Analysis On Interpolation Problem Of Missing High Dimensional Data

With the development of modern science and information technology,the lack of high-dimensional data has become more and more common.In addition,China is about to enter a new 5g era,so we will also be exposed to more and higher-dimensional data.Tensor is the data used to describe high-dimensional structure,which is the main object of this paper.However,in practical application,the data often appears missing,noise and pollution in the transmission process,so the data we get is often incomplete.Therefore,it is necessary to infer the unknown elements according to the known information so as to complete the missing high-dimensional data.The tensor completion studied in this paper is a data science method,which is modern machine learning and data integration Mining,recommendation algorithm,computer vision and many other fields of important research topics,but also the main content of this paper.In Chapter 1,we firstly present the introduction of the research object and influence on the interpolation of high-dimensional missing data,and then propose the method of tensor completion and the research status,as well as the innovation of this paper.In Chapter 2,we expand the tensor completion problem in detail,discuss the decomposition types of the tensor,and introduce the general model of the tensor completion problem.In addition,this paper improves the model,takes the log function as the objective function instead of the rank function,and constructs a nonconvex tensor completion model based on the log function,which can increase the penalty for the smaller singular value and reduce the penalty for the larger singular value,so as to get a more accurate solution.In Chapter 3,the optimal solution of the model is obtained by using DC programming and DC algorithm,and an iterative algorithm,Log-TC algorithm,is proposed to solve the tensor completion problem.In Chapter 4,a numerical experiment is carried out to test the effectiveness of the algorithm,which is based on the tensor generated by artificial random.The algorithm in this paper is compared with the fast low rank tensor completion algorithm(FaLRTC)and the FP-LRTC.Then the algorithm Log-TC is applied to color image restoration to test the effectiveness of the algorithm,which ability to process high dimensional missing data in real life.Chapter 5 summarizes the main work of this paper,puts forward the conclusion and prospects the improvement goal.In this paper,we use the tensor completion model based on log function to solve the interpolation problem of high-dimensional missing data,and finally come to a conclusion.In the experiment of synthesizing random tensors,it is found that for the same size tensors,when their ranks are the same,with the increase of sampling rate,the accuracy of the algorithm is getting better and the time is getting faster.Then when the sampling rate is the same,the problem becomes more and more difficult with the increasing rank,but the accuracy of Log-TC algorithm is still better than the other two algorithms.Secondly,when the tensors are different in size,whether for simple problems or complex problems,from the calculation results,it can be seen that the Log-TC algorithm has the best relative error in restoring the tensors,and the time is the least.Finally,when using the improved model to solve the problem of image restoration,no matter whether the color image is a picture with prominent texture structure or a picture with weak texture structure,Log-TC algorithm can restore with relatively ideal accuracy,which further proves that the new algorithm proposed in this paper is effective.Therefore,the empirical study shows that the algorithm proposed in this paper can be a new choice in the future practical application.