Algorithms And Applications Of Nonnegative Matrix And Tensor Factorization

With the advent of the era of big data,high-dimensional data processing technology has been widely studied.Tensors,the extension of matrices to higher-dimensional arrays,are very common in our real life.For example,a color image with an RGB channel can be regarded as a third-order tensor,and a color video stream can be regarded as a fourthorder tensor.The traditional method usually converts the high-dimensional data into lowdimensional array(such as unfolding a tensor to a matrix or a vector)for processing easily,which undoubtedly destroys the spatial structure inside the high-dimensional data and makes the result inaccurate.The utilize of tensors in processing high-dimensional data could ensure that the spatial structure of the data would not be destroyed,thus improving the calculation effect.Since many real data sets have the characteristics of nonnegative and sparse,in this paper,we studied the algorithm model of nonnegative and sparse tensor decomposition.Then,we studied the tensor completion(TC)by using tensor factorization.At last,we applied the TC algorithm to recovery missing real data.The main work of the paper is summarized as follows:A nonnegative tensor factorization method was introduced in this paper,this algorithm is called PGNTF algorithm,and the process of PGNTF is to limit the target tensor to be nonnegative and sparse.We used projected gradient to obtain optimal iteration of every factor matrix.Basing on CP decomposition of the tensor,this method optimized the factor matrices that are obtained by CP decomposition,and made an approximate substitution for the sparse penalty terms.It can be seen from the numerical experiments that the factorization result is better than that of updating the factor matrix of the tensor simultaneously and computing the gradient directly.This article also introduced a tensor completion method based on tensor factorization(be called TFFTC algorithm).In this method,the target tensor is factorized into the product of two smaller tensors.For the description of low tubal rank of the tensor,we not only used the low tubal rank of tensors which are obtained in the factorization,but also added the classical Nuclear norm to describe the low-rank property of the target tensor.Because the computation of Nuclear norm could be transformed into the computation of Frobenius norm in mild conditions,this skill makes the process of solving easily.Numerical experiments showed that the TFFTC algorithm is effective on the recovery of missing images or videos,especially for recovering the images or videos stream with high missing rate.