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arXiv 2017年

作者： Andén, Joakim Singer, Amit Center for Computational Biology Flatiron Institute New YorkNY United States Department of Mathematics and Program in Applied and Computational Mathematics Princeton University NJ United States

In cryo-electron microscopy, the 3D electric potentials of an ensemble of molecules are projected along arbitrary viewing directions to yield noisy 2D images. The volume maps representing these potentials typically exhibit a great deal of structural variability, which is described by their 3D covariance matrix. Typically, this covariance matrix is approximately low-rank and can be used to cluster the volumes or estimate the intrinsic geometry of the conformation space. We formulate the estimation of this covariance matrix as a linear inverse problem, yielding a consistent least-squares estimator. For n images of size N-by-N pixels, we propose an algorithm for calculating this covariance estimator with computational complexity O(nN4 + √κN6 log N), where the condition number κ is empirically in the range 10-200. Its efficiency relies on the observation that the normal equations are equivalent to a deconvolution problem in 6D. This is then solved by the conjugate gradient method with an appropriate circulant preconditioner. The result is the first computationally efficient algorithm for consistent estimation of the 3D covariance from noisy projections. It also compares favorably in runtime with respect to previously proposed non-consistent estimators. Motivated by the recent success of eigenvalue shrinkage procedures for high-dimensional covariance matrix estimation, we incorporate a shrinkage procedure that improves accuracy at lower signal-to-noise ratios. We evaluate our methods on simulated datasets and achieve classification results comparable to state-of-the-art methods in shorter running time. We also present results on clustering volumes in an experimental dataset, illustrating the power of the proposed algorithm for practical determination of structural variability.92C55, 68U10, 44A12, 65R32, 62G05, 62H30, 62J10, 62J07 Copyright © 2017, The Authors. All rights reserved.

关键词： Conjugate gradient method

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Manifold learning-based methods for analyzing single-cell RNA-sequencing data

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Current Opinion in Systems Biology 2018年 7卷 36-46页

作者： Moon, Kevin R. Stanley, Jay S. Burkhardt, Daniel van Dijk, David Wolf, Guy Krishnaswamy, Smita Department of Genetics Yale University New Haven CT United States Applied Mathematics Program Yale University New Haven CT United States Computational Biology and Bioinformatics Program Yale University New Haven CT United States Department of Computer Science Yale University New Haven CT United States

Recent advances in single-cell RNA sequencing technologies enable deep insights into cellular development, gene regulation, and phenotypic diversity by measuring gene expression for thousands of cells in a single experiment. While these technologies hold great potential for improving our understanding of cellular states and progression, they also pose new challenges and require advanced mathematical and algorithmic tools to extract underlying biological signals. In this review, we cover one of the most promising avenues of research into unlocking the potential of scRNA-seq data: the field of manifold learning, and the related manifold assumption in data analysis. Manifold learning provides a powerful structure for algorithmic approaches to process the data, extract its dynamics, and infer patterns in it. In particular, we cover manifold learning-based methods for denoising the data, revealing gene interactions, extracting pseudotime progressions with model fitting, visualizing the cellular state space via dimensionality reduction, and clustering the data. © 2017 Published by Elsevier Ltd.

关键词： Data mining Gene regulatory networks Imputation Manifold learning Pseudotime trends Single cell RNA sequencing

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MOMENT-BASED METRICS FOR MOLECULES COMPUTABLE FROM CRYO-EM IMAGES

arXiv

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arXiv 2024年

作者： Zhang, Andy Mickelin, Oscar Kileel, Joe Verbeke, Eric J. Marshall, Nicholas F. Gilles, Marc Aurèle Singer, Amit Program in Applied and Computational Mathematics Princeton University PrincetonNJ08544 United States Department of Mathematics Oden Institute University of Texas At Austin AustinTX78712 United States Department of Mathematics Oregon State University CorvallisOR97331 United States Department of Mathematics and Program in Applied and Computational Mathematics Princeton University PrincetonNJ08544 United States

Single particle cryogenic electron microscopy (cryo-EM) is an imaging technique capable of recovering the high-resolution 3-D structure of biological macromolecules from many noisy and randomly oriented projection images. One notable approach to 3-D reconstruction, known as Kam's method, relies on the moments of the 2-D images. Inspired by Kam's method, we introduce a rotationally invariant metric between two molecular structures, which does not require 3-D alignment. Further, we introduce a metric between a stack of projection images and a molecular structure, which is invariant to rotations and reflections and does not require performing 3-D reconstruction. Additionally, the latter metric does not assume a uniform distribution of viewing angles. We demonstrate uses of the new metrics on synthetic and experimental datasets, highlighting their ability to measure structural similarity. Copyright © 2024, The Authors. All rights reserved.

关键词： Method of moments

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Diffusion curvature for estimating local curvature in high dimensional data 22

Diffusion curvature for estimating local curvature in high d...

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Proceedings of the 36th International Conference on Neural Information Processing Systems

作者： Dhananjay Bhaskar Kincaid MacDonald Oluwadamilola Fasina Dawson Thomas Bastian Rieck Ian Adelstein Smita Krishnaswamy Department of Genetics Yale University Department of Mathematics Yale University Applied Mathematics Program Yale University Department of Mathematics Department of Physics Yale University Institute of AI for Health Helmholtz Pioneer Campus Helmholtz Munich Department of Computer Science Department of Genetics Applied Mathematics Program Program for Computational Biology and Bioinformatics Yale University

ISBN: (纸本)9781713871088

We introduce a new intrinsic measure of local curvature on point-cloud data called diffusion curvature. Our measure uses the framework of diffusion maps, including the data diffusion operator, to structure point cloud data and define local curvature based on the laziness of a random walk starting at a point or region of the data. We show that this laziness directly relates to volume comparison results from Riemannian geometry. We then extend this scalar curvature notion to an entire quadratic form using neural network estimations based on the diffusion map of point-cloud data. We show applications of both estimations on toy data, single-cell data and on estimating local Hessian matrices of neural network loss landscapes.

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Nonlocal orbital-free kinetic energy density functional for semiconductors

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Physical Review B 2010年第4期81卷 045206-045206页

作者： Chen Huang Emily A. Carter Department of Mechanical and Aerospace Engineering and Program in Applied and Computational Mathematics Princeton University Princeton New Jersey 08544-5263 USA

We propose a nonlocal kinetic energy density functional (KEDF) for semiconductors based on the expected asymptotic behavior of its susceptibility function. The KEDF’s kernel depends on both the electron density and the reduced density gradient, with an internal parameter formally related to the material’s static dielectric constant. We determine the accuracy of the KEDF within orbital-free density functional theory (DFT) by applying it to a variety of common semiconductors. With only two adjustable parameters, the KEDF reproduces quite well the exact noninteracting KEDF (i.e., Kohn-Sham DFT) predictions of bulk moduli, equilibrium volumes, and equilibrium energies. The two parameters in our KEDF are sensitive primarily to changes in the local crystal structure (such as atomic coordination number) and exhibit good transferability between different tetrahedrally-bonded phases. This local crystal structure dependence is rationalized by considering Thomas-Fermi dielectric screening theory.

关键词： .Phases orbital-free KEDF bulk moduli

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High-dimensional data fusion via joint manifold learning

High-dimensional data fusion via joint manifold learning

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2010 AAAI Fall Symposium

作者： Davenport, Mark A. Hegde, Chinmay Duarte, Marco F. Baraniuk, Richard G. Department of Statistics Stanford University Stanford CA 94305 United States Department of Electrical and Computer Engineering Rice University Houston TX 77005 United States Program in Applied and Computational Mathematics Princeton University Princeton NJ 08544 United States

ISBN: (纸本)9781577354888

The emergence of low-cost sensing architectures for diverse modalities has made it possible to deploy sensor networks that acquire large amounts of very high-dimensional data. To cope with such a data deluge, manifold models are often developed that provide a powerful theoretical and algorithmic framework for capturing the intrinsic structure of data governed by a low-dimensional set of ***, these models do not typically take into account dependencies among multiple sensors. We thus propose a new joint manifold framework for data ensembles that exploits such dependencies. We show that joint manifold structure can lead to improved performance for manifold learning. Additionally, we leverage recent results concerning random projections of manifolds to formulate a universal, network-scalable dimensionality reduction scheme that efficiently fuses the data from all sensors. Copyright © 2010, Association for the Advancement of Artificial Intelligence. All rights reserved.

关键词： Sensor networks

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MEAN FIELD GAMES WITH COMMON NOISE VIA MALLIAVIN CALCULUS

arXiv

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arXiv 2024年

作者： Tangpi, Ludovic Wang, Shichun Princeton University Department of Operations Research and Financial Engineering and Program in Computational and Applied Mathematics United States

We present a simpler proof of the existence of equilibria for a class of mean field games with common noise, where players interact through the conditional law given the current value of the common noise rather than its entire path. By extending a compactness criterion for Malliavin-differentiable random variables to processes, we establish existence of strong equilibria, where the conditional law and optimal control are adapted to the common noise filtration and defined on the original probability space. Notably, our approach only requires measurability of the drift and cost functionals with respect to the state *** Codes 91A16, 60H07 Copyright © 2024, The Authors. All rights reserved.

关键词： Cost functions

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Ab initio evaluation of Coulomb and exchange parameters for DFT+U calculations

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Physical Review B 2007年第15期76卷 155123-155123页

作者： Nicholas J. Mosey Emily A. Carter []Department of Mechanical and Aerospace Engineering and Program in Applied and Computational Mathematics Princeton University Princeton New Jersey 08544-5263 USA

Conventional density functional theory (DFT) fails for materials with strongly correlated electrons, such as late transition metal oxides. Large errors in the intra-atomic Coulomb and exchange interactions are the source of this failure. The DFT+U method has provided a means, through empirical parameters, to correct these errors. Here, we present a systematic ab initio approach in evaluating the intra-atomic Coulomb and exchange terms, U and J, respectively, in order to make the DFT+U method a fully first-principles technique. The method is based on a relationship between these terms and the Coulomb and exchange integrals evaluated in the basis of unrestricted Hartree-Fock molecular orbitals that represent localized states of the extended system. We used this ab initio scheme to evaluate U and J for chromia (Cr2O3). The resulting values are somewhat higher than those determined earlier either empirically or in constrained DFT calculations but have the advantage of originating from an ab initio theory containing exact exchange. Subsequent DFT+U calculations on chromia using the ab initio derived U and J yield properties consistent with experiment, unlike conventional DFT. Overall, the technique developed and tested in this work holds promise in enabling accurate and fully predictive DFT+U calculations of strongly correlated electron materials.

关键词： .electrons Empirical evaluation of Coulomb DFT calculations chromia

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Inferring System and Optimal Control Parameters of Closed-Loop Systems from Partial Observations 63

Inferring System and Optimal Control Parameters of Closed-Lo...

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63rd IEEE Conference on Decision and Control, CDC 2024

作者： Geadah, Victor Arbelaiz, Juncal Ritz, Harrison Daw, Nathaniel D. Cohen, Jonathan D. Pillow, Jonathan W. Princeton University Department of Mechanical and Aerospace Engineering PrincetonNJ United States Princeton University Program in Applied and Computational Mathematics PrincetonNJ United States Princeton Neuroscience Institute Princeton University PrincetonNJ United States

ISBN: (纸本)9798350316339

We consider the joint problem of system identification and inverse optimal control for discrete-time stochastic Linear Quadratic Regulators. We analyze finite and infinite time horizons in a partially observed setting, where the state is observed noisily. To recover closed-loop system parameters, we develop inference methods based on probabilistic statespace model (SSM) techniques. First, we show that the system parameters exhibit non-identifiability in the infinite-horizon from closed-loop measurements, and we provide exact and numerical methods to disentangle the parameters. Second, to improve parameter identifiability, we show that we can further enhance recovery by either (1) incorporating additional partial measurements of the control signals or (2) moving to the finitehorizon setting. We further illustrate the performance of our methodology through numerical examples. © 2024 IEEE.

关键词： Stochastic systems

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Understanding Rollover Data

Understanding Rollover Data

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IEEE Conference on Computer Communications Workshops

作者： Liang Zheng Carlee Joe-Wong Department of Electrical Engineering Princeton University USA Program in Applied and Computational Mathematics Princeton University USA

ISBN: (纸本)9781467399562

Internet service providers (ISPs) have been facing heavy competition to attract more users in the mobile data market, along with growing operational costs. Most mobile data plans charge users a fixed fee for a monthly data quota, and any unused data at the end of each month will be wasted. In the beginning of 2015, both AT&T and T-Mobile reinstated rollover data plans with constrained eligibility. Users could then keep the unused portion of their monthly data quotas for use in a future month(s). In this work, we evaluate the benefits of rollover data for both ISPs and users as well as identify the types of users who would upgrade to rollover data plans. To do so, we consider two alternatives: a smaller data plan without rollover data, and a more expensive data plan with larger monthly quota that does include rollover data. We find that moderately price-sensitive users benefit from rollover data: price-insensitive users would not have any data to rollover, while extremely price-sensitive users would not find the rollover data worth its higher cost. Under specific bounds on the size of the rollover data plan's monthly quota, we find that users can obtain more utility and ISPs can profit from rollover data, which is verified on one year of usage data from a U.S. ISP.

关键词： Mobile communication Data models Pricing Conferences Switches Aggregates Computers

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