Research On Nonnegative Tensor Ring Decomposition Algorithm With Applications In Feature Extraction And Clustering

Nonnegative tensor decomposition is a multilinear extension of nonnegative matrix factorization.It inherits the advantages of nonnegative matrix factorization which is "the whole based on perception of its parts".Nonnegative tensor decomposition can discover the latent information in various dimensions of the nonnegative data using the form of tensors,performing dimension reduction and feature extraction more efficiently,and has been widely used in computer vision,signal processing,recommendation system and other fields.With rapid increasing of data dimensions,while traditional nonnegative tensor decomposition models and algorithms are theoretically applicable to arbitrary higher-order tensor,the model parameters will increase exponentially under high-order tensors,and the computational cost will increase dramatically.In order to extend the traditional tensor decomposition model to ultra-high-order tensors,tensor networks that represent ultra-high-order tensors as a set of loworder tensors has attracted more and more attentions in recent years.However,there are few studies related to the research of non-negative tensor network which focuses on nonnegative data.In the article,based on the popular Tensor Ring Decomposition(TR)model,we propose a ring-structure nonnegative tensor network,i.e.,nonnegative tensor ring decomposition(NTR)model.The NTR model inherits the better data representation ability of the TR model and the characteristics of avoiding the curse of dimension,and can additionally extract local lowdimensional features of data to improve the performance of features in clustering experiments.We propose four different NTR algorithms.First,the NTR algorithm based on alternating least squares is developed,which is simple and can effectively achieve nonnegative constraints of decomposition parameters,but it is difficult to prove its convergence from a theoretical perspective.Secondly,the NTR algorithm based on the multiplicative update rule is proposed,while its convergence is guaranteed but the iteration speed is relatively slow.Thirdly,the NTR algorithm based on layered least squares is proposed to speed up the iterative process,but its efficiency is still relatively low in high-order tensor processing.By proving that the derivative of the NTR model subproblem is Lipschitz continuous,we demonstrate that the accelerated proximate gradient(APG)method can be used to efficiently optimize the NTR model,hence the NTR algorithm based on the APG optimization method is proposed.We also proposed four variants based on the NTR model,and proved that the model variants can also be efficiently optimized using the APG optimization method.In this way,four variant algorithms are developed,i.e.,sparse nonnegative tensor ring decomposition algorithm which can extract more local features of natural images,smooth nonnegative tensor ring decomposition algorithm that is more suitable for text clustering,graph regularized nonnegative tensor ring decomposition that can learn manifold geometric information of nonnegative ultra-high-order tensor data,and semi-nonnegative tensor ring decomposition that is suitable for processing high-order tensor with negative values.There four variant algorithms greatly expand the application range of the NTR algorithm.The effectiveness of the NTR algorithm has been demonstrated in tensor data feature extraction and unsupervised clustering experiments.As compared with existing algorithms of the same type,the NTR algorithm can extract more local features in tensor data.In clustering experiments,the features extracted by the NTR algorithm are also the most effective.We compared the difference in data representation ability of NTR algorithm and nonnegative tensor network algorithm with similar structure on different order tensor data,which shows that NTR algorithm can more efficiently represent tensor data.Finally,the future research direction of the NTR algorithm is prospected.It is expected that the NTR algorithm will be able to further expand its application range by improving the discriminating ability,robustness,and computational efficiency an so on.It can be expected that NTR algorithm is able to provide a new way for nonnegative tensor data processing in the era of big data.