This work considers the problem of computing the \textit{canonical polyadic decomposition} (CPD) of large tensors. Prior works mostly leverage data sparsity to handle this problem, which are not suitable for handling dense tensors that often arise in applications such as medical imaging, computer vision, and remote sensing. Stochastic optimization is known for its low memory cost and per-iteration complexity when handling dense data. However, existing stochastic CPD algorithms are hard to incorporate a variety of constraints and regularizations that are of interest in signal and data analytics. Convergence properties of many such algorithms are also unclear. In this work, we propose a stochastic optimization framework for large-scale CPD with constraints/regularizations. The framework works under a doubly randomized fashion, and can be regarded as a judicious combination of \textit{randomized block coordinate descent} (BCD) and \textit{stochastic proximal gradient} (SPG). The algorithm enjoys lightweight updates and small memory footprint, and thus scales well. In addition, this framework entails considerable flexibility---many frequently used regularizers and constraints can be readily handled under the proposed scheme. The approach is also supported by convergence analysis. Numerical results on large-scale dense tensors are employed to showcase the effectiveness of the proposed approach.

本文提出了一种用于大规模张量分解的随机优化框架，结合了随机块坐标下降和随机近端梯度等算法，支持常用的约束和正则化，并给出了收敛性分析。

基于块随机化的随机近端梯度下降方法用于低秩张量分解