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Optimal bayesian minimax rates for unconstrained large covariance matrices

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arXiv 2017年

作者： Lee, Kyoungjae Lee, Jaeyong Department of Applied and Computational Mathematics and Statistics University of Notre Dame Department of Statistics Seoul National University

We obtain the optimal Bayesian minimax rate for the unconstrained large covariance matrix of multivariate normal sample with mean zero, when both the sample size, n, and the dimension, p, of the covariance matrix tend to infinity. Traditionally the posterior convergence rate is used to compare the frequentist asymptotic performance of priors, but defining the optimality with it is elusive. We propose a new decision theoretic framework for prior selection and define Bayesian minimax rate. Under the proposed framework, we obtain the optimal Bayesian minimax rate for the spectral norm for all rates of p. We also considered Frobenius norm, Bregman divergence and squared log-determinant loss and obtain the optimal Bayesian minimax rate under certain rate conditions on p. A simulation study is conducted to support the theoretical results. Copyright © 2017, The Authors. All rights reserved.

关键词： Covariance matrix

On singularity formation in a Hele-Shaw model

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arXiv 2017年

作者： Constantin, Peter Elgindi, Tarek Nguyen, Huy Vicol, Vlad Department of Mathematics Princeton University PrincetonNJ08544 United States Department of Mathematics Uc San Diego San DiegoCA92093 United States Program In Applied and Computational Mathematics Princeton University PrincetonNJ08544 United States

We discuss a lubrication approximation model of the interface between two immiscible fluids in a Hele-Shaw cell, derived in [CDG+93] and widely studied since. The model consists of a single one dimensional evolution equation for the thickness 2h = 2h(x, t) of a thin neck of fluid, ∂th + ∂x(h ∂x3h) = 0, for x ∈ (−1, 1) and t ≥ 0. The boundary conditions fix the neck height and the pressure jump: h(±1, t) = 1, ∂x2h(±1, t) = P > 0. We prove that starting from smooth and positive h, as long as h(x, t) > 0, for x ∈ [−1, 1], t ∈ [0, T ], no singularity can arise in the solution up to time T. As a consequence, we prove for any P > 2 and any smooth and positive initial datum that the solution pinches off in either finite or infinite time, i.e., inf[−1,1]×[0,T∗) h = 0, for some T∗ ∈ (0, ∞]. These facts have been long anticipated on the basis of numerical and theoretical studies. August 30, 2017.35Q35, 35Q86 Copyright © 2017, The Authors. All rights reserved.

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Predicting Available Computer Resources in Supercomputers

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Predicting Available Computer Resources in Supercomputers

New York Scientific Data Summit (NYSDS)

作者： Peggy Yin Chenxiao Xu Sergey Panitkin Shinjae Yoo Earl L. Vandermeulen High School Port Jefferson New York USA Department of Applied Mathematics and Statistics Stony Brook University New York USA Physics Department Brookhaven National Lab New York USA Computational Sciencie Initiative Brookhaven National Lab New York USA

To maximize efficiency, the workload management system of a supercomputer must be able to process a multitude of job requests at a given time. The application of data processing techniques shows it is possible to predict node availability. The high error and variance of such predictions can be eliminated through the utilization of machine learning methods. Implementation of accurate predictions leads to more productive supercomputing systems, accelerating the pace of groundbreaking scientific discovery.

关键词： Predictive models Correlation Training Data models Supercomputers Time series analysis Multilayer perceptrons

Classical many-particle systems with unique disordered ground states

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Physical Review E 2017年第4期96卷 042146-042146页

作者： G. Zhang F. H. Stillinger S. Torquato Department of Chemistry Princeton University Princeton New Jersey 08544 USA 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 08544 USA

Classical ground states (global energy-minimizing configurations) of many-particle systems are typically unique crystalline structures, implying zero enumeration entropy of distinct patterns (aside from trivial symmetry operations). By contrast, the few previously known disordered classical ground states of many-particle systems are all high-entropy (highly degenerate) states. Here we show computationally that our recently proposed “perfect-glass” many-particle model [Sci. Rep. 6, 36963 (2016)] possesses disordered classical ground states with a zero entropy: a highly counterintuitive situation . For all of the system sizes, parameters, and space dimensions that we have numerically investigated, the disordered ground states are unique such that they can always be superposed onto each other or their mirror image. At low energies, the density of states obtained from simulations matches those calculated from the harmonic approximation near a single ground state, further confirming ground-state uniqueness. Our discovery provides singular examples in which entropy and disorder are at odds with one another. The zero-entropy ground states provide a unique perspective on the celebrated Kauzmann-entropy crisis in which the extrapolated entropy of a supercooled liquid drops below that of the crystal. We expect that our disordered unique patterns to be of value in fields beyond glass physics, including applications in cryptography as pseudorandom functions with tunable computational complexity.

关键词： Nonequilibrium statistical mechanics Pattern formation

Local-order metric for condensed-phase environments

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Physical Review B 2018年第6期97卷 064105-064105页

作者： Fausto Martelli Hsin-Yu Ko Erdal C. Oğuz Roberto Car Department of Chemistry Princeton University Princeton New Jersey 08544 USA Institut für Theoretische Physik II Heinrich-Heine-Universität D-40225 Düsseldorf Germany Department of Physics Princeton University Princeton New Jersey 08544 USA Princeton Institute for the Science and Technology of Materials Princeton University Princeton New Jersey 08544 USA Program in Applied and Computational Mathematics Princeton University Princeton New Jersey 08544 USA

We introduce a local order metric (LOM) that measures the degree of order in the neighborhood of an atomic or molecular site in a condensed medium. The LOM maximizes the overlap between the spatial distribution of sites belonging to that neighborhood and the corresponding distribution in a suitable reference system. The LOM takes a value tending to zero for completely disordered environments and tending to one for environments that perfectly match the reference. The site-averaged LOM and its standard deviation define two scalar order parameters, S and δS, that characterize with excellent resolution crystals, liquids, and amorphous materials. We show with molecular dynamics simulations that S, δS, and the LOM provide very insightful information in the study of structural transformations, such as those occurring when ice spontaneously nucleates from supercooled water or when a supercooled water sample becomes amorphous upon progressive cooling.

关键词： Glass transition Nucleation Order parameters Phase transitions

Cross-correlation of Dark Energy Survey Year 3 lensing data with ACT and Planck thermal Sunyaev-Zel’dovich effect observations. II. Modeling and constraints on halo pressure profiles

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Physical Review D 2022年第12期105卷 123526-123526页

作者： S. Pandey M. Gatti E. Baxter J. C. Hill X. Fang C. Doux G. Giannini M. Raveri J. DeRose H. Huang E. Moser N. Battaglia A. Alarcon A. Amon M. Becker A. Campos C. Chang R. Chen A. Choi K. Eckert J. Elvin-Poole S. Everett A. Ferte I. Harrison N. Maccrann J. Mccullough J. Myles A. Navarro Alsina J. Prat R. P. Rollins C. Sanchez T. Shin M. Troxel I. Tutusaus B. Yin M. Aguena S. Allam F. Andrade-Oliveira G. M. Bernstein E. Bertin B. Bolliet J. R. Bond D. Brooks E. Calabrese A. Carnero Rosell M. Carrasco Kind J. Carretero R. Cawthon M. Costanzi M. Crocce L. N. da Costa M. E. S. Pereira J. De Vicente S. Desai H. T. Diehl J. P. Dietrich P. Doel J. Dunkley A. E. Evrard S. Ferraro I. Ferrero B. Flaugher P. Fosalba J. García-Bellido E. Gaztanaga D. W. Gerdes T. Giannantonio D. Gruen R. A. Gruendl J. Gschwend G. Gutierrez K. Herner A. D. Hincks S. R. Hinton D. L. Hollowood K. Honscheid J. P. Hughes D. Huterer B. Jain D. J. James T. Jeltema E. Krause K. Kuehn O. Lahav M. Lima M. Lokken M. S. Madhavacheril M. A. G. Maia J. J. Mcmahon P. Melchior F. Menanteau R. Miquel J. J. Mohr K. Moodley R. Morgan F. Nati M. D. Niemack L. Page A. Palmese F. Paz-Chinchón A. Pieres A. A. Plazas Malagón M. Rodriguez-Monroy A. K. Romer E. Sanchez V. Scarpine E. Schaan S. Serrano I. Sevilla-Noarbe E. Sheldon B. D. Sherwin C. Sifón M. Smith M. Soares-Santos D. Spergel E. Suchyta M. E. C. Swanson G. Tarle D. Thomas C. To T. N. Varga J. Weller E. J. Wollack Z. Xu Department of Physics and Astronomy University of Pennsylvania Philadelphia Pennsylvania 19104 USA Institute for Astronomy University of Hawai’i 2680 Woodlawn Drive Honolulu Hawai’i 96822 USA Department of Physics Columbia University New York New York 10027 USA Center for Computational Astrophysics Flatiron Institute New York New York 10010 USA Department of Astronomy/Steward Observatory University of Arizona 933 North Cherry Avenue Tucson Arizona 85721-0065 USA Institut de Física d’Altes Energies (IFAE) The Barcelona Institute of Science and Technology Campus UAB 08193 Bellaterra (Barcelona) Spain Lawrence Berkeley National Laboratory 1 Cyclotron Road Berkeley California 94720 USA Department of Physics University of Arizona Tucson Arizona 85721 USA Department of Astronomy Cornell University Ithaca New York 14853 USA Argonne National Laboratory 9700 South Cass Avenue Lemont Illinois 60439 USA Kavli Institute for Particle Astrophysics and Cosmology P. O. Box 2450 Stanford University Stanford California 94305 USA Department of Physics Carnegie Mellon University Pittsburgh Pennsylvania 15312 USA Department of Astronomy and Astrophysics University of Chicago Chicago Illinois 60637 USA Kavli Institute for Cosmological Physics University of Chicago Chicago Illinois 60637 USA Department of Physics Duke University Durham North Carolina 27708 USA Center for Cosmology and Astro-Particle Physics The Ohio State University Columbus Ohio 43210 USA Department of Physics The Ohio State University Columbus Ohio 43210 USA Santa Cruz Institute for Particle Physics Santa Cruz California 95064 USA Jet Propulsion Laboratory California Institute of Technology 4800 Oak Grove Drive Pasadena California 91109 USA Department of Physics University of Oxford Denys Wilkinson Building Keble Road Oxford OX1 3RH United Kingdom Jodrell Bank Center for Astrophysics School of Physics and Astronomy University of Manchester Oxford Road Manchester M13 9PL United Kin

Hot, ionized gas leaves an imprint on the cosmic microwave background via the thermal Sunyaev-Zel’dovich (tSZ) effect. The cross-correlation of gravitational lensing (which traces the projected mass) with the tSZ effect (which traces the projected gas pressure) is a powerful probe of the thermal state of ionized baryons throughout the Universe and is sensitive to effects such as baryonic feedback. In a companion paper (Gatti et al. Phys. Rev. D 105, 123525 (2022)), we present tomographic measurements and validation tests of the cross-correlation between Galaxy shear measurements from the first three years of observations of the Dark Energy Survey and tSZ measurements from a combination of Atacama Cosmology Telescope and Planck observations. In this work, we use the same measurements to constrain models for the pressure profiles of halos across a wide range of halo mass and redshift. We find evidence for reduced pressure in low-mass halos, consistent with predictions for the effects of feedback from active Galactic nuclei. We infer the hydrostatic mass bias (B≡M500c/MSZ) from our measurements, finding B=1.8±0.1 when adopting the Planck-preferred cosmological parameters. We additionally find that our measurements are consistent with a nonzero redshift evolution of B, with the correct sign and sufficient magnitude to explain the mass bias necessary to reconcile cluster count measurements with the Planck-preferred cosmology. Our analysis introduces a model for the impact of intrinsic alignments (IAs) of galaxy shapes on the shear-tSZ correlation. We show that IA can have a significant impact on these correlations at current noise levels.

关键词： Cosmic microwave background Cosmological parameters Gravitational lenses Large scale structure of the Universe Optical, UV, & IR astronomy

Modeling global dynamics from local snapshots with deep generative neural networks

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arXiv 2018年

作者： Gigante, Scott Van Dijk, David Moon, Kevin R. Strzalkowski, Alexander Wolf, Guy Krishnaswamy, Smita Computational Biology and Bioinformatics Program Yale University New HavenCT United States Department of Genetics Yale University New HavenCT United States Department of Computer Science Yale University New HavenCT United States Department of Mathematics and Statistics Utah State University LoganUT United States Department of Computer Science Princeton University PrincetonNJ United States Department of Mathematics and Statistics Universitéde Montreacuteal MontéalQC Canada

Complex high dimensional stochastic dynamic systems arise in many applications in the natural sciences and especially biology. However, while these systems are difficult to describe analytically, "snapshot" measurements that sample the output of the system are often available. In order to model the dynamics of such systems given snapshot data, or local transitions, we present a deep neural network framework we call Dynamics Modeling Network or DyMoN. DyMoN is a neural network framework trained as a deep generative Markov model whose next state is a probability distribution based on the current state. DyMoN is trained using samples of current and nextstate pairs, and thus does not require longitudinal measurements. We show the advantage of DyMoN over shallow models such as Kalman filters and hidden Markov models, and other deep models such as recurrent neural networks in its ability to embody the dynamics (which can be studied via perturbation of the neural network) and generate longitudinal hypothetical trajectories. We perform three case studies in which we apply DyMoN to different types of biological systems and extract features of the dynamics in each case by examining the learned model. Copyright © 2018, The Authors. All rights reserved.

关键词： Deep neural networks

Low-Frequency variability in massive stars: Core generation or surface phenomenon?

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arXiv 2019年

作者： Lecoanet, Daniel Cantiello, Matteo Quataert, Eliot Couston, Louis-Alexandre Burns, Keaton J. Pope, Benjamin J.S. Jermyn, Adam S. Favier, Benjamin Le Bars, Michael Princeton Center for Theoretical Science Princeton University PrincetonNJ08544 United States Department of Astrophysical Sciences Princeton Univesity PrincetonNJ08544 United States Center for Computational Astrophysics Flatiron Institute New YorkNY10010 United States Astronomy Department Theoretical Astrophysics Center University of California BerkeleyCA94720 United States Cnrs Aix Marseille Université Centrale Marseille Irphe Umr 7342 Marseille13013 France British Antarctic Survey Cambridge CB3 0ET United Kingdom Department of Applied Mathematics and Theoretical Physics University of Cambridge CambridgeCB3 0WA United Kingdom Department of Mathematics Massachusetts Institute of Technology CambridgeMA02139 United States Center for Cosmology and Particle Physics Department of Physics New York University New YorkNY10003 United States Center for Data Science New York University New YorkNY10011 United States

Bowman et al. (2019a) reported low-frequency photometric variability in 164 O- and B-type stars observed with K2 and TESS. They interpret these motions as internal gravity waves, which could be excited stochastically by convection in the cores of these stars. The detection of internal gravity waves in massive stars would help distinguish between massive stars with convective or radiative cores, determine core size, and would provide important constraints on massive star structure and evolution. In this work, we study the observational signature of internal gravity waves generated by core convection. We calculate the wave transfer function, which links the internal gravity wave amplitude at the base of the radiative zone to the surface luminosity variation. This transfer function varies by many orders of magnitude for frequencies 1, and has regularly-spaced peaks near 1d"1due to standing modes. This is inconsistent with the observed spectra which have smooth "red noise" profiles, without the predicted regularly-spaced peaks. The wave transfer function is only meaningful if the waves stay predominatly linear. We next show that this is the case: low frequency traveling waves do not break unless their luminosity exceeds the radiative luminosity of the star;and, the observed luminosity fluctuations at high frequencies are so small that standing modes would be stable to nonlinear instability. These simple calculations suggest that the observed low-frequency photometric variability in massive stars is not due to internal gravity waves generated in the core of these stars. We finish with a discussion of (sub)surface convection, which produces low-frequency variability in low-mass stars, very similar to that observed in Bowman et al. (2019a) in higher mass stars. Copyright © 2019, The Authors. All rights reserved.

关键词： Luminance

Inversion problems for fourier transforms of particle distributions

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arXiv 2018年

作者： Kim, Jaeuk Zhang, Ge Stillinger, Frank H. Torquato, Salvatore Department of Physics Princeton University PrincetonNJ08544 United States Department of Physics and Astronomy University of Pennsylvania PhiladelphiaPA19104-6396 United States Department of Chemistry Princeton University PrincetonNJ08544 United States Princeton Institute for the Science and Technology of Materials Princeton University PrincetonNJ08544 United States Program in Applied and Computational Mathematics Princeton University PrincetonNJ08544 United States

Collective coordinates in a many-particle system are complex Fourier components of the particle density n(x) ≡ PNj=1 δ(x − rj), and often provide useful physical insights. However, given collective coordinates, it is desirable to infer the particle coordinates via inverse transformations. In principle, a sufficiently large set of collective coordinates are equivalent to particle coordinates, but the nonlinear relation between collective and particle coordinates makes the inversion procedure highly nontrivial. Given a "target" configuration in one-dimensional Euclidean space, we investigate the minimal set of its collective coordinates that can be uniquely inverted into particle coordinates. For this purpose, we treat a finite number M of the real and/or the imaginary parts of collective coordinates of the target configuration as constraints, and then reconstruct "solution" configurations whose collective coordinates satisfy these constraints. Both theoretical and numerical investigations reveal that the number of numerically distinct solutions depends sensitively on the chosen collective-coordinate constraints and target configurations. From detailed analysis, we conclude that collective coordinates at the dN2 e smallest wavevectors is the minimal set of constraints for unique inversion, where d·e represents the ceiling function. This result provides useful groundwork to the inverse transform of collective coordinates in higher-dimensional systems. Copyright © 2018, The Authors. All rights reserved.

关键词： Inverse transforms