A direct method to Frobenius norm-based matrix regression

Abstract

Regression analysis has been widely used for face recognition. This paper mainly discuss the following regularized matrix regression problem: Given a set of k image matrices $A_1,A_2,\dots ,A_k\in {\mathbf{R}}^{m\times n}$ and an image matrix $B\in {\mathbf{R}}^{m\times n},$ find $x=(x_1,x_2,\dots ,x_k{)}^{\mathrm{T}}$ $\in {\mathbf{R}}^k$ such that $\underset{x\in {\mathbf{R}}^{\mathbf{k}}}{min}\parallel x_1{A}_1+x_2{A}_2+\cdots +x_k{A}_k-B{\parallel }_F^2+{\lambda }^2\parallel x{\parallel }_2^2,$ where $x_1,x_2,\dots ,x_k$ are also a set of representation coefficients, λ is the model parameter, and $\parallel A{\parallel }_F$ represents the Frobenius norm of matrix A.

Yuan and Liao [S.F. Yuan, A.P. Liao, Least squares Hermitian solution of the complex matrix equation AXB + CXD = E with the least norm. J. Frankl. Inst. 351 (2014), pp. 4978–4997] introduced a new product for matrices and vectors, and solved the least squares Hermitian problem of complex matrix equation $AXB+CXD=E.$ In this paper, we deeply investigate this product and its relative properties about matrix trace, norm, and determinant. We then provide a direct method to get the close form solution for solving the regularized image matrix regression problem in face recognition.

Keywords:

• Matrix regression
• least squares problem
• face recognition
• Moore–Penrose generalized inverse

2010 Mathematics Subject Classifications:

• 65F05
• 65F35

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

The research is supported by Guangdong Natural Science Fund of China [grant numbers 2015A030313646, 2018A030313063], and the Innovation Project of Department of Education of Guangdong Province [grant number 2018KTSCX231].