Dimensionality reduction by collaborative preserving Fisher discriminant analysis

Sparse representation-based classifier (SRC) and collaborative representation-based classifier (CRC) are two commonly used classifiers. There has been pointed out that the utilization of all the training samples in representing a query sample (i.e. the least square part), which reflects the collaborative representation mechanism of SRC and CRC, is more important than the norm constraint on the coding coefficients for classification. From this perspective, both SRC and CRC can be viewed as collaborative representation (CR) but with different norm (i.e. L1 and L2) constraints on the coding coefficients. In this paper, two collaborative preserving Fisher discriminant analysis approaches are proposed for linear dimensionality reduction, in which both the local geometric information hidden in the CR coefficients and the global discriminant information inherited from Fisher/linear discriminant analysis (FDA/LDA) are effectively fused. Specifically, a datum adaptive graph is first built via CR with L1 or L2 norm constraint (corresponding to L1CPFDA and L2CPFDA, respectively), and then incorporated into the LDA framework to seek a powerful projection subspace with analytic solution. Both theoretical and experimental analysis of L1CPFDA and L2CPFDA show that they can best preserve the collaborative reconstruction relationship of the data and discriminate samples of different classes as well. Moreover, LDA is a special case of L1CPFDA and L2CPFDA and the available number of projection directions of them are twice that of LDA empirically. Experimental results on ORL, AR and FERET face databases and COIL-20 object database demonstrate their effectiveness, especially in low dimensions and small training sample size. (C) 2019 Elsevier B.V. All rights reserved.