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Robust algorithm to generate a diverse class of dense disordered and ordered sphere packings via linear programming

Physical Review E 2010年第6期82卷 061302-061302页

作者： S. Torquato Y. Jiao Department of Chemistry Department of Physics Princeton Center for Theoretical Science Princeton Institute for the Science and Technology of Materials Program in Applied and Computational Mathematics Princeton University Princeton New Jersey 08544 USA Department of Mechanical and Aerospace Engineering Princeton University Princeton New Jersey 08544 USA

We have formulated the problem of generating dense packings of nonoverlapping, nontiling nonspherical particles within an adaptive fundamental cell subject to periodic boundary conditions as an optimization problem called the adaptive-shrinking cell (ASC) formulation [S. Torquato and Y. Jiao, Phys. Rev. E 80, 041104 (2009)]. Because the objective function and impenetrability constraints can be exactly linearized for sphere packings with a size distribution in d-dimensional Euclidean space Rd, it is most suitable and natural to solve the corresponding ASC optimization problem using sequential-linear-programming (SLP) techniques. We implement an SLP solution to produce robustly a wide spectrum of jammed sphere packings in Rd for d=2, 3, 4, 5, and 6 with a diversity of disorder and densities up to the respective maximal densities. A novel feature of this deterministic algorithm is that it can produce a broad range of inherent structures (locally maximally dense and mechanically stable packings), besides the usual disordered ones (such as the maximally random jammed state), with very small computational cost compared to that of the best known packing algorithms by tuning the radius of the influence sphere. For example, in three dimensions, we show that it can produce with high probability a variety of strictly jammed packings with a packing density anywhere in the wide range [0.6, 0.7408…], where π/18=0.7408… corresponds to the density of the densest packing. We also apply the algorithm to generate various disordered packings as well as the maximally dense packings for d=2, 4, 5, and 6. Our jammed sphere packings are characterized and compared to the corresponding packings generated by the well-known Lubachevsky-Stillinger (LS) molecular-dynamics packing algorithm. Compared to the LS procedure, our SLP protocol is able to ensure that the final packings are truly jammed, produces disordered jammed packings with anomalously low densities, and is appreciably more robust an

关键词： Packing

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The interplay of competition and cooperation among service providers

arXiv

引用

arXiv 2017年

作者： Lotfi, Mohammad Hassan Chen, Xingran Sarkar, Saswati Department of Electrical and System Engineering University of Pennsylvania PhiladelphiaPA19104 United States Applied Mathematics and Computational Science University of Pennsylvania PhiladelphiaPA19104 United States

—We consider the economics of the interaction between Mobile Virtual Network Operators (MVNOs) and Mobile Network Operators (MNOs). We investigate the incentives of an MNO for offering some of her resources to an MVNO instead of using the resources for her own End-Users (EUs). We consider a market with one MNO and one MVNO, and a continuum of undecided EUs. Two cases for EUs are considered: (i) when EUs need to choose either the MNO or the MVNO, and (ii) when EUs have an outside option. In each of these cases, we consider a non-cooperative framework of sequential game and a cooperative framework of bargaining game. We characterize the Subgame Perfect Nash Equilibria (SPNE) and Nash Bargaining Solution (NBS) of the sequential and bargaining games, respectively. We show that in the non-cooperative framework, SPNE assumes two forms: (i) the MNO invests minimally on her infrastructure and the MVNO leases all the newly invested resources and (ii) the MNO invests more on her infrastructure, the MVNO leases only part of these resources to the MVNO, and the MNO uses the rest herself to attract EUs. Thus, in both, the MNO generates revenue indirectly through the MVNO. In addition to that, in (ii), the MNO generates revenue directly from the EUs. We also prove that in the bargaining framework, the MVNO either reserves all the resources or no resources from the MNO, and the MNO’s investments are guided by whether EUs have an outside option. If they don’t, then the MNO invests as little as possible on her infrastructure. If they do, then the MNO invests more. Copyright © 2017, The Authors. All rights reserved.

关键词： Game theory

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Entropic Optimal Transport Eigenmaps for Nonlinear Alignment and Joint Embedding of High-Dimensional Datasets

arXiv

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arXiv 2024年

作者： Landa, Boris Kluger, Yuval Ma, Rong Department of Electrical Engineering Yale University United States Program in Applied Mathematics Yale University United States Department of Biostatistics Harvard University United States Interdepartmental Program in Computational Biology and Bioinformatics Yale University United States Department of Pathology Yale University School of Medicine United States

Embedding high-dimensional data into a low-dimensional space is an indispensable component of data analysis. In numerous applications, it is necessary to align and jointly embed multiple datasets from different studies or experimental conditions. Such datasets may share underlying structures of interest but exhibit individual distortions, resulting in misaligned embeddings using traditional techniques. In this work, we propose Entropic Optimal Transport (EOT) eigenmaps, a principled approach for aligning and jointly embedding a pair of datasets with theoretical guarantees. Our approach leverages the leading singular vectors of the EOT plan matrix between two datasets to extract their shared underlying structure and align the datasets accordingly in a common embedding space. We interpret our approach as an inter-data variant of the classical Laplacian eigenmaps and diffusion maps embeddings, showing that it enjoys many favorable analogous properties. We then analyze a data-generative model where two observed high-dimensional datasets share latent variables on a common low-dimensional manifold, but each dataset is subject to data-specific translation, scaling, nuisance structures, and noise. We show that in a high-dimensional asymptotic regime, the EOT plan recovers the shared manifold structure by approximating a kernel function evaluated at the locations of the latent variables. Subsequently, we provide a geometric interpretation of our embedding by relating it to the eigenfunctions of population-level operators encoding the density and geometry of the shared manifold. Finally, we showcase the performance of our approach for data integration and embedding through simulations and analyses of real-world biological data, demonstrating its advantages over alternative methods in challenging scenarios. Copyright © 2024, The Authors. All rights reserved.

关键词： Embeddings

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Sparse neural additive model: Interpretable deep learning with feature selection via group sparsity

arXiv

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arXiv 2022年

作者： Xu, Shiyun Bu, Zhiqi Chaudhari, Pratik Barnett, Ian J. Department Of Applied Mathematics And Computational Science University Of Pennsylvania United States Department Of Electrical And Systems Engineering University Of Pennsylvania United States Department Of Biostatistics Epidemiology And Informatics University Of Pennsylvania United States

Interpretable machine learning has demonstrated impressive performance while preserving explainability. In particular, neural additive models (NAM) offer the interpretability to the black-box deep learning and achieve state-of-the-art accuracy among the large family of generalized additive models. In order to empower NAM with feature selection and improve the generalization, we propose the sparse neural additive models (SNAM) that employ the group sparsity regularization (e.g. Group LASSO), where each feature is learned by a sub-network whose trainable parameters are clustered as a group. We study the theoretical properties for SNAM with novel techniques to tackle the non-parametric truth, thus extending from classical sparse linear models such as the LASSO, which only works on the parametric truth. Specifically, we show that SNAM with subgradient and proximal gradient descents provably converges to zero training loss as t → ∞, and that the estimation error of SNAM vanishes asymptotically as n → ∞. We also prove that SNAM, similar to LASSO, can have exact support recovery, i.e. perfect feature selection, with appropriate regularization. Moreover, we show that the SNAM can generalize well and preserve the 'identifiability', recovering each feature's effect. We validate our theories via extensive experiments and further testify to the good accuracy and efficiency of SNAM. Copyright © 2022, The Authors. All rights reserved.

关键词： Feature extraction

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A Riemannian ADMM

arXiv

引用

arXiv 2022年

作者： Li, Jiaxiang Ma, Shiqian Srivastava, Tejes Department of Electrical and Computer Engineering University of Minnesota United States Computational Applied Mathematics and Operations Research Rice University United States Department of Computer Science University of Chicago United States

We consider a class of Riemannian optimization problems where the objective is the sum of a smooth function and a nonsmooth function, considered in the ambient space. This class of problems finds important applications in machine learning and statistics such as the sparse principal component analysis, sparse spectral clustering, and orthogonal dictionary learning. We propose a Riemannian alternating direction method of multipliers (ADMM) to solve this class of problems. Our algorithm adopts easily computable steps in each iteration. The iteration complexity of the proposed algorithm for obtaining an ϵ-stationary point is analyzed under mild assumptions. Existing ADMM for solving nonconvex problems either does not allow nonconvex constraint set, or does not allow nonsmooth objective function. Our algorithm is the first ADMM type algorithm that minimizes a nonsmooth objective over manifold – a particular nonconvex set. Numerical experiments are conducted to demonstrate the advantage of the proposed *** Codes 90C30 Copyright © 2022, The Authors. All rights reserved.

关键词： Principal component analysis

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INTRINSIC AND EXTRINSIC DEEP LEARNING ON MANIFOLDS

arXiv

引用

arXiv 2023年

作者： Fang, Yihao Gupta, Vijay Ohn, Ilsang Lin, Lizhen Department of Applied and Computational Mathematics Statistics University of Notre Dame United States School of Electrical and Computer Engineering Purdue University United States Department of Statistics Inha University Korea Republic of

We propose extrinsic and intrinsic deep neural network architectures as general frameworks for deep learning on manifolds. Specifically, extrinsic deep neural networks (eDNNs) preserve geometric features on manifolds by utilizing an equivariant embedding from the manifold to its image in the Euclidean space. Moreover, intrinsic deep neural networks (iDNNs) incorporate the underlying intrinsic geometry of manifolds via exponential and log maps with respect to a Riemannian structure. Consequently, we prove that the empirical risk of the empirical risk minimizers (ERM) of eDNNs and iDNNs converge in optimal rates. Overall, The eDNNs framework is simple and easy to compute, while the iDNNs framework is accurate and fast converging. To demonstrate the utilities of our framework, various simulation studies, and real data analyses are presented with eDNNs and iDNNs. © 2023, CC BY.

关键词： Deep neural networks

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Ground state phases of the two-dimension electron gas with a unified variational approach

arXiv

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arXiv 2024年

作者： Smith, Conor Chen, Yixiao Levy, Ryan Yang, Yubo Morales, Miguel A. Zhang, Shiwei Center for Computational Quantum Physics Flatiron Institute New YorkNY10010 United States Department of Electrical and Computer Engineering University of New Mexico AlbuquerqueNM87131 United States Program in Applied and Computational Mathematics Princeton University PrincetonNJ08544 United States

The two-dimensional electron gas (2DEG) is a fundamental model, which is drawing increasing interest because of recent advances in experimental and theoretical studies of 2D materials. Current understanding of the ground state of the 2DEG relies on quantum Monte Carlo calculations, based on variational comparisons of different ansatze for different phases. We use a single variational ansatz, a general backflow-type wave function using a message-passing neural quantum state architecture, for a unified description across the entire density range. The variational optimization consistently leads to lower ground-state energies than previous best results. Transition into a Wigner crystal (WC) phase occurs automatically at rs = 37±1, a density lower than currently believed. Between the liquid and WC phases, the same ansatz and variational search strongly suggest the existence of intermediate states in a broad range of densities, with enhanced short-range nematic spin correlations. Copyright © 2024, The Authors. All rights reserved.

关键词： Message passing

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Bayesian Parameter Inference for Partially Observed SDEs driven by Fractional Brownian Motion

arXiv

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arXiv 2022年

作者： Maama, Mohamed Jasra, Ajay Ombao, Hernando Applied Mathematics and Computational Science Program Computer Electrical and Mathematical Sciences and Engineering Division King Abdullah University of Science and Technology Thuwal23955 Saudi Arabia Statistics Program Computer Electrical and Mathematical Sciences and Engineering Division King Abdullah University of Science and Technology Thuwal23955 Saudi Arabia

In this paper we consider Bayesian parameter inference for partially observed fractional Brownian motion (fBM) models. The approach we follow is to time-discretize the hidden process and then to design Markov chain Monte Carlo (MCMC) algorithms to sample from the posterior density on the parameters given data. We rely on a novel representation of the time discretization, which seeks to sample from an approximation of the posterior and then corrects via importance sampling;the approximation reduces the time (in terms of total observation time T) by O(T). This method is extended by using a multilevel MCMC method which can reduce the computational cost to achieve a given mean square error (MSE) versus using a single time discretization. Our methods are illustrated on simulated and real data. Copyright © 2022, The Authors. All rights reserved.

关键词： Mean square error

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Importance of reference Hamiltonians containing exact exchange for accurate one-shot GW calculations of Cu2O

引用

Physical Review B 2012年第23期85卷 235142-235142页

作者： Leah Y. Isseroff Emily A. Carter Department of Chemical and Biological Engineering Princeton University Princeton New Jersey 08544-5263 USA Department of Mechanical and Aerospace Engineering Program in Applied and Computational Mathematics and Andlinger Center for Energy and the Environment Princeton University Princeton New Jersey 08544-5263 USA

We show that a “one-shot” GW approach (denoted G0W0) can accurately calculate the photoemission/inverse-photoemission properties of Cu2O. As the results of any perturbative method are heavily dependent on the reference state, the appropriate reference Hamiltonian for G0W0 is identified by evaluating the performance of density-functional-theory-based input wave functions and eigenvalues generated with selected exchange-correlation functionals. It is shown that a reference Hamiltonian employing the hybrid Heyd-Scuseria-Ernzerhof functional used in conjunction with G0W0 produces an accurate photoemission/inverse-photoemission band gap and photoemission spectrum whose character is then further analyzed. The physical origin of why a hybrid functional is required for the zeroth-order wave function is discussed, giving insight into the unique electronic structure of Cu2O in comparison to other transition-metal oxides.

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PiRL: Participant-Invariant Representation Learning for Healthcare

arXiv

引用

arXiv 2022年

作者： Cao, Zhaoyang Yu, Han Yang, Huiyuan Sano, Akane Department of Computational Applied Mathematics and Operations Research Rice University HoustonTX77030 United States Department of Electrical and Computer Engineering Rice University HoustonTX77030 United States

Due to individual heterogeneity, performance gaps are observed between generic (one-size-fits-all) models and person-specific models in data-driven health applications. However, in real-world applications, generic models are usually more favorable due to new-user-adaptation issues and system complexities, etc. To improve the performance of the generic model, we propose a representation learning framework that learns participant-invariant representations, named PiRL. The proposed framework utilizes maximum mean discrepancy (MMD) loss and domain-adversarial training to encourage the model to learn participant-invariant representations. Further, a triplet loss, which constrains the model for inter-class alignment of the representations, is utilized to optimize the learned representations for downstream health applications. We evaluated our frameworks on two public datasets related to physical and mental health, for detecting sleep apnea and stress, respectively. As preliminary results, we found the proposed approach shows around a 5% increase in accuracy compared to the baseline. Copyright © 2022, The Authors. All rights reserved.

关键词： Machine learning

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