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Linear discriminant analysis (LDA) might be the most widely used linear feature extraction method in pattern recognition. Based on the analysis on the several limitations of traditional LDA, this paper makes an effort to propose a new computational paradigm named optimal discriminatory projection pursuit (ODPP), which is totally different from the traditional LDA and its variants. Only two simple steps are involved in the proposed ODPP: one is the construction of candidate projection set; the other is the optimal discriminatory projection pursuit. For the former step, candidate projections are generated as the difference vectors between nearest between-class boundary samples with redundancy well-controlled, while the latter is efficiently achieved by classifiability-based AdaBoost learning from the large candidate projection set. We show that the new ldquoprojection pursuitrdquo paradigm not only does not suffer from the limitations of the traditional LDA but also inherits good generalizability from the boundary attribute of candidate projections. Extensive experimental comparisons with LDA and its variants on synthetic and real data sets show that the proposed method consistently has better performances.