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 only apply to reduce the dimension of high dimensional vector-form data. For high dimensional SPD matrices, it is impossible to directly use manifold learning algorithms to reduce the dimension of matrix-form data, but we need first transform the matrix into a long vector and then reduce the dimension of this vector. This however 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 the high dimensional SPD matrices to lower dimensional SPD matrices. Our work is based on the fact that the set of all SPD matrices with the same size is known to have a Lie group structure and we aims to transform the manifold learning algorithm to SPD matrix Lie group. We make use of the basic idea of manifold learning algorithm LPP (locality preserving projection) to construct the corresponding Laplacian matrix on SPD matrix Lie group. Thus we call our approach Lie-LPP to emphasize its Lie group character. Finally our method gets a lower dimensional and more discriminable SPD matrix Lie group. We also show by experiments that our approach achieves effective results on Human action recognition and Human face recognition.