Analyzing facial expressions with fuzzy quantification theory II: indefinite generalized eigenvalue problem

Machine understanding of human interactive behaviour is a challenging task. In this paper, facial expressions are used to induce emotions on a group of subjects and sense fuzzy human interactive behaviour. When fuzzy quantification theory II is used to analyze the emotions induced by facial expressions, the relationship between induced emotions and facial features is linearized by solving a dense generalized eigenvalue problem. As the matrices are ill-conditioned and indefinite when small data set of facial expressions is used, the theory describing the possible solutions of the eigenvalue problem gets complicated. This paper proposes a perturbation of the eigenvalue problem by using the Moore–Penrose pseudoinverse that keeps the error bounds of the solution within the error bounds of the widely used GUPTRI algorithm. Experimental results show that GUPTRI algorithm fails in some cases where the number of samples is very small while the solution provided by proposed approach never fails and is more accurate.