An Approximate Projection onto the Tangent Cone to the Variety of Third-Order Tensors of Bounded Tensor-Train Rank

TL;DR

This paper introduces an improved approximate projection method onto the tangent cone of third-order tensors with bounded tensor-train rank, enhancing gradient computation for tensor completion algorithms.

Contribution

It proposes a new approximate projection satisfying a better angle condition than previous methods, with theoretical proof and practical validation.

Findings

The new projection satisfies a superior angle condition.

Numerical experiments show improved practical performance.

The method benefits tensor completion tasks.

Abstract

An approximate projection onto the tangent cone to the variety of third-order tensors of bounded tensor-train rank is proposed and proven to satisfy a better angle condition than the one proposed by Kutschan (2019). Such an approximate projection enables, e.g., to compute gradient-related directions in the tangent cone, as required by algorithms aiming at minimizing a continuously differentiable function on the variety, a problem appearing notably in tensor completion. A numerical experiment is presented which indicates that, in practice, the angle condition satisfied by the proposed approximate projection is better than both the one satisfied by the approximate projection introduced by Kutschan and the proven theoretical bound.