We study the tensor robust principal component analysis (TRPCA) problem, a tensorial extension of matrix robust principal component analysis (RPCA), that aims to split the given tensor into an underlying low-rank component and a sparse outlier component. This work proposes a fast algorithm, called Robust Tensor CUR Decompositions (RTCUR), for large-scale non-convex TRPCA problems under the Tucker rank setting. RTCUR is developed within a framework of alternating projections that projects between the set of low-rank tensors and the set of sparse tensors. We utilize the recently developed tensor CUR decomposition to substantially reduce the computational complexity in each projection. In addition, we develop four variants of RTCUR for different application settings. We demonstrate the effectiveness and computational advantages of RTCUR against state-of-the-art methods on both synthetic and real-world datasets.

翻译：本文研究张量鲁棒主成分分析（TRPCA）问题，该问题是矩阵鲁棒主成分分析（RPCA）的张量扩展，旨在将给定张量分解为潜在低秩分量与稀疏异常分量。针对Tucker秩设定下的大规模非凸TRPCA问题，本文提出一种快速算法——鲁棒张量CUR分解（RTCUR）。该算法基于交替投影框架，在低秩张量集合与稀疏张量集合之间进行投影。通过利用近期发展的张量CUR分解技术，每次投影的计算复杂度得到显著降低。此外，针对不同应用场景，我们开发了四种RTCUR变体。在合成数据集与真实世界数据集上的实验表明，RTCUR相较于现有最优方法具有有效性与计算优势。