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We propose a new algorithm for the computation of a singular valuedecomposition (SVD) low-rank approximation of a matrix in the Matrix ProductOperator (MPO) format, also called the Tensor Train Matrix format. Our tensornetwork randomized SVD (TNrSVD) algorithm is an MPO implementation of therandomized SVD algorithm that is able to compute dominant singular values andtheir corresponding singular vectors. In contrast to the state-of-the-arttensor-based alternating least squares SVD (ALS-SVD) and modified alternatingleast squares SVD (MALS-SVD) matrix approximation methods, TNrSVD can be up to17 times faster while achieving the same accuracy. In addition, our TNrSVDalgorithm also produces accurate approximations in particular cases where bothALS-SVD and MALS-SVD fail to converge. We also propose a new algorithm for thefast conversion of a sparse matrix into its corresponding MPO form, which is upto 509 times faster than the standard Tensor Train SVD (TT-SVD) method whileachieving machine precision accuracy. The efficiency and accuracy of bothalgorithms are demonstrated in numerical experiments.

Computing low-rank approximations of large-scale matrices with the Tensor Network randomized SVD