Semi-definite embedding (SDE) has been a recently proposed to maximize the sum of pair wise squared distances between outputs while the input data and outputs are locally isometric, i.e. it pulls the outputs as far apart as possible, subject to unfolding a manifold without any furling or fold for unsupervised nonlinear dimensionality reduction. The extensions of SDE to supervised feature extraction, named as supervised Semi-definite embedding (SSDE) was proposed by the authors of this paper. Here, the method is unified in a mathematical framework and applied to a number of benchmark data sets. Results show that SSDE performs very well on high-dimensional data, which exhibits a manifold structure.