Tensor p-shrinkage nuclear norm for low-rank tensor completion

Smooth Tensor Product for Tensor Completion

Low-rank tensor completion (LRTC) has shown promise in processing incomplete visual data, yet it often overlooks the inherent local smooth structures in images and videos. Recent advances in LRTC, integrating total variation regularization to capitalize on the local smoothness, have yielded notable improvements. Nonetheless, these methods are limited to exploiting local smoothness within the original data space, neglecting the latent factor space of tensors. More seriously, there is a lack of theoretical backing for the role of local smoothness in enhancing recovery performance. In response, this paper introduces an innovative tensor completion model that concurrently leverages the global low-rank structure of the original tensor and the local smooth structure of its factor tensors. Our objective is to learn a low-rank tensor that decomposes into two factor tensors, each exhibiting sufficient local smoothness. We propose an efficient alternating direction method of multipliers to optimize our model. Further, we establish generalization error bounds for smooth factor-based tensor completion methods across various decomposition frameworks. These bounds are significantly tighter than existing baselines. We conduct extensive inpainting experiments on color images, multispectral images, and videos, which demonstrate the efficacy and superiority of our method. Additionally, our approach shows a low sensitivity to hyper-parameter settings, enhancing its convenience and reliability for practical applications.

Tensor decomposition based on the potential low-rank and [Formula: see text]-shrinkage generalized threshold algorithm for analyzing cancer multiomics data

Tensor Robust Principal Component Analysis (TRPCA) has achieved promising results in the analysis of genomics data. However, the TRPCA model under the existing tensor singular value decomposition ([Formula: see text]-SVD) framework insufficiently extracts the potential low-rank structure of the data, resulting in suboptimal restored components. Simultaneously, the tensor nuclear norm (TNN) defined based on [Formula: see text]-SVD uses the same standard to handle various singular values. TNN ignores the difference of singular values, leading to the failure of the main information that needs to be well preserved. To preserve the heterogeneous structure in the low-rank information, we propose a novel TNN and extend it to the TRPCA model. Potential low-rank space may contain important information. We learn the low-rank structural information from the core tensor. The singular value space contains the association information between genes and cancers. The [Formula: see text]-shrinkage generalized threshold function is utilized to preserve the low-rank properties of larger singular values. The optimization problem is solved by the alternating direction method of the multiplier (ADMM) algorithm. Clustering and feature selection experiments are performed on the TCGA data set. The experimental results show that the proposed model is more promising than other state-of-the-art tensor decomposition methods.