A rank estimation method for third-order tensor completion in the tensor-train format

A numerical method to obtain an adequate value for the upper bound on the rank for the tensor completion problem on the variety of third-order tensors of bounded tensor-train rank is proposed. The method is inspired by the parametrization of the tangent cone derived by Kutschan (2018). The adequacy of the upper bound for a related low-rank tensor approximation problem is shown and an estimated rank is defined to extend the result to the low-rank tensor completion problem. Some experiments on synthetic data are given to illustrate the approach and show that the method is robust, e.g., to noise on the data.