  Tensor networks have in recent years emerged as the powerful tools for
solving the large-scale optimization problems. One of the most popular tensor
network is tensor train (TT) decomposition that acts as the building blocks for
the complicated tensor networks. However, the TT decomposition highly depends
on permutations of tensor dimensions, due to its strictly sequential
multilinear products over latent cores, which leads to difficulties in finding
the optimal TT representation. In this paper, we introduce a fundamental tensor
decomposition model to represent a large dimensional tensor by a circular
multilinear products over a sequence of low dimensional cores, which can be
graphically interpreted as a cyclic interconnection of 3rd-order tensors, and
thus termed as tensor ring (TR) decomposition. The key advantage of TR model is
the circular dimensional permutation invariance which is gained by employing
the trace operation and treating the latent cores equivalently. TR model can be
viewed as a linear combination of TT decompositions, thus obtaining the
powerful and generalized representation abilities. For optimization of latent
cores, we present four different algorithms based on the sequential SVDs, ALS
scheme, and block-wise ALS techniques. Furthermore, the mathematical properties
of TR model are investigated, which shows that the basic multilinear algebra
can be performed efficiently by using TR representaions and the classical
tensor decompositions can be conveniently transformed into the TR
representation. Finally, the experiments on both synthetic signals and
real-world datasets were conducted to evaluate the performance of different
algorithms.
