Provably finding stationary points on bounded-rank tensors turns out to be an open problem [E. Levin, J. Kileel, and N. Boumal, Math. Program., 199 (2023), pp. 831--864] due to the inherent non-smoothness of the set of bounded-rank tensors. We resolve this problem by proposing two first-order methods with guaranteed convergence to stationary points. Specifically, we revisit the variational geometry of bounded-rank tensors and explicitly characterize its normal cones. Moreover, we propose gradient-related approximate projection methods that are provable to find stationary points, where the decisive ingredients are gradient-related vectors from tangent cones, line search along approximate projections, and rank-decreasing mechanisms near rank-deficient points. Numerical experiments on tensor completion validate that the proposed methods converge to stationary points across various rank parameters.

翻译：由于有界秩张量集合固有的非光滑性，可证明地在其上寻找稳定点仍是一个开放性问题[E. Levin, J. Kileel, and N. Boumal, Math. Program., 199 (2023), pp. 831--864]。本文通过提出两种保证收敛到稳定点的一阶方法解决了该问题。具体而言，我们重新审视了有界秩张量的变分几何结构，并显式刻画了其法锥。此外，我们提出了梯度相关近似投影方法，该方法可证明能找到稳定点，其中的关键要素包括：来自切锥的梯度相关向量、沿近似投影的线搜索，以及在秩亏缺点附近的降秩机制。在张量补全任务上的数值实验验证了所提方法能在不同秩参数下收敛到稳定点。