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Tensor decompositions such as the canonical format and the tensor trainformat have been widely utilized to reduce storage costs and operationalcomplexities for high-dimensional data, achieving linear scaling with the inputdimension instead of exponential scaling. In this paper, we investigate evenlower storage-cost representations in the tensor ring format, which is anextension of the tensor train format with variable end-ranks. Firstly, weintroduce two algorithms for converting a tensor in full format to tensor ringformat with low storage cost. Secondly, we detail a rounding operation fortensor rings and show how this requires new definitions of common linearalgebra operations in the format to obtain storage-cost savings. Lastly, weintroduce algorithms for transforming the graph structure of graph-based tensorformats, with orders of magnitude lower complexity than existing literature.The efficiency of all algorithms is demonstrated on a number of numericalexamples, and in certain cases, we demonstrate significantly higher compressionratios when compared to previous approaches to using the tensor ring format.

On Algorithms for and Computing with the Tensor Ring Decomposition