  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.
