Low Rank Approximation Of Matrices And Tensors With Applications

The emergence of high-dimensional data provides a large number of potential resources for the acquisition of information.Also,it presents a greater challenge to data mining technology.The expression of data is very important to information mining.The traditional form of data expression is vectors at present.Although its form is simple and easy to operate,it is often difficult to describe the complex structure of high-dimensional data in an all-round way.In contrast,the matrices and tensors,as the higher-order extension of vectors,can better maintain the internal information of the data.It provides convenience for the highdimensional data analysis and mining.Especially,in recent years,low rank approximation theory of matrices and tensors play a more and more important role in high-dimensional data analysis.And they have become important tools for missing data completion and high-order correlation learning.However,how to apply many low rank algorithms to more practical problems and test their effectiveness,so as to improve the existing algorithms or design new algorithms? It is still an open problem to be thoroughly studied.And it is also the focus of this paper.To this end,this paper mainly focuses on the missing exam scores completion and brain functional network construction etc.to study low-rank learning.The main contributions are as follows:1)We propose a new method for completing missing exam scores,which improves the accuracy of the traditional method and the low rank method.For the problem of missing score completion,through a comparative study of some mainstream methods,we find that although low rank approximation has been successfully applied in many data completion tasks,it cannot work well in the exam score dataset.Therefore,we propose a new method to complete exam scores.By effectively using the dual correlation of subjects and students,we explicitly model the structural information in the exam scores data.The best estimation results are obtained on two real score datasets.2)Based on the low rank approximation theory of higher-order tensors,a new estimation approach of brain functional network is proposed.Considering that the brain functional network has similar but not necessarily identical topology,a two-step learning framework is used to construct the brain network.Firstly,according to the traditional method,Pearson's correlation and sparse representation,are used to independently estimate each individual brain network,which ensures that each brain network can capture the specific properties of the corresponding subjects.Then,the estimated brain network(matrix form)of all individuals is stacked into a third-order tensor.And the topology of each brain network is made as similar as possible by low rank tensor approximation.Finally,the improved brain network is applied to identify subjects with mild cognitive impairment.And the accuracy is significantly improved compared with the traditional methods.