Tensor Decompositions With Auxiliary Information And Applications

Nowadays,the data we acquired often have multi-dimensional structure.For example,a grayscale video clip has the row-column-frame structure.It is a hot and challenging topic that how to process the multi-dimensional data effectively and mine the information.Tensor decompositions are higher-order generalizations of the matrix singular value decomposition.They are potential tools for multi-dimensional data processing since they can consider the structural information of the multi-dimensional data simultaneously along every modes.In recent years,theories and algorithms of tensor decompositions have been developing rapidly as well as their applications.But there are still many problems worth studying.In fact,many types of multi-dimensional data come with auxiliary information.For example,the local geometrical structure between different spectral bands of multispectral image,the user's age and item specifications in the shopping score data,and so on.However,most of the existing tensor methods do not consider the use of such auxiliary information.Therefore,this thesis studies the method of using auxiliary information in the existing tensor methods.We hope to improve the performance of the model by using auxiliary information reasonably.Our main results are as follows.1.This thesis proposes a sparse and graph-Laplacian Tucker decomposition model,and gives an algorithm to solve the model.For the model with auxiliary information,we use weighted graphs to characterize the intrinsic local geometrical structure.The graph-Laplacian regularization is introduced to make similar data have similar low dimensional representation.That is,to reduce the dimensions of data while preserving the structure.2.Based on the Tucker decomposition,a graph-Laplacian regularized tensor robust principal component analysis is proposed.It takes into account both the sparsity of the core tensor and the local geometrical structure of modes with auxiliary information.Furthermore,an algorithm using the alternating direction method of multipliers method is proposed to solve the model.3.We apply the proposed tensor decomposition methods with auxiliary information to multispectral image denoising,and video background modeling and subtraction,respectively.Many experiments and analysis are carried out.Since the native structure and auxiliary information are considered simultaneously,our methods outperform the state-of-the-arts,both in quantitative and qualitative aspects.These results not only verify the effectiveness of the proposed method,but also provide some evidences for the idea that problem solving can be improved by the reasonable use of auxiliary information.