Locally preserving projection on symmetric positive definite matrix lie group

Symmetric Positive Definite (SPD) matrices have been widely used as feature descriptors in image recognition. However, the dimension of an SPD matrix built by image feature descriptors is usually high. So SPD matrices oriented dimensionality reduction techniques are needed. The existing manifold learning algorithms just reduce the dimension of high dimensional vector-form data. Our work is based on the fact that the set of all SPD matrices is known to have a Lie group structure. Our method aims to transform the manifold learning algorithm to SPD matrix Lie group. We first construct the corresponding Laplacian matrix on SPD matrix Lie group, then make use of the basic idea of manifold learning algorithm LPP (locality preserving projection). Thus we call our approach Lie-LPP to emphasize its Lie group character. We also do some experiments on human face recognition and achieve effective results.