Tensor p-shrinkage nuclear norm for low-rank tensor completion

Abstract

In this paper, a new definition of tensor p-shrinkage nuclear norm (p-TNN) is proposed based on tensor singular value decomposition (t-SVD). In particular, it can be proved that p-TNN is a better approximation of the tensor average rank than the tensor nuclear norm when −∞ < p < 1. Therefore, by employing the p-shrinkage nuclear norm, a novel low-rank tensor completion (LRTC) model is proposed to estimate a tensor from its partial observations. Statistically, the upper bound of recovery error is provided for the LRTC model. Furthermore, an efficient algorithm, accelerated by the adaptive momentum scheme, is developed to solve the resulting nonconvex optimization problem. It can be further guaranteed that the algorithm enjoys a global convergence rate under the smoothness assumption. Numerical experiments conducted on both synthetic and real-world data sets verify our results and demonstrate the superiority of our p-TNN in LRTC problems over several state-of-the-art methods.

Individual traffic prediction in cellular networks based on tensor completion

“…Utilizing the TC technique, 25,26 we simultaneously recover and decompose the incomplete individual traffic data as shown in Figure 4. Generally, we consider the TC problems of the form:…”

“…Various low-rank regularizers based on Tucker decomposition 28,30 and tensor-train decomposition 29 have been previously proposed to approximate the tensor rank; however, these methods involve the unfolding operation of the tensor, which will not only destroy the internal structure of the tensor but also increase the computational complexity. Referring to our previous work, 25 without loss of generality, let gðSÞ ¼ kSk 1 and rðLÞ ¼ kLk p , where k Á k p denotes the p-shrinkage tensor nuclear norm (p-TNN). Consequently, model (3) can be rewritten as follows:…”

“…Motivated by these challenges, in this paper, we investigate the problem of individual traffic prediction with incomplete data in cellular networks. Different from the "three-step" strategy, we propose a novel "two-step" prediction strategy based on tensor completion (TC) theory, [25][26][27][28][29][30] which can estimate the missing data and decompose the recovered per-user traffic data into two components simultaneously. Compared with the "three-step" scheme, the proposed "two-step" strategy reduces the risk of amplifying the approximation error.…”

Data Collection for Mobile Crowd Sensing Based on Tensor Completion

“…The original data can be reconstructed by extracting the Ndimensional data sequence from the measurement L=Ψc. The reconstruction can be completed by solving (10), that is, the optimization problem (11):…”

“…Compared with the various sensing methods of traditional networks, mobile crowd sensing boasts huge data scale, low cost, and high sampling speed, and has broad application prospects in many fields, namely, public security, intelligent transport, environmental monitoring, social recommendation, and public facility management [10][11][12][13]. Capponi et al [14] developed an iOS app for monitoring the water quality of rivers, which allows users to upload photos on the states (e.g.…”