The trouble with tensor ring decompositions

The tensor train decomposition decomposes a tensor into a "train" of 3-way tensors that are interconnected through the summation of auxiliary indices. The decomposition is stable, has a well-defined notion of rank and enables the user to perform various linear algebra operations on vectors and matrices of exponential size in a computationally efficient manner. The tensor ring decomposition replaces the train by a ring through the introduction of one additional auxiliary variable. This article discusses a major issue with the tensor ring decomposition: its inability to compute an exact minimal-rank decomposition from a decomposition with sub-optimal ranks. Both the contraction operation and Hadamard product are motivated from applications and it is shown through simple examples how the tensor ring-rounding procedure fails to retrieve minimal-rank decompositions with these operations. These observations, together with the already known issue of not being able to find a best low-rank tensor ring approximation to a given tensor indicate that the applicability of tensor rings is severely limited.