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A direct method to Frobenius norm-based matrix regression

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Pages 1767-1780 | Received 23 Feb 2019, Accepted 19 Aug 2019, Published online: 26 Sep 2019
 

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 A1,A2,,AkRm×n and an image matrix BRm×n, find x=(x1,x2,,xk)T Rk such that minxRkx1A1+x2A2++xkAkBF2+λ2x22, where x1,x2,,xk are also a set of representation coefficients, λ is the model parameter, and AF 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.

2010 Mathematics Subject Classifications:

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].

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