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randomized algorithms for Computation of Tucker Decomposition and Higher Order SVD (HOSVD)

IEEE ACCESS 2021年 9卷 28684-28706页

作者： Ahmadi-Asl, Salman Abukhovich, Stanislav Asante-Mensah, Maame G. Cichocki, Andrzej Phan, Anh Huy Tanaka, Tohishisa Oseledets, Ivan Skolkovo Inst Sci & Technol SKOLTECH CDISE Moscow 121205 Russia Tokyo Univ Agr & Technol Inst Global Innovat & Res Tokyo 1848588 Japan

Big data analysis has become a crucial part of new emerging technologies such as the internet of things, cyber-physical analysis, deep learning, anomaly detection, etc. Among many other techniques, dimensionality reduction plays a key role in such analyses and facilitates feature selection and feature extraction. randomized algorithms are efficient tools for handling big data tensors. They accelerate decomposing large-scale data tensors by reducing the computational complexity of deterministic algorithms and the communication among different levels of memory hierarchy, which is the main bottleneck in modern computing environments and architectures. In this article, we review recent advances in randomization for computation of Tucker decomposition and Higher Order SVD (HOSVD). We discuss random projection and sampling approaches, single-pass and multi-pass randomized algorithms and how to utilize them in the computation of the Tucker decomposition and the HOSVD. Simulations on synthetic and real datasets are provided to compare the performance of some of best and most promising algorithms.

关键词： Tensors Matrix decomposition Signal processing algorithms Approximation algorithms Memory management Licenses Probability distribution randomized algorithm tensor decomposition random projection sampling unfolding Tucker decomposition HOSVD

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Tracking tensor ring decompositions of streaming tensors

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COMPUTATIONAL & APPLIED MATHEMATICS 2025年第1期44卷 1-30页

作者： Yu, Yajie Li, Hanyu Chongqing Univ Coll Math & Stat Chongqing 401331 Peoples R China Chongqing Univ Key Lab Nonlinear Anal & its Applicat Minist Educ Chongqing Peoples R China

Tensor ring (TR) decomposition is an efficient approach to discover the hidden low-rank patterns in higher-order tensors, and streaming tensors are becoming highly prevalent in real-world applications. In this paper, we investigate how to track TR decompositions of streaming tensors. An efficient algorithm is first proposed. Then, based on this algorithm and randomized sketching techniques, we present a randomized streaming TR decomposition. The proposed algorithms make full use of the structure of TR decomposition, and the randomized version can allow any sketching type. Theoretical results on sketch size are provided. In addition, the complexity analyses for the obtained algorithms are also given. We compare our proposals with the existing batch methods using both real and synthetic data. Numerical results show that they have better performance in computing time with maintaining similar accuracy.

关键词： Tensor ring decomposition Streaming tensor randomized algorithm Alternating least squares Kronecker sub-sampled randomized Fourier transform Uniform sampling Importance sampling Leverage score

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An L-DEIM induced high order tensor interpolatory decomposition

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JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS 2025年 453卷

作者： Cao, Zhengbang Wei, Yimin Xie, Pengpeng Ocean Univ China Sch Math Sci Qingdao 266100 Peoples R China Fudan Univ Sch Math Sci Shanghai 200433 Peoples R China Fudan Univ Key Lab Math Nonlinear Sci Shanghai 200433 Peoples R China

This paper derives the CUR-type factorization for tensors in the Tucker format based on a new variant of the discrete empirical interpolation method known as L-DEIM. This novel sampling technique allows us to construct an efficient algorithm for computing the structure-preserving decomposition, which significantly reduces the computational cost. For large-scale datasets, we incorporate the random sampling technique with the L-DEIM procedure to further improve efficiency. Moreover, we propose randomized algorithms for computing a hybrid decomposition, which yield interpretable factorization and provide a smaller approximation error than the tensor CUR factorization. We provide comprehensive analysis of probabilistic errors associated with our proposed algorithms, and present numerical results that demonstrate the effectiveness of our methods.

关键词： CUR decomposition L-DEIM Low-rank approximation Tucker decomposition randomized algorithm

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Double-Tucker Decomposition and Its Computations

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NUMERICAL LINEAR ALGEBRA WITH APPLICATIONS 2025年第1期32卷

作者： Wang, Mengyu Cui, Honghua Li, Hanyu Xiamen Univ Wang Yanan Inst Studies Econ Xiamen Fujian Peoples R China Chongqing Univ Coll Math & Stat Chongqing Peoples R China

The famous Tucker decomposition has been widely and successfully used in many fields. However, it often suffers from the curse of dimensionality due to the core tensor and large ranks. To tackle this issue, we introduce an additional core tensor into Tucker decomposition and propose the so-called double-Tucker (dTucker) decomposition. The additional core can share the ranks of the original Tucker decomposition and hence make the parameters of the new decomposition be reduced greatly. We employ the alternating least squares (ALS) method with explicit structures on coefficient matrices of the ALS subproblems to compute the dTucker decomposition. To figure out the structures, a new tensor product is defined. Its properties and the aforementioned structures together motivate an ALS-based randomized algorithm built on Kronecker sub-sampled randomized Fourier transform for our new decomposition. A special case of the algorithm leads to a more efficient leverage-based random sampling algorithm. These randomized algorithms can avoid forming the full coefficient matrices of ALS subproblems by implementing projecting and sampling on factor tensors. Numerical experiments including tensor reconstruction and multi-view subspace clustering are presented to test our decomposition and algorithms, which show that dTucker decomposition can effectively decrease the ranks of the classical one and hence the total parameters, and the randomized algorithms reduce running time greatly while maintaining similar accuracy. Moreover, the numerical results also show that our decomposition can even outperform the popular tensor train decomposition and the newly developed tensor wheel decomposition on compressing parameters.

关键词： alternating least squares double-Tucker decomposition Kronecker sub-sampled randomized Fourier transform leverage sampling multi-view subspace clustering randomized algorithm tensor reconstruction Tucker decomposition

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randomized algorithms and upper bounds for multiple domination in graphs and networks

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DISCRETE APPLIED MATHEMATICS 2013年第4-5期161卷 604-611页

作者： Gagarin, Andrei Poghosyan, Anush Zverovich, Vadim Acadia Univ Dept Math & Stat Wolfville NS B4P 2R6 Canada Univ W England Dept Math & Stat Bristol BS16 1QY Avon England

We consider four different types of multiple domination and provide new improved upper bounds for the k- and k-tuple domination numbers. They generalize two classical bounds for the domination number and are better than a number of known upper bounds for these two multiple domination parameters. Also, we explicitly present and systematize randomized algorithms for finding multiple dominating sets, whose expected orders satisfy new and recent upper bounds. The algorithms for k- and k-tuple dominating sets are of linear time in terms of the number of edges of the input graph, and they can be implemented as local distributed algorithms. Note that the corresponding multiple domination problems are known to be NP-complete. (C) 2011 Elsevier B.V. All rights reserved.

关键词： randomized algorithm k-domination k-tuple domination alpha-domination alpha-rate domination

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An Effective randomized QoS Routing algorithm on Networks with Inaccurate Parameters

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Journal of Computer Science & Technology 2002年第1期17卷 38-46页

作者：王建新陈建二陈松乔 Department of Computer Science Central South University Changsha 410083 P.R. China

This paper develops an effective randomized on-demand QoS routing algorithm on networks with inaccurate link-state information. Several new techniques are proposed in the algorithm. First, the maximum safety rate and the minimum delay for each node in the network are pre-computed, which simplify the network complexity and provide the routing process with useful information. The routing process is dynamically directed by the safety rate and delay of the partial routing path developed so far and by the maximum safety rate and the minimum delay of the next node. Randomness is used at the link level and depends dynamically on the routing configuration. This provides great flexibility for the routing process, prevents the routing process from overusing certain fixed routing paths, and adequately balances the safety rate and delay of the routing path. A network testing environment has been established and five parameters are introduced to measure the performance of QoS routing *** results demonstrate that in terms of the proposed parameters, the algorithm outperforms existing QoS algorithms appearing in the literature.

关键词： QoS network routing randomized algorithm link-state information

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randomized algorithms for Feedforward Neural Networks 35

Randomized Algorithms for Feedforward Neural Networks

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第35届中国控制会议

作者： LI Fan-jun LI Ying School of Mathematical Science University of Jinan School of Science Qilu University of Technology

ISBN: (纸本)9781509009107

randomized algorithm for feedforward neural networks have been summarized and evaluated in this paper,Firstly,a simplified model of feedforward neural networks with random weights is proposed,which consists of a randomized layer and an output ***,randomized algorithms for different network structures are summarized on the basis of the simplified ***,several feedforward neural networks with different randomized layers are evaluated on ten UCI data *** results show that random vector Functional-link neural network is more powerful than multilayer perceptron structure with random weights.

关键词： randomized algorithm feedforward neural network functional-link neural network

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randomized algorithms for robust feasibility problems with general nonconvex constraints

Randomized algorithms for robust feasibility problems with g...

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Annual Conference on the Society-of-Instrument-and-Control-Engineers

作者： Wada, Takayuki Fujisaki, Yasumasa Kobe Univ Grad Sch Sci & Technol Kobe Hyogo 6578501 Japan Kobe Univ Grad Sch Engn Kobe Hyogo 6578501 Japan

ISBN: (纸本)9784907764289

A class of robust feasibility problems is considered, which is to find a design variable satisfying a parameter-dependent constraint for all parameter values. A randomized algorithm for solving the problem with a general nonconvex constraint is proposed, where random samples of candidates of the design variable and uncertain parameters are used. The algorithm stops in a finite number of iterations. Then, it gives a design variable satisfying the constraint for almost all parameter values with a prescribed confidence or says that the problem is infeasible in a probabilistic sense.

关键词： robust control robust feasibility problem randomized algorithm probabilistic solution probabilistic infeasibility

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Enlarged GKB and RSVD for KP approximations illustrated for mixed precision ℓ1 image restoration

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Numerical algorithms 2025年 1-23页

作者： Alsubhi, Abdulmajeed Renaut, Rosemary A. School of Mathematical and Statistical Sciences Arizona State University Tempe 85287 AZ United States Department of Mathematics Faculty of Science Islamic University of Madinah Madinah 42351 Saudi Arabia

The singular value decomposition (SVD) of a reordering of a matrix A can be used to determine an efficient Kronecker product (KP) sum approximation to A. We present the use of an approximate truncated SVD (TSVD) to find the KP approximation, and contrast using a randomized singular value decomposition algorithm (RSVD), a new enlarged Golub Kahan Bidiagonalization algorithm (EGKB) and the exact TSVD. The EGKB algorithm enlarges the Krylov subspace beyond a given rank for the desired approximation. A suitable rank is determined using an automatic stopping test. We also contrast the use of single and double precision arithmetic to find the approximate TSVDs. To illustrate the accuracy and efficiency in terms of memory and computational cost of these approximate KPs, we consider the solution of the total variation regularized image deblurring problem using the split Bregman algorithm implemented in double precision. Together with an efficient implementation for the reordering of A we demonstrate that the approximate KP sum can be obtained using a TSVD, and that the new EGKB algorithm contrasts favorably with the use of the RSVD. These results verify that it is feasible to use single precision when estimating a KP sum from an approximate TSVD. © The Author(s), under exclusive licence to Springer Science+Business Media, LLC, part of Springer Nature 2025.

关键词： Golub Kahan Bidiagonalization Isotropic Kronecker Product randomized algorithm Single and double precision Split Bregman ℓ1 regularization

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FAST AND ACCURATE randomized algorithmS FOR LINEAR SYSTEMS AND EIGENVALUE PROBLEMS

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SIAM JOURNAL ON MATRIX ANALYSIS AND APPLICATIONS 2024年第2期45卷 1183-1214页

作者： Nakatsukasa, Yuji Tropp, Joel A. Univ Oxford Math Inst Oxford OX2 6GG England Caltech Comp & Math Sci Pasadena CA 91125 USA

This paper develops a class of algorithms for general linear systems and eigenvalue problems. These algorithms apply fast randomized dimension reduction (``sketching"") to accelerate standard subspace projection methods, such as GMRES and Rayleigh--Ritz. This modification makes it possible to incorporate nontraditional bases for the approximation subspace that are easier to construct. When the basis is numerically full rank, the new algorithms have accuracy similar to classic methods but run faster and may use less storage. For model problems, numerical experiments show large advantages over the optimized MATLAB routines, including a 70 \times speedup over gmres and a 10 \times speedup over eigs.

关键词： eigenvalue problem linear system numerical linear algebra Petrov-Galerkin method projection method randomized algorithm Rayleigh--Ritz sketching subspace embedding

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