  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.
