  This work developed novel complex matrix factorization methods for face
recognition; the methods were complex matrix factorization (CMF), sparse
complex matrix factorization (SpaCMF), and graph complex matrix factorization
(GraCMF). After real-valued data are transformed into a complex field, the
complex-valued matrix will be decomposed into two matrices of bases and
coefficients, which are derived from solutions to an optimization problem in a
complex domain. The generated objective function is the real-valued function of
the reconstruction error, which produces a parametric description. Factorizing
the matrix of complex entries directly transformed the constrained optimization
problem into an unconstrained optimization problem. Additionally, a complex
vector space with N dimensions can be regarded as a 2N-dimensional real vector
space. Accordingly, all real analytic properties can be exploited in the
complex field. The ability to exploit these important characteristics motivated
the development herein of a simpler framework that can provide better
recognition results. The effectiveness of this framework will be clearly
elucidated in later sections in this paper.
