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On algorithms for and computing with the tensor ring decomposition

Author(s)
Mickelin, Oscar; Karaman, Sertac
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Abstract
© 2020 John Wiley & Sons, Ltd. 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.
Date issued
2020
URI
https://hdl.handle.net/1721.1/135948
Department
Massachusetts Institute of Technology. Department of Mathematics; Massachusetts Institute of Technology. Department of Aeronautics and Astronautics
Journal
Numerical Linear Algebra with Applications
Publisher
Wiley

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