Tensor Ring Decomposition with Rank Minimization on Latent Space: An Efficient Approach for Tensor Completion

Abstract

In tensor completion tasks, the traditional low-rank tensor decomposition models suffer from the laborious model selection problem due to their high model sensitivity. In particular, for tensor ring (TR) decomposition, the number of model possibilities grows exponentially with the tensor order, which makes it rather challenging to find the optimal TR decomposition. In this paper, by exploiting the low-rank structure of the TR latent space, we propose a novel tensor completion method which is robust to model selection. In contrast to imposing the low-rank constraint on the data space, we introduce nuclear norm regularization on the latent TR factors, resulting in the optimization step using singular value decomposition (SVD) being performed at a much smaller scale. By leveraging the alternating direction method of multipliers (ADMM) scheme, the latent TR factors with optimal rank and the recovered tensor can be obtained simultaneously. Our proposed algorithm is shown to effectively alleviate the burden of TR-rank selection, thereby greatly reducing the computational cost. The extensive experimental results on both synthetic and real-world data demonstrate the superior performance and efficiency of the proposed approach against the state-of-the-art algorithms.

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

“…The TRNNM algorithm is summarized in Algorithm 1. ) respectively, the computational complexity of our method is may much smaller when I ≤ R. In practice, as shown in [3,5], the suitable TR-rank R is always much higher than the dimension I in the high-order form of visual data, which is also found in our experiment. In addition, due to the immense TR-rank selection possibilities, the computational complexity of TR-ALS and TRLRF exponentially increases by using cross validation.…”

“…Low rank tensor completion (LRTC) problem aims to recover the incomplete tensor from the observed entries by assuming different low-rank tensor structures, and it has attracted a lot of attentions in the past decades [1][2][3][4][5]. Most recently, Zhao et al proposed tensor ring decomposition [6], which achieves the state-of-the-art performance in the LRTC problem [3][4][5].…”

“…One of the drawbacks is that the existing TR-based methods are quite time-consuming. For example, TR-ALS [3] has the computational complexity of O(P N R 4 I N + N R 6 ) and the one of tensor ring lowrank factors (TRLRF) [5] equals O(N R 2 I N + N R 6 ), where N denotes the order of the tensor, I denotes the dimension of each mode, R represents the rank of the model and P is a constant between 0 and 1. It can be seen that the computational cost of the two methods increase with the sixth power of TR rank R. It implies that a large TR-rank chosen in practice leads to terribly low efficiency, and sometimes Fig.…”

Provable tensor ring completion

“…This method reduces computational cost but still requires a pre-defined TR rank. By exploiting the low rank structure of the TR latent space, a nuclear norm regularization model is propounded to alleviate the burden of TR rank selection [20], which greatly reduces the computational cost. In [21], a TR nuclear norm minimization model with tensor circular unfolding scheme is proposed for tensor completion.…”

High-order tensor completion via gradient-based optimization under tensor train format

“…There are also tensor completion algorithms which are based on the other tensor decomposition models, i.e., tensor ring (TR) decomposition [23,24] and hierarchical Tucker (HT) decomposition. Based on TR decomposition, works in [25,26,27] propose algorithms named TR-ALS, TR-WOPT and TRLRF which apply ALS, gradient descent and nuclear norm minimization methods to solve various tensor completion problems. Moreover, by total variations (TV) and HT decomposition, [28] proposes a completion algorithm named STTC, which explores the global low-rank tensor structure and the local correlation structure of the data simultaneously.…”