  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.
