In the realm of tensor optimization, low-rank tensor decomposition, particularly Tucker decomposition, stands as a pivotal technique for reducing the number of parameters and for saving storage. We embark on an exploration of Tucker tensor varieties -- the set of tensors with bounded Tucker rank -- in which the geometry is notably more intricate than the well-explored geometry of 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. The search directions are computed from approximate projections of antigradient onto the tangent cone, which circumvents the calculation of intractable metric projections. 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 is capable of identifying an appropriate rank during iterations while the convergence is also guaranteed. Numerical experiments on tensor completion with synthetic and real-world datasets reveal that the proposed methods are in favor of recovering performance over other state-of-the-art methods. Moreover, the rank-adaptive method performs the best across various rank parameter selections and is indeed able to find an appropriate rank.

翻译：在张量优化领域中，低秩张量分解（尤其是Tucker分解）是减少参数数量和节省存储的关键技术。本文探索了Tucker张量簇——即具有有界Tucker秩的张量集合——其几何结构比研究较为成熟的矩阵簇几何更为复杂。我们给出了Tucker张量簇切锥的显式参数化，并利用其几何结构开发了可证明的梯度相关线搜索方法，用于Tucker张量簇上的优化。搜索方向通过反梯度在切锥上的近似投影计算得出，从而避免了难以处理的度量投影计算。据我们所知，这是首个涉及Tucker张量簇几何与优化的研究工作。在实际应用中，低秩张量优化面临可靠秩参数选择困难的挑战。为此，我们结合已建立的几何框架，提出了一种Tucker秩自适应方法，能够在迭代过程中识别合适的秩，同时保证收敛性。在合成数据集和真实数据集上的张量补全数值实验表明，所提方法在恢复性能上优于其他现有方法。此外，秩自适应方法在各种秩参数选择下表现最佳，且确实能够找到合适的秩。