Rank Estimation for Third-Order Tensor Completion in the Tensor-Train Format

TL;DR

This paper introduces a numerical method to estimate an appropriate upper bound on the tensor rank for third-order tensor completion in the tensor-train format, improving robustness and applicability.

Contribution

It presents a novel approach inspired by tangent cone parametrization to determine rank bounds for tensor completion, extending previous low-rank approximation results.

Findings

Method effectively estimates tensor rank bounds.

Approach demonstrates robustness to data noise.

Experiments validate the method's accuracy and stability.

Abstract

We propose 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. The method is inspired by the parametrization of the tangent cone derived by Kutschan (2018). A proof of the adequacy of the upper bound for a related low-rank tensor approximation problem is given and an estimated rank is defined to extend the result to the low-rank tensor completion problem. Some experiments on synthetic data illustrate the approach and show that the method is very robust, e.g., to noise on the data.