The recently proposed tensor robust principal component analysis (TRPCA) methods based on tensor singular value decomposition (t-SVD) have achieved numerous successes in many fields. However, most of these methods are only applicable to third-order tensors, whereas the data obtained in practice are often of higher order, such as fourth-order color videos, fourth-order hyperspectral videos, and fifth-order light-field images. Additionally, in the t-SVD framework, the multi-rank of a tensor can describe more fine-grained low-rank structure in the tensor compared with the tubal rank. However, determining the multi-rank of a tensor is a much more difficult problem than determining the tubal rank. Moreover, most of the existing TRPCA methods do not explicitly model the noises except the sparse noise, which may compromise the accuracy of estimating the low-rank tensor. In this work, we propose a novel high-order TRPCA method, named as Low-Multi-rank High-order Bayesian Robust Tensor Factorization (LMH-BRTF), within the Bayesian framework. Specifically, we decompose the observed corrupted tensor into three parts, i.e., the low-rank component, the sparse component, and the noise component. By constructing a low-rank model for the low-rank component based on the order-$d$ t-SVD and introducing a proper prior for the model, LMH-BRTF can automatically determine the tensor multi-rank. Meanwhile, benefiting from the explicit modeling of both the sparse and noise components, the proposed method can leverage information from the noises more effectivly, leading to an improved performance of TRPCA. Then, an efficient variational inference algorithm is established for parameters estimation. Empirical studies on synthetic and real-world datasets demonstrate the effectiveness of the proposed method in terms of both qualitative and quantitative results.

翻译：近期基于张量奇异值分解（t-SVD）的张量鲁棒主成分分析（TRPCA）方法已在诸多领域取得显著成功。然而，此类方法大多仅适用于三阶张量，而实际获取的数据往往具有更高阶结构，例如四阶彩色视频、四阶高光谱视频及五阶光场图像。此外，在t-SVD框架下，张量的多秩相较于管状秩能更精细地刻画张量中的低秩结构，但多秩的确定比管状秩更具挑战性。同时，现有TRPCA方法除稀疏噪声外通常未显式建模其他噪声，这可能导致低秩张量估计精度的下降。本文在贝叶斯框架下提出一种新型高阶TRPCA方法——低多秩高阶贝叶斯鲁棒张量分解（LMH-BRTF）。具体而言，我们将观测到的含噪张量分解为低秩分量、稀疏分量与噪声分量三部分。通过基于d阶张量奇异值分解构建低秩分量的低秩模型并引入合理先验，LMH-BRTF可自动确定张量多秩。同时，得益于对稀疏与噪声分量的显式建模，所提方法能更有效利用噪声信息，从而提升TRPCA性能。进而建立高效变分推理算法进行参数估计。在合成数据集与真实数据集上的定性及定量实验结果均验证了所提方法的有效性。