  Tensor decompositions such as the canonical format and the tensor train
format have been widely utilized to reduce storage costs and operational
complexities for high-dimensional data, achieving linear scaling with the input
dimension instead of exponential scaling. In this paper, we investigate even
lower storage-cost representations in the tensor ring format, which is an
extension of the tensor train format with variable end-ranks. Firstly, we
introduce two algorithms for converting a tensor in full format to tensor ring
format with low storage cost. Secondly, we detail a rounding operation for
tensor rings and show how this requires new definitions of common linear
algebra operations in the format to obtain storage-cost savings. Lastly, we
introduce algorithms for transforming the graph structure of graph-based tensor
formats, with orders of magnitude lower complexity than existing literature.
The efficiency of all algorithms is demonstrated on a number of numerical
examples, and in certain cases, we demonstrate significantly higher compression
ratios when compared to previous approaches to using the tensor ring format.
