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Global small solutions to heat conductive compressible nematic liquid crystal system: Smallness on a scaling invariant quantity

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arXiv 2021年

作者： Li, Jinkai Tao, Qiang South China Research Center for Applied Mathematics and Interdisciplinary Studies School of Mathematical Sciences South China Normal University Guangzhou510631 China School of Mathematics and Statistics Shenzhen University Shenzhen518060 China Shenzhen Key Laboratory of Advanced Machine Learning and Applications Shenzhen University Shenzhen518060 China

In this paper, we consider the Cauchy problem to the three dimensional heat conducting compressible nematic liquid crystal system in the presence of vacuum and with vacuum far fields. Global well-posedness of strong solutions is established under the condition that the scaling invariant quantity (||ρ0||∞ + 1)[||ρ0||3 + (||ρ0||∞ + 1)2(||√ρ0u0||22 + ||∇d0||22)][||∇u0||22 + (||ρ0||∞ + 1)(||√ρ0E0||22 + ||∇2d0||22)] is sufficiently small with the smallness depending only on the parameters appeared in the system. Copyright © 2021, The Authors. All rights reserved.

关键词： Heat conduction

A Class-wise Non-salient Region Generalized Framework for Video Semantic Segmentation

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arXiv 2022年

作者： Zhang, Yuhang Tian, Shishun Liao, Muxin Zhang, Zhengyu Zou, Wenbin Xu, Chen Guangdong Key Laboratory of Intelligent Information Processing College of Electronics and Information Engineering Shenzhen University Shenzhen518060 China Univ Rennes INSA Rennes CNRS IETR - UMR 6164 RennesF-35000 France Guangdong Key Laboratory of Intelligent Information Processing Shenzhen Key Laboratory of Advanced Machine Learning and Applications Institute of Artificial Intelligence and Advanced Communication College of Electronics and Information Engineering Shenzhen University Shenzhen518060 China College of Mathematics and Statistics Shenzhen University Shenzhen518060 China

Video semantic segmentation (VSS) is beneficial for dealing with dynamic scenes due to the continuous property of the real-world environment. On the one hand, some methods alleviate the predicted inconsistent problem between continuous frames. On the other hand, other methods employ the previous frame as the prior information to assist in segmenting the current frame. Although the previous methods achieve superior performances on the independent and identically distributed (i.i.d) data, they can not generalize well on other unseen domains. Thus, we explore a new task, the video generalizable semantic segmentation (VGSS) task that considers both continuous frames and domain generalization. In this paper, we propose a class-wise non-salient region generalized (CNSG) framework for the VGSS task. Concretely, we first define the class-wise non-salient feature, which describes features of the class-wise non-salient region that carry more generalizable information. Then, we propose a class-wise non-salient feature reasoning strategy to select and enhance the most generalized channels adaptively. Finally, we propose an inter-frame non-salient centroid alignment loss to alleviate the predicted inconsistent problem in the VGSS task. We also extend our video-based framework to the image-based generalizable semantic segmentation (IGSS) task. Experiments demonstrate that our CNSG framework yields significant improvement in the VGSS and IGSS tasks. Copyright © 2022, The Authors. All rights reserved.

关键词： Semantic Segmentation

Multiplicative noise removal: Nonlocal low-rank model and its proximal alternating reweighted minimization algorithm

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arXiv 2020年

作者： Liu, Xiaoxia Lu, Jian Shen, Lixin Xu, Chen Xu, Yuesheng Shenzhen Key Laboratory of Advanced Machine Learning and Applications College of Mathematics and Statistics Shenzhen University Shenzhen518060 China Department of Mathematics Syracuse University SyracuseNY13244 United States Department of Mathematics and Statistics Old Dominion University NorfolkVA23529 United States

The goal of this paper is to develop a novel numerical method for efficient multiplicative noise removal. The nonlocal self-similarity of natural images implies that the matrices formed by their nonlocal similar patches are low-rank. By exploiting this low-rank prior with application to multiplicative noise removal, we propose a nonlocal low-rank model for this task and develop a proximal alternating reweighted minimization (PARM) algorithm to solve the optimization problem resulting from the model. Specifically, we utilize a generalized nonconvex surrogate of the rank function to regularize the patch matrices and develop a new nonlocal low-rank model, which is a nonconvex nonsmooth optimization problem having a patchwise data fidelity and a generalized nonlocal low-rank regularization term. To solve this optimization problem, we propose the PARM algorithm, which has a proximal alternating scheme with a reweighted approximation of its subproblem. A theoretical analysis of the proposed PARM algorithm is conducted to guarantee its global convergence to a critical point. Numerical experiments demonstrate that the proposed method for multiplicative noise removal significantly outperforms existing methods such as the benchmark SAR-BM3D method in terms of the visual quality of the denoised images, and the PSNR (the peak-signal-to-noise ratio) and SSIM (the structural similarity index measure) values. Copyright © 2020, The Authors. All rights reserved.

关键词： Image reconstruction

Semi-supervised Symmetric Non-negative Matrix Factorization with Low-Rank Tensor Representation

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arXiv 2024年

作者： Jia, Yuheng Li, Jia-Nan Wu, Wenhui Wang, Ran The School of Computer Science and Engineering Southeast University Nanjing211189 China Ministry of Education China The College of Electronics and Information Engineering Shenzhen University Shenzhen518060 China The School of Mathematical Sciences Shenzhen University Shenzhen518060 China The Guangdong Provincial Key Laboratory of Intelligent Information Processing Shenzhen University Shenzhen518060 China The Shenzhen Key Laboratory of Advanced Machine Learning and Applications Shenzhen University Shenzhen518060 China

Semi-supervised symmetric non-negative matrix factorization (SNMF) utilizes the available supervisory information (usually in the form of pairwise constraints) to improve the clustering ability of SNMF. The previous methods introduce the pairwise constraints from the local perspective, i.e., they either directly refine the similarity matrix element-wisely or restrain the distance of the decomposed vectors in pairs according to the pairwise constraints, which overlook the global perspective, i.e., in the ideal case, the pairwise constraint matrix and the ideal similarity matrix possess the same low-rank structure. To this end, we first propose a novel semi-supervised SNMF model by seeking low-rank representation for the tensor synthesized by the pairwise constraint matrix and a similarity matrix obtained by the product of the embedding matrix and its transpose, which could strengthen those two matrices simultaneously from a global perspective. We then propose an enhanced SNMF model, making the embedding matrix tailored to the above tensor low-rank representation. We finally refine the similarity matrix by the strengthened pairwise constraints. We repeat the above steps to continuously boost the similarity matrix and pairwise constraint matrix, leading to a high-quality embedding matrix. Extensive experiments substantiate the superiority of our method. The code is available at https://***/JinaLeejnl/TSNMF. Copyright © 2024, The Authors. All rights reserved.

关键词： Non-negative matrix factorization

A mathematical model for efficient extraction of key locations from point-cloud data in track area

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Industrial Artificial Intelligence 2023年第1期1卷 1-14页

作者： Chen, Shuyue Wu, Jiaolv Lu, Jian Wang, Xizhao College of Mathematics and Statistics Shenzhen University Shenzhen China School of Software Engineering Shenzhen Institue of Information Technology Shenzhen China Shenzhen No. 3 Vocational School of Technology Shenzhen China College of Engineering Huaqiao University Quanzhou China Shenzhen Key Lab. of Advanced Machine Learning and Applications Shenzhen University Shenzhen China Guangdong Key Lab. of Intelligent Information Process Shenzhen University Shenzhen China College of Computer Science and Software Engineering Shenzhen University Shenzhen China

During the construction of a metro system, it is inevitable that deviations will occur between the excavated tunnel and the original designed scheme. As such, it is necessary to adjust the designed scheme to accommodate these discrepancies. Specifically, the adjustment of the designed scheme involves a rigorous process of repeatedly selecting and verifying the feasibility of the proposed modifications using point-cloud data obtained from the tunnel. However, this process can be considerably time-consuming due to the large-scale and potentially redundant nature of the point-cloud data. This paper proposes a mathematical model for point-cloud data acquired in measuring a mined tunnel, which may deviate from the originally designed one. The modeling, which mainly includes determining its normal plane, and building the equation of tunnel point-cloud data, is to quickly extract several key locations in the tunnel surface for modifying the original design in order to achieve a minimum error between the modified design and the mined tunnel. In comparison with the conventional processing of extracting several key locations directly from point-cloud data, our model shows a significant promotion of extraction efficiency under an acceptable error bound. The model is tested in a real tunnel point-cloud data and the testing results confirm the increase of fitting accuracy and the decrease of computational load.

关键词：

Analysis of adaptive synchrosqueezing transform with a time-varying parameter

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arXiv 2020年

作者： Lu, Jian Jiang, Qingtang Li, Lin Shenzhen Key Laboratory of Advanced Machine Learning and Applications College of Mathematics and Statistics Shenzhen University Shenzhen518060 China Department of Mathematics and Statistics University of Missouri-St. Louis St. LouisMO63121 United States School of Electronic Engineering Xidian University Xi'an710071 China

The synchrosqueezing transform (SST) was developed recently to separate the components of non-stationary multicomponent signals. The continuous wavelet transform-based SST (WSST) reassigns the scale variable of the continuous wavelet transform of a signal to the frequency variable and sharpens the time-frequency representation. The WSST with a time-varying parameter, called the adaptive WSST, was introduced very recently in the paper "Adaptive synchrosqueezing transform with a time-varying parameter for non-stationary signal separation". The well-separated conditions of non-stationary multicomponent signals with the adaptive WSST and a method to select the time-varying parameter were proposed in that paper. In addition, simulation experiments in that paper show that the adaptive WSST is very promising in estimating the instantaneous frequency of a multicomponent signal, and in accurate component recovery. However the theoretical analysis of the adaptive WSST has not been studied. In this paper, we carry out such analysis and obtain error bounds for component recovery with the adaptive WSST and the 2nd-order adaptive WSST. These results provide a mathematical guarantee to non-stationary multicomponent signal separation with the adaptive WSST. Copyright © 2020, The Authors. All rights reserved.

关键词： Separation

A three-dimensional dual-domain deep network for high-pitch and sparse helical CT reconstruction

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arXiv 2022年

作者： Wang, Wei Xia, Xiang-Gen He, Chuanjiang Ren, Zemin Lu, Jian The School of Biomedical Engineering Shenzhen University Shenzhen China The Department of Electrical and Computer Engineering University of Delaware NewarkDE19716 United States The College of Mathematics and Statistics Chongqing University Chongqing China The College of Mathematics and Physics Chongqing University of Science and Technology Chongqing China The Shenzhen Key Laboratory of Advanced Machine Learning and Applications Shenzhen University Shenzhen China

In this paper, we propose a new GPU implementation of the Katsevich algorithm for helical CT reconstruction. Our implementation divides the sinograms and reconstructs the CT images pitch by pitch. By utilizing the periodic properties of the parameters of the Katsevich algorithm, our method only needs to calculate these parameters once for all the pitches and so has lower GPU-memory burdens and is very suitable for deep learning. By embedding our implementation into the network, we propose an end-to-end deep network for the high pitch helical CT reconstruction with sparse detectors. Since our network utilizes the features extracted from both sinograms and CT images, it can simultaneously reduce the streak artifacts caused by the sparsity of sinograms and preserve fine details in the CT images. Experiments show that our network outperforms the related methods both in subjective and objective evaluations. Copyright © 2022, The Authors. All rights reserved.

关键词： Computerized tomography

Nonconvex and nonsmooth sparse optimization via adaptively iterative reweighted methods

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arXiv 2018年

作者： Wang, Hao Zhang, Fan Shi, Yuanming Hu, Yaohua School of Information Science and Technology ShanghaiTech University Shanghai201210 China College of Mathematics and Statistics Shenzhen Key Laboratory of Advanced Machine Learning and Applications Shenzhen University Shenzhen518061 China University of Chinese Academy of Sciences Beijing100049 China Shanghai Institute of Microsystem and Information Technology Chinese Academy of Sciences Shanghai200050 China

We propose a general formulation of nonconvex and nonsmooth sparse optimization problems with convex set constraint, which can take into account most existing types of nonconvex sparsity-inducing terms, bringing strong applicability to a wide range of applications. We design a general algorithmic framework of iteratively reweighted algorithms for solving the proposed nonconvex and nonsmooth sparse optimization problems, which solves a sequence of weighted convex regularization problems with adaptively updated weights. First-order optimality condition is derived and global convergence results are provided under loose assumptions, making our theoretical results a practical tool for analyzing a family of various reweighted algorithms. The effectiveness and efficiency of our proposed formulation and the algorithms are demonstrated in numerical experiments on various sparse optimization problems. Copyright © 2018, The Authors. All rights reserved.

关键词： Iterative methods

Sparse estimation via `q optimization method in high-dimensional linear regression

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arXiv 2019年

作者： Li, Xin Hu, Yaohua Li, Chong Yang, Xiaoqi Jiang, Tianzi School of Mathematical Sciences Zhejiang University Hangzhou310027 China Shenzhen Key Laboratory of Advanced Machine Learning and Applications College of Mathematics and Statistics Shenzhen University Shenzhen518060 China Department of Applied Mathematics Hong Kong Polytechnic University Kowloon Hong Kong Brainnetome Center Institute of Automation Chinese Academy of Sciences Beijing100190 China

In this paper, we discuss the statistical properties of the `q optimization methods (0 q minimization method and the `q regularization method, for estimating a sparse parameter from noisy observations in high-dimensional linear regression with either a deterministic or random design. For this purpose, we introduce a general q-restricted eigenvalue condition (REC) and provide its sufficient conditions in terms of several widely-used regularity conditions such as sparse eigenvalue condition, restricted isometry property, and mutual incoherence property. By virtue of the q-REC, we exhibit the stable recovery property of the `q optimization methods for either deterministic or random designs by showing that the `2 recovery bound O(2) for the `q minimization method and the oracle inequality and `2 recovery bound O(λ2−2 q s) for the `q regularization method hold respectively with high probability. The results in this paper are nonasymptotic and only assume the weak q-REC. The preliminary numerical results verify the established statistical property and demonstrate the advantages of the `q regularization method over some existing sparse optimization methods. Copyright © 2019, The Authors. All rights reserved.

关键词： Recovery