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Unbiased Estimation Using a Class of Diffusion Processes

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SSRN

SSRN 2022年

作者： Ruzayqat, Hamza Beskos, Alexandros Crisan, Dan Jasra, Ajay Kantas, Nikolas Applied Mathematics and Computational Science Program Computer Electrical and Mathematical Sciences and Engineering Division King Abdullah University of Science and Technology Thuwal23955-6900 Saudi Arabia Department of Statistical Science University College London LondonWC1E 6BT United Kingdom Department of Mathematics Imperial College London LondonSW7 2AZ United Kingdom

We study the problem of unbiased estimation of expectations with respect to (w.r.t.) π a given, general probability measure on (Rd,B(Rd)) that is absolutely continuous with respect to a standard Gaussian measure. We focus on simulation associated to a particular class of diffusion processes, sometimes termed the Schrödinger-Föllmer Sampler, which is a simulation technique that approximates the law of a particular diffusion bridge process {Xt}t∈[0,1] on Rd, d ∈ N0. This latter process is constructed such that, starting at X0 = 0, one has X1 ∼ π. Typically, the drift of the diffusion is intractable and, even if it were not, exact sampling of the associated diffusion is not possible. As a result, [10, 15] consider a stochastic Euler-Maruyama scheme that allows the development of biased estimators for expectations w.r.t. π. We show that for this methodology to achieve a mean square error of O(Ε2), for arbitrary Ε > 0, the associated cost is O(Ε−5). We then introduce an alternative approach that provides unbiased estimates of expectations w.r.t. π, that is, it does not suffer from the time discretization bias or the bias related with the approximation of the drift function. We prove that to achieve a mean square error of O(Ε2), the associated cost is, with high probability, O(Ε−2|log(Ε)|2+δ), for any δ > 0. We implement our method on several examples including Bayesian inverse problems. © 2022, The Authors. All rights reserved.

关键词： Diffusion

Weighted majority voting by statistical performance analysis on ensemble multiclassifier 5

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Weighted majority voting by statistical performance analysis...

5th International Conference on Informatics and Computing, ICIC 2020

作者： Wardoyo, Retantyo Musdholifah, Aina Angga Pradipta, Gede Hariyasa Sanjaya, I. Nyoman Faculty of Mathematics and Natural Sciences Universitas Gadjah Mada Department of Computer Science and Electronics Yogyakarta Indonesia Faculty of Mathematics and Natural Sciences Universitas Gadjah Mada Doctoral Program Department of Computer Science and Electronics Yogyakarta Indonesia Faculty of Medicine Udayana University/Sanglah General Hospital Department of Obstettics and Gynecology Bali Indonesia

ISBN: (纸本)9781728180472

Ensemble classifier method uses several base classifiers to predict a new test instance, while weighted majority voting has a scheme providing different weight values using several measurement parameters. However, the determination of the appropriate weight value to obtain an adequate ensemble model is a critical issue. This study, therefore, proposed a novel weighted majority voting scheme involving five base classifiers based on ensemble learning, including Random Forest, Decision Tree (C.45), Gradient Boosting Machine, XGBosst, and Bagging. The weighting scheme was formulated by analyzing the base classifier performance measured from the parameters of accuracy, recall, precision, and F Measure. The experiments were conducted using public datasets and umbilical cord data owned and the results showed the proposed method has the ability to improve performance in comparison with the base classifier and methods from previous studies with the best recorded in umbilical cord dataset with an average accuracy of 86.1%, a precision of 86%, a recall of 86%, and an F measure of 86%. © 2020 IEEE.

关键词： Decision trees

Exponential Asymptotics using Numerical Rational Approximation in Linear Differential Equations

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arXiv 2023年

作者： Lustri, Christopher J. Crew, Samuel Jonathan Chapman, S. School of Mathematics and Statistics The University of Sydney NSW2006 Australia Faculty of Computer Science Ruhr Universität Bochum Universitätsstraße Bochum44799 Germany Theoretical Sciences Visiting Program Okinawa Institute of Science and Technology Graduate University Onna904-0495 Japan Mathematical Institute University of Oxford OxfordOX2 6GG United Kingdom

MSC Codes 30E15, 34E15, 34M30, 41A20Singularly-perturbed ordinary differential equations often exhibit Stokes’ phenomenon, which describes the appearance and disappearance of oscillating exponentially small terms across curves in the complex plane known as Stokes curves. These curves originate at singular points in the leading-order solution to the differential equation. In many important problems, it is impossible to obtain a closed-form expression for these leading-order solutions, and it is therefore challenging to locate these singular points. We present evidence that the analytic leading-order solution of a linear differential equation can be replaced with a rational approximation based on a numerical leading-order solution using the adaptive Antoulas-Anderson (AAA) method. We show that the subsequent exponential asymptotic analysis accurately predicts the exponentially small behaviour present in the solution. We explore the limitations of this approach, and show that for sufficiently small values of the asymptotic parameter, this approach breaks down;however, the range of validity may be extended by increasing the number of poles in the rational approximation. We finish by presenting a related nonlinear problem and discussing the challenges that arise when attempting to apply this method to nonlinear problems. © 2023, CC BY.

关键词： Ordinary differential equations

Bayesian multinomial logistic normal models through marginally latent matrix-T processes

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The Journal of Machine Learning Research

The Journal of Machine Learning Research 2022年第1期23卷 255-296页

作者： Justin D. Silverman Kimberly Roche Zachary C. Holmes Lawrence A. David Sayan Mukherjee College of Information Science and Technology Department of Statistics and Institute for Computational and Data Science Penn State University University Park PA Program in Computational Biology and Bioinformatics Duke University Durham NC Department of Molecular Genetics and Microbiology Duke University Durham NC Department of Molecular Genetics and Microbiology and Center for Genomic and Computational Biology Duke University Durham NC Departments of Statistical Science Mathematics Computer Science Biostatistics & Bioinformatics Duke University Durham NC

Bayesian multinomial logistic-normal (MLN) models are popular for the analysis of sequence count data (e.g., microbiome or gene expression data) due to their ability to model multivariate count data with complex covariance structure. However, existing implementations of MLN models are limited to small datasets due to the non-conjugacy of the multinomial and logistic-normal distributions. Motivated by the need to develop efficient inference for Bayesian MLN models, we develop two key ideas. First, we develop the class of Marginally Latent Matrix-T Process (Marginally LTP) models. We demonstrate that many popular MLN models, including those with latent linear, non-linear, and dynamic linear structure are special cases of this class. Second, we develop an efficient inference scheme for Marginally LTP models with specific accelerations for the MLN subclass. Through application to MLN models, we demonstrate that our inference scheme are both highly accurate and often 4-5 orders of magnitude faster than MCMC.

关键词： Bayesian statistics multivariate analysis count data microbiome gene expression

Accelerating Real-Time Imaging for Radiotherapy: Leveraging Multi-GPU Training with PyTorch

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Accelerating Real-Time Imaging for Radiotherapy: Leveraging ...

International Conference on Machine Learning and Applications (ICMLA)

作者： Ruth Obe Brandt Kaufmann Kaelen Baird Sam Kadel Yasmin Soltani Mostafa Cham Matthias K. Gobbert Carlos A. Barajas Zhuoran Jiang Vijay R. Sharma Lei Ren Stephen W. Peterson Jerimy C. PoIf Dept. of Computer Science and of Software Engineering U. of Houston-Clear Lake USA Dept. of Mathematics and Statistics U. of San Francisco USA Dept. of Computer Science and of Mathematics Skidmore College USA Dept. of Computer Science and of Psychology Mount Holyoke College USA Dept. of Biomedical Engineering U. of Houston USA Dept. of Information Systems U. of Maryland Baltimore County Dept. of Mathematics and Statistics U. of Maryland Baltimore County Medical Physics Graduate Program Duke University USA Dept. of Radiation Oncology U. of Maryland School of Medicine USA Department of Radiation Oncology U. of Maryland School of Medicine USA Dept. of Physics U. of Cape Town South Africa H3D Inc. USA

Proton beam therapy is an advanced form of cancer radiotherapy that uses high-energy proton beams to deliver precise and targeted radiation to tumors. This helps to mit-igate unnecessary radiation exposure in healthy tissues. Real-time imaging of prompt gamma rays with Compton cameras has been suggested to improve therapy efficacy. However, the camera's non-zero time resolution leads to incorrect interaction classifications and noisy images that are insufficient for accurately assessing proton delivery in patients. To address the challenges posed by the Compton camera's image quality, machine learning techniques are employed to classify and refine the generated data. These machine-learning techniques include recurrent and feedforward neural networks. A PyTorch model was designed to improve the data captured by the Compton camera. This decision was driven by PyTorch's flexibility, powerful capabilities in handling sequential data, and enhanced G PU usage. This accelerates the model's computations on large-scale radiotherapy data. Through hyperparameter tuning, the validation accuracy of our PyTorch model has been improved from an initial 7 % to over 60 %. Moreover, the PyTorch Distributed Data Parallelism strategy was used to train the RNN models on multiple G PU s, which significantly reduced the training time with a minor impact on model accuracy.

关键词：

Developing a medical artificial intelligence course for high school students

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Developing a medical artificial intelligence course for high...

International Forum on Medical Imaging in Asia 2021, IFMIA 2021

作者： Huang, Chao-Jung Wu, Tinghui Lu, Jui-Ting Lin, Beatrice Chang, Dawei Wang, Pochuan Wang, Mei-Chi Lee, Peijung Wang, Weichung National Taiwan University-Stanford Joint Program Office of AI in Biotechnology Ministry of Science and Technology Joint Research Center for Artificial Intelligence Technology and All Vista Healthcare Taipei Taiwan Institute of Applied Mathematical Sciences National Taiwan University Taipei Taiwan Department of Mathematics Ecole Normale Supérieure de Lyon Lyon France Department of Computer Science and Information Engineering National Taiwan University Taipei Taiwan Department of Electronic and Computer Engineering National Taiwan University of Science and Technology Taipei Taiwan Department of Mathematics National Taiwan University Taipei Taiwan

ISBN: (数字)9781510644212

ISBN: (纸本)9781510644205

Artificial intelligence (AI) training courses often require prerequisites such as calculus or statistics. It is hence challenging to design and develop an introductory AI course for students of secondary education. This research intends to develop a medical AI course, provide high school students with an overview of deep learning applications in medical image analysis, and inspire them to pursue careers in the field of medical AI. We designed a 20-hour course, including lectures and two hands-on projects based on medical image classification. The proposed courses provided medical AI disciplines and built up their knowledge from basic to advanced levels. During the ten-day online courses, all the students were fully engaged and gave us positive feedback. The students endeavored to complete the experimental study in training, testing, and hypothesis of medical images application in the course. Their performance exceeded all expectations, for they did further analysis by tuning different hyperparameters. We designed a course evaluation form, which suggested that the students found it essential and expected to interact with the instructors. The results indicate that combining lectures with hands-on sessions would lead to evidently better achievement in terms of high school students' medical AI knowledge and positive attitudes while addressing real-world problems in the projects. Through this innovative education model, high school students regained their enthusiasm and were encouraged to cultivate their medical AI skills through self-learning while finishing the project. We conclude that this course could be successfully applied to interdisciplinary education in high school. © 2021 SPIE.

关键词： Students

Unbiased Estimation using a Class of Diffusion Processes

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arXiv 2022年

作者： Ruzayqat, Hamza Beskos, Alexandros Crisan, Dan Jasra, Ajay Kantas, Nikolas Applied Mathematics and Computational Science Program Computer Electrical and Mathematical Sciences and Engineering Division King Abdullah University of Science and Technology Thuwal23955-6900 KSA Saudi Arabia Department of Statistical Science University College London LondonWC1E 6BT United Kingdom Department of Mathematics Imperial College London LondonSW7 2AZ United Kingdom

We study the problem of unbiased estimation of expectations with respect to (w.r.t.) π a given, general probability measure on (d, B(d)) that is absolutely continuous with respect to a standard Gaussian measure. We focus on simulation associated to a particular class of diffusion processes, sometimes termed the Schrödinger-Föllmer Sampler, which is a simulation technique that approximates the law of a particular diffusion bridge process {Xt}tϵ[0,1]on d, d ϵ 0. This latter process is constructed such that, starting at X0= 0, one has X1∼ π. Typically, the drift of the diffusion is intractable and, even if it were not, exact sampling of the associated diffusion is not possible. As a result, [10, 16] consider a stochastic Euler-Maruyama scheme that allows the development of biased estimators for expectations w.r.t. π. We show that for this methodology to achieve a mean square error of O(ϵ2), for arbitrary ϵ > 0, the associated cost is O(ϵ-5). We then introduce an alternative approach that provides unbiased estimates of expectations w.r.t. π, that is, it does not suffer from the time discretization bias or the bias related with the approximation of the drift function. We prove that to achieve a mean square error of O(ϵ2), the associated cost (which is random) is, with high probability, O(ϵ-2| log(ϵ)|2+δ), for any δ > 0. We implement our method on several examples including Bayesian inverse *** Codes 60J60, 62D05, 65C40 © 2022, CC BY.

关键词： Diffusion

Embedding and trajectories of temporal networks

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arXiv 2022年

作者： Thongprayoon, Chanon Livi, Lorenzo Masuda, Naoki Department of Mathematics State University of New York at Buffalo BuffaloNY United States Department of Computer Science University of Manitoba WinnipegMB Canada Computational and Data-Enabled Science and Engineering Program State University of New York at Buffalo BuffaloNY United States Faculty of Science and Engineering Waseda University Shinjuku Tokyo Japan

Temporal network data are increasingly available in various domains, and often represent highly complex systems with intricate structural and temporal evolutions. Due to the difficulty of processing such complex data, it may be useful to coarse grain temporal network data into a numeric trajectory embedded in a low-dimensional space. We refer to such a procedure as temporal network embedding, which is distinct from procedures that aim at embedding individual nodes. Temporal network embedding is a challenging task because we often have access only to discrete time-stamped events between node pairs, and, in general, the events occur with irregular intervals, making the construction of the network at a given time a nontrivial question already. We propose a method to generate trajectories of temporal networks embedded in a low-dimensional space given a sequence of time-stamped events as input. We realize this goal by combining the landmark multidimensional scaling, which is an out-of-sample extension of the well-known multidimensional scaling method, and the framework of tie-decay temporal networks. This combination enables us to obtain a continuous-time trajectory describing the evolution of temporal networks. We then study mathematical properties of the proposed temporal network embedding framework. Finally, we showcase the method with empirical data of social contacts to find temporal organization of contact events and loss of them over a single day and across different days. © 2022, CC BY.

关键词： Trajectories

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