The Statistic Models Based On Tensor Decompositions And Their Applications

As the higher-order extension of matrices,a tensor can be considered as a member of the tensor product of some vector spaces.There are several kinds of tensor decompositions,including the singular value decomposition of a tensor and the rank-one decomposition of a tensor.They are respectively the higher order extension of the singular value decomposition and the rank-one decomposition of matrices.Tensor representation is a powerful tool for the analysis of high dimensional data since a high order tensor can not only represent the large-scale high dimensional data,but can also preserve the structure of high dimensional data points.In this paper,we first introduce the basic concepts related to tensor,the basic operations on tensors,tensor decomposition algorithms and some special tensors such as symmetric tensors,positive semidefinite(definite)tensor,Hankel tensor and Vandermonde tensor.By Kronecker product of a matrix and Tucker decomposition,we factorize a 3-order tensor into the product of a core tensor and some orthogonal matrices.The low rank approximation of a high order tensor is also investigated.The 2-dim principal component analysis is also employed together with the Tucker decomposition of a tensor to fulfill the 3D tensor feature extractions.Finally under the constraint of the mean square errors,HOSVD,Tucker decompositions,3DPCA and tensor slicing method,we obtain the nonlinear statistical model as an improvement of linear regressions.Some well-known concepts such as PCA,2DPCA,SVD and their applications in matrix case is introduced.Also defined are tensor slicing,singular values of tensors and the applications of tensors in Statistics such as the regression tensor models.The orthogonality of the tensor slices in a core tensor is also verified,and a tensor statistical model is established by Tucker decomposition and HOSVD.Finally we use the tensor model into the recognition of some pictures of handcrafts of some Arabic numbers as 0,1,2,...,9.Our experiment shows that our statistical model based on tensor decomposition is more accurate than the one on the traditional matrix decompositions.