Tensor Decomposition And Its Applications Based On Graph And Low Rank Representation

We know that most of the existing high-dimensional image and video data has a natural tensor structure, or can be organized into a tensor structure. Moreover, the tensor structure representation possesses good presentation skills and calculation features, thus on the basis of summarizing and inheriting the predecessors’ research results. This thesis studies the related algorithms based on tensor. The main contents are as follows:(1) Initialization method has been proposed for support tensor machine classifier. We point out that the disadvantages of the initialization method for the non-negative matrix dimensionality reduction method of traditional support tensor machine via randomization. On one hand, in the absence of the data, it needs an assumption of the data distribution, such as Gaussian distribution, uniform distribution, etc; it is also not easy to estimate the parameters of the distribution only by repeatedly veritification on test data. On the other hand, it is difficult to capture the characteristics of image itself by using the method of random initialization. Therefore, the random way will ultimately affect the classification results of the classifiers and the performances of the dimension reduction results.In order to address these two problems of random initialization method, this paper put forwards the image content based initialization method and makes use of the image content to initialize the support tensor machine and the non-negative matrix decomposition method. First of all, the raw data is processed by support tensor machine into tensor form. Specifically, each image is represented by a third-order tensor structure, and the collections of images become a fourth-order tensor. Secondly, this paper proposed a weighted higher order singular value decomposition algorithm for support tensor machine initialization. This initialization method combines graph theory with manifold learning algorithm, and initializes the support tensor machine by the image data set to avoid the influence caused by the random initialization. Moreover, in terms of subspace dimension reduction method, this paper adopts the non-negative matrix decomposition method for third-order image characteristics tensor dimensionality reduction, and proposes to initialize the non-negative matrix decomposition method via the two-dimensional principal component analysis method, which makes full use of the correlated information of the collections of image contents. At last, the dimensionality of the input data of support tensor machine is reduced by the method of the improved non-negative matrix decomposition algorithm. After that, support tensor machine has been trained in the subspace and the image classification has been performed by a dimension reduction method with the improved classifier of support tensor machine. Experimental results show that the classification results are better compared with other algorithms.(2) Non-negative tensor decomposition based on graph and low-rank representation has been proposed. We point out that, in the field of image processing, if the non-negative matrix decomposition method is adopted, each image is required to be straightened into a vector form. In the procedure of conversion, the structural information of the image content is lost and the space geometry structure of image is damaged. In order to avoid these problems, two improvement methods of the non-negative tensor decomposition algorithm have been proposed, and the two subspace dimension reduction methods have been used for image classification. At first, we put forward the new non-negative tensor decomposition algorithm based on graph. Based on the non-negative tensor decomposition algorithm for graph, the non-negative tensor decomposition algorithm has been further expanded. By learning lessons from graph theory and the advantages of manifold learning algorithm, we introduce the structural information of data sets into the non-negative tensor decomposition algorithm. Then, considering the construction of neighborhood graph for big data consuming too much time on the calculation, this paper presents a non-negative tensor decomposition algorithm based on low-rank representation. As the extension and the development of compressed sensing theory, the low-rank representation denotes that the rank of the matrix can be used as a measurement of sparsity. Since the rank of a matrix reflects the inherent property of the matrix, the low-rank analysis can effectively analyze and process the matrix data. This paper introduces the low-rank representation into tensor model, namely to introduce it into non-negative tensor decomposition algorithm and to further expand the non-negative tensor decomposition algorithm. Experimental results show that the classification accuracy of the two algorithms proposed in this paper is better compared to other existing algorithms.(3) A multistage nonnegative low-rank sparse matrix decomposition algorithm based on high-order singular value decomposition has been proposed. Firstly, the details of the calculation method for the low-rank sparse matrix decomposition algorithm are introduced. Then, the importance of the data arrangement for the image sequence is pointed out, as well as the impact to the data sorting after using high-order singular value decomposition. The method, which is to combine the high-order singular value decomposition and low-rank sparse matrix decomposition, has been introduced. It further manifests that the video image sequence data can be separated by low-rank of tensor representation into foreground and background. Inspired by these, we proposed a higher order singular value decomposition method via a multistage nonnegative low-rank sparse matrix decomposition algorithm. In order to ensure that the characteristics of the video image sequence data are not undermined, and to implement that the pure additive description of the original video data, this method introduces the non-negative constraints and decomposes the data information into time and space information at each step. It is worth mentioning that, because of this gradual decomposition method, the non-negative constraints are particularly of great importance. Secondary or higher decomposition process of low-rank matrix is only for sparse matrix and the decomposition results of the sparse matrix is corresponding to the time information (motion information), and the result of the low-rank matrix is the space information (background information). From the experiments of two image sequences, they show the promising results of the proposed method in this part for extracting the foreground and background information.