Tensor completion via multi-directional partial tensor nuclear norm with total variation regularization

This paper addresses the tensor completion problem, whose task is to estimate missing values with limited information. However, the crux of this problem is how to reasonably represent the low-rank structure embedded in the underlying data. In this work, we consider a new low-rank tensor completion model combined with the multi-directional partial tensor nuclear norm and the total variation (TV) regularization. Specifically, the partial sum of the tensor nuclear norm (PSTNN) is used to narrow the gap between the tensor tubal rank and its lower convex envelop [i.e. tensor nuclear norm (TNN)], and the TV regularization is adopted to maintain the smooth structure along the spatial dimension. In addition, the weighted sum of the tensor nuclear norm (WSTNN) is introduced to replace the traditional TNN to extend the PSTNN to the high-order tensor, which also can flexibly handle different correlations along different modes, resulting in an improved low d-tubal rank approximation. To tackle this new model, we develop the alternating directional method of multipliers (ADMM) algorithm tailored for the proposed optimization problem. Theoretical analysis of the ADMM is conducted to prove the Karush–Kuhn–Tucker (KKT) conditions. Numerical examples demonstrate the proposed method outperforms some state-of-the-art methods in qualitative and quantitative aspects.