Low-Rank Tensor Completion by Truncated Nuclear Norm Regularization

Abstract

Currently, low-rank tensor completion has gained cumulative attention in recovering incomplete visual data whose partial elements are missing. By taking a color image or video as a three-dimensional (3D) tensor, previous studies have suggested several definitions of tensor nuclear norm. However, they have limitations and may not properly approximate the real rank of a tensor. Besides, they do not explicitly use the low-rank property in optimization. It is proved that the recently proposed truncated nuclear norm (TNN) can replace the traditional nuclear norm, as a better estimation to the rank of a matrix. Thus, this paper presents a new method called the tensor truncated nuclear norm (T-TNN), which proposes a new definition of tensor nuclear norm and extends the truncated nuclear norm from the matrix case to the tensor case. Beneficial from the low rankness of TNN, our approach improves the efficacy of tensor completion. We exploit the previously proposed tensor singular value decomposition and the alternating direction method of multipliers in optimization. Extensive experiments on real-world videos and images demonstrate that the performance of our approach is superior to those of existing methods.

SURE Based Truncated Tensor Nuclear Norm Regularization for Low Rank Tensor Completion

“…Xue et al [8], [9] exploited and proved the symmetry properties of the trace of the tensor product to demonstrate as follows:…”

“…III. TENSOR TRUNCATED NUCLEAR NORM Extending the Hu et al [4] truncated nuclear norm from the matrix case to the tensor case Xue et al [8], [9] adapted the approach within the t-SVD case by aiming to minimize the smallest min(m, n) − r singular values. Thus incorporating the above reduced complexity Fourier based tensor nuclear norm Xue et al defined the tensor truncated nuclear norm as follows:…”

“…Within this paper the work of Xue et al [8], [9] which extended the truncated nuclear norm method to the tensor case is further developed and improved to utilize an enhanced singular value thresholding, based on the Stein Unbiased Risk Estimator (SURE) [10]. It is demonstrated that the improved SURE based singular value thresholding improves upon the previous method by experimental evaluation.…”

Infrared Small Target Detection via Non-Convex Rank Approximation Minimization Joint l2,1 Norm

“…For instance, Gu et al [33] utilized ADMM to solve the non-convex weighted nuclear norm minimization (WNNM) problem efficiently, which was successfully applied to image inpainting. Xue et al [34] proposed a non-convex low-rank tensor completion model using ADMM to obtain the best recovery result in color images. Based on total variation regularized tensor RPCA, Cao et al [35] designed a non-convex and non-separable model to complete background subtraction with an ADMM solver.…”

Image recovery and recognition: a combining method of matrix norm regularisation

“…Therefore, the effect of large singular values can be removed when the nuclear norm is replaced by TNN. Recently, the TNN has been applied in RPCA to solve some related problems in [18, 22, 23], and the problem based on TNN regulation can be represented as However, problem (3) is often constrained in actual application, because the solution of matrix recovery is unstable when predictors are highly correlated. This narrows the practical scope of the algorithm.…”