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.

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.