Tensor-ring Nuclear Norm Minimization and Application for Visual : Data Completion

Abstract

Tensor ring (TR) decomposition has been successfully used to obtain the state-of-the-art performance in the visual data completion problem. However, the existing TR-based completion methods are severely non-convex and computationally demanding. In addition, the determination of the optimal TR rank is a tough work in practice. To overcome these drawbacks, we first introduce a class of new tensor nuclear norms by using tensor circular unfolding. Then we theoretically establish connection between the rank of the circularlyunfolded matrices and the TR ranks. We also develop an efficient tensor completion algorithm by minimizing the proposed tensor nuclear norm. Extensive experimental results demonstrate that our proposed tensor completion method outperforms the conventional tensor completion methods in the image/video in-painting problem with striped missing values.

Provable tensor ring completion

“…Lemma 2 shows that any TR unfolding X {i,l} obeys the strong matrix incoherence condition if the TR is strong incoherent. Note that [21] states that a TR unfolding obeys rank X {i,l} ≤ r i r i+l . We emphasize this inequality becomes equality under specific conditions.…”

“…Eight algorithms are benchmarked on real-world data, including tensor ring nuclear norm minimization for tensor completion (TRNNM) [21], low rank tensor completion via alternating least square (TR-ALS) [18], simple low rank tensor completion via tensor train (SiLRTC-TT) [14], high accuracy low rank tensor completion algorithm (HaLRTC) [7], low rank tensor completion via tensor nuclear norm minimization (LRTC-TNN) [25], Bayesian CP Factorization (FBCP) for image recovery [26], smooth low rank tensor tree completion (STTC) [15] and the proposed one. These methods are based on different tensor decompositions, including CP, Tucker, t-SVD, HT, TT and TR decompositions.…”

An Efficient Tensor Completion Method Via New Latent Nuclear Norm

“…Our work is somewhat related to latent-norm based completion methods [37], [38] and Tensor-Ring based completion methods [32], [36], [39]. In [37], Tomioka et al proposed the latent nuclear norm by mode-k unfolding scheme (one mode versus the rest), and shown that it generalizes better than the overlapped nuclear norm [34] when only several modes are low-rank.…”

“…Rank-minimization based method is another type of approach to exploit the low-rank structure of incompleted tensor. Since the tensor rank minimization rank(•) is an NP-hard problem, a number of norms are defined as the convex surrogates of tensor rank, and the most commonly used ones are overlapped nuclear norm [34]- [36] and latent nuclear norm [37], [38]. In [34], the overlapped nuclear norm via Tucker rank was first proposed by assuming all modes are low-rank, while it performs poorly when the target tensor is only lowrank in a certain mode.…”

“…However, these two norms are still based on the unbalanced unfolding scheme, i.e., k-mode unfolding scheme (the first k modes versus the rest). Inspired by the powerful ablitity of TR decomposition in representing high-order tensors, in our prevent work [36], an convex TRbased overlapped nuclear norm is proposed by building the rank relation between TR decomposition and a balanced unfolding scheme called Tensor Circular Unfolding (TCU).…”