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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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Robust camera location estimation by convex programming

Robust camera location estimation by convex programming

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Conference on Computer Vision and Pattern Recognition (CVPR)

作者： Onur Özyeşil Amit Singer Program in Applied and Computational Mathematics Princeton University Princeton NJ USA Department of Mathematics Princeton University Princeton NJ USA

3D structure recovery from a collection of 2D images requires the estimation of the camera locations and orientations, i.e. the camera motion. For large, irregular collections of images, existing methods for the location estimation part, which can be formulated as the inverse problem of estimating n locations t 1 , t 2 , ..., tn in ℝ 3 from noisy measurements of a subset of the pairwise directions t i -t j /∥t i -t j ∥, are sensitive to outliers in direction measurements. In this paper, we firstly provide a complete characterization of well-posed instances of the location estimation problem, by presenting its relation to the existing theory of parallel rigidity. For robust estimation of camera locations, we introduce a two-step approach, comprised of a pairwise direction estimation method robust to outliers in point correspondences between image pairs, and a convex program to maintain robustness to outlier directions. In the presence of partially corrupted measurements, we empirically demonstrate that our convex formulation can even recover the locations exactly. Lastly, we demonstrate the utility of our formulations through experiments on Internet photo collections.

关键词： Estimation Cameras Robustness Noise measurement Three-dimensional displays Accuracy Minimization

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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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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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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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Algorithms and data structures for truncated hierarchical B-splines

Algorithms and data structures for truncated hierarchical B-...

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8th International Conference on Mathematical Methods for Curves and Surfaces, MMCS 2012

作者： Kiss, Gábor Giannelli, Carlotta Juẗtler, Bert Doctoral Program Computational Mathematics Johannes Kepler University Linz Altenberger Str. 69 4040 Linz Austria Institute of Applied Geometry Johannes Kepler University Linz Altenberger Str. 69 4040 Linz Austria

ISBN: (纸本)9783642543814

Tensor-product B-spline surfaces are commonly used as standard modeling tool in Computer Aided Geometric Design and for numerical simulation in Isogeometric Analysis. However, when considering tensor-product grids, there is no possibility of a localized mesh refinement without propagation of the refinement outside the region of interest. The recently introduced truncated hierarchical B-splines (THB-splines) [5] provide the possibility of a local and adaptive refinement procedure, while simultaneously preserving the partition of unity property. We present an effective implementation of the fundamental algorithms needed for the manipulation of THB-spline representations based on standard data structures. By combining a quadtree data structure - which is used to represent the nested sequence of subdomains - with a suitable data structure for sparse matrices, we obtain an efficient technique for the construction and evaluation of THB-splines. © Springer-Verlag 2014.

关键词： Data structures

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Geometry based data generation 32

Geometry based data generation

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32nd Conference on Neural Information Processing Systems, NeurIPS 2018

作者： Lindenbaum, Ofir Wolf, Guy Stanley, Jay S. Krishnaswamy, Smita Applied Mathematics Program Yale University New HavenCT06511 United States Computational Biology and Bioinformatics Program Yale University New HavenCT06510 United States Departments of Genetics and Computer Science Yale University New HavenCT06510 United States

We propose a new type of generative model for high-dimensional data that learns a manifold geometry of the data, rather than density, and can generate points evenly along this manifold. This is in contrast to existing generative models that represent data density, and are strongly affected by noise and other artifacts of data collection. We demonstrate how this approach corrects sampling biases and artifacts, thus improves several downstream data analysis tasks, such as clustering and classification. Finally, we demonstrate that this approach is especially useful in biology where, despite the advent of single-cell technologies, rare subpopulations and gene-interaction relationships are affected by biased sampling. We show that SUGAR can generate hypothetical populations, and it is able to reveal intrinsic patterns and mutual-information relationships between genes on a single-cell RNA sequencing dataset of hematopoiesis. © 2018 Curran Associates Inc..All rights reserved.

关键词： Genes

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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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An L2 Analysis of Reinforcement Learning in High Dimensions with Kernel and Neural Network Approximation

arXiv

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arXiv 2021年

作者： Long, Jihao Han, Jiequn Weinan, E. Program of Applied and Computational Mathematics Princeton University United States Department of Mathematics Princeton University United States Center for Computational Mathematics Flatiron Institute School of Mathematical Sciences Peking University

Reinforcement learning (RL) algorithms based on high-dimensional function approximation have achieved tremendous empirical success in large-scale problems with an enormous number of states. However, most analysis of such algorithms gives rise to error bounds that involve either the number of states or the number of features. This paper considers the situation where the function approximation is made either using the kernel method or the two-layer neural network model, in the context of a fitted Q-iteration algorithm with explicit regularization. We establish an Õ(H3|A|1/4 n- 1/4) bound for the optimal policy with Hn samples, where H is the length of each episode and |A| is the size of action space. Our analysis hinges on analyzing the L2 error of the approximated Q-function using n data points. Even though this result still requires a finite-sized action space, the error bound is independent of the dimensionality of the state space. Copyright © 2021, The Authors. All rights reserved.

关键词： Error analysis

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Comment on ‘‘Bifurcation structure and the Eckhaus instability’’

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Physical Review Letters 1991年第8期67卷 1051-1051页

作者： Laurette S. Tuckerman Dwight Barkley Program in Applied and Computational Mathematics Princeton University Princeton New Jersey 08544

A Comment on the Letter by Tsiveriotis, and Brown, Phys. Rev. Lett. 63, 2048 (1989).

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