  Tensor ring (TR) decomposition is a simple but effective tensor network for
analyzing and interpreting latent patterns of tensors. In this work, we propose
a doubly randomized optimization framework for computing TR decomposition. It
can be regarded as a sensible mix of randomized block coordinate descent and
stochastic gradient descent, and hence functions in a double-random manner and
can achieve lightweight updates and a small memory footprint. Further, to
improve the convergence, especially for ill-conditioned problems, we propose a
scaled version of the framework that can be viewed as an adaptive
preconditioned or diagonally-scaled variant. Four different probability
distributions for selecting the mini-batch and the adaptive strategy for
determining the step size are also provided. Finally, we present the
theoretical properties and numerical performance for our proposals.
