Equivalence between LDA/QR and direct LDA

Singularity problems of scatter matrices in Linear Discriminant Analysis (LDA) are challenging and have obtained attention during the last decade. Linear Discriminant Analysis via QR decomposition (LDA/QR) and Direct Linear Discriminant analysis (DLDA) are two popular algorithms to solve the singularity problem. This paper establishes the equivalent relationship between LDA/QR and DLDA. They can be regarded as special cases of pseudo-inverse LDA. Similar to LDA/QR algorithm, DLDA can also be considered as a two-stage LDA method. Interestingly, the first stage of DLDA can act as a dimension reduction algorithm. The experiment compares LDA/QR and DLDA algorithms in terms of classification accuracy, computational complexity on several benchmark datasets and compares their first stages. The results confirm the established equivalent relationship and verify their capabilities in dimension reduction.