A Randomized Tensor Train Singular Value Decomposition
نویسندگان
چکیده
The hierarchical SVD provides a quasi-best low rank approximation of high dimensional data in the hierarchical Tucker framework. Similar to the SVD for matrices, it provides a fundamental but expensive tool for tensor computations. In the present work we examine generalizations of randomized matrix decomposition methods to higher order tensors in the framework of the hierarchical tensors representation. In particular we present and analyze a randomized algorithm for the calculation of the hierarchical SVD (HSVD) for the tensor train (TT) format.
منابع مشابه
Computing low-rank approximations of large-scale matrices with the Tensor Network randomized SVD
We propose a new algorithm for the computation of a singular value decomposition (SVD) low-rank approximation of a matrix in the Matrix Product Operator (MPO) format, also called the Tensor Train Matrix format. Our tensor network randomized SVD (TNrSVD) algorithm is an MPO implementation of the randomized SVD algorithm that is able to compute dominant singular values and their corresponding sin...
متن کاملA constructive arbitrary-degree Kronecker product decomposition of matrices
We propose a constructive algorithm, called the tensor-based Kronecker product (KP) singular value decomposition (TKPSVD), that decomposes an arbitrary real matrix A into a finite sum of KP terms with an arbitrary number of d factors, namely A = ∑R j=1 σj A dj ⊗ · · · ⊗A1j . The algorithm relies on reshaping and permuting the original matrix into a d-way tensor, after which its tensor-train ran...
متن کاملA Constructive Algorithm for Decomposing a Tensor into a Finite Sum of Orthonormal Rank-1 Terms
Abstract. We propose a novel and constructive algorithm that decomposes an arbitrary tensor into a finite sum of orthonormal rank-1 outer factors. The algorithm, named TTr1SVD, works by converting the tensor into a rank-1 tensor train (TT) series via singular value decomposition (SVD). TTr1SVD naturally generalizes the SVD to the tensor regime and delivers elegant notions of tensor rank and err...
متن کاملEstimating a Few Extreme Singular Values and Vectors for Large-Scale Matrices in Tensor Train Format
We propose new algorithms for singular value decomposition (SVD) of very large-scale matrices based on a low-rank tensor approximation technique called the tensor train (TT) format. The proposed algorithms can compute several dominant singular values and corresponding singular vectors for large-scale structured matrices given in a TT format. The computational complexity of the proposed methods ...
متن کاملA Randomized Tensor Singular Value Decomposition based on the t-product
The tensor Singular Value Decomposition (t-SVD) for third order tensors that was proposed by Kilmer and Martin [30] has been applied successfully in many fields, such as computed tomography, facial recognition, and video completion. In this paper, we propose a method that extends a well-known randomized matrix method to the t-SVD. This method can produce a factorization with similar properties ...
متن کامل