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Estimation in the group action channel

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arXiv 2018年

作者： Abbe, Emmanuel Pereira, João M. Singer, Amit Program in Applied and Computational Mathematics Princeton University PrincetonNJ United States Electrical Engineering Department Princeton University PrincetonNJ United States Department of Mathematics Princeton University PrincetonNJ United States

MSC Codes 94A15, 62B10We analyze the problem of estimating a signal from multiple measurements on a group action channel that linearly transforms a signal by a random group action followed by a fixed projection and additive Gaussian noise. This channel is motivated by applications such as multi-reference alignment and cryoelectron microscopy. We focus on the large noise regime prevalent in these applications. We give a lower bound on the mean square error (MSE) of any asymptotically unbiased estimator of the signal's orbit in terms of the signal's moment tensors, which implies that the MSE is bounded away from 0 when N=σ2dis bounded from above, where N is the number of observations, σ is the noise standard deviation, and d is the so-called moment order cutoff. In contrast, the maximum likelihood estimator is shown to be consistent if N=σ2ddiverges. Copyright © 2018, The Authors. All rights reserved.

关键词： Mean square error

The Athena++ Adaptive Mesh Refinement Framework: Design and Magnetohydrodynamic Solvers

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Astrophysical Journal, Supplement Series 2020年第1期249卷

作者： Stone, James M. Tomida, Kengo White, Christopher J. Felker, Kyle G. Department of Astrophysical Sciences Princeton University Princeton 08544 NJ United States Program in Applied and Computational Mathematics Princeton University Princeton 08544 NJ United States Department of Earth and Space Science Osaka University Toyonaka Osaka 560-0043 Japan Kavli Institute for Theoretical Physics University of California Santa Barbara Santa Barbara 93107 CA United States School of Natural Sciences Institute for Advanced Study Princeton 08544 NJ United States Astronomical Institute Tohoku University Sendai Miyagi 980-8578 Japan Argonne National Laboratory Lemont 60439 IL United States

The design and implementation of a new framework for adaptive mesh refinement calculations are described. It is intended primarily for applications in astrophysical fluid dynamics, but its flexible and modular design enables its use for a wide variety of physics. The framework works with both uniform and nonuniform grids in Cartesian and curvilinear coordinate systems. It adopts a dynamic execution model based on a simple design called a "task list"that improves parallel performance by overlapping communication and computation, simplifies the inclusion of a diverse range of physics, and even enables multiphysics models involving different physics in different regions of the calculation. We describe physics modules implemented in this framework for both nonrelativistic and relativistic magnetohydrodynamics (MHD). These modules adopt mature and robust algorithms originally developed for the Athena MHD code and incorporate new extensions: support for curvilinear coordinates, higher-order time integrators, more realistic physics such as a general equation of state, and diffusion terms that can be integrated with super-time-stepping algorithms. The modules show excellent performance and scaling, with well over 80% parallel efficiency on over half a million threads. The source code has been made publicly available. © 2020. The American Astronomical Society. All rights reserved..

关键词： Astronomy software Magnetohydrodynamics

Communication-computation efficient gradient coding

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arXiv 2018年

作者： Ye, Min Abbe, Emmanuel Department of Electrical Engineering Princeton University PrincetonNJ United States Program in Applied and Computational Mathematics Department of Electrical Engineering Princeton University School of Mathematics Institute for Advanced Study PrincetonNJ08544 United States

This paper develops coding techniques to reduce the running time of distributed learning tasks. It characterizes the fundamental tradeoff to compute gradients (and more generally vector summations) in terms of three parameters: computation load, straggler tolerance and communication cost. It further gives an explicit coding scheme that achieves the optimal tradeoff based on recursive polynomial constructions, coding both across data subsets and vector components. As a result, the proposed scheme allows to minimize the running time for gradient computations. Implementations are made on Amazon EC2 clusters using Python with mpi4py package. Results show that the proposed scheme maintains the same generalization error while reducing the running time by 32% compared to uncoded schemes and 23% compared to prior coded schemes focusing only on stragglers (Tandon et al., ICML 2017). Copyright © 2018, The Authors. All rights reserved.

关键词： Python

Solving many-electron Schrödinger equation using deep neural networks

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arXiv 2018年

作者： Han, Jiequn Zhang, Linfeng Weinan, E. Program in Applied and Computational Mathematics Princeton University PrincetonNJ08544 United States Department of Mathematics Princeton University PrincetonNJ08544 United States Beijing Institute of Big Data Research Beijing100871 China

We introduce a new family of trial wave-functions based on deep neural networks to solve the many-electron Schrödinger equation. The Pauli exclusion principle is dealt with explicitly to ensure that the trial wave-functions are physical. The optimal trial wave-function is obtained through variational Monte Carlo and the computational cost scales quadratically with the number of electrons. The algorithm does not make use of any prior knowledge such as atomic orbitals. Yet it is able to represent accurately the ground-states of the tested systems, including He, H2, Be, B, LiH, and a chain of 10 hydrogen atoms. This opens up new possibilities for solving large-scale many-electron Schrödinger equation. Copyright © 2018, The Authors. All rights reserved.

关键词： Wave functions

Recovering Signals from their FROG Trace

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Recovering Signals from their FROG Trace

IEEE International Conference on Acoustics, Speech and Signal Processing

作者： Tamir Bendory Dan Edidin Yonina C. Eldar The Program in Applied and Computational Mathematics Princeton University Princeton NJ USA Department of Mathematics University of Missouri Columbia Missouri USA The Andrew and Erna Viterbi Faculty of Electrical Engineering Technion - Israel Institute of Technology Haifa Israel

ISBN: (纸本)9781538646595

The problem of recovering a signal from its power spectrum is called phase retrieval. This problem appears in a variety of scientific applications, such as ultra-short laser pulse characterization and diffraction imaging. However, the problem for one-dimensional signals is ill-posed as there is no one-to-one mapping between a one-dimensional signal and its power spectrum. In the field of ultra-short laser pulse characterization, it is common to overcome this ill-posedness by using a technique called Frequency-Resolved Optical Gating (FROG). In FROG, the measured data, referred to as FROG trace, is the Fourier magnitude of the product of the underlying signal with several translated versions of itself. Therefore, in order to recover a signal from its FROG trace, one needs to invert a system of phaseless quartic equations. In this paper, we explore the symmetries and uniqueness of the FROG mapping. Our main result states that a signal bandlimited to B is determined uniquely, up to symmetries, by only 3B FROG measurements.

关键词： phase retrieval phaseless quartic system of equations ultra-short laser pulse characterization FROG signals Ultrashort laser pulses power spectra frogs Amphibians RECOVERING Trace One dimensional quartic equations

Comparing demographics of signatories to public letters on diversity in the mathematical sciences

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arXiv 2019年

作者： Topaz, Chad M. Cart, James Eaton, Carrie Diaz Shrout, Anelise Hanson Higdon, Jude A. İnce, Kenan Katz, Brian Lewis, Drew Libertini, Jessica Smith, Christian Michael Department of Mathematics and Statistics Williams College WilliamstownMA United States Institute for the Quantitative Study of Inclusion Diversity and Equity WilliamstownMA United States Office of Institutional Research Williams College WilliamstownMA United States Digital and Computational Studies Program Bates College LewistonME United States Department of Mathematics Westminster College Salt Lake CityUT United States Department of Mathematics and Statistics Smith College NorthamptonMA United States Department of Mathematics and Statistics University of South Alabama MobileAL United States Department of Applied Mathematics Virginia Military Institute LexingtonVA United States Department of Sociology University of Wisconsin-Madison MadisonWI United States

In its December 2019 edition, the Notices of the American Mathematical Society published an essay critical of the use of diversity statements in academic hiring. The publication of this essay prompted many responses, including three public letters circulated within the mathematical sciences community. Each letter was signed by hundreds of people and was published online, also by the American Mathematical Society. We report on a study of the signatories’ demographics, which we infer using a crowdsourcing approach. Letter A highlights diversity and social justice. The pool of signatories contains relatively more individuals inferred to be women and/or members of underrepresented ethnic groups. Moreover, this pool is diverse with respect to the levels of professional security and types of academic institutions represented. Letter B does not comment on diversity, but rather, asks for discussion and debate. This letter was signed by a strong majority of individuals inferred to be white men in professionally secure positions at highly research intensive universities. Letter C speaks out specifically against diversity statements, calling them "a mistake," and claiming that their usage during early stages of faculty hiring "diminishes mathematical achievement." Individuals who signed both Letters B and C, that is, signatories who both privilege debate and oppose diversity statements, are overwhelmingly inferred to be tenured white men at highly research intensive universities. Our empirical results are consistent with theories of power drawn from the social sciences. Copyright © 2019, The Authors. All rights reserved.

关键词： Population statistics

Plane-wave analysis of a hyperbolic system of equations with relaxation in Rd

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arXiv 2018年

作者： de Hoop, Maarten V. Liu, Jian-Guo Markowich, Peter A. Ussembayev, Nail S. Department of Computational and Applied Mathematics Rice University HoustonTX77005 United States Department of Mathematics Department of Physics Duke University DurhamNC27708 United States Applied Mathematics and Computer Science Program CEMSE Division King Abdullah University of Science and Technology Thuwal23955-6900 Saudi Arabia Faculty of Mathematics University of Vienna ViennaA-1090 Austria Applied Mathematics and Computer Science Program CEMSE Division King Abdullah University of Science and Technology Thuwal23955-6900 Saudi Arabia

We consider a multi-dimensional scalar wave equation with memory corresponding to the viscoelastic material described by a generalized Zener model. We deduce that this relaxation system is an example of a non-strictly hyperbolic system satisfying Majda's block structure condition. Well-posedness of the associated Cauchy problem is established by showing that the symbol of the spatial derivatives is uniformly diagonalizable with real eigenvalues. A long-time stability result is obtained by plane-wave analysis when the memory term allows for dissipation of energy.35L40 (Primary) 74D05, 35B35 (Secondary) Copyright © 2018, The Authors. All rights reserved.

关键词： Viscoelasticity

Blind phaseless short-time fourier transform recovery

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arXiv 2018年

作者： Bendory, Tamir Edidin, Dan Eldar, Yonina C. Program in Applied and Computational Mathematics Princeton University PrincetonNJ United States Department of Mathematics University of Missouri ColumbiaMO United States Andrew and Erna Viterbi Faculty of Electrical Engineering Technion - Israel Institute of Technology Haifa Israel

The problem of recovering a pair of signals from their blind phaseless short-time Fourier transform measurements arises in several important phase retrieval applications, including ptychography and ultra-short pulse characterization. In this paper, we prove that in order to determine a pair of generic signals uniquely, up to trivial ambiguities, the number of phaseless measurements one needs to collect is, at most, five times the number of parameters required to describe the signals. This result improves significantly upon previous papers, which required the number of measurements to be quadratic in the number of parameters rather than linear. In addition, we consider the simpler problem of recovering a pair of generic signals from their blind short-time Fourier transform, when the phases are known. In this setting, which can be understood as a special case of the blind deconvolution problem, we show that the number of measurements required to determine the two signals, up to trivial ambiguities, equals exactly the number of parameters to be recovered. As a side result, we study the classical phase retrieval problem-that is, recovering a signal from its Fourier magnitudes-when some entries of the signal are known a priori. We derive a bound on the number of required measurements as a function of the size of the set of known entries. Specifically, we show that if most of the signal's entries are known, then only a few Fourier magnitudes are necessary to determine a signal uniquely. Copyright © 2018, The Authors. All rights reserved.

关键词： Convolution

Numerical Algorithms for the Computation of Generalized Prolate Spheroidal Functions

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arXiv 2017年

作者： Lederman, Roy R. Program in Applied and Computational Mathematics Princeton University

Generalized Prolate Spheroidal Functions (GPSF) are the eigenfunctions of the truncated Fourier transform, restricted to D-dimensional balls in the spatial domain and frequency domain. Despite their useful properties in many applications, GPSFs are often replaced by crude approximations. The purpose of this paper is to review the elements of computing GPSFs and associated eigenvalues. This paper is accompanied by open-source code.42A38, 65T99, 33F05, 65D20 Copyright © 2017, The Authors. All rights reserved.

关键词： Eigenvalues and eigenfunctions