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.

提出了一种名为 RTCUR 的快速算法，用于在 Tucker 秩设置下，将给定的张量分成基础低秩分量和稀疏异常分量的大规模非凸 TRPCA 问题中，利用交替投影框架在低秩张量集和稀疏张量集之间进行投影，并利用最近开发的张量 CUR 分解大大降低了每个投影的计算复杂度，通过在合成和真实数据集上的实验，证明了 RTCUR 算法的有效性和计算优势，是一种研究 tensor robust principal component analysis 问题的有效方法。

稳健张量 CUR 分解：用稀疏矩阵破损快速恢复低 Tucker 秩张量

Robust Tensor CUR Decompositions: Rapid Low-Tucker-Rank Tensor Recovery  with Sparse Corruption

This paper is about a curious phenomenon. Suppose we have a data matrix,
which is the superposition of a low-rank component and a sparse component. Can
we recover each component individually? We prove that under some suitable
assumptions, it is possible to recover both the low-rank and the sparse
components exactly by solving a very convenient convex program called Principal
Component Pursuit; among all feasible decompositions, simply minimize a
weighted combination of the nuclear norm and of the L1 norm. This suggests the
possibility of a principled approach to robust principal component analysis
since our methodology and results assert that one can recover the principal
components of a data matrix even though a positive fraction of its entries are
arbitrarily corrupted. This extends to the situation where a fraction of the
entries are missing as well. We discuss an algorithm for solving this
optimization problem, and present applications in the area of video
surveillance, where our methodology allows for the detection of objects in a
cluttered background, and in the area of face recognition, where it offers a
principled way of removing shadows and specularities in images of faces.

本文介绍了一种名为主成分追踪的凸型优化方法，能在有噪声或缺损情况下准确分离一个 $ m * n $ 数据矩阵的低秩和稀疏成分，该方法有望应用于视频监控和人脸识别等领域。