Anh-Huy Phan - Skolkovo Institute of Science and Technology Tensor chain, a looped tensor network, was introduced to overcome the main issue with the opened straight Tensor Train/MPS network: the intermediate ranks are often unbalanced and grow dramatically with the tensor dimension. Since there are no first and last core tensors within the loop of the network, the TC network becomes more balanced. However, the loop in TC leads to severe numerical instability issues in finding the best approximation. Existing algorithms cannot find the exact TC decomposition of a small tensor of size 7 x 7 x 7 and rank-(3-3-3), and very often get stack in false local minima. We proposed a novel method to deal with instability in TC. The proposed algorithms can achieve perfect decomposition for tensors that admits the models and outperform the estimation using existing methods. We will demonstrate our methods in the decomposition of color images and in the application of CNN compression.