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A Likelihood Approach to Nonparametric Estimation of a Singular Distribution Using Deep Generative Models

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arXiv 2021年

作者： Chae, Minwoo Kim, Dongha Kim, Yongdai Lin, Lizhen Department of Industrial and Management Engineering Pohang University of Science and Technology Gyeongbuk Pohang37673 Korea Republic of School of Mathematics Statistics and Data Science Data Science Center Sungshin Women’s University Seoul02844 Korea Republic of Department of Statistics Seoul National University Seoul08826 Korea Republic of Department of Applied and Computational Mathematics and Statistics University of Notre Dame South BendIN46556 United States

We investigate statistical properties of a likelihood approach to nonparametric estimation of a singular distribution using deep generative models. More specifically, a deep generative model is used to model high-dimensional data that are assumed to concentrate around some low-dimensional structure. Estimating the distribution supported on this low-dimensional structure, such as a low-dimensional manifold, is challenging due to its singularity with respect to the Lebesgue measure in the ambient space. In the considered model, a usual likelihood approach can fail to estimate the target distribution consistently due to the singularity. We prove that a novel and effective solution exists by perturbing the data with an instance noise, which leads to consistent estimation of the underlying distribution with desirable convergence rates. We also characterize the class of distributions that can be efficiently estimated via deep generative models. This class is sufficiently general to contain various structured distributions such as product distributions, classically smooth distributions and distributions supported on a low-dimensional manifold. Our analysis provides some insights on how deep generative models can avoid the curse of dimensionality for nonparametric distribution estimation. We conduct a thorough simulation study and real data analysis to empirically demonstrate that the proposed data perturbation technique improves the estimation performance *** Codes 62G05 (Primary), 62G20 (Secondary) © 2021, CC BY-NC-SA.

关键词： Perturbation techniques

An adaptive moving mesh discontinuous Galerkin method for the radiative transfer equation

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arXiv 2018年

作者： Zhang, Min Cheng, Juan Huang, Weizhang Qiu, Jianxian School of Mathematical Sciences Xiamen University XiamenFujian361005 China Institute of Applied Physics and Computational Mathematics Beijing100094 China Department of Mathematics University of Kansas LawrenceKS66045 United States School of Mathematical Sciences Fujian Provincial Key Laboratory of Mathematical Modeling and High-Performance Scientific Computing Xiamen University XiamenFujian361005 China

The radiative transfer equation models the interaction of radiation with scattering and absorbing media and has important applications in various fields in science and engineering. It is an integro-differential equation involving time, space and angular variables and contains an integral term in angular directions while being hyperbolic in space. The challenges for its numerical solution include the needs to handle with its high dimensionality, the presence of the integral term, and the development of discontinuities and sharp layers in its solution along spatial directions. Its numerical solution is studied in this paper using an adaptive moving mesh discontinuous Galerkin method for spatial discretization together with the discrete ordinate method for angular discretization. The former employs a dynamic mesh adaptation strategy based on moving mesh partial differential equations to improve computational accuracy and efficiency. Its mesh adaptation ability, accuracy, and efficiency are demonstrated in a selection of one- and two-dimensional numerical *** Codes 65M50, 65M60, 65M70, 65R05, 65.75 Copyright © 2018, The Authors. All rights reserved.

关键词： Radiative transfer

Nonlinear Geometric Effects in Mechanical Bistable Morphing Structures

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Physical Review Letters 2012年第11期109卷 114302-114302页

作者： Zi Chen Qiaohang Guo Carmel Majidi Wenzhe Chen David J. Srolovitz Mikko P. Haataja Department of Mechanical and Aerospace Engineering Princeton University Princeton New Jersey 08544 USA Princeton Institute for the Science and Technology of Materials (PRISM) Princeton University Princeton New Jersey 08544 USA Department of Biomedical Engineering Washington University St. Louis Missouri 63130 USA College of Materials Science and Engineering Fuzhou University Fuzhou Fujian 350108 China Department of Mathematics and Physics Fujian University of Technology Fuzhou Fujian 350108 China Department of Mechanical Engineering Carnegie Mellon University Pittsburgh Pennsylvania 15213 USA Department of Materials Science and Engineering Fujian University of Technology Fuzhou Fujian 350108 China Institute for High Performance Computing 1 Fusionopolis Way Singapore 138632 Program in Applied and Computational Mathematics Princeton University Princeton New Jersey 08544 USA School of Mathematics Institute for Advanced Study Princeton New Jersey 08540 USA

Bistable structures associated with nonlinear deformation behavior, exemplified by the Venus flytrap and slap bracelet, can switch between different functional shapes upon actuation. Despite numerous efforts in modeling such large deformation behavior of shells, the roles of mechanical and nonlinear geometric effects on bistability remain elusive. We demonstrate, through both theoretical analysis and tabletop experiments, that two dimensionless parameters control bistability. Our work classifies the conditions for bistability, and extends the large deformation theory of plates and shells.

关键词： NONLINEAR theories DEFORMATIONS (Mechanics) PARAMETER estimation OPTICAL bistability STRUCTURAL shells PLATES (Engineering) CRYSTAL structure

Complexity for superconformal primaries from BCH techniques

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arXiv 2022年

作者： Rabambi, Phumudzo van Zyl, Hendrik J.R. Mandelstam Institute for Theoretical Physics School of Physics NITheCS CoE-MaSS University of the Witwatersrand JohannesburgWITS 2050 South Africa The Laboratory for Quantum Gravity & Strings Department of Mathematics & Applied Mathematics University of Cape Town Cape Town South Africa The National Institute for Theoretical and Computational Sciences Private Bag X1 Matieland South Africa

We extend existing results for the Nielsen complexity of scalar primaries and spinning primaries in four dimensions by including supersymmetry. Specifically, we study the Nielsen complexity of circuits that transform a superconformal primary with definite scaling dimension, spin and R-charge by means of continuous unitary gates from the su(2,2|N) group. Our analysis makes profitable use of Baker-Campbell-Hausdorff formulas including a special class of BCH formulas we conjecture and motivate. With this approach we are able to determine the super-Kähler potential characterizing the circuit complexity geometry and obtain explicit expressions in the case of N = 1 and N = 2 supersymmetry. © 2022, CC BY.

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Ground-state preparation and energy estimation on early fault-tolerant quantum computers via quantum eigenvalue transformation of unitary matrices

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arXiv 2022年

作者： Dong, Yulong Lin, Lin Tong, Yu Department of Mathematics University of California BerkeleyCA94720 United States Applied Mathematics and Computational Research Division Lawrence Berkeley National Laboratory BerkeleyCA94720 United States Challenge Institute of Quantum Computation University of California BerkeleyCA94720 United States Berkeley Center for Quantum Information and Computation BerkeleyCA94720 United States

Under suitable assumptions, the algorithms in [Lin, Tong, Quantum 2020] can estimate the ground-state energy and prepare the ground state of a quantum Hamiltonian with near-optimal query complexities. However, this is based on a block encoding input model of the Hamiltonian, whose implementation is known to require a large resource overhead. We develop a tool called quantum eigenvalue transformation of unitary matrices with real polynomials (QET-U), which uses a controlled Hamiltonian evolution as the input model, a single ancilla qubit and no multi-qubit control operations, and is thus suitable for early fault-tolerant quantum devices. This leads to a simple quantum algorithm that outperforms all previous algorithms with a comparable circuit structure for estimating the ground-state energy. For a class of quantum spin Hamiltonians, we propose a new method that exploits certain anti-commutation relations and further removes the need of implementing the controlled Hamiltonian evolution. Coupled with a Trotter-based approximation of the Hamiltonian evolution, the resulting algorithm can be very suitable for early fault-tolerant quantum devices. We demonstrate the performance of the algorithm using IBM Qiskit for the transverse field Ising model. If we are further allowed to use multi-qubit Toffoli gates, we can then implement amplitude amplification and a new binary amplitude estimation algorithm, which increases the circuit depth but decreases the total query complexity. The resulting algorithm saturates the near-optimal complexity for ground-state preparation and energy estimating using a constant number of ancilla qubits (no more than 3). © 2022, CC BY.

关键词： Hamiltonians

Digital Audio Watermarking by Amplitude Embedding System 15th

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Digital Audio Watermarking by Amplitude Embedding System

15th International Conference on Intelligent Information Hiding and Multimedia Signal Processing, IIH-MSP 2019, held in conjunction with the 12th International Conference on Frontiers of Information Technology, Applications and Tools, FITAT 2019

作者： Lin, Meng-Ju Zhao, Ming Chen, Shuo-Tsung Chen, Ching-Ju Department of Applied Mathematics Tunghai University No. 1727 Sec.4 Taiwan Boulevard Taichung Taiwan40704 Taiwan School of Computer Science Yangtze University Jingzhou CityHubei Province China College of Future Bachelor Program in Interdisciplinary Studies National Yunlin University of Science and Technology TaiwanYunlin64002 Taiwan National Yunlin University of Science and Technology TaiwanYunlin64002 Taiwan

ISBN: (纸本)9789811397134

This paper proposed an amplitude embedding system for digital audio watermarking in the wavelet domain. In this paper, the relationship between the watermarked coefficients and the original coefficients is first presented and then amplitude-based embedding rules are combined into an amplitude-based embedding system that incorporates a system state. For the embedding in this system, the information includes synchronization codes and the watermarks are embedded into the DWT lowest frequency sub-band coefficients which can be described by a matrix form. Moreover, the SNR is rewritten as a performance index which is a quadratic form. Then, a nonlinear function connecting the performance index and the embedding system is derived. Accordingly, an optimization-based embedding system is proposed. Finally, the Karush–Kuhn–Tucker (KKT) Theorem is used to obtain the optimal solution, which is an important formula for watermarking. In addition, the hidden information can be extracted without knowledge of the original audio. In the experiments, the performance of the proposed system is tested and the results verify the high SNR and strong robustness against multimedia signal processing or malicious attacks. © 2020, Springer Nature Singapore Pte Ltd.

关键词： Embeddings

The Role of Heterogeneity in Autonomous Perimeter Defense Problems

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arXiv 2022年

作者： Adler, Aviv Mickelin, Oscar Ramachandran, K. Sukhatme, Gaurav S. Karaman, Sertac Department of Electrical Engineering and Computer Science Massachusetts Institute of Technology CambridgeMA United States Program in Applied and Computational Mathematics Princeton University Princeton NJ08544 United States Department of Computer Science University of Southern California Los AngelesCA90089 United States Department of Aeronautics and Astronautics Massachusetts Institute of Technology CambridgeMA02139 United States

When is heterogeneity in the composition of an autonomous robotic team beneficial and when is it detrimental? We investigate and answer this question in the context of a minimally viable model that examines the role of heterogeneous speeds in perimeter defense problems, where defenders share a total allocated speed budget. We consider two distinct problem settings and develop strategies based on dynamic programming and on local interaction rules. We present a theoretical analysis of both approaches and our results are extensively validated using simulations. Interestingly, our results demonstrate that the viability of heterogeneous teams depends on the amount of information available to the defenders. Moreover, our results suggest a universality property: across a wide range of problem parameters the optimal ratio of the speeds of the defenders remains nearly constant. Copyright © 2022, The Authors. All rights reserved.

关键词： Dynamic programming

Combination chemotherapy optimization with discrete dosing

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arXiv 2021年

作者： Ajayi, Temitayo Hosseinian, Seyedmohammadhossein Schaefer, Andrew J. Fuller, Clifton D. Nature Source Improved Plants IthacaNY14850 United States Department of Computational and Applied Mathematics Rice University HoustonTX77005 United States Department of Radiation Oncology The University of Texas MD Anderson Cancer Center HoustonTX77030 United States

Chemotherapy is one of the primary modalities of cancer treatment. Chemotherapy drug administration is a complex problem that often requires expensive clinical trials to evaluate potential regimens. One way to alleviate this burden and better inform future trials is to build reliable models for drug administration. Previous chemotherapy optimization models have mainly relied on optimal control, which does not lend itself to capturing complex and vital operational constraints in chemotherapy planning involving discrete decisions, such as doses via pills and rest periods. In addition, most of the existing models for chemotherapy optimization lack an explicit toxicity measure and impose toxicity constraints primarily through (fixed) limits on drug concentration. The existing stochastic optimization models also focus on maximizing the probability of cure when tumor heterogeneity is uncertain. In this paper, we develop a mixed-integer program for combination chemotherapy (utilization of multiple drugs) optimization that incorporates various important operational constraints and, besides dose and concentration limits, controls treatment toxicity based on its effect on the count of white blood cells. To address the uncertainty of tumor heterogeneity, we propose chance constraints that guarantee reaching an operable tumor size with a high probability in a neoadjuvant setting. We present analytical results pertinent to the accuracy of the model in representing biological processes of chemotherapy and establish its merit for clinical applications through a numerical study of breast cancer. © 2021, CC BY-NC-ND.

关键词： Chemotherapy

Extreme localization of rare events on photonic chips

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Extreme localization of rare events on photonic chips

European Conference on Lasers and Electro-Optics, CLEO 2015

作者： Liu, C. Van der Wel, R. Rotenberg, N. Kuipers, L. Krauss, T.F. Di Falco, A. Fratalocchi, A. PRIMALIGHT Electrical Engineering and Applied Mathematics and Computational Science KAUST Thuwal23955-6900 Saudi Arabia Center for Nanophotonics FOM Institute AMOLF Science Park 104 Amsterdam1098 XG Netherlands Department of Physics University of York Heslington YorkYO10 5DD United Kingdom School of Physics and Astronomy University of St. Andrews North Haugh St. Andrews KY16 9SS United Kingdom