Robust High-Order Tensor Recovery Via Nonconvex Low-Rank Approximation

Wenjin Qin, Hailin Wang, Weijun Ma, Jianjun Wang

Abstract: The latest tensor recovery methods based on tensor Singular Value Decomposition (t-SVD) mainly utilize the tensor nuclear norm (TNN) as a convex surrogate of the rank function. However, TNN minimization treats each rank component equally and tends to over-shrink the dominant ones, thereby usually leading to biased solutions. To handle this critical issue, we put forward a weighted tensor Schantten-p (0 < p ≤ 1) norm (WTSN) as high-order tensor rank’s more flexible nonconvex relaxation. Furthermore, another nonconvex ℓ <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">q</inf> (0 < q ≤ 1) sparse regularization item on the extensively existed noises/outliers is incorporated into the WSTN minimization to enhance its robustness in the impulsive scenarios. Finally, we propose an efficient and scalable robust high-order tensor recovery method solving a double nonconvex optimization with convergence guarantees. Synthetic and real experiments demonstrate that the proposed approach outperforms the state-of-the-art ones in terms of both accuracy and computational complexity.