Tensor ring decomposition: Optimization landscape and one-loop convergence of alternating least squares
In this work, we study the tensor ring decomposition and its associated numerical algorithms. We establish a sharp transition of algorithmic difficulty of the optimization problem as the bond dimension increases: On one hand, we show the existence of spurious local minima for the optimization landscape even when the tensor ring format is much overparameterized, i.e., with bond dimension much larger than that of the true target tensor. On the other hand, when the bond dimension is further increased, we establish one-loop convergence for the alternating least squares algorithm for the tensor ring decomposition. The theoretical results are complemented by numerical experiments for both local minima and the one-loop convergence for the alternating least squares algorithm.
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