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.

中文翻译：

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张量环分解：交替最小二乘法的优化景观和单环收敛

在这项工作中，我们研究了张量环分解及其相关的数值算法。随着键维数的增加，我们建立了优化问题的算法难度的急剧转变：一方面，即使张量环格式被过度参数化，我们也表明优化景观存在虚假局部最小值，即具有键维度远大于真实目标张量的维度。另一方面，当键维数进一步增加时，我们为张量环分解的交替最小二乘算法建立单环收敛。理论结果由交替最小二乘算法的局部最小值和单环收敛的数值实验补充。