Tensor-Ring Decomposition with Index-Splitting

Tensor-ring decomposition of tensors plays a key role in various applications of tensor network representation in physics as well as in other fields. In most heuristic algorithms for the tensor-ring decomposition, one encounters the problem of local-minima trapping. Particularly, the minima related to the topological structure in the correlation are hard to escape. Therefore, identification of the correlation structure, somewhat analogous to finding matching ends of entangled strings, is the task of central importance. We show how this problem naturally arises in physical applications, and present a strategy for winning this "string-pull" game.