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.

翻译：提出了一种有界张量列车秩三阶张量簇切锥上的近似投影方法，并证明其满足比Kutschan（2019）提出的方法更优的角度条件。该近似投影使得能够在切锥内计算梯度相关方向，例如在最小化簇上连续可微函数的算法中所需，这类问题在张量补全中尤为突出。数值实验表明，在实际应用中，该近似投影满足的角度条件既优于Kutschan所提近似投影满足的条件，也优于理论证明的上界。