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A prototype-based spd matrix network for domain adaptation EEG emotion recognition

PATTERN RECOGNITION 2021年 110卷 107626-107626页

作者： Wang, Yixin Qiu, Shuang Ma, Xuelin He, Huiguang Chinese Acad Sci Res Ctr Brain Inspired Intelligence Beijing Peoples R China Chinese Acad Sci Natl Lab Pattern Recognit Inst Automat Beijing Peoples R China Univ Chinese Acad Sci Beijing Peoples R China Chinese Acad Sci Ctr Excellence Brain Sci & Intelligence Technol Beijing Peoples R China

Emotion plays a vital role in human daily life, and EEG signals are widely used in emotion recognition. Due to individual variability, training a generic emotion recognition model across different subjects is difficult. The conventional method involves the collection of a large amount of calibration data to build subject-specific models. Recently, developing an effective brain-computer interface with a short calibra-tion time has become a challenge. To solve this problem, we propose a domain adaptation spd matrix network (daspdnet) that can successfully capture an intrinsic emotional representation shared between different subjects. Our method jointly exploits feature adaptation with distribution confusion and sample adaptation with centroid alignment. We compute the spd matrix based on the covariance as a feature and make a novel attempt to combine prototype learning with the Riemannian metric. Extensive experiments are conducted on the DREAMER and DEAP datasets, and the results show the superiority of our proposed method. (c) 2020 Elsevier Ltd. All rights reserved.

关键词： EEG Emotion recognition Domain adaptation spd matrix Riemannian manifold Prototype learning

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EEG emotion recognition based on enhanced spd matrix and manifold dimensionality reduction

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COMPUTERS IN BIOLOGY AND MEDICINE 2022年 146卷 105606-105606页

作者： Gao, Yunyuan Sun, Xinyu Meng, Ming Zhang, Yingchun Hangzhou Dianzi Univ Coll Automat Hangzhou Peoples R China Univ Houston Dept Biomed Engn Houston TX USA

Recently, Riemannian geometry-based pattern recognition has been widely employed to brain computer interface (BCI) researches, providing new idea for emotion recognition based on electroencephalogram (EEG) signals. Although the symmetric positive definite (spd) matrix manifold constructed from the traditional covariance matrix contains large amount of spatial information, these methods do not perform well to classify and recognize emotions, and the high dimensionality problem still unsolved. Therefore, this paper proposes a new strategy for EEG emotion recognition utilizing Riemannian geometry with the aim of achieving better classification performance. The emotional EEG signals of 32 healthy subjects were from an open-source dataset (DEAP). The wavelet packets were first applied to extract the time-frequency features of the EEG signals, and then the features were used to construct the enhanced spd matrix. A supervised dimensionality reduction algorithm was then designed on the Riemannian manifold to reduce the high dimensionality of the spd matrices, gather samples of the same labels together, and separate samples of different labels as much as possible. Finally, the samples were mapped to the tangent space, and the K-nearest neighbors (KNN), Random Forest (RF) and Support Vector Machine (SVM) method were employed for classification. The proposed method achieved an average accuracy of 91.86%, 91.84% on the valence and arousal recognition tasks. Furthermore, we also obtained the superior accuracy of 86.71% on the four-class recognition task, demonstrated the superiority over state-of-the-art emotion recognition methods.

关键词： EEG Emotion recognition Riemannian manifold spd matrix Dimensionality reduction

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On the parameterized two-step iteration method for solving the matrix equation A X B = C

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APPLIED MATHEMATICS AND COMPUTATION 2024年 464卷

作者： Tian, Zhaolu Wang, Yudong Wu, Nian-Ci Liu, Zhongyun Shanxi Univ Finance & Econ Coll Appl Math Taiyuan 030006 Peoples R China Guangxi Univ Sci & Technol Coll Biol & Chem Engn Liuzhou 545006 Peoples R China South Cent Minzu Univ Sch Math & Stat Wuhan 430074 Peoples R China Changsha Univ Sci & Technol Sch Math & Stat Changsha 410076 Peoples R China

In this paper, based on the iteration framework [10], by introducing two tuning parameters alpha,beta in the splittings of the matrices A and B, a parameterized two-step iteration (PTSI) method is presented for solving the matrix equation A X B = C. In the sequel, the convergence property and choices of the parameters alpha,beta are analyzed in detail. For some special cases of the matrices A and B, the corresponding PTSI methods are also investigated, and the optimal parameters alpha,beta can be obtained for the symmetric positive definite (spd) matrices A and B. In addition, some comparison results of the PTSI method are given for the M-matrices alpha,beta compared with the TSI method [10]. Finally, several numerical examples are performed to verify the efficiencies of the PTSI method and choices of the optimal parameters.

关键词： spd matrix M-matrix Two-step iteration Optimal parameter Regular splitting

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A randomized algorithm for approximating the log determinant of a symmetric positive definite matrix

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LINEAR ALGEBRA AND ITS APPLICATIONS 2017年 533卷 95-117页

作者： Boutsidis, Christos Drineas, Petros Kambadur, Prabhanjan Kontopoulou, Eugenia-Maria Zouzias, Anastasios Purdue Univ W Lafayette IN 47907 USA Bloomberg LP New York NY USA

We introduce a novel algorithm for approximating the logarithm of the determinant of a symmetric positive definite (spd) matrix. The algorithm is randomized and approximates the traces of a small number of matrix powers of a specially constructed matrix, using the method of Avron and Toledo [1]. From a theoretical perspective, we present additive and relative error bounds for our algorithm. Our additive error bound works for any spd matrix, whereas our relative error bound works for spd matrices whose eigenvalues lie in the interval (theta(1),1), with 0 < theta(1) < 1;the latter setting was proposed in [16]. From an empirical perspective, we demonstrate that a C++ implementation of our algorithm can approximate the logarithm of the determinant of large matrices very accurately in a matter of seconds. (C) 2017 Elsevier Inc. All rights reserved.

关键词： Logdeterminant approximation spd matrix Randomized algorithm Power method Trace estimation

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ROBUST AND EFFECTIVE eSIF PRECONDITIONING FOR GENERAL DENSE spd MATRICES

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SIAM JOURNAL ON SCIENTIFIC COMPUTING 2021年第5期43卷 S767-S790页

作者： Xia, Jianlin Purdue Univ Dept Math W Lafayette IN 47907 USA

We propose an unconditionally robust and highly effective preconditioner for general dense symmetric positive definite matrices based on structured incomplete factorization (SIF), called the enhanced SIF (eSIF) preconditioner. The original SIF strategy proposed recently derives a structured preconditioner by applying block diagonal preprocessing to the matrix and then compressing appropriate scaled off-diagonal blocks. Here, we use an enhanced scaling-and-compression strategy to design the new eSIF preconditioner. Some subtle modifications are made, such as the use of two-sided block triangular preprocessing. A practical multilevel eSIF scheme is then designed. We give rigorous analysis for both the enhanced scaling-and-compression strategy and the multilevel eSIF preconditioner. The new eSIF framework has some significant advantages and overcomes some major limitations of the SIF strategy. (i) With the same tolerance for compressing the off-diagonal blocks, the eSIF preconditioner can approximate the original matrix to a much higher accuracy. (ii) The new preconditioner leads to much more significant reductions of condition numbers due to an accelerated magnification effect for the decay in the singular values of the scaled off-diagonal blocks. (iii) With the new preconditioner, the eigenvalues of the preconditioned matrix are much better clustered around 1. (iv) The multilevel eSIF preconditioner is further unconditionally robust or is guaranteed to be positive definite without the need of extra stabilization, while the multilevel SIF preconditioner has a strict requirement in order to preserve positive definiteness. Comprehensive numerical tests are used to show the advantages of the eSIF preconditioner in accelerating the convergence of iterative solutions.

关键词： eSIF preconditioning spd matrix enhanced scaling-and-compression strategy effectiveness unconditional robustness multilevel scheme

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A dual-parameter double-step splitting iteration method for solving complex symmetric linear equations

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APPLICATIONS OF MATHEMATICS 2024年第3期69卷 311-337页

作者： Li, Beibei Cui, Jingjing Huang, Zhengge Xie, Xiaofeng Guangxi Minzu Univ Coll Math & Phys Ctr Appl Math Guangxi 188 East Daxue Rd Nanning 530006 Peoples R China

We multiply both sides of the complex symmetric linear system Ax = b by 1 - i omega to obtain a new equivalent linear system, then a dual-parameter double-step splitting (DDSS) method is established for solving the new linear system. In addition, we present an upper bound for the spectral radius of iteration matrix of the DDSS method and obtain its quasi-optimal parameter. Theoretical analyses demonstrate that the new method is convergent when some conditions are satisfied. Some tested examples are given to illustrate the effectiveness of the proposed method.

关键词： DDSS iteration method linear equations spd matrix SPSD matrix convergence property

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Discriminative block-diagonal covariance descriptors for image set classification

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PATTERN RECOGNITION LETTERS 2020年 136卷 230-236页

作者： Ren, Jieyi Wu, Xiao-jun Kittler, Josef Jiangnan Univ Sch Internet Things Engn Wuxi 214122 Jiangsu Peoples R China Univ Surrey Ctr Vis Speech & Signal Proc Guildford GU2 7XH Surrey England

Image set classification has recently received much attention due to its various applications in pattern recognition and computer vision. To compare and match image sets, the major challenges are to devise an effective and efficient representation and to define a measure of similarity between image sets. In this paper, we propose a method for representing image sets based on block-diagonal Covariance Descriptors (CovDs). In particular, the proposed image set representation is in the form of non-singular covariance matrices, also known as Symmetric Positive Definite (spd) matrices, that lie on Riemannian manifold. By dividing each image of an image set into square blocks of the same size, we compute the corresponding block CovDs instead of the global one. Taking the relative discriminative power of these block CovDs into account, a block-diagonal spd matrix can be constructed to achieve a better discriminative capability. We extend the proposed approach to work with bidirectional CovDs and achieve a further boost in performance. The resulting block-diagonal spd matrices combined with Riemannian metrics are shown to provide a powerful basis for image set classification. We perform an extensive evaluation on four datasets for several image set classification tasks. The experimental results demonstrate the effectiveness and efficiency of the proposed method. (C) 2020 Elsevier B.V. All rights reserved.

关键词： Covariance descriptor spd matrix Riemannian manifold Image set classification

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AN ADAPTIVE FAST SOLVER FOR A GENERAL CLASS OF POSITIVE DEFINITE MATRICES VIA ENERGY DECOMPOSITION

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MULTISCALE MODELING & SIMULATION 2018年第2期16卷 615-678页

作者： Hou, Thomas Y. Huang, De Lam, Ka Chun Zhang, Pengchuan CALTECH Appl & Computat Math Pasadena CA 91125 USA CALTECH Comp & Math Sci Pasadena CA 91125 USA Microsoft Res Redmond Deep Learning Team Redmond WA 98052 USA

In this paper, we propose an adaptive fast solver for a general class of symmetric positive definite (spd) matrices which include the well-known graph Laplacian. We achieve this by developing an adaptive operator compression scheme and a multiresolution matrix factorization algorithm which achieve nearly optimal performance on both complexity and well-posedness. To develop our adaptive operator compression and multiresolution matrix factorization methods, we first introduce a novel notion of energy decomposition for spd matrix A using the representation of energy elements. The interaction between these energy elements depicts the underlying topological structure of the operator. This concept of decomposition naturally reflects the hidden geometric structure of the operator which inherits the localities of the structure. By utilizing the intrinsic geometric information under this energy framework, we propose a systematic operator compression scheme for the inverse operator A. In particular, with an appropriate partition of the underlying geometric structure, we can construct localized basis by using the concept of interior and closed energy. Meanwhile, two important localized quantities are introduced, namely, the error factor and the condition factor. Our error analysis results show that these two factors will be the guidelines for finding the appropriate partition of the basis functions such that prescribed compression error and acceptable condition number can be achieved. By virtue of this insight, we propose the patch pairing algorithm to realize our energy partition framework for operator compression with controllable compression error and condition number.

关键词： energy decomposition graph Laplacian spd matrix fast solver operator compression multiresolution matrix decomposition

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GRSIM: A FORTRAN subroutine for the solution of non-symmetric linear systems

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COMMUNICATIONS IN NUMERICAL METHODS IN ENGINEERING 2002年第11期18卷 803-815页

作者： Shah, AA MTSSR Dept Math & Comp Jalan Muara BE2313 BSB Brunei

A FORTRAN subroutine called GRSIM subroutine is presented for the iterative solution of a set of non-symmetric linear equations, Ax = b, where the coefficient matrix A is a sparse nearly symmetric structured M-matrix. These matrices occur repeatedly in the finite-difference solution of partial differential equations. The method solves non-symmetric systems of linear equations, but uses highly developed techniques for the solution of symmetric systems of linear equations. A general description of the method, which is based on particular class of regular splitting, is given. The GRSIM subroutine uses a regular splitting and the convergence is, therefore, guaranteed. Copyright 2002 John Wiley Sons, Ltd.

关键词： nearly-symmetric regular-splitting spd matrix discretization Cholesky factors diagonally dominant

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Learning a robust representation via a deep network on symmetric positive definite manifolds

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PATTERN RECOGNITION 2019年 92卷 1-12页

作者： Gao, Zhi Wu, Yuwei Bu, Xingyuan Yu, Tan Yuan, Junsong Jia, Yunde Beijing Inst Technol Sch Comp Sci Beijing Lab Intelligent Informat Technol Beijing 100081 Peoples R China Nanyang Technol Univ Sch EEE Singapore 639798 Singapore SUNY Buffalo Dept Comp Sci & Engn Buffalo NY USA

Recent studies have shown that aggregating convolutional features of a Convolutional Neural Network (CNN) can obtain impressive performance for a variety of computer vision tasks. The Symmetric Positive Definite (spd) matrix becomes a powerful tool due to its remarkable ability to learn an appropriate statistic representation to characterize the underlying structure of visual features. In this paper, we propose a method of aggregating deep convolutional features into a robust representation through the spd generation and the spd transformation under an end-to-end deep network. To this end, several new layers are introduced in our method, including a nonlinear kernel generation layer, a matrix transformation layer, and a vector transformation layer. The nonlinear kernel generation layer is employed to aggregate convolutional features into a kernel matrix which is guaranteed to be an spd matrix. The matrix transformation layer is designed to project the original spd representation to a more compact and discriminative spd manifold. The vectorization and normalization operations are performed in the vector transformation layer to take the upper triangle elements of the spd representation and carry out the power normalization and l(2) normalization to reduce the redundancy and accelerate the convergence. The spd matrix in our network can be considered as a mid-level representation bridging convolutional features and high-level semantic features. Results of extensive experiments show that our method notably outperforms state-of-the-art methods. (C) 2019 Elsevier Ltd. All rights reserved.

关键词： Feature aggregation spd matrix Riemannian manifold Deep convolutional network

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