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Covariance-Based Activity Detection in Cooperative Multi-Cell Massive MIMO: Scaling Law and Efficient Algorithms

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arXiv 2023年

作者： Wang, Ziyue Liu, Ya-Feng Wang, Zhaorui Yu, Wei The State Key Laboratory of Scientific and Engineering Computing Institute of Computational Mathematics and Scientific/Engineering Computing Academy of Mathematics and Systems Science Chinese Academy of Sciences Beijing100190 China The School of Mathematical Sciences University of Chinese Academy of Sciences Beijing100049 China Guangdong Provincial Key Laboratory of Future Networks of Intelligence The Chinese University of Hong Kong Shenzhen518172 China The Edward S. Rogers Sr. Department of Electrical and Computer Engineering University of Toronto TorontoONM5S 3G4 Canada

This paper focuses on the covariance-based activity detection problem in a multi-cell massive multiple-input multiple-output (MIMO) system. In this system, active devices transmit their signature sequences to multiple base stations (BSs), and the BSs cooperatively detect the active devices based on the received signals. While the scaling law for the covariance-based activity detection in the single-cell scenario has been extensively analyzed in the literature, this paper aims to analyze the scaling law for the covariance-based activity detection in the multi-cell massive MIMO system. Specifically, this paper demonstrates a quadratic scaling law in the multi-cell system, under the assumption that the path-loss exponent of the fading channel γ > 2. This finding shows that, in the multi-cell massive MIMO system, the maximum number of active devices that can be correctly detected in each cell increases quadratically with the length of the signature sequence and decreases logarithmically with the number of cells (as the number of antennas tends to infinity). Moreover, in addition to analyzing the scaling law for the signature sequences randomly and uniformly distributed on a sphere, the paper also establishes the scaling law for signature sequences based on a finite alphabet, which are easier to generate and store. Finally, this paper proposes two efficient accelerated coordinate descent (CD) algorithms with a convergence guarantee for solving the device activity detection problem. The first algorithm reduces the complexity of CD by using an inexact coordinate update strategy. The second algorithm avoids unnecessary computations of CD by using an active set selection strategy. Simulation results show that the proposed algorithms exhibit excellent performance in terms of computational efficiency and detection error probability. Copyright © 2023, The Authors. All rights reserved.

关键词： Fading channels

Deepstsnet: Reconstructing the Quantum state-Resolved Thermochemical Nonequilibrium Flowfield Using Deep Neural Operator Learning With Scarce Data

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SSRN

SSRN 2023年

作者： Lv, Jiaqi Hong, Qizhen Wang, Xiaoyong Mao, Zhiping Sun, Quanhua State Key Laboratory of High Temperature Gas Dynamics Institute of Mechanics Chinese Academy of Sciences Beijing100190 China School of Engineering Science University of Chinese Academy of Sciences Beijing100049 China School of Mathematical Sciences Fujian Provincial Key Laboratory of Mathematical Modeling and High-Performance Scientific Computing Xiamen University Xiamen361005 China

The hypersonic flow is in a thermochemical nonequilibrium state due to the high-temperature caused by the strong shock compression. In a thermochemical nonequilibrium flow, the distribution of molecular internal energy levels strongly deviates from the equilibrium distribution (i.e., the Boltzmann distribution). It is intractable to directly obtain the microscopic nonequilibrium distribution from existed experimental measurements usually described by macroscopic field variables such as temperature or velocity. Motivated by the idea of deep multi-scale multi-physics neural network (DeepMMNet) proposed in [Mao et al, J. Comput. Phys., 2021], we develop in this paper a data assimilation framework called DeepStSNet to accurately reconstruct the quantum state-resolved thermochemical nonequilibrium flowfield by using sparse experimental measurements of vibrational temperature and pre-trained deep neural operator networks (DeepONets). In particular, we first construct several DeepONets to express the coupled dynamics between field variables in the thermochemical nonequilibrium flow and to approximate the state-to-state (StS) approach, which traces the variation of each vibrational level of molecule accurately. These proposed DeepONets are then trained by using the numerical simulation data, and would later be served as building blocks for the DeepStSNet. We demonstrate the effectiveness and accuracy of DeepONets with different test cases showing that the density and energy of vibrational groups as well as the temperature and velocity fields are predicted with high accuracy. We then extend the architectures of DeepMMNet by considering a simplified thermochemical nonequilibrium model, i.e., the 2T model, showing that the entire thermochemical nonequilibrium flowfield is well predicted by using scattered measurements of full or even partial field variables. We next consider a more accurate and complex thermochemical nonequilibrium model, i.e., the StS-CGM model, and develop a

关键词： Multiphysics

Some Sharp Error Bounds for Multivariate Linear Interpolation and Extrapolation

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arXiv 2022年

作者： Cao, Liyuan Wen, Zaiwen Yuan, Ya-Xiang Beijing International Center for Mathematical Research Peking University Beijing China State Key Laboratory of Scientific and Engineering Computing Academy of Mathematics and Systems Science Chinese Academy of Sciences Beijing China

We study in this paper the function approximation error of linear interpolation and extrapolation. Several upper bounds are presented along with the conditions under which they are sharp. All results are under the assumptions that the function has Lipschitz continuous gradient and is interpolated on an affinely independent sample set. Errors for quadratic functions and errors of bivariate linear extrapolation are analyzed in *** Codes 41A05, 41A10, 41A80, 46N10 Copyright © 2022, The Authors. All rights reserved.

关键词： Extrapolation

Symplectic Stiefel manifold: tractable metrics, second-order geometry and Newton’s methods

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arXiv 2024年

作者： Gao, Bin Son, Nguyen Thanh Stykel, Tatjana State Key Laboratory of Scientific and Engineering Computing Academy of Mathematics and Systems Science Chinese Academy of Sciences Beijing100190 China Department of Mathematics and Informatics Thai Nguyen University of Sciences Thai Nguyen24118 Viet Nam Universität Augsburg Universitätsstraße 12a Augsburg86159 Germany

Optimization under the symplecticity constraint is an approach for solving various problems in quantum physics and scientific computing. Building on the results that this optimization problem can be transformed into an unconstrained problem on the symplectic Stiefel manifold, we construct geometric ingredients for Riemannian optimization with a new family of Riemannian metrics called tractable metrics and develop Riemannian Newton schemes. The newly obtained ingredients do not only generalize several existing results but also provide us with freedom to choose a suitable metric for each problem. To the best of our knowledge, this is the first try to develop the explicit second-order geometry and Newton’s methods on the symplectic Stiefel manifold. For the Riemannian Newton method, we first consider novel operator-valued formulas for computing the Riemannian Hessian of a cost function, which further allows the manifold to be endowed with a weighted Euclidean metric that can provide a preconditioning effect. We then solve the resulting Newton equation, as the central step of Newton’s methods, directly via transforming it into a saddle point problem followed by vectorization, or iteratively via applying any matrix-free iterative method either to the operator Copyright © 2024, The Authors. All rights reserved.

关键词： Geometry

Phase-field modeling of dendritic growth with gas bubbles in the solidification of binary alloys

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arXiv 2024年

作者： Zhan, Chengjie Chai, Zhenhua Sun, Dongke Shi, Baochang Geng, Shaoning Jiang, Ping School of Mathematics and Statistics Huazhong University of Science and Technology Wuhan430074 China Institute of Interdisciplinary Research for Mathematics and Applied Science Huazhong University of Science and Technology Wuhan430074 China Hubei Key Laboratory of Engineering Modeling and Scientific Computing Huazhong University of Science and Technology Wuhan430074 China School of Mechanical Engineering Southeast University Nanjing211189 China The State Key Laboratory of Digital Manufacturing Equipment and Technology School of Mechanical Science and Engineering Huazhong University of Science and Technology Wuhan430074 China

In this work, a phase-field model is developed for the dendritic growth with gas bubbles in the solidification of binary alloys. In this model, a total free energy for the complex gas-liquid-dendrite system is proposed through considering the interactions of gas bubbles, liquid melt and solid dendrites, and it can reduce to the energy for gas-liquid flows in the region far from the solid phase, while degenerate to the energy for thermosolutal dendritic growth when the gas bubble disappears. The governing equations are usually obtained by minimizing the total free energy, but here some modifications are made to improve the capacity of the conservative phase-field equation for gas bubbles and convection-diffusion equation for solute transfer. Additionally, through the asymptotic analysis of the thin-interface limit, the present general phase-field model for alloy solidification can match the corresponding free boundary problem, and it is identical to the commonly used models under a specific choice of model parameters. Furthermore, to describe the fluid flow, the incompressible Navier-Stokes equations are adopted in the entire domain including gas, liquid, and solid regions, where the fluid-structure interaction is considered by a simple diffuse-interface method. To test the present phase-field model, the lattice Boltzmann method is used to study several problems of gas-liquid flows, dendritic growth as well as the solidification in presence of gas bubbles, and a good performance of the present model for such complex problems is observed. © 2024, CC BY-NC-ND.

关键词： Solidification

Learning-rate-free Momentum SGD with Reshuffling Converges in Nonsmooth Nonconvex Optimization

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arXiv 2024年

作者： Hu, Xiaoyin Xiao, Nachuan Liu, Xin Toh, Kim-Chuan School of Computer and Computing Science Hangzhou City University Hangzhou310015 China Institute of Operational Research and Analytics National University of Singapore Singapore State Key Laboratory of Scientific and Engineering Computing Academy of Mathematics and Systems Science Chinese Academy of Sciences University of Chinese Academy of Sciences China Department of Mathematics Institute of Operations Research and Analytics National University of Singapore Singapore119076 Singapore

In this paper, we propose a generalized framework for developing learning-rate-free momentum stochastic gradient descent (SGD) methods in the minimization of nonsmooth nonconvex functions, especially in training nonsmooth neural networks. Our framework adaptively generates learning rates based on the historical data of stochastic subgradients and iterates. Under mild conditions, we prove that our proposed framework enjoys global convergence to the stationary points of the objective function in the sense of the conservative field, hence providing convergence guarantees for training nonsmooth neural networks. Based on our proposed framework, we propose a novel learning-rate-free momentum SGD method (LFM). Preliminary numerical experiments reveal that LFM performs comparably to the state-of-the-art learning-rate-free methods (which have not been shown theoretically to be convergence) across well-known neural network training benchmarks. Copyright © 2024, The Authors. All rights reserved.

关键词： Momentum

Proximal Subgradient Norm Minimization of Ista and Fista

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SSRN

SSRN 2022年

作者： Li, Bowen Shi, Bin Yuan, Ya-Xiang State Key Laboratory of Scientific and Engineering Computing Academy of Mathematics and Systems Science Chinese Academy of Sciences Beijing Beijing100190 China University of Chinese Academy of Sciences Beijing Beijing100049 China

For first-order smooth optimization, the research on the acceleration phenomenon has a long-time history. Until recently, the mechanism leading to acceleration was not successfully uncovered by the gradient correction term and its equivalent implicit-velocity form. Furthermore, based on the high-resolution differential equation framework with the corresponding emerging techniques, phase-space representation and Lyapunov function, the squared gradient norm of Nesterov's accelerated gradient descent (NAG) method at an inverse cubic rate is discovered. However, this result cannot be directly generalized to composite optimization widely used in practice, e.g., the linear inverse problem with sparse representation. In this paper, we meticulously observe a pivotal inequality used in composite optimization about the step size s and the Lipschitz constant L and find that it can be improved tighter. We apply the tighter inequality discovered in the well-constructed Lyapunov function and then obtain the proximal subgradient norm minimization by the phase-space representation, regardless of gradient-correction or implicit-velocity. Furthermore, we demonstrate that the squared proximal subgradient norm for the class of iterative shrinkage-thresholding algorithms (ISTA) converges at an inverse square rate, and the squared proximal subgradient norm for the class of faster iterative shrinkage-thresholding algorithms (FISTA) is accelerated to convergence at an inverse cubic rate. © 2022, The Authors. All rights reserved.

关键词： Lyapunov functions

On the Hyperparameters in Stochastic Gradient Descent with Momentum

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arXiv 2021年

作者： Shi, Bin State Key Laboratory of Scientific and Engineering Computing Academy of Mathematics and Systems Science Chinese Academy of Sciences Beijing China

Following the same routine as [SSJ20], we continue to present the theoretical analysis for stochastic gradient descent with momentum (SGD with momentum) in this paper. Differently, for SGD with momentum, we demonstrate that the two hyperparameters together, the learning rate and the momentum coefficient, play a significant role in the linear convergence rate in nonconvex optimizations. Our analysis is based on using a hyperparameters-dependent stochastic differential equation (hp-dependent SDE) that serves as a continuous surrogate for SGD with momentum. Similarly, we establish the linear convergence for the continuous-time formulation of SGD with momentum and obtain an explicit expression for the optimal linear rate by analyzing the spectrum of the Kramers-Fokker-Planck operator. By comparison, we demonstrate how the optimal linear rate of convergence and the final gap for SGD only about the learning rate varies with the momentum coefficient increasing from zero to one when the momentum is introduced. Then, we propose a mathematical interpretation of why, in practice, SGD with momentum converges faster and is more robust in the learning rate than standard stochastic gradient descent (SGD). Finally, we show the Nesterov momentum under the presence of noise has no essential difference from the traditional momentum. © 2021, CC BY-NC-SA.

关键词： Momentum

The Sampling Method for Optimal Precursors of ENSO Events

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arXiv 2023年

作者： Shi, Bin Ma, Junjie State Key Laboratory of Scientific and Engineering Computing Academy of Mathematics and Systems Science Chinese Academy of Sciences Beijing100190 China School of Mathematical Sciences University of Chinese Academy of Sciences Beijing100049 China School of Mathematics North University of China Taiyuan030051 China

El Niño-Southern Oscillation (ENSO) is one of the significant climate phenomena, which appears periodically in the tropic Pacific. The intermediate coupled ocean-atmosphere Zebiak-Cane (ZC) model is the first and classical one designed to numerically forecast the ENSO events. Traditionally, the conditional nonlinear optimal perturbation (CNOP) approach has been used to capture optimal precursors in practice. In this paper, based on state-of-the-art statistical machine learning techniques,1 we investigate the sampling algorithm proposed in [Shi and Sun, 2023] to obtain optimal precursors via the CNOP approach in the ZC model. For the ZC model, or more generally, the numerical models with dimension O(104−105), the numerical performance, regardless of the statically spatial patterns and the dynamical nonlinear time evolution behaviors as well as the corresponding quantities and indices, shows the high efficiency of the sampling method by comparison with the traditional adjoint method. The sampling algorithm does not only reduce the gradient (first-order information) to the objective function value (zeroth-order information) but also avoids the use of the adjoint model, which is hard to develop in the coupled ocean-atmosphere models and the parameterization models. In addition, based on the key characteristic that the samples are independently and identically distributed, we can implement the sampling algorithm by parallel computation to shorten the computation time. Meanwhile, we also show in the numerical experiments that the important features of optimal precursors can be still captured even when the number of samples is reduced sharply. © 2023, CC BY-NC-SA.

关键词： Numerical methods