  Symmetric positive definite (SPD) matrices used as feature descriptors in
image recognition are usually high dimensional. Traditional manifold learning
is only applicable for reducing the dimension of high-dimensional vector-form
data. For high-dimensional SPD matrices, directly using manifold learning
algorithms to reduce the dimension of matrix-form data is impossible. The SPD
matrix must first be transformed into a long vector, and then the dimension of
this vector must be reduced. However, this approach breaks the spatial
structure of the SPD matrix space. To overcome this limitation, we propose a
new dimension reduction algorithm on SPD matrix space to transform
high-dimensional SPD matrices into low-dimensional SPD matrices. Our work is
based on the fact that the set of all SPD matrices with the same size has a Lie
group structure, and we aim to transform the manifold learning to the SPD
matrix Lie group. We use the basic idea of the manifold learning algorithm
called locality preserving projection (LPP) to construct the corresponding
Laplacian matrix on the SPD matrix Lie group. Thus, we call our approach
Lie-LPP to emphasize its Lie group character. We present a detailed algorithm
analysis and show through experiments that Lie-LPP achieves effective results
on human action recognition and human face recognition.
