Adaptive singular value shrinkage estimate for low rank tensor denoising

Recently, tensors are widely used to represent higher-order data with internal spatial or temporal relations, e.g. images, videos, hyperspectral images (HSIs). While the true signals are usually corrupted by noises, it is of interest to study tensor recovery problems. To this end, many models have been established based on tensor decompositions. Traditional tensor decomposition models, such as the CP and Tucker factorization, treat every mode of tensors equally. However, in many real applications, some modes of the data act differently from the other modes, e.g. channel mode of images, time mode of videos, band mode of HSIs. The recently proposed model called t-SVD aims to tackle such problems. In this paper, we focus on tensor denoising problems. Specifically, in order to obtain low-rank estimators of true signals, we propose to use different shrinkage functions to shrink the tensor singular values based on the t-SVD. We derive Stein's unbiased risk estimate (SURE) of the proposed model and develop adaptive SURE-based tuning parameter selection procedure, which is totally data-driven and simultaneous with the estimation process. The whole modeling procedure requires only one round of t-SVD. To demonstrate our model, we conduct experiments on simulation data, images, videos and HSIs. The results show that the proposed SURE approximates the true risk function accurately. Moreover, the proposed model selection procedure picks good tuning parameters out. We show the superiority of our model by comparing with state-of-the-art denoising models. The experiments manifest that our model outperforms in both quantitative metrics (e.g. RSE, PSNR) and visualizing results.