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Geometry-based data generation

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arXiv 2018年

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

We propose a new type of generative model of 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, and 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 reveal intrinsic patterns and mutual-information relationships between genes on a single cell dataset of hematopoiesis. Copyright © 2018, The Authors. All rights reserved.

关键词： Clustering algorithms

Global weak solutions for generalized SQG in bounded domains

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arXiv 2017年

作者： Nguyen, Huy Quang PROGRAM IN APPLIED AND COMPUTATIONAL MATHEMATICS PRINCETON UNIVERSITY PRINCETONNJ08544 United States

We prove the existence of global L2 weak solutions for a family of generalized inviscid surface-quasi geostrophic (SQG) equations in bounded domains of the plane. In these equations, the active scalar is transported by a velocity field which is determined by the scalar through a more singular nonlocal operator compared to the SQG equation. The result is obtained by establishing appropriate commutator representations for the weak formulation together with good bounds for them in bounded *** Classification: 35Q35, 35Q86.35Q35, 35Q86 Copyright © 2017, The Authors. All rights reserved.

关键词： Velocity

CEMRACS 2017: numerical probabilistic approach to MFG

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arXiv 2018年

作者： Angiuli, Andrea Graves, Christy V. Li, Houzhi Chassagneux, Jean-François Delarue, François Carmona, René Department of Statistics and Applied Probability University of California Santa Barbara Program in Applied and Computational Mathematics Princeton University Laboratoire de Probabilités Modélisation et Statistiques Université Paris Diderot Laboratoire Jean-Alexandre Dieudonné Université de Nice Sophia-Antipolis Operations Research and Financial Engineering Bendheim Center for Finance Princeton University

This project investigates numerical methods for solving fully coupled forward-backward stochastic differential equations (FBSDEs) of McKean-Vlasov type. Having numerical solvers for such mean field FBSDEs is of interest because of the potential application of these equations to optimization problems over a large population, say for instance mean field games (MFG) and optimal mean field control problems. Theory for this kind of problems has met with great success since the early works on mean field games by Lasry and Lions, see [27], and by Huang, Caines, and Malhamé, see [24]. Generally speaking, the purpose is to understand the continuum limit of optimizers or of equilibria (say in Nash sense) as the number of underlying players tends to infinity. When approached from the probabilistic viewpoint, solutions to these control problems (or games) can be described by coupled mean field FBSDEs, meaning that the coefficients depend upon the own marginal laws of the solution. In this note, we detail two methods for solving such FBSDEs which we implement and apply to five benchmark problems. The first method uses a tree structure to represent the pathwise laws of the solution, whereas the second method uses a grid discretization to represent the time marginal laws of the solutions. Both are based on a Picard scheme;importantly, we combine each of them with a generic continuation method that permits to extend the time horizon (or equivalently the coupling strength between the two equations) for which the Picard iteration converges. Copyright © 2018, The Authors. All rights reserved.

关键词： Numerical methods

Benchmark problems for transcranial ultrasound simulation: Intercomparison of compressional wave models

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arXiv 2022年

作者： Aubry, Jean-Francois Bates, Oscar Boehm, Christian Pauly, Kim Butts Christensen, Douglas Cueto, Carlos Gélat, Pierre Guasch, Lluis Jaros, Jiri Jing, Yun Jones, Rebecca Li, Ningrui Marty, Patrick Montanaro, Hazael Neufeld, Esra Pichardo, Samuel Pinton, Gianmarco Pulkkinen, Aki Stanziola, Antonio Thielscher, Axel Treeby, Bradley Van't Wout, Elwin Physics for Medicine Paris INSERM U1273 ESPCI Paris PSL University CNRS UMR 8063 France Department of Bioengineering Imperial College London Exhibition Road LondonSW7 2AZ United Kingdom Institute of Geophysics ETH Zürich Sonneggstrasse 5 Zürich8092 Switzerland Department of Radiology Stanford University StanfordCA United States Department of Biomedical Engineering Department of Electrical and Computer Engineering University of Utah United States Department of Surgical Biotechnology Division of Surgery and Interventional Science University College London LondonNW3 2PF United Kingdom Earth Science and Engineering Department Imperial College London London United Kingdom Centre of Excellence IT4Innovations Faculty of Information Technology Brno University of Technology Bozetechova 2 Brno612 00 Czech Republic Graduate Program in Acoustics The Pennsylvania State University University Park PA16802 United States Joint Department of Biomedical Engineering University of North Carolina at Chapel Hill North Carolina State University NC United States Department of Electrical Engineering Stanford University StanfordCA United States Zurich Switzerland Laboratory for Acoustics / Noise control Empa Swiss Federal Laboratories for Materials Science and Technology Dubendorf Switzerland Zurich Switzerland Radiology and Clinical Neurosciences Departments Cumming School of Medicine University of Calgary Canada Department of Applied Physics University of Eastern Finland Kuopio70211 Finland Department of Medical Physics and Biomedical Engineering University College London Gower Street LondonWC1E 6BT United Kingdom Technical University of Denmark Denmark Danish Research Center for Magnetic Resonance Copenhagen University Hospital Hvidovre Denmark Institute for Mathematical and Computational Engineering School of Engineering and Faculty of Mathematics Pontificia Universidad Catolica de Chile Santiago Chile

computational models of acoustic wave propagation are frequently used in transcranial ultrasound therapy, for example, to calculate the intracranial pressure field or to calculate phase delays to correct for skull distortions. To allow intercomparison between the different modeling tools and techniques used by the community, an international working group was convened to formulate a set of numerical benchmarks. Here, these benchmarks are presented, along with intercomparison results. Nine different benchmarks of increasing geometric complexity are defined. These include a single-layer planar bone immersed in water, a multi-layer bone, and a whole skull. Two transducer configurations are considered (a focused bowl and a plane piston), giving a total of 18 permutations of the benchmarks. Eleven different modeling tools are used to compute the benchmark results. The models span a wide range of numerical techniques, including the finite-difference time-domain method, angular-spectrum method, pseudospectral method, boundary-element method, and spectral-element method. Good agreement is found between the models, particularly for the position, size, and magnitude of the acoustic focus within the skull. When comparing results for each model with every other model in a cross comparison, the median values for each benchmark for the difference in focal pressure and position are less than 10% and 1 mm, respectively. The benchmark definitions, model results, and intercomparison codes are freely available to facilitate further comparisons. Copyright © 2022, The Authors. All rights reserved.

关键词： Bone

Semidefinite programming approach for the quadratic assignment problem with a sparse graph

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arXiv 2017年

作者： Bravo Ferreira, José F.S. Khoo, Yuehaw Singer, Amit Program in Applied and Computational Mathematics Princeton University PrincetonNJ08544 United States Department of Mathematics Stanford University StanfordCA94305 United States Department of Mathematics and Program in Applied and Computational Mathematics Princeton University PrincetonNJ08544 United States

The matching problem between two adjacency matrices can be formulated as the NP-hard quadratic assignment problem (QAP). Previous work on semidefinite programming (SDP) relaxations to the QAP have produced solutions that are often tight in practice, but such SDPs typically scale badly, involving matrix variables of dimension n2 where n is the number of nodes. To achieve a speed up, we propose a further relaxation of the SDP involving a number of positive semidefinite matrices of dimension O(n) no greater than the number of edges in one of the graphs. The relaxation can be further strengthened by considering cliques in the graph, instead of edges. The dual problem of this novel relaxation has a natural three-block structure that can be solved via a convergent Augmented Direction Method of Multipliers (ADMM) in a distributed manner, where the most expensive step per iteration is computing the eigendecomposition of matrices of dimension O(n). The new SDP relaxation produces strong bounds on quadratic assignment problems where one of the graphs is sparse with reduced computational complexity and running times, and can be used in the context of nuclear magnetic resonance spectroscopy (NMR) to tackle the assignment problem. Copyright © 2017, The Authors. All rights reserved.

关键词： Combinatorial optimization

Effect of imperfections on the hyperuniformity of many-body systems

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Physical Review B 2018年第5期97卷 054105-054105页

作者： Jaeuk Kim Salvatore Torquato Department of Physics Princeton University Princeton New Jersey 08544 USA Department of Chemistry 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

A hyperuniform many-body system is characterized by a structure factor Sk that vanishes in the small-wave-number limit or equivalently by a local number variance σN2R associated with a spherical window of radius R that grows more slowly than Rd in the large-R limit. Thus, the hyperuniformity implies anomalous suppression of long-wavelength density fluctuations relative to those in typical disordered systems, i.e., σN2R∼Rd as R→∞. Hyperuniform systems include perfect crystals, quasicrystals, and special disordered systems. Disordered hyperuniform systems are amorphous states of matter that lie between a liquid and crystal [S. Torquato et al., Phys. Rev. X 5, 021020 (2015)], and have been the subject of many recent investigations due to their novel properties. In the same way that there is no perfect crystal in practice due to the inevitable presence of imperfections, such as vacancies and dislocations, there is no “perfect” hyperuniform system, whether it is ordered or not. Thus, it is practically and theoretically important to quantitatively understand the extent to which imperfections introduced in a perfectly hyperuniform system can degrade or destroy its hyperuniformity and corresponding physical properties. This paper begins such a program by deriving explicit formulas for Sk in the small-wave-number regime for three types of imperfections: (1) uncorrelated point defects, including vacancies and interstitials, (2) stochastic particle displacements, and (3) thermal excitations in the classical harmonic regime. We demonstrate that our results are in excellent agreement with numerical simulations. We find that “uncorrelated” vacancies or interstitials destroy hyperuniformity in proportion to the defect concentration p. We show that “uncorrelated” stochastic displacements in perfect lattices can never destroy the hyperuniformity but it can be degraded such that the perturbed lattices fall into class III hyperuniform systems, where σN2R∼Rd−α as R→∞ and 0<α<1. By cont

关键词： Interstitials Structural properties Vacancies

Beyond trees: Classification with sparse pairwise dependencies

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arXiv 2018年

作者： Tenzer, Yaniv Moscovich, Amit Dorn, Mary Frances Nadler, Boaz Spiegelman, Clifford Department of Computer Science and Applied Mathematics Weizmann Institute of Science Rehovot76100 Israel Program in Applied and Computational Mathematics Princeton University PrincetonNJ08544 United States Los Alamos National Laboratory P.O. Box 1663 MS F600 Los AlamosNM87545 United States Department of Statistics Texas A&M University College StationTX77843 United States

Several classification methods assume that the underlying distributions follow tree-structured graphical models. Indeed, trees capture statistical dependencies between pairs of variables, which may be crucial to attain low classification errors. The resulting classifier is linear in the log-transformed univariate and bivariate densities that correspond to the tree edges. In practice, however, observed data may not be well approximated by trees. Yet, motivated by the importance of pairwise dependencies for accurate classification, here we propose to approximate the optimal decision boundary by a sparse linear combination of the univariate and bivariate log-transformed densities. Our proposed approach is semi-parametric in nature: we non-parametrically estimate the univariate and bivariate densities, remove pairs of variables that are nearly independent using the Hilbert-Schmidt independence criteria, and finally construct a linear SVM on the retained log-transformed densities. We demonstrate using both synthetic and real data that our resulting classifier, denoted SLB (Sparse Log-Bivariate density), is competitive with popular classification *** Codes 62H30 Copyright © 2018, The Authors. All rights reserved.

关键词： Binary trees

Community Detection and Stochastic Block Models

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arXiv 2017年

作者： Abbe, Emmanuel Program in Applied and Computational Mathematics Department of Electrical Engineering Princeton University Princeton United States

The stochastic block model (SBM) is a random graph model with different group of vertices connecting differently. It is widely employed as a canonical model to study clustering and community detection, and provides a fertile ground to study the information-theoretic and computational tradeoffs that arise in combinatorial statistics and more generally data science. This monograph surveys the recent developments that establish the fundamental limits for community detection in the SBM, both with respect to information-theoretic and computational tradeoffs, and for various recovery requirements such as exact, partial and weak recovery. The main results discussed are the phase transitions for exact recovery at the Chernoff-Hellinger threshold, the phase transition for weak recovery at the Kesten-Stigum threshold, the optimal SNR-mutual information tradeoff for partial recovery, and the gap between information-theoretic and computational thresholds. The monograph gives a principled derivation of the main algorithms developed in the quest of achieving the limits, in particular two-round algorithms via graph-splitting, semi-definite programming, (linearized) belief propagation, classical/nonbacktracking spectral methods and graph powering. Extensions to other block models, such as geometric block models, and a few open problems are also discussed. Copyright © 2017, The Authors. All rights reserved.

关键词： Recovery

Nonbacktracking bounds on the influence in independent cascade models

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arXiv 2017年

作者： Abbe, Emmanuel Kulkarni, Sanjeev Lee, Eun Jee Program in Applied and Computational Mathematics Department of Electrical Engineering Princeton University Princeton United States Department of Electrical Engineering Princeton University Princeton United States Program in Applied and Computational Mathematics Princeton University Princeton United States

This paper develops upper and lower bounds on the influence measure in a network, more precisely, the expected number of nodes that a seed set can influence in the independent cascade model. In particular, our bounds exploit nonbacktracking walks, Fortuin–Kasteleyn–Ginibre (FKG) type inequalities, and are computed by message passing implementation. Nonbacktracking walks have recently allowed for headways in community detection, and this paper shows that their use can also impact the influence computation. Further, we provide a knob to control the trade-off between the efficiency and the accuracy of the bounds. Finally, the tightness of the bounds is illustrated with simulations on various network models. Copyright © 2017, The Authors. All rights reserved.

关键词： Economic and social effects