Tensor ring (TR) decomposition is a simple but effective tensor network for analyzing and interpreting latent patterns of tensors. In this work, we propose a doubly randomized optimization framework for computing TR decomposition. It can be regarded as a sensible mix of randomized block coordinate descent and stochastic gradient descent, and hence functions in a double-random manner and can achieve lightweight updates and a small memory footprint. Further, to improve the convergence, especially for ill-conditioned problems, we propose a scaled version of the framework that can be viewed as an adaptive preconditioned or diagonally-scaled variant. Four different probability distributions for selecting the mini-batch and the adaptive strategy for determining the step size are also provided. Finally, we present the theoretical properties and numerical performance for our proposals.

翻译：张量环（TR）分解是一种简单而有效的张量网络，用于分析和解释张量的潜在模式。本文提出一种用于计算张量环分解的双随机优化框架。该框架可视为随机块坐标下降法与随机梯度下降法的合理融合，因此以双重随机方式运行，能够实现轻量级更新和较小的内存占用。此外，为改善收敛性（尤其针对病态问题），我们提出该框架的缩放版本，可视为自适应预条件或对角缩放变体。文中还提供了四种用于选择小批次数据的概率分布，以及确定步长的自适应策略。最后，我们给出了所提方案的理论性质与数值性能结果。