This work presents a new approach which derives a learned data representation method through matrix factorization on the complex domain. In particular, we introduce an encoding matrix-a new representation of data-that satisfies the simplicial constraint of the projective basis matrix on the field of complex numbers. A complex optimization framework is provided. It employs the gradient descent method and computes the derivative of the cost function based on Wirtinger's calculus.

In this work, two new proposed NMF models are developed for facial expression recognition. They are called maximum volume constrained nonnegative matrix factorization (MV_NMF) and maximum volume constrained graph nonnegative matrix factorization (MV_GNMF). They achieve sparseness from a larger simplicial cone constraint and the extracted features preserve the topological structure of the original images.