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Exponential convergence of the deep neural network approximation for analytic functions Dedicated to Professor Ta Tsien Li on the Occasion of His 80th Birthday

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Science China mathematics 2018年第10期61卷 1733-1740页

作者： Weinan E Qingcan Wang Department of Mathematics and PACM Princeton University Princeton NJ 08544 USA Center for Big Data Research Peking University Beijing 100871 China Beijing Institute of Big Data Research Beijing 100871 China Program in Applied and Computational Mathematics Princeton University Princeton NJ 08544 USA

We prove that for analytic functions in low dimension, the convergence rate of the deep neural network approximation is exponential.

关键词： neural networks approximation theory analytic functions

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

Spectral neighbor joining for reconstruction of latent tree models

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arXiv 2020年

作者： Jaffe, Ariel Amsel, Noah Nadler, Boaz Chang, Joseph T. Kluger, Yuval Program in Applied Mathematics Yale University New HavenCT06511 United States Department of Computer Science Weizmann Institute of Science Rehovot76100 Israel Department of Statistics Yale University New HavenCT06520 United States Interdepartmental Program in Computational Biology and Bioinformatics New HavenCT06511 Singapore Department of Pathology New HavenCT06511 United States

A key assumption in multiple scientific applications is that the distribution of observed data can be modeled by a latent tree graphical model. An important example is phylogenetics, where the tree models the evolutionary lineages of various organisms. Given a set of independent realizations of the random variables at the leaves of the tree, a common task is to infer the underlying tree topology. In this work we develop Spectral Neighbor Joining (SNJ), a novel method to recover latent tree graphical models. In contrast to distance based methods, SNJ is based on a spectral measure of similarity between all pairs of observed variables. We prove that SNJ is consistent, and derive a sufficient condition for correct tree recovery from an estimated similarity matrix. Combining this condition with a concentration of measure result on the similarity matrix, we bound the number of samples required to recover the tree with high probability. We illustrate via extensive simulations that SNJ requires fewer samples to accurately recover trees in regimes where the tree contains a large number of leaves or long edges. We provide theoretical support for this observation by analyzing the model of a perfect binary tree. Copyright © 2020, The Authors. All rights reserved.

关键词： Recovery

Effective Elastic Wave Characteristics of Composite Media

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arXiv 2020年

作者： Kim, Jaeuk Torquato, S. Department of Physics Princeton University PrincetonNJ08544 United States Department of Chemistry Princeton University PrincetonNJ08544 United States Princeton Institute for the Science and Technology of Materials Princeton University PrincetonNJ08544 United States Program in Applied and Computational Mathematics Princeton University PrincetonNJ08544 United States

We derive exact expressions for effective elastodynamic properties of two-phase composites in the long-wavelength (quasistatic) regime via homogenized constitutive relations that are local in space. This is accomplished by extending the "strong-contrast" expansion formalism that was previously applied to the static problem. These strong-contrast expansions explicitly incorporate complete microstructural information of the composite via an infinite set of n-point correlation functions. Utilizing the rapid-convergence properties of these series expansions (even for extreme contrast ratios), we extract accurate approximations that depend on the microstructure via the spectral density, which is easy to compute or measure for any composite. We also investigate the predictive power of modifications of such approximation formulas postulated elsewhere [J. Kim and S. Torquato, Proc. Nat. Acad. Sci. 117, 8764 (2020)] to extend their applicability beyond the quasistatic regime. The accuracy of these nonlocal microstructure-dependent approximations is validated by comparison to full-waveform simulation results for certain models of dispersions. We apply our formulas to a variety of models of nonhyperuniform and hyperuniform disordered composites. We demonstrate that hyperuniform systems are less lossy than their nonhyperuniform counterparts in the quasistatic regime, and stealthy hyperuniform media can be perfectly transparent for a wide range of wavenumbers. Finally, we discuss how to utilize our approximations for engineering composites with prescribed elastic wave characteristics. Copyright © 2020, The Authors. All rights reserved.

关键词： Microstructure

Quantum phase transitions in long-range interacting hyperuniform spin chains in a transverse field

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arXiv 2020年

作者： Bose, Amartya Torquato, Salvatore Department of Chemistry Princeton University PrincetonNJ08544 United States Department of Physics Princeton University PrincetonNJ08544 United States Princeton Institute for the Science and Technology of Materials Princeton University PrincetonNJ08544 United States Program in Applied and Computational Mathematics Princeton University PrincetonNJ08544 United States

Hyperuniform states of matter are characterized by anomalous suppression of long-wavelength density fluctuations. While most of the interesting cases of disordered hyperuniformity are provided by complex many-body systems like liquids or amorphous solids, classical spin chains with certain long-range interactions have been shown to demonstrate the same phenomenon. Such systems involving spin chains are ideal models for exploring the effects of quantum mechanics on hyperuniformity. It is well-known that the transverse field Ising model shows a quantum phase transition (QPT) at zero temperature. Under the quantum effects of a transverse magnetic field, classical hyperuniform spin chains are expected to lose their hyperuniformity. High-precision simulations of these cases are complicated because of the presence of highly nontrivial long-range interactions. We perform extensive analysis of these systems using density matrix renormalization group (DMRG) to study the possibilities of phase transitions and the mechanism by which they lose hyperuniformity. Even for a spin chain of length 30, we see discontinuous changes in properties like the "τ order metric" of the ground state, the measure of hyperuniformity and the second cumulant of the total magnetization along the x-direction, all suggestive of first-order QPTs. An interesting feature of the phase transitions in these disordered hyperuniform spin chains is that, depending on the parameter values, the presence of transverse magnetic field may remarkably lead to increase in the order of the ground state as measured by the "τ order metric," even if hyperuniformity is lost. Therefore, it would be possible to design materials to target specific novel quantum behaviors in the presence of a transverse magnetic field. Our numerical investigations suggest that these spin chains can show no more than two QPTs. We further analyze the long-range interacting spin chains via the Jordan-Wigner mapping on to a system of spinless ferm

关键词： Phase transitions

Nonlocal effective electromagnetic wave characteristics of composite media: Beyond the quasistatic regime

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arXiv 2020年

作者： Torquato, Salvatore Kim, Jaeuk Department of Physics Princeton University PrincetonNJ08544 United States Department of Chemistry Princeton University PrincetonNJ08544 United States Princeton Institute Science and Technology of Materials Princeton University PrincetonNJ08544 United States Program in Applied and Computational Mathematics Princeton University PrincetonNJ08544 United States

We derive exact nonlocal homogenized constitutive relations for the effective electromagnetic wave properties of disordered two-phase composites and metamaterials from first principles. This exact formalism enables us to extend the long-wavelength limitations of conventional homogenization estimates of the effective dynamic dielectric constant tensor Εe(kI, ω) for arbitrary microstructures so that it can capture spatial dispersion well beyond the quasistatic regime (where ω and kI are the frequency and wave vector of the incident radiation). We accomplish this task by deriving nonlocal strong-contrast expansions that exactly account for complete microstructural information (infinite set of n-point correlation functions) and hence multiple scattering to all orders for the range of wave numbers for which our extended homogenization theory applies, i.e., 0 ≤ |kI|. 1 (where is a characteristic heterogeneity length scale). Because of the fast-convergence properties of such expansions, their lower-order truncations yield accurate closed-form approximate formulas for Εe(kI, ω) that apply for a wide class of microstructures. These nonlocal formulas are resummed representations of the strong-contrast expansions that still accurately capture multiple scattering to all orders via the microstructural information embodied in the spectral density, which is easy to compute for any composite. The accuracy of these microstructure-dependent approximations is validated by comparison to full-waveform simulation computations for both 2D and 3D ordered and disordered models of composite media. Thus, our closed-form formulas enable one to predict accurately and efficiently the effective wave characteristics well beyond the quasistatic regime for a wide class of composite microstructures without having to perform full-blown simulations. We find that disordered hyperuniform media are generally less lossy than their nonhyperuniform counterparts. We also show that certain disordered hyperunif

关键词： Microstructure

Characterizing the hyperuniformity of ordered and disordered two-phase media

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arXiv 2020年

作者： Kim, Jaeuk Torquato, Salvatore Department of Physics Princeton University PrincetonNJ08544 United States Department of Chemistry Princeton University PrincetonNJ08544 United States Princeton Institute for the Science and Technology of Materials Princeton University PrincetonNJ08544 United States Program in Applied and Computational Mathematics Princeton University PrincetonNJ08544 United States

The hyperuniformity concept provides a unified means to classify all perfect crystals, perfect quasicrystals, and exotic amorphous states of matter according to their capacity to suppress large-scale density fluctuations. While the classification of hyperuniform point configurations has received considerable attention, much less is known about the classification of hyperuniform heterogeneous two-phase media, which include composites, porous media, foams, cellular solids, colloidal suspensions and polymer blends. The purpose of this article is to begin such a program for certain two-dimensional models of hyperuniform two-phase media by ascertaining their local volume-fraction variances σV2 (R) and the associated hyperuniformity order metrics BV . This is a highly challenging task because the geometries and topologies of the phases are generally much richer and more complex than point-configuration arrangements and one must ascertain a broadly applicable length scale to make key quantities dimensionless. Therefore, we purposely restrict ourselves to a certain class of two-dimensional periodic cellular networks as well as periodic and disordered/irregular packings, some of which maximize their effective transport and elastic properties. Among the cellular networks considered, the honeycomb networks have the minimal value of the hyperuniformity order metrics BV across all volume fractions. On the other hand, among all packings considered, the triangular-lattice packings have the smallest values of BV for the possible range of volume fractions. Among all structures studied here, the triangular-lattice packing has the minimal order metric for almost all volume fractions. Our study provides a theoretical foundation for the establishment of hyperuniformity order metrics for general two-phase media and a basis to discover new hyperuniform two-phase systems by inverse design procedures. Copyright © 2020, The Authors. All rights reserved.

关键词： Wireless networks

Nearest-neighbor functions for disordered stealthy hyperuniform many-particle systems

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arXiv 2020年

作者： Middlemas, Timothy M. Torquato, Salvatore Department of Chemistry Princeton University PrincetonNJ08544 United States Department of Physics Princeton University PrincetonNJ08544 United States Princeton Institute for the Science and Technology of Materials Princeton University PrincetonNJ08544 United States Program in Applied and Computational Mathematics Princeton University PrincetonNJ08544 United States

Disordered stealthy many-particle systems in d-dimensional Euclidean space Rd are exotic amorphous states of matter that suppress any single scattering events for a finite range of wavenumbers around the origin in reciprocal space. They are currently the subject of intense fundamental and practical interest. We derive analytical formulas for the nearest-neighbor functions of disordered stealthy many-particle systems. First, we analyze asymptotic small-r approximations and expansions of the nearest-neighbor functions based on the pseudo-hard-sphere ansatz. We then consider the problem of determining how many of the standard n-point correlation functions are needed to determine the nearest neighbor functions, and find that a finite number suffice. Via theoretical and computational methods, we are able to compare the large-r behavior of these functions for disordered stealthy systems to those belonging to crystalline lattices. Such ordered and disordered stealthy systems have bounded hole sizes, and thus compact support for their nearest-neighbor functions. However, we find that the approach to the critical-hole size can be quantitatively different, emphasizing the importance of hole statistics in distinguishing ordered and disordered stealthy configurations. We argue that the probability of finding a hole close to the critical-hole size should decrease as a power law with an exponent only dependent on the space dimension d for ordered systems, but that this probability decays asymptotically faster for disordered systems, with either an increase in the exponent of the power law or a crossover into a decay faster than any power law. This implies that holes close to the critical-hole size are rarer in disordered systems. The rarity of observing large holes in disordered systems creates substantial numerical difficulties in sampling the nearest neighbor distributions near the critical-hole size. This motivates both the need for new computational methods for efficient samp

关键词： computational methods

Localization transition in three-dimensional stealthy hyperuniform disordered systems

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arXiv 2021年

作者： Sgrignuoli, F. Torquato, S. Dal Negro, L. Department of Electrical and Computer Engineering & Photonics Center Boston University 8 Saint Mary’s Street BostonMA02215 United States Department of Chemistry Department of Physics Princeton Institute for the Science and Technology of Materials Program in Applied and Computational Mathematics Princeton University PrincetonNJ08540 United States Division of Material Science and Engineering Boston University 15 Saint Mary’s Street BrooklineMA02446 United States Department of Physics Boston University 590 Commonwealth Avenue BostonMA02215 United States

The purpose of this work is to understand the fundamental connection between structural correlations and light localization in three-dimensional (3D) open scattering systems of finite size. We numerically investigate the transport of vector electromagnetic waves scattered by resonant electric dipoles spatially arranged in 3D space by stealthy hyperuniform disordered point patterns. Three-dimensional stealthy hyperuniform disordered systems (3D-SHDS) are engineered with different structural correlation properties determined by their degree of stealthiness χ. Such fine control of exotic states of amorphous matter enables the systematic design of optical media that interpolate in a tunable fashion between uncorrelated random structures and crystalline materials. By solving the electromagnetic multiple scattering problem using Green’s matrix spectral method, we establish a transport phase diagram that demonstrates a clear delocalization-localization transition beyond a critical scattering density that depends on χ. The transition is characterized by studying the Thouless number and the spectral statistics of the scattering resonances. In particular, by tuning the χ parameter, we demonstrate large spectral gaps and suppressed sub-radiant proximity resonances, facilitating light localization. Moreover, consistently with previous studies, our results show a region of the transport phase diagram where the investigated scattering systems become transparent. Our work provides a systematic description of the transport and localization properties of light in stealthy hyperuniform structures and motivates the engineering of novel photonic systems with enhanced light-matter interactions for applications to both classical and quantum devices. Copyright © 2021, The Authors. All rights reserved.

关键词： Phase diagrams