Tensor Low Multi-tubal Rank Decomposition With Applications

Tensor completion problems are of great significance in various fields,including machine learning,image completion,and many others,and have been widely studied in recent years.In practice,tensor datasets are often affected by noise,missing data,or other factors that render them incomplete or corrupted.To address this issue,low-rank tensor completion（LRTC）models have been increasingly popular,particularly in the fields of hyperspectral images（HSIs）,multispectral images（MSIs）,and gray video recovery,as they preserve the intricate structure of high-order data.This thesis presents a novel tensor completion model based on tensor low multi-tubal rank decomposition using a unitary transform.In addition,we propose an improved LRTC model that replaces the ₁norm of the regular term with a class of non-convex functions.We also introduce a tensor-tensor product in the transformation domain to enhance the model’s performance.The main contributions of this thesis can be summarized as follows:（1）In this chapter,we introduce a tensor completion method that is based on unitary transformation tensor low multi-tubal rank decomposition.By incorporating low-rank constraints,we are able to effectively capture the global structure of the data,while tensor multi-tubal rank allows us to make use of the multi-dimensional features present in image and video data.To further improve the accuracy of our model,we have included a spatial smoothing regular term（TV）,which is inspired by the spatial spectral smoothness feature in multispectral images.The resulting UTCTFTV model is solved using a proximal alter-nating minimization（PAM）algorithm,which has been shown to have global convergence.Our simulation experiments,which were conducted on multispectral images（MSIs）data,demonstrate that the UTCTFTV model outperforms other comparable algorithms in terms of both recovery accuracy and visual effect.（2）The UTCTFTV model proposed above has shown good performance in recover-ing both hyperspectral images（HSIs）and multispectral images（MSIs）.However,the reg-ular terms in this model are not accurate enough for approximating the 0-norm.To address this issue,we propose an improved model,MTRTC-NRTV,which uses a non-convex and non-smooth function as the regular term.This allows us to fully utilize the low-rank and smooth prior of different modes of tensor data.We still use the PAM algorithm to solve the completion model.Through a large number of experiments on various types of visual data,such as gray videos,HSIs,and MSIs,we have shown that the MTRTC-NRTV model can accurately and flexibly capture the low-rankness and smoothness on any mode of the third-order tensor,resulting in better recovery performance.