Local Discriminant Embedding (LDE) was recently proposed to overcome
some limitations of the global Linear Discriminant Analysis (LDA) method. Whenever
a small training data set is used, LDE cannot directly be applied to high-dimensional
data. This case is the so-called small-sample-size (SSS) problem. The classic solution
to this problem was applying dimensionality reduction on the raw data (e.g., using
Principal Component Analysis (PCA)). This chapter introduces a novel discriminant
technique called “Exponential Local Discriminant Embedding” (ELDE). The proposed
ELDE can be seen as an extension of LDE framework in two directions. Firstly, the
proposed framework overcomes the SSS problem without discarding the discriminant
information that was contained in the null space of the locality preserving scatter
matrices associated with LDE. Secondly, the proposed ELDE is equivalent to
transforming original data into a new space by distance diffusion mapping (similar to
Kernel-based non-linear mapping), and then, LDE is applied in such a new space. As a
result of diffusion mapping, the margin between samples belonging to different classes
is enlarged, which is helpful in improving classification accuracy. The experiments are
conducted on four public face databases, Extended Yale, PF01, PIE and FERET. The
results show that the performances of the proposed ELDE are better than those of LDE
and many state-of-the-art discriminant analysis techniques.
Keywords: Complete Kernel Fisher discriminant method, Distance diffusion
mapping, Distance metric learning, Exponential discriminant analysis,
Exponential locality preserving projections, Face recognition, Feature extraction,
Generalized eigenvectors, Intrinsic graph, Kernel Fisher discriminant analysis,
Kernel Principal component analysis, Linear discriminant analysis, Local
discriminant embedding, Matrix exponential, Penalty graph, Principal component analysis, Regularization, Regularized Kernel discriminant analysis, Singular
matrix, Small sample size problem.
