A direct method to Frobenius norm-based matrix regression
S. J. Yuan, Yibin Yu, Ming‐Zhao Li, Hua Bing Jiang · International Journal of Computer Mathematics · 2019
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,…,Ak∈Rm×n and an image matrix B∈Rm×n, find x=(x1,x2,…,xk)T ∈Rk such that minx∈Rk∥x1A1+x2A2+⋯+xkAk−B∥F2+λ2∥x∥22, where x1,x2,…,xk are also a set of representation coefficients, λ is the model parameter, and ∥A∥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.