Regularization-Based Approaches For Tensor Completion

Firstly,we propose a mixture model for tensor completion by combining the nu-clear norm with the low-rank matrix factorization.To solve this model,we develop two algorithms:non-smooth low-rank tensor completion(NS-LRTC),smooth low-rank tensor completion(S-LRTC).When the sampling rate(SR)is high,our experiments on real-world data show that the NS-LRTC algorithm outperforms other tested methods in running time and recovery quality.In addition,whatever the SR is,the proposed S-LRTC algorithm delivers state-of-art recovery performance compared with other tested approaches.Although the objective function in our model is non-convex and non-differentiable,we prove that every cluster point of the sequence generated by NS-LRTC or S-LRTC is a stationary point.Secondly,we propose a balancing scheme called BS to obtain new matricizations for an unbalanced tensor,and develop an effcient algorithm,which employs the idea of Orthogonal-Matehing-Pursuit,to implement the BS,Then,we propose a new model for tensor completion based on the BS,and deveelop an algorithm called BS-TMae,which is rooted from a well-known algorithm TMac,to solve the proposed model.Finally,we test our algorithms on synthetic and real world data to show the robustness of the BS-based model in reconstruction for unbalanced tensors.The numerical experiments show that BS-TMac outperforms compared methods in recovery quality.Finally,the matrix nuclear norm has been widely applied to approximate the matrix rank for low-rank tensor completion because of its convexity.However,this relaxation makes the solution seriously deviate from the original solution for real-world data recov-ery.In this paper,using a nonconvex approximation of rank,i.e.,the Schatten p-norm,we propose a novel model for tensor completion.It’ s hard to solve this model directly because the objective function of the model is nonconvex.To solve the model,we de-velop a variant of this model via the classical quadric penalty method,and propose an algorithm,i.e.,SpBCD,based on the block coordinate descent method.Although the objective function of the variant is nonconvex,we show that the sequence generated by SpBCD is convergent to a stationary point of the function.Our numerical experiments on real-world data show that SpBCD delivers state-of-art performance in recovering missing data.