Research On Algorithm And Application Of Matrix Completion

In many machine learning applications,data can be represented by a matrix X ? Rm*n.In some cases,some entries are missing or unavailable,which results in only partial entries being observed.The matrix completion problem aims to study how to recover the unknown entries of the matrix from the known entries of the low-rank or approximately low-rank matrix reasonably and accurately.In recent years,the algorithms and theories of the problem have been developed rapidly,especially in image processing,recommender system and other fields have significantly practical value.Although a lot of classical algorithms can be applied to solve the matrix completion problem,the accuracy is often not ideal when processing real datasets.In this thesis,we propose a systematic study of matrix completion algorithms from the theoretical analysis to specific applications.In summary,the contribution of this thesis is as follows:(1)We propose a high accuracy algorithm,called matrix completion via truncated schatten p-norm regularization,which combines the advantages of truncated nuclear norm and schatten p-norm to improve the flexibility of the nuclear norm.We further apply the alternating direction method of multipliers to solve the optimization problem.Finally,we use the proposed algorithm for image inpainting and conduct a series of experiments on real visual datasets.The experimental results have demonstrated the good performance of the proposed algorithm.(2)We propose a score similarity based matrix factorization recommendation algorithm with group sparsity(SSMF-GS)for exploiting the group structure of the rating data.Firstly,according to users' rating behavior,the rating data matrix was divided into groups,similar users' rating matrixes were obtained simultaneously;Then,SSMF-GS for similar users' rating matrixes was used to matrix decomposition with group sparsity;Finally,the alternating optimization algorithm was employed to solve the proposed model.The model could select the favorite item latent factors of different user groups out,which improved the interpretation of latent factors.We test the proposed method on MovieLens datasets provided by GroupLens websites.The experimental results show that the proposed method can improve recommendation accuracy significantly,the Mean Absolute Error(MAE)and Root Mean Squared Error(RMSE)both have good performance.