A fuzzy supervised learning method with dynamical parameter estimation for nonlinear discriminant analysis

In this paper, we develop a novel fuzzy supervised learning algorithm based on the dynamical parameter estimation. First, a reformative supervised fuzzy LDA algorithm (RF-LDA) for the training samples is proposed. Compared with the conventional fuzzy LDA algorithm, the presented algorithm computes the discriminant vectors associated with the membership grade from each training sample, which is theoretically effective in overcoming the classification limitation originating from the imprecise samples. Second, when a novel fuzzy LDA model is required in order to take some decision about the feature extraction and classification, the dynamical parameter estimation method of the fuzzy LDA model should recursively process the measured data as they become available. In the line of previous arguments, we approach the problem of control parameter estimation of RF-LDA by considering the formulation of a Hopfield Neural Network (HNN), which is named HRF-LDA. Third, considering the fact that the Kernel Fisher Discriminant (KFD) is effective in extracting the nonlinear discriminative information of the feature space by using kernel trick, a kernel version of HRF-LDA is presented subsequently, which has the potential to outperform the traditional fuzzy learning algorithms, especially in the cases of nonlinear small sample sizes. The advantage of this learning algorithm is that it successfully utilizes the improved kernel fuzzy LDA algorithm as a supervised feature extraction tool. Meanwhile, by means of the control parameter estimation, we address the problem that the particular value of offset in the calculation of the grade of membership is dynamically assigned. Extensive experimental studies conducted on the ORL, NUST603, FERET, Yale and XM2VTS face images show the effectiveness of the proposed fuzzy integrated algorithm.