Relaxed 2-D Principal Component Analysis by LpL_p Norm for Face Recognition

A relaxed two dimensional principal component analysis (R2DPCA) approach is proposed for face recognition. Different to the 2DPCA, 2DPCA-$L_1$ and G2DPCA, the R2DPCA utilizes the label information (if known) of training samples to calculate a relaxation vector and presents a weight to each subset of training data. A new relaxed scatter matrix is defined and the computed projection axes are able to increase the accuracy of face recognition. The optimal $L_p$-norms are selected in a reasonable range. Numerical experiments on practical face databased indicate that the R2DPCA has high generalization ability and can achieve a higher recognition rate than state-of-the-art methods.Comment: 19 pages, 11 figure

A structure-preserving one-sided Jacobi method for computing the SVD of a quaternion matrix

Abstract(#br)In this paper, we propose a structure-preserving one-sided cyclic Jacobi method for computing the singular value decomposition of a quaternion matrix. In our method, the columns of the quaternion matrix are orthogonalized in pairs by using a sequence of orthogonal JRS-symplectic Jacobi matrices to its real counterpart. We establish the quadratic convergence of our method specially. We also give some numerical examples to illustrate the effectiveness of the proposed method

A Two-Sided Quaternion Higher-Order Singular Value Decomposition

Higher-order singular value decomposition (HOSVD) is one of the most celebrated tensor decompositions that generalizes matrix SVD to higher-order tensors. It was recently extended to the quaternion domain \cite{miao2023quat} (we refer to it as L-QHOSVD in this work). However, due to the non-commutativity of quaternion multiplications, L-QHOSVD is not consistent with matrix SVD when the order of the quaternion tensor reduces to $2$; moreover, theoretical guaranteed truncated L-QHOSVD was not investigated. To derive a more natural higher-order generalization of the quaternion matrix SVD, we first utilize the feature that left and right multiplications of quaternions are inconsistent to define left and right quaternion tensor unfoldings and left and right mode-$k$ products. Then, by using these basic tools, we propose a two-sided quaternion higher-order singular value decomposition (TS-QHOSVD). TS-QHOSVD has the following two main features: 1) it computes two factor matrices at a time from SVDs of left and right unfoldings, inheriting certain parallel properties of the original HOSVD; 2) it is consistent with matrix SVD when the order of the tensor is $2$. In addition, we study truncated TS-QHOSVD and establish its error bound measured by the tail energy; correspondingly, we also present truncated L-QHOSVD and its error bound. Deriving the error bounds is nontrivial, as the proofs are more complicated than their real counterparts, again due to the non-commutativity of quaternion multiplications. %Numerical experiments on synthetic and color video data show the efficacy of the proposed TS-QHOSVD. Finally, we illustrate the derived properties of TS-QHOSVD and its efficacy via some numerical examples