We study the best low-rank Tucker decomposition of symmetric tensors. The motivating application is decomposing higher-order multivariate moments. Moment tensors have special structure and are important to various data science problems. We advocate for projected gradient descent (PGD) method and higher-order eigenvalue decomposition (HOEVD) approximation as computation schemes. Most importantly, we develop scalable adaptations of the basic PGD and HOEVD methods to decompose sample moment tensors. With the help of implicit and streaming techniques, we evade the overhead cost of building and storing the moment tensor. Such reductions make computing the Tucker decomposition realizable for large data instances in high dimensions. Numerical experiments demonstrate the efficiency of the algorithms and the applicability of moment tensor decompositions to real-world datasets. Finally we study the convergence on the Grassmannian manifold, and prove that the update sequence derived by the PGD solver achieves first- and second-order criticality.

翻译：我们研究对称张量的最佳低秩Tucker分解。该研究的主要动机是分解高阶多元矩。矩张量具有特殊结构，对各类数据科学问题具有重要意义。我们推荐采用投影梯度下降（PGD）方法和高阶特征值分解（HOEVD）近似作为计算方案。最重要的是，我们开发了基本PGD和HOEVD方法的可扩展改进版本，用于分解样本矩张量。借助隐式技术和流式技术，我们避免了构建和存储矩张量的开销成本。这种缩减使得高维大样本数据的Tucker分解计算切实可行。数值实验证明了算法的效率以及矩张量分解在真实数据集上的适用性。最后，我们研究了Grassmann流形上的收敛性，并证明由PGD求解器推导的更新序列达到了一阶和二阶临界点。