  We propose a novel algorithm for supervised dimensionality reduction named
Manifold Partition Discriminant Analysis (MPDA). It aims to find a linear
embedding space where the within-class similarity is achieved along the
direction that is consistent with the local variation of the data manifold,
while nearby data belonging to different classes are well separated. By
partitioning the data manifold into a number of linear subspaces and utilizing
the first-order Taylor expansion, MPDA explicitly parameterizes the connections
of tangent spaces and represents the data manifold in a piecewise manner. While
graph Laplacian methods capture only the pairwise interaction between data
points, our method capture both pairwise and higher order interactions (using
regional consistency) between data points. This manifold representation can
help to improve the measure of within-class similarity, which further leads to
improved performance of dimensionality reduction. Experimental results on
multiple real-world data sets demonstrate the effectiveness of the proposed
method.
