Research On The Hierarchical Tensor Decomposition Algorithm

Tensor decomposition is widely used in image and data mining.Existing tensor decomposition algorithms,such as CANDECOMP/PARAFAC(CP)decomposition algorithm and Tucker decomposition algorithm,have been well applied,but there are still some problems that have not been solved.For example,when the CP decomposition algorithm is applied to feature extraction,if the user chooses an inappropriate rank,it will lead to inaccurate feature extraction.However,there is no good way to choose the right rank at present.The problem with the Tucker decomposition algorithm is that the decomposition results are not unique,which may confuse users.Therefore,this paper aims to propose a simple,effective and unique method of tensor decomposition-hierarchical tensor decomposition.Centering on the study of the hierarchical tensor decomposition and its application,the following aspects are studied in this thesis:1)A framework for Hierarchical Tensor Decomposition(HTD)is proposed.This algorithm introduces low-rank decomposition into tensor decomposition structure.The decomposition result is unique,the structure is simple,and it can be applied to the fusion of various low-rank decomposition.2)In the framework of hierarchical tensor decomposition,a hierarchical singular value decomposition(HSVD)method based on tensor decomposition is proposed.A large number of experimental results show that HSVD can effectively decompose and reconstruct tensors with a small minimum mean square error.Compared with the CP decomposition,HSVD is less affected by data rank.In addition,HSVD can obtain similar features to the Tucker decomposition algorithm,but the time complexity of HSVD is much lower.HSVD is not sensitive to data shuffling.That is,when the order of data points in one dimension changes,the decomposed features between other dimensions are not affected.The results also demonstrate that HSVD performs well on real-world dataset with significantly different expressions and ORL dataset.3)In the framework of hierarchical tensor decomposition,a hierarchical nonnegative matrix factorization(HNMF)method based on tensor decomposition is proposed.Experiments show that the HNMF algorithm has a better denoising effect than the CP decomposition algorithm,and can extract more data features.4)Moreover,two kinds of synthetic data,a spot data following the Gaussian distribution and a grid data with obvious visual characteristics,are designed for experiments.These two kinds of data not only have the generality of real data,but also can clearly see some laws and characteristics of experimental results visually.