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Optimal rank selection is an important issue in tensor decomposition problems, especially for Tensor Train (TT) and Tensor Ring (TR) (also known as Tensor Chain) decompositions. In this paper, a new rank selection method for TR decomposition has been proposed for automatically finding near-optimal TR ranks, which result in a lower storage cost, especially for tensors with inexact TT or TR structures. In many of the existing approaches, TR ranks are determined in advance or by using truncated Singular Value Decomposition (t-SVD). There are also other approaches for selecting TR ranks adaptively. In our approach, the TR ranks are not determined in advance, but are increased gradually in each iteration until the model achieves a desired approximation accuracy. For this purpose, in each iteration, the sensitivity of the approximation error to each of the core tensors is measured and the core tensors with the highest sensitivity measures are selected and their sizes are increased. Simulation results confirmed that the proposed approach reduces the storage cost considerably and allows us to find optimal model in TR format, while preserving the desired accuracy of the approximation.
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