that are amenable, e.g., for use with planewave basis sets. Copyright © 2019, The Authors. All rights reserved.

关键词： Electronic structure

Low Back Muscle Electrical Activity in Pregnant Women of the Second and Third Trimesters

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Journal of Physics: Conference Series 2019年第1期1179卷

作者： K D Purnamasari M N Widyawati S Suryono Postgraduate Applied Science Midwifery program Poltekkes Kemenkes Semarang Indonesia Department of Physics Faculty of Science and Mathematics Diponegoro University Semarang Indonesia

Low back pain during pregnancy causes axial or parasagittal discomfort. These problems are experienced by more than two-thirds of pregnancies. Clinically the measurement of pain was carried out using the Visual Analogue Scale (VAS) method. Unfortunately, the method has low accuracy and is subjective. This paper proposes a method of measuring pain by using electromyography tools so that pain level measurement has a high level of accuracy and is non-invasive to patients. The study was carried out by installing electromyography electrodes on L4 and L5 muscles. The signal is recorded on a digital storage oscilloscope to find out the electrical activity. The results of the recording were evaluated for the action potential and frequency *** study was conducted on 15 respondents with lower back muscle pain in mothers monitored for 1 minute with electromyography. The monitoring results are calculated on average frequency and action potential and then presented in the form of trends. The results of the analysis of pain level measurement were compared with the results of VAS measurements. From the results of this study indicate that the trend of the action potential and signal frequency using electromyography shows different results when respondents experience the same pain scale on VAS. The value generated in the measurement using VAS is only a pain number scale that is felt by the respondent where the value is subjective.

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E055 Circulatory levels of RANKL, OPG and oxidative stress markers in postmenopausal women with normal and low BMD

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Rheumatology 2019年第SUPPLEMENT_3期58卷

作者： Aljarallah, Khaled Azizieh, Fawaz Y Shehab, Dia Gupta, Renu Raghupathy, Raj Department of Medicine Faculty of Medicine Kuwait University KUWAIT Department of Mathematics and Natural Sciences International Centre for Applied Mathematics and Computational Bioengineering Gulf University for Science and Technology KUWAIT Department of Radiology Faculty of Medicine Kuwait University KUWAIT Department of Microbiology Faculty of Medicine Kuwait University KUWAIT