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arXiv 2020年

作者： Weinan, E. Wojtowytsch, Stephan Department of Mathematics Program for Applied and Computational Mathematics Princeton University PrincetonNJ08544 United States Program for Applied and Computational Mathematics Princeton University PrincetonNJ08544 United States

We use explicit representation formulas to show that solutions to certain partial differential equations lie in Barron spaces or multilayer spaces if the PDE data lie in such function spaces. Consequently, these solutions can be represented efficiently using artificial neural networks, even in high dimension. Conversely, we present examples in which the solution fails to lie in the function space associated to a neural network under *** Codes 68T07, 35C15, 65M80 Copyright © 2020, The Authors. All rights reserved.

关键词： Neural networks

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A priori estimates for classification problems using neural networks

arXiv

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arXiv 2020年

作者： Weinan, E. Wojtowytsch, Stephan Program for Applied and Computational Mathematics Department of Mathematics Princeton University PrincetonNJ08544 United States Program for Applied and Computational Mathematics Princeton University PrincetonNJ08544 United States

We consider binary and multi-class classification problems using hypothesis classes of neural networks. For a given hypothesis class, we use Rademacher complexity estimates and direct approximation theorems to obtain a priori error estimates for regularized loss *** Codes 68T07, 60-08 Copyright © 2020, The Authors. All rights reserved.

关键词： Classification (of information)

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TSP cuts which do not conform to the template paradigm 5th

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5th International Conference on Mobile Agents, MA 2001

作者： Applegate, David Bixby, Robert Chvátal, Vašek Cook, William Algorithms and Optimization Department AT and T Labs – Research Florham ParkNJ07932 United States Computational and Applied Mathematics Rice University HoustonTX77005 United States Department of Computer Science Rutgers University PiscatawayNJ08854 United States Program in Applied and Computational Mathematics Princeton University PrincetonNJ08544 United States

ISBN: (纸本)3540428771

The first computer implementation of the Dantzig-Fulkerson- Johnson cutting-plane method for solving the traveling salesman problem, written by Martin, used subtour inequalities as well as cutting planes of Gomory’s type. The practice of looking for and using cuts that match prescribed templates in conjunction with Gomory cuts was continued in computer codes of Miliotis, Land, and Fleischmann. Gr¨otschel, Padberg, and Hong advocated a different policy, where the template paradigm is the only source of cuts;furthermore, they argued for drawing the templates exclusively from the set of linear inequalities that induce facets of the TSP polytope. These policies were adopted in the work of Crowder and Padberg, in the work of Gr¨otschel and Holland, and in the work of Padberg and Rinaldi;their computer codes produced the most impressive computational TSP successes of the nineteen eighties. Eventually, the template paradigm became the standard frame of reference for cutting planes in the TSP. The purpose of this paper is to describe a technique for finding cuts that disdains all understanding of the TSP polytope and bashes on regardless of all prescribed templates. Combining this technique with the traditional template approach was a crucial step in our solutions of a 13,509-city TSP instance and a 15,112-city TSP instance. © Springer-Verlag Berlin Heidelberg 2001.

关键词： Traveling salesman problem

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Perturbational Complexity by Distribution Mismatch: A Systematic Analysis of Reinforcement Learning in Reproducing Kernel Hilbert Space

arXiv

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arXiv 2021年

作者： Long, Jihao Han, Jiequn Program of Applied and Computational Mathematics Princeton University United States Center for Computational Mathematics Flatiron Institute United States

Most existing theoretical analysis of reinforcement learning (RL) is limited to the tabular setting or linear models due to the difficulty in dealing with function approximation in high dimensional space with an uncertain environment. This work offers a fresh perspective into this challenge by analyzing RL in a general reproducing kernel Hilbert space (RKHS). We consider a family of Markov decision processes M of which the reward functions lie in the unit ball of an RKHS and transition probabilities lie in a given arbitrary set. We define a quantity called perturbational complexity by distribution mismatch ∆M(ǫ) to characterize the complexity of the admissible state-action distribution space in response to a perturbation in the RKHS with scale ǫ. We show that ∆M(ǫ) gives both the lower bound of the error of all possible algorithms and the upper bound of two specific algorithms (fitted reward and fitted Q-iteration) for the RL problem. Hence, the decay of ∆M(ǫ) with respect to ǫ measures the difficulty of the RL problem on M. We further provide some concrete examples and discuss whether ∆M(ǫ) decays fast or not in these examples. As a byproduct, we show that when the reward functions lie in a high dimensional RKHS, even if the transition probability is known and the action space is finite, it is still possible for RL problems to suffer from the curse of dimensionality. Copyright © 2021, The Authors. All rights reserved.

关键词： Iterative methods

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Orthogonal matrix retrieval in cryo-electron microscopy

Orthogonal matrix retrieval in cryo-electron microscopy

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IEEE International Symposium on Biomedical Imaging

作者： Tejal Bhamre Teng Zhang Amit Singer Program in Applied and Computational Mathematics Princeton University Princeton NJ USA

In single particle reconstruction (SPR) from cryo-electron microscopy (EM), the 3D structure of a molecule needs to be determined from its 2D projection images taken at unknown viewing directions. Zvi Kam showed already in 1980 that the autocorrelation function of the 3D molecule over the rotation group SO(3) can be estimated from 2D projection images whose viewing directions are uniformly distributed over the sphere. The autocorrelation function determines the expansion coefficients of the 3D molecule in spherical harmonics up to an orthogonal matrix of size (2l+1)×(2l +1) for each l = 0, 1, 2, · · ·. In this paper we show how techniques for solving the phase retrieval problem in X-ray crystallography can be modified for the cryo-EM setup for retrieving the missing orthogonal matrices. Specifically, we present two new approaches that we term Orthogonal Extension and Orthogonal Replacement, in which the main algorithmic components are the singular value decomposition and semidefinite programming. We demonstrate the utility of these approaches through numerical experiments on simulated data.

关键词： Three-dimensional displays Image reconstruction Covariance matrices Microscopy Crystallography Noise measurement Matrix decomposition

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Approximating algebraic space curves by circular arcs 1

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7th International Conference on Curves and Surfaces, Curves and Surfaces 2010

作者： Béla, Szilvia Jüttler, Bert Doctoral Program in Computational Mathematics Johannes Kepler University 4040 Linz Austria Institute of Applied Geometry Johannes Kepler University Altenberger Str. 69 4040 Linz Austria

ISBN: (数字)9783642274138

ISBN: (纸本)9783642274121

We introduce a new method to approximate algebraic space curves. The algorithm combines a subdivision technique with local approximation of piecewise regular algebraic curve segments. The local technique computes pairs of polynomials with modified Taylor expansions and generates approximating circular arcs. We analyze the connection between the generated approximating arcs and the osculating circles of the algebraic curve. © 2012 Springer-Verlag.

关键词： Approximation algorithms

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ON THE CONVERGENCE OF GRADIENT DESCENT TRAINING FOR TWO-LAYER RELU-NETWORKS IN THE MEAN FIELD REGIME

arXiv

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arXiv 2020年

作者： Wojtowytsch, Stephan Program in Applied and Computational Mathematics Princeton University PrincetonNJ08544

We describe a necessary and sufficient condition for the convergence to minimum Bayes risk when training two-layer ReLU-networks by gradient descent in the mean field regime with omni-directional initial parameter distribution. This article extends recent results of Chizat and Bach to ReLU-activated networks and to the situation in which there are no parameters which exactly achieve MBR. The condition does not depend on the initalization of parameters and concerns only the weak convergence of the realization of the neural network, not its parameter *** Codes 35Q68, 68T07, 49Q22, 35F20 Copyright © 2020, The Authors. All rights reserved.

关键词： Gradient methods

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Mesoscopic model for microscale hydrodynamics and interfacial phenomena: Slip, films, and contact-angle hysteresis

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Physical Review E 2013年第1期87卷 013302-013302页

作者： Carlos E. Colosqui Michail E. Kavousanakis Athanasios G. Papathanasiou Ioannis G. Kevrekidis Department of Chemical and Biological Engineering and Program in Applied and Computational Mathematics Princeton University Princeton New Jersey 08544 USA

We present a model based on the lattice Boltzmann equation that is suitable for the simulation of dynamic wetting. The model is capable of exhibiting fundamental interfacial phenomena such as weak adsorption of fluid on the solid substrate and the presence of a thin surface film within which a disjoining pressure acts. Dynamics in this surface film, tightly coupled with hydrodynamics in the fluid bulk, determine macroscopic properties of primary interest: the hydrodynamic slip; the equilibrium contact angle; and the static and dynamic hysteresis of the contact angles. The pseudo-potentials employed for fluid-solid interactions are composed of a repulsive core and an attractive tail that can be independently adjusted. This enables effective modification of the functional form of the disjoining pressure so that one can vary the static and dynamic hysteresis on surfaces that exhibit the same equilibrium contact angle. The modeled fluid-solid interface is diffuse, represented by a wall probability function that ultimately controls the momentum exchange between solid and fluid phases. This approach allows us to effectively vary the slip length for a given wettability (i.e., a given static contact angle) of the solid substrate.

关键词： HYDRODYNAMICS INTERFACES (Physical sciences) THIN films HYSTERESIS MATHEMATICAL models LATTICE Boltzmann methods DYNAMICAL systems WETTING

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Spectral bounds for independent cascade model with sensitive edges

Spectral bounds for independent cascade model with sensitive...

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Annual Conference on Information Sciences and Systems (CISS)

作者： Eun Jee Lee Sudeep Kamath Emmanuel Abbe Sanjeev R. Kulkarni Program in Applied and Computational Mathematics Department of Electrical Engineering Princeton University

This paper studies independent cascade models where influence propagates from seed-nodes along edges with independent probabilities. Upper-bounds for the expected number of influenced nodes were previously proposed using the spectral norm of a Hazard matrix. However, these bounds turn out loose in many cases, in particular with respect to sensitive edges such as bottlenecks, seed adjacent, and high probability edges. This paper proposes a similar bound that improves in such cases by handling sensitives edges more carefully.

关键词： Hazards Upper bound Stochastic processes Information science Merging computational modeling Mathematical model

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Pseudo prolate spheroidal functions 11

Pseudo prolate spheroidal functions

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11th International Conference on Sampling Theory and Applications, SampTA 2015

作者： Daniel Abreu, Luis Pereira, Joao M. Austrian Academy of Sciences Acoustics Research Institute Wohllebengasse 12-14 ViennaA-1040 Austria Program in Applied and Computational Mathematics Princeton University PrincetonNJ08544 United States

ISBN: (纸本)9781467373531

Let Dt and BΩ denote the operators which cut the time content outside T and the frequency content outside Ω, respectively. The prolate spheroidal functions are the eigen-functions of the operator Ρτ,Ω = DtBΩDt. With the aim of formulating in precise mathematical terms the notion of Nyquist rate, Landau and Pollack have shown that, asymptotically, the number of such functions with eigenvalue close to one is ≈ TΩ/2π. We have recently revisited this problem with a new approach: instead of counting the number of eigenfunctions with eigenvalue close to one, we count the maximum number of orthogonal Ε-pseudoeigenfunctions with Ε-pseudoeigenvalue one. Precisely, we count how many orthogonal functions have a maximum of energy e outside the domain T x Ω, in the sense that PT, Ωf - f2 ≤ Ε. We have recently discovered that the sharp asymptotic number is ≈ (1 - Ε)-1TΩ/2π. The proof involves an explicit construction of the pseudoeigenfunctions of PT, Ω. When T and Ω are intervals we call them pseudo prolate spheroidal functions. In this paper we explain how they are constructed. © 2015 IEEE.

关键词： Orthogonal functions

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