In the realm of tensor optimization, the low-rank Tucker decomposition is crucial for reducing the number of parameters and for saving storage. We explore the geometry of Tucker tensor varieties -- the set of tensors with bounded Tucker rank -- which is notably more intricate than the well-explored matrix varieties. We give an explicit parametrization of the tangent cone of Tucker tensor varieties and leverage its geometry to develop provable gradient-related line-search methods for optimization on Tucker tensor varieties. To the best of our knowledge, this is the first work concerning geometry and optimization on Tucker tensor varieties. In practice, low-rank tensor optimization suffers from the difficulty of choosing a reliable rank parameter. To this end, we incorporate the established geometry and propose a Tucker rank-adaptive method that aims to identify an appropriate rank with guaranteed convergence. Numerical experiments on tensor completion reveal that the proposed methods are in favor of recovering performance over other state-of-the-art methods. The rank-adaptive method performs the best across various rank parameter selections and is indeed able to find an appropriate rank.

翻译：在张量优化领域，低秩Tucker分解对于减少参数数量和节省存储空间至关重要。我们研究了Tucker张量簇——即具有有界Tucker秩的张量集合——的几何结构，该结构明显比已深入研究的矩阵簇更为复杂。我们给出了Tucker张量簇切锥的显式参数化，并利用其几何特性开发了可证明的梯度相关线搜索方法，用于在Tucker张量簇上进行优化。据我们所知，这是首个关于Tucker张量簇几何与优化问题的研究。在实际应用中，低秩张量优化面临选择可靠秩参数的困难。为此，我们结合已建立的几何理论，提出了一种Tucker秩自适应方法，旨在以可保证的收敛性确定合适的秩。在张量补全上的数值实验表明，所提方法在恢复性能上优于其他先进方法。秩自适应方法在各种秩参数选择下均表现最佳，且确实能够找到合适的秩。