Low-rank tensors appear to be prosperous in many applications. However, the sets of bounded-rank tensors are non-smooth and non-convex algebraic varieties, rendering the low-rank optimization problems to be challenging. To this end, we delve into the geometry of bounded-rank tensor sets, including Tucker and tensor train formats. We propose a desingularization approach for bounded-rank tensor sets by introducing slack variables, resulting in a low-dimensional smooth manifold embedded in a higher-dimensional space while preserving the structure of low-rank tensor formats. Subsequently, optimization on tensor varieties can be reformulated to optimization on smooth manifolds, where the methods and convergence are well explored. We reveal the relationship between the landscape of optimization on varieties and that of optimization on manifolds. Numerical experiments on tensor completion illustrate that the proposed methods are in favor of others under different rank parameters.

翻译：低秩张量在众多应用中展现出广阔前景。然而，有界秩张量集是非光滑且非凸的代数簇，这使得低秩优化问题具有挑战性。为此，我们深入研究了有界秩张量集的几何结构，包括Tucker格式和张量链格式。我们提出了一种通过引入松弛变量对有界秩张量集进行奇点消解的方法，从而在保持低秩张量格式结构的同时，得到一个嵌入高维空间的低维光滑流形。随后，张量簇上的优化问题可转化为光滑流形上的优化问题，而后者已有成熟的方法与收敛性研究。我们揭示了簇上优化与流形上优化的景观关系。张量补全的数值实验表明，在不同秩参数下，所提方法均优于其他方法。