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.

提出一种名为低多秩高阶贝叶斯鲁棒张量分解 (LMH-BRTF) 的新型高阶 TRPCA 方法，在贝叶斯框架内对观测到的受损张量进行分解，结合了明确建模稀疏和噪声成分的优势，实现了对张量的多秩自动确定，并采用高效的变分推断算法进行参数估计，通过对合成和现实世界数据集的实证研究，在定性和定量结果方面证明了该方法的有效性。

低多秩高阶贝叶斯鲁棒张量分解

Low-Multi-Rank High-Order Bayesian Robust Tensor Factorization

Tensor decompositions are powerful tools for analyzing multi-dimensional data
in their original format. Besides tensor decompositions like Tucker and CP,
Tensor SVD (t-SVD) which is based on the t-product of tensors is another
extension of SVD to tensors that recently developed and has found numerous
applications in analyzing high dimensional data. This paper offers a new
insight into the t-Product and shows that this product is a block convolution
of two tensors with periodic boundary conditions. Based on this viewpoint, we
propose a new tensor-tensor product called the $\star_c{}\text{-Product}$ based
on Block convolution with reflective boundary conditions. Using a tensor
framework, this product can be easily extended to tensors of arbitrary order.
Additionally, we introduce a tensor decomposition based on our
$\star_c{}\text{-Product}$ for arbitrary order tensors. Compared to t-SVD, our
new decomposition has lower complexity, and experiments show that it yields
higher-quality results in applications such as classification and compression.

此篇论文介绍了一种新的 tensor-tensor 乘积方法 —— 基于带有反射边界条件的块卷积，同时提出了一种基于该乘积的任意阶 tensor 的分解方式，与 t-SVD 相比，新的分解具有更低的复杂度且在分类和压缩等应用中获得更高质量的结果。

基于块卷积和反射边界条件的任意阶张量分解方法用于多维数据分析

A Novel Tensor Decomposition of arbitrary order based on Block  Convolution with Reflective Boundary Conditions for Multi-Dimensional Data  Analysis

This study aims to solve the over-reliance on the rank estimation strategy in
the standard tensor factorization-based tensor recovery and the problem of a
large computational cost in the standard t-SVD-based tensor recovery. To this
end, we proposes a new tensor norm with a dual low-rank constraint, which
utilizes the low-rank prior and rank information at the same time. In the
proposed tensor norm, a series of surrogate functions of the tensor tubal rank
can be used to achieve better performance in harness low-rankness within tensor
data. It is proven theoretically that the resulting tensor completion model can
effectively avoid performance degradation caused by inaccurate rank estimation.
Meanwhile, attributed to the proposed dual low-rank constraint, the t-SVD of a
smaller tensor instead of the original big one is computed by using a sample
trick. Based on this, the total cost at each iteration of the optimization
algorithm is reduced to $\mathcal{O}(n^3\log n +kn^3)$ from $\mathcal{O}(n^4)$
achieved with standard methods, where $k$ is the estimation of the true tensor
rank and far less than $n$. Our method was evaluated on synthetic and
real-world data, and it demonstrated superior performance and efficiency over
several existing state-of-the-art tensor completion methods.

本文提出了一种新的张量范数，同时利用低秩先验和秩信息，包括一系列张量管秩的代理函数，可在张量数据中更好地利用低秩，通过使用样本技巧计算更小张量的 t-SVD 而不是原始张量来计算提出的双低秩约束的张量范数。随后，优化算法的每个迭代的计算成本从标准方法的 O (n^4) 降低到 O (n^3 log n + kn^3)，其中 k 为真实张量秩的估计值，远小于 n。本方法在合成和现实数据上得到了评估，并表现出优于现有 STOA 张量完成方法的性能和效率。