A New Tensor Factorization Based on the Discrete Simplified Fractional Fourier Transform

Tensor analysis approaches are of great importance in various fields such as computa-tion vision and signal processing.Thereinto,the definitions of tensor-tensor product(t-product)and tensor singular value decomposition(t-SVD)are significant in practice.This work presents new t-product and t-SVD definitions based on the discrete simplified fractional Fourier transform(DSFRFT).The proposed definitions can effectively deal with special complex tenors,which fur-ther motivates the transform based tensor analysis approaches.Then,we define a new tensor nucle-ar norm induced by the DSFRFT based t-SVD.In addition,we analyze the computational complex-ity of the proposed t-SVD,which indicates that the proposed t-SVD can improve the computation-al efficiency.

Identification of Secondary Dislocations by Singular Value Decomposition of the Nye Tensor

Recently, we proposed the use of singular values and singular vectors of the Nye tensor to specify the features of dislocations. To further validate the method in identifying dislocations, the algorithm is applied to detect secondary dislocations in this paper. Different from the perfect and partial dislocations where the reference state is a single crystal, the reference state for detecting secondary dislocations should be a bi-crystal in a coincident coherent state. It is demonstrated that the secondary dislocations in an interface can be correctly identified when the reference state is appropriately selected.

FAST ALGORITHMS FOR HIGHER-ORDER SINGULAR VALUE DECOMPOSITION FROM INCOMPLETE DATA

Higher-order singular value decomposition （HOSVD） is an efficient way for data reduction and also eliciting intrinsic structure of multi-dimensional array data. It has been used in many applications, and some of them involve incomplete data. To obtain HOSVD of the data with missing values, one can first impute the missing entries through a certain tensor completion method and then perform HOSVD to the reconstructed data. However, the two-step procedure can be inefficient and does not make reliable decomposition. In this paper, we formulate an incomplete HOSVD problem and combine the two steps into solving a single optimization problem, which simultaneously achieves imputation of missing values and also tensor decomposition. We also present one algorithm for solving the problem based on block coordinate update （BCU）. Global convergence of the algorithm is shown under mild assumptions and implies that of the popular higher-order orthogonality iteration （HOOI） method, and thus we, for the first time, give global convergence of HOOI. In addition, we compare the proposed method to state-of-the-art ones for solving incom- plete HOSVD and also low-rank tensor completion problems and demonstrate the superior performance of our method over other compared ones. Furthermore, we apply it to face recognition and MRI image reconstruction to show its practical performance.

High-Order Supervised Discriminant Analysis for Visual Data

In practical applications, we often have to deal with high-order data, for example, a grayscale image and a video clip are intrinsically a 2nd-order tensor and a 3rd-order tensor, respectively. In order to satisty these high-order data, it is conventional to vectorize these data in advance, which often destroys the intrinsic structures of the data and includes the curse of dimensionality. For this reason, we consider the problem of high-order data representation and classification, and propose a tensor based fisher discriminant analysis （FDA）, which is a generalized version of FDA, named as GFDA. Experimental results show our GFDA outperforms the existing methods, such as the 2-directional 2-dimensional principal component analysis （（2D）2pCA）, 2-directional 2-dimensional linear discriminant analysis （（2D）2LDA）, and multilinear discriminant analysis （MDA）, in high-order data classification under a lower compression ratio.