  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 over-parameterized, 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 alternating least square algorithm for tensor ring decomposition. The
theoretical results are complemented by numerical experiments for both local
minimum and one-loop convergence for the alternating least square algorithm.
