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Theoretical Analysis of Direct LDA under Small Sample Conditions

2013-09-19
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Direct Linear Discriminant Analysis (DLDA) is an extended LDA method proposed to overcome the small sample problem. It is claimed to utilize all information outside the null space containing the intra-class scatter matrix. However, many counterexamples show that this is not the case. In order to gain a deeper understanding of the characteristics of DLDA, this paper analyzes it theoretically and concludes that DLDA based on the traditional Fisher criterion hardly utilizes the null space and will lose some useful discriminant information; while DLDA based on the generalized Fisher criterion is equivalent to null space LDA and orthogonal LDA if it meets certain conditions (generally satisfied in high-dimensional small sample data applications) and the optimal discriminant vector is orthogonal. The comparative experimental results on the face databases ORL and YALE are also consistent with the theoretical analysis.

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