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Double coupled canonical polyadic decomposition of third-order tensors: algebraic algorithm and relaxed uniqueness conditions

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SIGNAL PROCESSING-IMAGE COMMUNICATION 2019年 73卷 22-36页

作者： Gong, Xiao-Feng Lin, Qiu-Hua Cong, Feng-Yu De Lathauwer, Lieven Dalian Univ Technol Sch Informat & Commun Engn Dalian 116024 Peoples R China Dalian Univ Technol Dept Biomed Engn Dalian 116024 Peoples R China Katholieke Univ Leuven Dept Elect Engn ESAT STADIUS Ctr Dynam Syst Signal Proc & Data Analyt BE-3001 Leuven Belgium Katholieke Univ Leuven Grp Sci Engn & Technol B-8500 Kortrijk Belgium

Double coupled canonical polyadic decomposition (DC-CPD) decomposes multiple tensors with coupling in the first two modes, into minimal number of rank-1 tensors that also admit the double coupling structure. It has a particular interest in joint blind source separation (J-BSS) applications. In a preceding paper, we proposed an algebraic algorithm for underdetermined DC-CPD, of which the factor matrices in the first two modes of each tensor may have more columns than rows. It uses a pairwise coupled rank-1 detection mapping to transform a possibly underdetermined DC-CPD into an overdetermined DC-CPD, which can be solved algebraically via generalized eigenvalue decomposition (GEVD). In this paper, we generalize the pairwise or second-order coupled rank-1 detection mapping to an arbitrary order K >= 2. Based on this generalized coupled rank-1 detection mapping, we propose a broad framework for the algebraic computation of DC-CPD, which consists of a series of algorithms with more relaxed working assumptions, each corresponding to a fixed order K >= 2. Deterministic and generic uniqueness conditions are provided. We will show through analysis and numerical results that our new uniqueness conditions for DC-CPD are more relaxed than the existing results for DC-CPD and CPD. We will further show, through simulation results, the performance of the proposed algebraic DC-CPD framework in approximate DC-CPD and a J-BSS application, in comparison with existing DC-CPD and CPD algorithms.

关键词： Tensor Canonical polyadic decomposition Double coupled algebraic algorithm Uniqueness

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Characterisation of an algebraic algorithm for Probabilistic Automata 33

Characterisation of an Algebraic Algorithm for Probabilistic...

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33rd Symposium on Theoretical Aspects of Computer Science (STACS)

作者： Fijalkow, Nathanael Univ Oxford Oxford England LIAFA Paris 7 France Univ Warsaw Warsaw Poland

ISBN: (纸本)9783959770019

We consider the value 1 problem for probabilistic automata over finite words: it asks whether a given probabilistic automaton accepts words with probability arbitrarily close to 1. This problem is known to be undecidable. However, different algorithms have been proposed to partially solve it;it has been recently shown that the Markov Monoid algorithm, based on algebra, is the most correct algorithm so far. The first contribution of this paper is to give a characterisation of the Markov Monoid algorithm. The second contribution is to develop a profinite theory for probabilistic automata, called the prostochastic theory. This new framework gives a topological account of the value 1 problem, which in this context is cast as an emptiness problem. The above characterisation is reformulated using the prostochastic theory, allowing to give a modular proof.

关键词： Probabilistic Automata Value 1 Problem Markov Monoid algorithm algebraic algorithm Profinite Theory Topology in Computer Science

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algebraic algorithms for eigen-problems of a reduced biquaternion matrix and applications

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APPLIED MATHEMATICS AND COMPUTATION 2024年 463卷

作者： Guo, Zhenwei Jiang, Tongsong Wang, Gang Vasil'ev, V. I. North Eastern Fed Univ Inst Math & Informat Sci Yakutsk 677000 Russia Shandong Xiandai Univ Sch Elect Informat Jinan 250104 Shandong Peoples R China Linyi Univ Sch Math & Stat Linyi 276005 Peoples R China

In recent years, the reduced biquaternion algebras have been widely used in color image processing problems and in the field of electromagnetism. This paper studies eigen-problems of reduced biquaternion matrices by means of a complex representation of a reduced biquaternion matrix and derives new algebraic algorithms to find the eigenvalues and eigenvectors of reduced biquaternion matrices. This paper also concludes that the number of eigenvalues of an n x n reduced biquaternion matrix is infinite. In addition, the proposed algebraic algorithms are shown to be effective in application to a color face recognition problem.

关键词： Reduced biquaternion matrix Complex representation Eigenvalue Eigenvector algebraic algorithm

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Recursive Forward-Backward EDMD: Guaranteed algebraic Search for Koopman Invariant Subspaces

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IEEE ACCESS 2025年 13卷 61006-61025页

作者： Haseli, Masih Cortes, Jorge Univ Calif San Diego Dept Mech & Aerosp Engn La Jolla CA 92093 USA

The implementation of the Koopman operator on digital computers often relies on the approximation of its action on finite-dimensional function spaces. This approximation is generally done by orthogonally projecting on the subspace. Extended Dynamic Mode Decomposition (EDMD) is a popular, special case of this projection procedure in a data-driven setting. Importantly, the accuracy of the model obtained by EDMD depends on the quality of the finite-dimensional space, specifically on how close it is to being invariant under the Koopman operator. This paper presents a data-driven algebraic search algorithm, termed Recursive Forward-Backward EDMD, for subspaces close to being invariant under the Koopman operator. Relying on the concept of temporal consistency, which measures the quality of the subspace, our algorithm recursively decomposes the search space into two subspaces with different prediction accuracy levels. The subspace with lower level of accuracy is removed if it does not reach a satisfactory threshold. The algorithm allows for tuning the level of accuracy depending on the underlying application and is endowed with convergence and accuracy guarantees.

关键词： Accuracy Eigenvalues and eigenfunctions Vectors Prediction algorithms Heuristic algorithms Hands Pollution measurement Complex systems Approximation algorithms Transmission line matrix methods Accuracy bound algebraic algorithm dynamic mode decomposition invariant subspace Koopman operator unknown nonlinear system

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TARGET LOCALIZATION BASED ON MULTISTATIC MIMO RADAR VIA DOUBLE COUPLED CANONICAL POLYADIC DECOMPOSITION 49

TARGET LOCALIZATION BASED ON MULTISTATIC MIMO RADAR VIA DOUB...

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49th IEEE International Conference on Acoustics, Speech, and Signal Processing (ICASSP)

作者： Liao, Guo-Zhao Gong, Xiao-Feng Lin, Qiu-Hua Dalian Univ Technol Sch Informat & Commun Engn Dalian Peoples R China

ISBN: (纸本)9798350344868;9798350344851

This paper considers target localization with a multistatic MIMO radar system of multiple transmit arrays and multiple receive arrays. We formulate the matched filtered output data into tensors that admit the double coupled canonical polyadic decomposition (DC-CPD) model, which efficiently characterizes the coupling between receive arrays and that between transmit arrays, and multilinear structure in each tensor. We propose a novel algebraic DC-CPD algorithm based on coupled rank-1 detection mapping, and provides analysis into the uniqueness conditions. A post-processing approach is also introduced to calculate and fuse the DOD and DOA information from the DC-CPD results to finally obtain the target locations. Simulations are provided to illustrate the merits of proposed method over CPD and C-CPD methods, with regards to accuracy and identifiability.

关键词： Multistatic MIMO radar target localization double couple canonical polyadic decomposition algebraic algorithm uniqueness

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On an inverse photoacoustic tomography problem of small absorbers with inhomogeneous sound speed

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IMA JOURNAL OF APPLIED MATHEMATICS 2022年第4期87卷 607-646页

作者： Al Jebawy, Hanin El Badia, Abdellatif Sorbonne Univ Univ Technol Compiegne LMAC Lab Appl Math Compiegne F-60205 Compiegne France Univ Libanaise Lab Math Appl Beirut 1003 Lebanon

This work is devoted to the study of the inverse photoacoustic tomography (PAT) problem. It is an imaging technique similar to TAT studied in El Badia & Ha-Duong (2000);however, in this case, a high-frequency radiation is delivered into the biological tissue to be imaged, such as visible or near infra red light that are characterized by their high frequency compared with that of radio waves that are used in TAT. As in the case of TAT El Badia & Ha-Duong (2000), the inverse problem we are concerned in is the reconstruction of small absorbers in an open, bounded and connected domain Omega subset of R-3. Again, we follow the algebraic algorithm, initially proposed in El Badia & Jebawy (2020), that allows us to resolve the problem from a single Cauchy data and without the knowledge of the Gritheisen's coefficient. However, the high-frequency radiation used in this case makes some changes in the context of the problem and allows us to give our results using partial boundary observations and in both cases of constant and variable acoustic speed. Finally, we establish the corresponding Holder stability result.

关键词： photo-acoustic tomography exact controllability algebraic algorithm

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Generalized Canonical Correlation Analysis: A Subspace Intersection Approach

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IEEE TRANSACTIONS ON SIGNAL PROCESSING 2021年 69卷 2452-2467页

作者： Sorensen, Mikael Kanatsoulis, Charilaos, I Sidiropoulos, Nicholas D. Univ Virginia Dept ECE Charlottesville VA 22904 USA Univ Minnesota Dept ECE Minneapolis MN 55455 USA

Generalized Canonical Correlation Analysis (GCCA) is an important tool that finds numerous applications in data mining, machine learning, and artificial intelligence. It aims at finding 'common' random variables that are strongly correlated across multiple feature representations (views) of the same set of entities. CCA and to a lesser extent GCCA have been studied from the statistical and algorithmic points of view, but not as much from the standpoint of linear algebra. This paper offers a fresh algebraic perspective on GCCA based on a (bi-)linear generative model that naturally captures its essence. It is shown that from a linear algebra point of view, GCCA is tantamount to subspace intersection;and conditions under which the common subspace of the different views is identifiable are provided. A novel GCCA algorithm is proposed based on subspace intersection, which scales up to handle large GCCA tasks. Synthetic as well as real data experiments are provided to showcase the effectiveness of the proposed approach.

关键词： Correlation Signal processing algorithms Tools Principal component analysis Linear algebra Machine learning Feature extraction Canonical correlation analysis generalized canonical correlation analysis subspace intersection multiview learning identifiability algebraic algorithm common subspace analysis

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Absolute reconstruction for sums of powers of linear forms: degree 3 and beyond

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COMPUTATIONAL COMPLEXITY 2023年第2期32卷 8-8页

作者： Koiran, Pascal Saha, Subhayan Univ Lyon EnsL UCBL CNRSLIP F-69342 Lyon 07 France

We study the decomposition of multivariate polynomials as sums of powers of linear forms. We give a randomised algorithm for the following problem: If a homogeneous polynomial f ? K[x(1), ..., x(n)] (where K ? C) of degree d is given as a blackbox, decide whether it can be written as a linear combination of d-th powers of linearly independent complex linear forms. The main novel features of the algorithm are:? For d = 3, we improve by a factor of n on the running time from the algorithm in Koiran & Skomra (2021). The price to be paid for this improvement is that the algorithm now has two-sided error.? For d > 3, we provide the first randomised blackbox algorithm for this problem that runs in time poly(n, d) (in an algebraic model where only arithmetic operations and equality tests are allowed). A previous algorithm for this problem due to Kayal (2011) as well as most of the existing reconstruction algorithms for other classes appeal to a polynomial factorisation subroutine. This requires extraction of complex polynomial roots at unit cost and in standard models such as the unit-cost RAM or the Turing machine this approach does not yield polynomial time algorithms.? For d > 3, when f has rational coefficients (i.e. K = Q), the running time of the blackbox algorithm is polynomial in n, d and the maximal bit size of any coefficient of f. This yields the first algorithm for this problem over C with polynomial running time in the bit model of *** results are true even when we replace C by R. We view the problem as a tensor decomposition problem and use linear algebraic methods such as checking the simultaneous diagonalisability of the slices of a tensor. The number of such slices is exponential in d. But surprisingly, we show that after a random change of variables, computing just 3 special slices is enough. We also show that our approach can be extended to the computation of the actual decomposition. In forthcoming work we plan to extend these results to o

关键词： reconstruction algorithms tensor decomposition sums of powers of linear forms simultaneous diagonalisation algebraic algorithm black box

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A three-dimensional immersed boundary method based on an algebraic forcing-point-searching scheme for water impact problems

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OCEAN ENGINEERING 2021年 233卷 109189-109189页

作者： Yan, Bin Bai, Wei Jiang, Sheng-Chao Cong, Peiwen Ning, Dezhi Qian, Ling Natl Univ Singapore Dept Civil & Environm Engn Singapore 117576 Singapore Manchester Metropolitan Univ Dept Comp & Math Chester St Manchester M1 5GD Lancs England Dalian Univ Technol State Key Lab Coastal & Offshore Engn Dalian 116024 Peoples R China

A new three-dimensional immersed boundary method combined with the level set method for the interface capturing is developed to simulate the interaction between the fixed\moving structure and the two-phase fluid flow. The concept of the forcing point searching scheme developed for the two-dimensional situations in Yan et al. (2018) is extended in the present study to three dimensions, where the determination of intersections between the arbitrary body surface and the Cartesian background grid system is the major issue. This problem can be converted to the prediction of triangle-triangle intersection, which was traditionally solved from the geometrical point of view. Here, an algebraic algorithm is adopted for the triangle-triangle intersection, based on which the forcing points can be determined in the three-dimensional immersed boundary method. This algebraic algorithm is robust for any body geometry and easy for implementation. To demonstrate the accuracy and capability of the developed numerical model, three benchmark testing cases for water impact problems are conducted, including dam break over a fixed obstacle, water entry of a wedge and free decay of a bobber. Extensive comparisons with the experimental data and the numerical results obtained by other immersed boundary methods suggest that the developed immersed boundary method is accurate and effective for both fixed and moving bodies with complex geometries.

关键词： Immersed boundary method Level set method Forcing point algebraic algorithm Triangle-triangle intersection Water impact

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Single neuron PID adaptive control for BLDCM based on algebraic algorithm

Single neuron PID adaptive control for BLDCM based on algebr...

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2nd International Conference on Advanced Design and Manufacturing Engineering (ADME 2012)

作者： Diao, Yan Jia, Hongping Geng, Tianjun JiangSu Univ Sch Elect & Informat Engn Zhenjiang Peoples R China

ISBN: (纸本)9783037855034

The brushless DC motor control system often adopts the classic PID control, the advantages of which are as follows: simple to control, easy to adjust the parameter and a certain degree of control precision. But it relies on accurate mathematical model. The permanent magnet brushless DC motor control system is a multi-variable and nonlinear. As to the deficiencies of the classic PID control method, this thesis proposes a method called artificial neural network PID adaptive control method, which is based on algebraic algorithm and overcomes the shortcomings such as the slow convergence of BP algorithm, easy to trap in local minimum, and etc.

关键词： Artificial Neural Networks (ANN) single neuron algebraic algorithm PID control

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