本文提出 Tucker 格式下张量的混合 CUR 型分解方法，通过保留纤维并优化矩阵来降低近似误差，且误差优势随维度增加和保留模式减少而扩大。
摘要：
本文介绍了一种在 Tucker 格式下进行张量 CUR 型分解的混合方法。混合算法的思想是将张量 $\mathcal{X}$ 写成核心张量 $\mathcal{S}$、由提取 $\mathcal{X}$ 的 mode-$k$ 纤维得到的矩阵 $C$，以及为最小化近似误差而选取的矩阵 $U_j$（$j=1,\ldots,k-1,k+1,\ldots,d$）的乘积。该近似可以轻松修改以在多个模式下保留纤维。通过这种方式获得的近似误差小于标准张量 CUR 型方法所得的误差。该差异随着张量维度的增加而增大，同时也随着原始纤维被保留的模式数量的减少而增大。
英文摘要：
The paper introduces a hybrid approach to the CUR-type decomposition of tensors in the Tucker format. The idea of the hybrid algorithm is to write a tensor $\mathcal{X}$ as a product of a core tensor $\mathcal{S}$, a matrix $C$ obtained by extracting mode-$k$ fibers of $\mathcal{X}$, and matrices $U_j$, $j=1,\ldots,k-1,k+1,\ldots,d$, chosen to minimize the approximation error. The approximation can easily be modified to preserve the fibers in more than one mode. The approximation error obtained this way is smaller than the one from the standard tensor CUR-type method. This difference increases as the tensor dimension increases. It also increases as the number of modes in which the original fibers are preserved decreases.
Approximating higher-order tensors by the Tucker format has been applied in many fields such as psychometrics, chemometrics, signal processing, pattern classification, and so on. In this paper, we propose some new Tucker-like approximations based on the modal semi-tensor product (STP), especially, a new singular value decomposition (SVD) and a new higher-order SVD (HOSVD) are derived. Algorithms for computing new decompositions are provided. We also give some numerical examples to illustrate our theoretical results.
Many applications in data science and scientific computing involve large-scale datasets that are expensive to store and compute with, but can be efficiently compressed and stored in an appropriate tensor format. In recent years, randomized matrix methods have been used to efficiently and accurately compute low-rank matrix decompositions. Motivated by this success, we focus on developing randomized algorithms for tensor decompositions in the Tucker representation. Specifically, we present randomized versions of two well-known compression algorithms, namely, HOSVD and STHOSVD. We present a detai…
This work is concerned with approximating a trivariate function defined on a tensor-product domain via function evaluations. Combining tensorized Chebyshev interpolation with a Tucker decomposition of low multilinear rank yields function approximations that can be computed and stored very efficiently. The existing Chebfun3 algorithm [Hashemi and Trefethen, SIAM J. Sci. Comput., 39 (2017)]uses a similar format but the construction of the approximation proceeds indirectly, via a so called slice-Tucker decomposition. As a consequence, Chebfun3 sometimes uses unnecessarily many function evaluation…
In the paper `On the Dirac-Frenkel Variational Principle on Tensor Banach Spaces', we provided a geometrical description of manifolds of tensors in Tucker format with fixed multilinear (or Tucker) rank in tensor Banach spaces, that allowed to extend the Dirac-Frenkel variational principle in the framework of topological tensor spaces. The purpose of this note is to extend these results to more general tensor formats. More precisely, we provide a new geometrical description of manifolds of tensors in tree-based (or hierarchical) format, also known as tree tensor networks, which are intersection…
Efficient Sketching-Based Summation of Tucker Tensors
Department of Mathematics, Virginia Tech（弗吉尼亚理工大学数学系）；Oden Institute for Computational Engineering and Sciences, The University of Texas at Austin（德克萨斯大学奥斯汀分校奥登计算工程与科学研究所）；Computational Mathematics Group, Rutherford Appleton Laboratory（拉瑟福德·阿普尔顿实验室计算数学组）
Tensor decomposition methodologies are proposed to reduce the memory requirement of translation operator tensors arising in the fast multipole method-fast Fourier transform (FMM-FFT)-accelerated surface integral equation (SIE) simulators. These methodologies leverage Tucker, hierarchical Tucker (H-Tucker), and tensor train (TT) decompositions to compress the FFT'ed translation operator tensors stored in three-dimensional (3D) and four-dimensional (4D) array formats. Extensive numerical tests are performed to demonstrate the memory saving achieved by and computational overhead introduced by the…
We derive a CUR-type factorization for tensors in the Tucker format based on interpolatory decomposition, which we will denote as Higher Order Interpolatory Decomposition (HOID). Given a tensor $\mathcal{X}$, the algorithm provides a set of column vectors $\{ \mathbf{C}_n\}_{n=1}^d$ which are columns extracted from the mode-$n$ tensor unfolding, along with a core tensor $\mathcal{G}$ and together, they satisfy some error bounds. Compared to the Higher Order SVD (HOSVD) algorithm, the HOID provides a decomposition that preserves certain important features of the original tensor such as sparsity…
The tensor-train (TT) decomposition expresses a tensor in a data-sparse format used in molecular simulations, high-order correlation functions, and optimization. In this paper, we propose four parallelizable algorithms that compute the TT format from various tensor inputs: (1) Parallel-TTSVD for traditional format, (2) PSTT and its variants for streaming data, (3) Tucker2TT for Tucker format, and (4) TT-fADI for solutions of Sylvester tensor equations. We provide theoretical guarantees of accuracy, parallelization methods, scaling analysis, and numerical results. For example, for a $d$-dimensi…
Fast multidimensional convolution in low-rank formats via cross approximation
Skolkovo Institute of Science and Technology（斯科尔科沃科学技术研究所）；Moscow Institute of Physics and Technology（莫斯科物理技术学院）；Institute of Numerical Mathematics, Russian Academy of Sciences（俄罗斯科学院数值数学研究所）
Geometry-aware training of factorized layers in tensor Tucker format
School of Mathematics, Gran Sasso Science Institute（格兰萨索科学研究所数学系）；Computer Science and Mathematics Division, Oak Ridge National Laboratory（橡树岭国家实验室计算机科学与数学部）；Department of Mathematics, University of Innsbruck（因斯布鲁克大学数学系）；Department of Data Science, Norwegian University of Life Sciences（挪威生命科学大学数据科学系）；School of Mathematics and Maxwell Institute, University of Edinburgh（爱丁堡大学数学系和麦克斯韦研究所）
Tucker Tensor analysis of Matern functions in spatial statistics
King Abdullah University of Science and Technology（阿卜杜拉国王科技大学）；Max-Planck Institute for Mathematics in the Sciences（马克斯·普朗克科学促进学会数学研究所）；Max-Planck Institute for Dynamics of Complex Technical Systems（马克斯·普朗克复杂技术系统动力学研究所）；Technische Universitaet Braunschweig（布伦瑞克工业大学）
There is an emerging interest in tensor factorization applications in big-data analytics and machine learning. To speed up the factorization of extra-large datasets, organized in multidimensional arrays (aka tensors), easy to compute compression-based tensor representations, such as Tucker and Tensor Train formats, are used to approximate the initial large-tensor. Further, tensor factorization is used to extract latent features that can facilitate discoveries of new mechanisms and signatures hidden in the data, where the explainability of the latent features is of principal importance. Nonnega…
In this article, we analyze tensor approximation schemes for continuous functions. We assume that the function to be approximated lies in an isotropic Sobolev space and discuss the cost when approximating this function in the continuous analogue of the Tucker tensor format or of the tensor train format. We especially show that the cost of both approximations are dimension-robust when the Sobolev space under consideration provides appropriate weights.
The use of fractional differential equations is a key tool in modeling non-local phenomena. Often, an efficient scheme for solving a linear system involving the discretization of a fractional operator is evaluating the matrix function $x = \mathcal A^{-α} c$, where $\mathcal A$ is a discretization of the classical Laplacian, and $α$ a fractional exponent between $0$ and $1$. In this work, we derive an exponential sum approximation for $f(z) =z^{-α}$ that is accurate over $[1, \infty)$ and allows to efficiently approximate the action of bounded and unbounded operators of this kind on tensors st…
Geometric Structures in Tensor Representations (Final Release)
Universidad CEU Cardenal Herrera（CEU卡迪纳尔埃雷拉大学）；Max-Planck-Institut Mathematik in den Naturwissenschaften（马克斯·普朗克自然科学数学研究所）；Ecole Centrale Nantes（南特中央理工学院）
In this paper, we introduce a method for multivariate function approximation using function evaluations, Chebyshev polynomials, and tensor-based compression techniques via the Tucker format. We develop novel randomized techniques to accomplish the tensor compression, provide a detailed analysis of the computational costs, provide insight into the error of the resulting approximations, and discuss the benefits of the proposed approaches. We also apply the tensor-based function approximation to develop low-rank matrix approximations to kernel matrices that describe pairwise interactions between …
FFT-based Kronecker product approximation to micromagnetic long-range interactions
University of Applied Sciences（应用科学大学）；Fachbereich Mathematik, Universität Hamburg（汉堡大学数学系）；Institut für Angewandte Physik und Zentrum für Mikrostrukturforschung, Universität Hamburg（汉堡大学应用物理与微结构研究中心）；Wolfgang Pauli Institute c/o Fakultät f. Mathematik, Universität Wien（维也纳大学沃尔夫冈·泡利研究所）；Institute of Solid State Physics, Vienna University of Technology（维也纳工业大学固体物理研究所）
Fast Fourier transform (FFT) based methods have turned out to be an effective computational approach for numerical homogenisation. In particular, Fourier-Galerkin methods are computational methods for partial differential equations that are discretised with trigonometric polynomials. Their computational effectiveness benefits from efficient FFT based algorithms as well as a favourable condition number. Here these kind of methods are accelerated by low-rank tensor approximation techniques for a solution field using canonical polyadic, Tucker, and tensor train formats. This reduced order model a…
