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mathematics for cryo-electron microscopy

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arXiv 2018年

作者： Singer, Amit Department of Mathematics Program in Applied and Computational Mathematics Princeton University

Single-particle cryo-electron microscopy (cryo-EM) has recently joined X-ray crystallography and NMR spectroscopy as a high-resolution structural method for biological macromolecules. Cryo-EM was selected by Nature Methods as Method of the Year 2015, large scale investments in cryo-EM facilities are being made all over the world, and the Nobel Prize in Chemistry 2017 was awarded to Jacques Dubochet, Joachim Frank and Richard Henderson "for developing cryo-electron microscopy for the high-resolution structure determination of biomolecules in solution". This paper focuses on the mathematical principles underlying existing algorithms for structure determination using single particle cryo-EM.92C55, 68U10, 44A12, 62H12, 33C55, 90C22 Copyright © 2018, The Authors. All rights reserved.

关键词： Electron microscopes

A FAST SIMPLE ALGORITHM FOR COMPUTING THE POTENTIAL OF CHARGES ON A LINE

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arXiv 2019年

作者： Gimbutas, Zydrunas Marshall, Nicholas F. Rokhlin, Vladimir National Institute of Standards and Technology Boulder CO80305 United States Department of Mathematics Princeton University Princeton NJ08540 United States Program in Applied Mathematics Yale University New HavenCT06511 United States

We present a fast method for evaluating expressions of the form (Equation presented). where αi are real numbers, and xi are points in a compact interval of . This expression can be viewed as representing the electrostatic potential generated by charges on a line in 3. While fast algorithms for computing the electrostatic potential of general distributions of charges in 3 exist, in a number of situations in computational physics it is useful to have a simple and extremely fast method for evaluating the potential of charges on a line;we present such a method in this paper, and report numerical results for several examples. Copyright © 2019, The Authors. All rights reserved.

关键词： Numerical methods

Inferring a network from dynamical signals at its nodes

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arXiv 2020年

作者： Weistuch, Corey Agozzino, Luca Mujica-Parodi, Lilianne R. Dill, Ken Laufer Center for Physical and Quantitative Biology Stony Brook University Department of Applied Mathematics and Statistics Stony Brook University Department of Physics and Astronomy Stony Brook University Department of Biomedical Engineering Stony Brook University Program in Neuroscience Stony Brook University Athinoula A. Martinos Center for Biomedical Imaging Massachusetts General Hospital Harvard Medical School Department of Chemistry Stony Brook University

We give an approximate solution to the difficult inverse problem of inferring the topology of an unknown network from given time-dependent signals at the nodes. For example, we measure signals from individual neurons in the brain, and infer how they are inter-connected. We use Maximum Caliber as an inference principle. The combinatorial challenge of high-dimensional data is handled using two different approximations to the pairwise couplings. We show two proofs of principle: in a nonlinear genetic toggle switch circuit, and in a toy neural network. Copyright © 2020, The Authors. All rights reserved.

关键词： Toys

Training of Physical Neural Networks

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arXiv 2024年

作者： Momeni, Ali Rahmani, Babak Scellier, Benjamin Wright, Logan G. McMahon, Peter L. Wanjura, Clara C. Li, Yuhang Skalli, Anas Berloff, Natalia G. Onodera, Tatsuhiro Oguz, Ilker Morichetti, Francesco del Hougne, Philipp Le Gallo, Manuel Sebastian, Abu Mirhoseini, Azalia Zhang, Cheng Marković, Danijela Brunner, Daniel Moser, Christophe Gigan, Sylvain Marquardt, Florian Ozcan, Aydogan Grollier, Julie Liu, Andrea J. Psaltis, Demetri Alù, Andrea Fleury, Romain Lausanne Switzerland Microsoft Research 198 Cambridge Science Park CambridgeCB4 0AB United Kingdom Rain AI San Francisco United States Department of Applied Physics Yale University CT United States School of Applied and Engineering Physics Cornell University IthacaNY14853 United States Max Planck Institute for the Science of Light Staudtstraße 2 Erlangen91058 Germany Department of Electrical and Computer Engineering University of California Los AngelesCA90095 United States FEMTO-ST Institute Optics Department CNRS University Bourgogne Franche-Comté Besançon25030 Cedex France Department of Applied Mathematics and Theoretical Physics University of Cambridge Cambridge United Kingdom NTT Physics and Informatics Laboratories NTT Research Inc. Sunnyvale United States Lausanne Switzerland Dipartimento di Elettronica Informazione e Bioingegneria Politecnico di Milano Milan Italy Univ Rennes CNRS IETR UMR 6164 RennesF-35000 France IBM Research Europe– Zurich Rüschlikon8803 Switzerland Department of Computer Science Stanford University United States Google DeepMind 1600 Amphitheatre Parkway Mountain ViewCA94043 United States Unité Mixte de Physique CNRS/Thales CNRS Thales Université Paris-Saclay Palaiseau France Laboratoire Kastler Brossel Sorbonne Université École Normale Supérieure Collège de France CNRS UMR 8552 Paris France Laboratoire Albert Fert CNRS Thales UniversitéParis-Saclay Palaiseau91767 France Department of Physics and Astronomy University of Pennsylvania PhiladelphiaPA19104 United States Lausanne Switzerland Photonics Initiative Advanced Science Research Center City University of New York New YorkNY10031 United States Physics Program Graduate Center City University of New York New YorkNY10016 United States

Physical neural networks (PNNs) are a class of neural-like networks that leverage the properties of physical systems to perform computation. While PNNs are so far a niche research area with small-scale laboratory demonstrations, they are arguably one of the most underappreciated important opportunities in modern artificial intelligence (AI). Could we train AI models 1000x larger than current ones? Could we do this and also have them perform inference locally and privately on edge devices, such as smartphones or sensors? Research over the past few years has shown that the answer to all these questions is likely "textityes, with enough research": PNNs could one day radically change what is possible and practical for AI systems. To do this will however require rethinking both how AI models work, and how they are trained – primarily by considering the problems through the constraints of the underlying hardware physics. To train PNNs at large scale, many methods including backpropagation-based and backpropagation-free approaches are now being explored. These methods have various trade-offs, and so far no method has been shown to scale to the same scale and performance as the backpropagation algorithm widely used in deep learning today. However, this is rapidly changing, and a diverse ecosystem of training techniques provides clues for how PNNs may one day be utilized to create both more efficient realizations of current-scale AI models, and to enable unprecedented-scale models. Copyright © 2024, The Authors. All rights reserved.

关键词： Deep learning

Self-isolation or borders closing: What prevents the spread of the epidemic better?

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Physical Review E 2020年第1期102卷 010401(R)-010401(R)页

作者： O. Valba V. Avetisov A. Gorsky S. Nechaev Department of Applied Mathematics National Research University Higher School of Economics 101000 Moscow Russia Federal Research Center of Chemical Physics RAS 119991 Moscow Russia Institute of Information Transmission Problems RAS 127051 Moscow Russia Moscow Institute of Physics and Technology Dolgoprudny 141700 Russia Interdisciplinary Scientific Center Poncelet CNRS UMI 2615 119002 Moscow Russia P.N. Lebedev Physical Institute RAS 119991 Moscow Russia

Pandemic propagation of COVID-19 motivated us to discuss the impact of the human network clustering on epidemic spreading. Today, there are two clustering mechanisms which prevent of uncontrolled disease propagation in a connected network: an “internal” clustering, which mimics self-isolation (SI) in local naturally arranged communities, and an “external” clustering, which looks like a sharp frontiers closing (FC) between cities and countries, and which does not care about the natural connections of network agents. SI networks are “evolutionarily grown” under the condition of maximization of small cliques in the entire network, while FC networks are instantly created. Running the standard SIR model on clustered SI and FC networks, we demonstrate that the evolutionary grown clustered network prevents the spread of an epidemic better than the instantly clustered network with similar parameters. We find that SI networks have the scale-free property for the degree distribution P(k)∼kη, with a small critical exponent −2<η<−1. We argue that the scale-free behavior emerges as a result of the randomness in the initial degree distributions.

关键词： Clustering Diseases Evolving networks

On Log-q-Gaussian Distribution

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Calcutta Statistical Association Bulletin 2018年第2期70卷 105-121页

作者： Koike, Ken-Ichi Shimegi, Yuya Department of Mathematics Faculty of Pure and Applied Sciences University of Tsukuba Ibaraki Tsukuba Japan Master’s Program in Education University of Tsukuba Ibaraki Tsukuba Japan

We propose the log-q-Gaussian distribution which is obtained as the distribution of a random variable whose logarithm is q-Gaussian. Various types of properties of the new distribution are given such as the moments, the cumulative distribution function, the extreme value distribution, the likelihood estimation, and so on. Some examples for real data are also given. AMS Classification: 60Exx 62Exx. © 2018 Calcutta Statistical Association, Kolkata.

关键词： extreme value distribution log-normal distribution Log-q-Gaussian distribution q-Gaussian distribution

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Physical Review E 2019年第5期99卷 052414-052414页

作者： Cody Baker Christopher Ebsch Ilan Lampl Robert Rosenbaum Department of Applied and Computational Mathematics and Statistics University of Notre Dame Notre Dame Indiana 46556 USA Department of Neurobiology Weizmann Institute of Science Rehovot 7610001 Israel Interdisciplinary Center for Network Science and Applications University of Notre Dame Notre Dame Indiana 46556 USA

Understanding the magnitude and structure of interneuronal correlations and their relationship to synaptic connectivity structure is an important and difficult problem in computational neuroscience. Early studies show that neuronal network models with excitatory-inhibitory balance naturally create very weak spike train correlations, defining the “asynchronous state.” Later work showed that, under some connectivity structures, balanced networks can produce larger correlations between some neuron pairs, even when the average correlation is very small. All of these previous studies assume that the local network receives feedforward synaptic input from a population of uncorrelated spike trains. We show that when spike trains providing feedforward input are correlated, the downstream recurrent network produces much larger correlations. We provide an in-depth analysis of the resulting “correlated state” in balanced networks and show that, unlike the asynchronous state, it produces a tight excitatory-inhibitory balance consistent with in vivo cortical recordings.

关键词： Neuronal dynamics Spiking neurons Cortical networks Exponential integrate-and-fire model Spiking neuron models

On the cross-validation bias due to unsupervised preprocessing

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arXiv 2019年

作者： Moscovich, Amit Rosset, Saharon Program in Applied and Computational Mathematics Princeton University PrincetonNJ United States Department of Statistics and Operations Research Tel-Aviv University Tel-Aviv Israel

Cross-validation is the de facto standard for predictive model evaluation and selection. In proper use, it provides an unbiased estimate of a model's predictive performance. However, data sets often undergo various forms of data-dependent preprocessing, such as mean-centering, rescaling, dimensionality reduction, and outlier removal. It is often believed that such preprocessing stages, if done in an unsupervised manner (that does not incorporate the class labels or response values) are generally safe to do prior to cross-validation. In this paper, we study three commonly-practiced preprocessing procedures prior to a regression analysis: (i) variance-based feature selection;(ii) grouping of rare categorical features;and (iii) feature rescaling. We demonstrate that unsupervised preprocessing can, in fact, introduce a substantial bias into cross-validation estimates and potentially hurt model selection. This bias may be either positive or negative and its exact magnitude depends on all the parameters of the problem in an intricate manner. Further research is needed to understand the real-world impact of this bias across different application domains, particularly when dealing with small sample sizes and high-dimensional *** Codes 62-07 Copyright © 2019, The Authors. All rights reserved.

关键词： Regression analysis

Universal approximation of symmetric and anti-symmetric functions

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arXiv 2019年

作者： Han, Jiequn Li, Yingzhou Lin, Lin Lu, Jianfeng Zhang, Jiefu Zhang, Linfeng Department of Mathematics Princeton University PrincetonNJ08544 United States Department of Mathematics Duke University DurhamNC27708 United States Department of Mathematics University of California BerkeleyCA94720 United States Computational Research Division Lawrence Berkeley National Laboratory BerkeleyCA94720 United States Department of Mathematics Department of Physics Department of Chemistry Duke University DurhamNC27708 United States Department of Mathematics University of California BerkeleyCA94720 United States Program in Applied and Computational Mathematics Princeton University PrincetonNJ08544 United States

We consider universal approximations of symmetric and anti-symmetric functions, which are important for applications in quantum physics, as well as other scientific and engineering computations. We give constructive approximations with explicit bounds on the number of parameters with respect to the dimension and the target accuracy ǫ. While the approximation still suffers from curse of dimensionality, to the best of our knowledge, these are first results in the literature with explicit error bounds. Moreover, we also discuss neural network architecture that can be suitable for approximating symmetric and anti-symmetric functions. Copyright © 2019, The Authors. All rights reserved.

关键词： Network architecture