  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 singular vectors. In contrast to the state-of-the-art
tensor-based alternating least squares SVD (ALS-SVD) and modified alternating
least squares SVD (MALS-SVD) matrix approximation methods, TNrSVD can be up to
17 times faster while achieving the same accuracy. In addition, our TNrSVD
algorithm also produces accurate approximations in particular cases where both
ALS-SVD and MALS-SVD fail to converge. We also propose a new algorithm for the
fast conversion of a sparse matrix into its corresponding MPO form, which is up
to 509 times faster than the standard Tensor Train SVD (TT-SVD) method while
achieving machine precision accuracy. The efficiency and accuracy of both
algorithms are demonstrated in numerical experiments.
