  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.
