arXiv subjects

Guangjing Song

Publications and source records attributed to Guangjing Song.

6 recordsLinked to original sources

arXiv2024

Moore Determinant of Dual Quaternion Hermitian Matrices

In this paper, we extend the Chen and Moore determinants of quaternion Hermitian} matrices to dual quaternion Hermitian matrices. We show the Chen determinant of dual quaternion Hermitian {matrices is invariant under addition, switching, multiplication, and unitary operations at the both hand sides. We then show the Chen and Moore determinants of dual quaternion Hermitian matrices are equal to each other, and they are also equal to the products of eigenvalues. The characteristic polynomial of a dual quaternion Hermitian matrix is also studied.

math.RA↗

arXiv2021

Nonnegative Low Rank Tensor Approximation and its Application to Multi-dimensional Images

The main aim of this paper is to develop a new algorithm for computing nonnegative low rank tensor approximation for nonnegative tensors that arise in many multi-dimensional imaging applications. Nonnegativity is one of the important property as each pixel value refers to nonzero light intensity in image data acquisition. Our approach is different from classical nonnegative tensor factorization (NTF) which requires each factorized matrix and/or tensor to be nonnegative. In this paper, we determine a nonnegative low Tucker rank tensor to approximate a given nonnegative tensor. We propose an alternating projections algorithm for computing such nonnegative low rank tensor approximation, which is referred to as NLRT. The convergence of the proposed manifold projection method is established. Experimental results for synthetic data and multi-dimensional images are presented to demonstrate the performance of NLRT is better than state-of-the-art NTF methods.

cs.CV↗

arXiv2020

Low Rank Pure Quaternion Approximation for Pure Quaternion Matrices

Quaternion matrices are employed successfully in many color image processing applications. In particular, a pure quaternion matrix can be used to represent red, green and blue channels of color images. A low-rank approximation for a pure quaternion matrix can be obtained by using the quaternion singular value decomposition. However, this approximation is not optimal in the sense that the resulting low-rank approximation matrix may not be pure quaternion, i.e., the low-rank matrix contains real component which is not useful for the representation of a color image. The main contribution of this paper is to find an optimal rank-$r$ pure quaternion matrix approximation for a pure quaternion matrix (a color image). Our idea is to use a projection on a low-rank quaternion matrix manifold and a projection on a quaternion matrix with zero real component, and develop an alternating projections algorithm to find such optimal low-rank pure quaternion matrix approximation. The convergence of the projection algorithm can be established by showing that the low-rank quaternion matrix manifold and the zero real component quaternion matrix manifold has a non-trivial intersection point. Numerical examples on synthetic pure quaternion matrices and color images are presented to illustrate the projection algorithm can find optimal low-rank pure quaternion approximation for pure quaternion matrices or color images.

math.NA↗

arXiv2020

Tangent Space Based Alternating Projections for Nonnegative Low Rank Matrix Approximation

In this paper, we develop a new alternating projection method to compute nonnegative low rank matrix approximation for nonnegative matrices. In the nonnegative low rank matrix approximation method, the projection onto the manifold of fixed rank matrices can be expensive as the singular value decomposition is required. We propose to use the tangent space of the point in the manifold to approximate the projection onto the manifold in order to reduce the computational cost. We show that the sequence generated by the alternating projections onto the tangent spaces of the fixed rank matrices manifold and the nonnegative matrix manifold, converge linearly to a point in the intersection of the two manifolds where the convergent point is sufficiently close to optimal solutions. This convergence result based inexact projection onto the manifold is new and is not studied in the literature. Numerical examples in data clustering, pattern recognition and hyperspectral data analysis are given to demonstrate that the performance of the proposed method is better than that of nonnegative matrix factorization methods in terms of computational time and accuracy.

cs.LG↗

arXiv2020

Fast Alternating Projections on Manifolds Based on Tangent Spaces

In this paper, we study alternating projections on nontangential manifolds based on the tangent spaces. The main motivation is that the projection of a point onto a manifold can be computational expensive. We propose to use the tangent space of the point in the manifold to approximate the projection onto the manifold in order to reduce the computational cost. We show that the sequence generated by alternating projections on two nontangential manifolds based on tangent spaces, converges linearly to a point in the intersection of the two manifolds where the convergent point is close to the optimal solution. Numerical examples for nonnegative low rank matrix approximation and low rank image quaternion matrix (color image) approximation, are given to demonstrate that the performance of the proposed method is better than that of the classical alternating projection method in terms of computational time.

math.NA↗

arXiv2019

Robust Tensor Completion Using Transformed Tensor SVD

In this paper, we study robust tensor completion by using transformed tensor singular value decomposition (SVD), which employs unitary transform matrices instead of discrete Fourier transform matrix that is used in the traditional tensor SVD. The main motivation is that a lower tubal rank tensor can be obtained by using other unitary transform matrices than that by using discrete Fourier transform matrix. This would be more effective for robust tensor completion. Experimental results for hyperspectral, video and face datasets have shown that the recovery performance for the robust tensor completion problem by using transformed tensor SVD is better in PSNR than that by using Fourier transform and other robust tensor completion methods.

cs.LG↗