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Hyperbolic procrustes analysis using riemannian geometry 21

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Hyperbolic procrustes analysis using riemannian geometry

Proceedings of the 35th International Conference on Neural Information Processing Systems

作者： Ya-Wei Eileen Lin Yuval Kluger Ronen Talmon Viterbi Faculty of Electrical and Computer Engineering Technion Program in Applied Mathematics Yale University and Interdepartmental Program in Computational Biology and Bioinformatics Yale University and Department of Pathology Yale University

ISBN: (纸本)9781713845393

Label-free alignment between datasets collected at different times, locations, or by different instruments is a fundamental scientific task. Hyperbolic spaces have recently provided a fruitful foundation for the development of informative representations of hierarchical data. Here, we take a purely geometric approach for label-free alignment of hierarchical datasets and introduce hyperbolic Procrustes analysis (HPA). HPA consists of new implementations of the three prototypical Procrustes analysis components: translation, scaling, and rotation, based on the Riemannian geometry of the Lorentz model of hyperbolic space. We analyze the proposed components, highlighting their useful properties for alignment. The efficacy of HPA, its theoretical properties, stability and computational efficiency are demonstrated in simulations. In addition, we showcase its performance on three batch correction tasks involving gene expression and mass cytometry data. Specifically, we demonstrate high-quality unsupervised batch effect removal from data acquired at different sites and with different technologies that outperforms recent methods for label-free alignment in hyperbolic spaces.

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Scaling limits of external multi-particle DLA on the plane and the supercooled stefan problem

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arXiv 2021年

作者： Nadtochiy, Sergey Shkolnikov, Mykhaylo Zhang, Xiling Department of Applied Mathematics Illinois Institute of Technology ChicagoIL60616 United States ORFE Department Bendheim Center for Finance Program in Applied & Computational Mathematics Princeton University PrincetonNJ08544 United States

We consider (a variant of) the external multi-particle diffusion-limited aggregation (MDLA) process of Rosenstock and Marquardt on the plane. Based on the recent findings of [11], [10] in one space dimension it is natural to conjecture that the scaling limit of the growing aggregate in such a model is given by the growing solid phase in a suitable "probabilistic" formulation of the single-phase supercooled Stefan problem for the heat equation. To address this conjecture, we extend the probabilistic formulation from [10] to multiple space dimensions. We then show that the equation that characterizes the growth rate of the solid phase in the supercooled Stefan problem is satisfied by the scaling limit of the external MDLA process with an inequality, which can be strict in general. In the course of the proof, we establish two additional results interesting in their own right: (i) the stability of a "crossing property" of planar Brownian motion and (ii) a rigorous connection between the probabilistic solutions to the supercooled Stefan problem and its classical and weak *** Codes 82C22, 80A22, 60H30 Copyright © 2021, The Authors. All rights reserved.

关键词： Supercooling

Hyper-molecules: On the representation and recovery of dynamical structures, with application to flexible macro-molecular structures in cryo-EM

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arXiv 2019年

作者： Lederman, Roy R. Andén, Joakim Singer, Amit Department of Statistics and Data Science Yale University New HavenCT United States Center for Computational Mathematics Flatiron Institute New YorkNY United States Department of Mathematics and Program in Applied and Computational Mathematics Princeton University PrincetonNJ United States

Cryo-electron microscopy (cryo-EM), the subject of the 2017 Nobel Prize in Chemistry, is a technology for determining the 3-D structure of macromolecules from many noisy 2-D projections of instances of these macromolecules, whose orientations and positions are unknown. The molecular structures are not rigid objects, but flexible objects involved in dynamical processes. The different conformations are exhibited by different instances of the macromolecule observed in a cryo-EM experiment, each of which is recorded as a particle image. The range of conformations and the conformation of each particle are not known a priori;one of the great promises of cryo-EM is to map this conformation space. Remarkable progress has been made in determining rigid structures from homogeneous samples of molecules in spite of the unknown orientation of each particle image and significant progress has been made in recovering a few distinct states from mixtures of rather distinct conformations, but more complex heterogeneous samples remain a major challenge. We introduce the "hyper-molecule" framework for modeling structures across different states of heterogeneous molecules, including continuums of states. The key idea behind this framework is representing heterogeneous macromolecules as high-dimensional objects, with the additional dimensions representing the conformation space. This idea is then refined to model properties such as localized heterogeneity. In addition, we introduce an algorithmic framework for recovering such maps of heterogeneous objects from experimental data using a Bayesian formulation of the problem and Markov chain Monte Carlo (MCMC) algorithms to address the computational challenges in recovering these high dimensional hyper-molecules. We demonstrate these ideas in a prototype applied to synthetic data. Copyright © 2019, The Authors. All rights reserved.

关键词： Medical imaging

Elastic and Thermodynamic Properties of Complex Mg-Al Intermetallic Compounds via Orbital-Free Density Functional Theory

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Physical Review applied 2016年第6期5卷 064021-064021页

作者： Houlong Zhuang Mohan Chen Emily A. Carter Department of Mechanical and Aerospace Engineering Princeton University Princeton New Jersey 08544 USA Program in Applied and Computational Mathematics and Andlinger Center for Energy and the Environment Princeton University Princeton New Jersey 08544 USA

Magnesium-aluminum (Mg-Al) alloys are important metal alloys with a wide range of engineering applications. We investigate the elastic and thermodynamic properties of Mg, Al, and four stoichiometric Mg-Al compounds including Mg17Al12, Mg13Al14, and Mg23Al30, and MgAl2 with orbital-free density-functional theory (OFDFT). We first calculate the lattice constants, zero-temperature formation energy, and independent elastic constants of these six materials and compare the results to those computed via Kohn-Sham DFT (KSDFT) benchmarks. We obtain excellent agreement between these two methods. Our calculated elastic constants of hexagonal close-packed Mg and face-centered-cubic Al are also consistent with available experimental data. We next compute their phonon spectra using the force constants extracted from the very fast OFDFT calculations, because such calculations are computationally challenging using KSDFT. This is especially the case for the Mg23Al30 compound, whose 3×3×3 supercell consists of 1431 atoms. We finally employ the quasiharmonic approximation to investigate temperature-dependent thermodynamic properties, including formation energies, heat capacities, and thermal expansion of the four Mg-Al intermetallic compounds. The calculated heat capacity and thermal expansion of both Mg and Al agree well with experimental data. We additionally find that Mg13Al14 and MgAl2 are both unstable, consistent with their absence from the equilibrium Mg-Al phase diagram. Our work demonstrates that OFDFT is an efficient and accurate quantum-mechanical computational tool for predicting elastic and thermodynamic properties of complicated Mg-Al alloys and also should be applicable to many other engineering alloys.

关键词： Elastic deformation Elastic modulus First-principles calculations Specific heat Thermal expansion Alloys

Quantifying behavioral data sets of criminal activity

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Quantifying behavioral data sets of criminal activity

2010 AAAI Spring Symposium

作者： Toole, Jameson Eagle, Nathan Plotkin, Joshua Department of Physics University of Michigan Ann Arbor MI 48109 United States Santa Fe Institute 1399 Hyde Park Rd Santa Fe NM 87501 United States Department of Biology and Program in Applied Mathematics and Computational Science University of Pennsylvania Philadelphia PA 19104 United States

ISBN: (纸本)9781577354550

With the increased availability of rich behavioral data sets, we present a novel combination of tools to analyze to analyze this information. Using criminal offense records as an example, we employ cross-correlation measures, eigenvalue spectrum analysis, and results from random matrix theory to identify spatiotemporal patterns. Finally, with multivariate autoregressive models, we demonstrate a possible source of structure within the data. instructions. © 2010, Association for the Advancement of Artificial Intelligence.

关键词： Spectrum analysis

Multi-target detection with application to cryo-electron microscopy

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arXiv 2019年

作者： Bendory, Tamir Boumal, Nicolas Leeb, William Levin, Eitan Singer, Amit Program in Applied and Computational Mathematics Princeton University PrincetonNJ United States Department of Mathematics Princeton University PrincetonNJ United States School of Mathematics University of Minnesota MinneapolisMN United States

We consider the multi-target detection problem of recovering a set of signals that appear multiple times at unknown locations in a noisy measurement. In the low noise regime, one can estimate the signals by first detecting occurrences, then clustering and averaging them. In the high noise regime, however, neither detection nor clustering can be performed reliably, so that strategies along these lines are destined to fail. Notwithstanding, using autocorrelation analysis, we show that the impossibility to detect and cluster signal occurrences in the presence of high noise does not necessarily preclude signal estimation. Specifically, to estimate the signals, we derive simple relations between the autocorrelations of the observation and those of the signals. These autocorrelations can be estimated accurately at any noise level given a sufficiently long measurement. To recover the signals from the observed autocorrelations, we solve a set of polynomial equations through nonlinear least-squares. We provide analysis regarding well-posedness of the task, and demonstrate numerically the effectiveness of the method in a variety of settings. The main goal of this work is to provide theoretical and numerical support for a recently proposed framework to image 3-D structures of biological macromolecules using cryo-electron microscopy in extreme noise levels. Copyright © 2019, The Authors. All rights reserved.

关键词： Autocorrelation

DeePMD-kit: A deep learning package for many-body potential energy representation and molecular dynamics

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arXiv 2017年

作者： Wang, Han Zhang, Linfeng Han, Jiequn Weinan, E. Institute of Applied Physics and Computational Mathematics Fenghao East Road 2 Beijing100094 China CAEP Software Center for High Performance Numerical Simulation Huayuan Road 6 Beijing100088 China Program in Applied and Computational Mathematics Princeton University PrincetonNJ08544 United States Department of Mathematics and Program in Applied and Computational Mathematics Princeton University PrincetonNJ08544 United States Center for Data Science Beijing International Center for Mathematical Research Peking University Beijing Institute of Big Data Research Beijing100871 China

Recent developments in many-body potential energy representation via deep learning have brought new hopes to addressing the accuracy-versus-efficiency dilemma in molecular simulations. Here we describe DeePMD-kit, a package written in Python/C++ that has been designed to minimize the effort required to build deep learning based representation of potential energy and force field and to perform molecular dynamics. Potential applications of DeePMD-kit span from finite molecules to extended systems and from metallic systems to chemically bonded systems. DeePMD-kit is interfaced with TensorFlow, one of the most popular deep learning frameworks, making the training process highly automatic and efficient. On the other end, DeePMD-kit is interfaced with high-performance classical molecular dynamics and quantum (path-integral) molecular dynamics packages, i.e., LAMMPS and the i-PI, respectively. Thus, upon training, the potential energy and force field models can be used to perform efficient molecular simulations for different purposes. As an example of the many potential applications of the package, we use DeePMD-kit to learn the interatomic potential energy and forces of a water model using data obtained from density functional theory. We demonstrate that the resulted molecular dynamics model reproduces accurately the structural information contained in the original model. Copyright © 2017, The Authors. All rights reserved.

关键词： Potential energy

Towards a mathematical understanding of neural network-based machine learning: What we know and what we don't

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arXiv 2020年

作者： Weinan, E. Ma, Chao Wojtowytsch, Stephan Wu, Lei Department of Mathematics Princeton University United States Program in Applied and Computational Mathematics Princeton University United States Department of Mathematics Stanford University United States Beijing Institute of Big Data Research China

Unified Variational Approach Description of Ground-State Phases of the Two-Dimensional Electron Gas

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Physical Review Letters 2024年第26期133卷 266504-266504页

作者： Conor Smith Yixiao Chen (陈一潇) Ryan Levy Yubo Yang (杨煜波) Miguel A. Morales Shiwei Zhang Center for Computational Quantum Physics Flatiron Institute New York New York 10010 USA Department of Electrical and Computer Engineering University of New Mexico Albuquerque New Mexico 87131 USA Program in Applied and Computational Mathematics Princeton University Princeton New Jersey 08544 USA

The two-dimensional electron gas (2DEG) is a fundamental model, which is drawing increasing interest because of recent advances in experimental and theoretical studies of 2D materials. Current understanding of the ground state of the 2DEG relies on quantum Monte Carlo calculations, based on variational comparisons of different Ansätze for different phases. We use a single variational ansatz, a general backflow-type wave function using a message-passing neural quantum state architecture, for a unified description across the entire density range. The variational optimization consistently leads to lower ground-state energies than previous best results. Transition into a Wigner crystal (WC) phase occurs automatically at rs=37±1, a density lower than currently believed. Between the liquid and WC phases, the same ansatz and variational search strongly suggest the existence of intermediate states in a broad range of densities, with enhanced short-range nematic spin correlations.

关键词： Wave functions