Locality preserving projection on SPD matrix Lie group: algorithm and analysis

摘要

Symmetric positive definite (SPD) matrices used as feature descriptors inimage recognition are usually high dimensional. Traditional manifold learningis only applicable for reducing the dimension of high-dimensional vector-formdata. For high-dimensional SPD matrices, directly using manifold learningalgorithms to reduce the dimension of matrix-form data is impossible. The SPDmatrix must first be transformed into a long vector, and then the dimension ofthis vector must be reduced. However, this approach breaks the spatialstructure of the SPD matrix space. To overcome this limitation, we propose anew dimension reduction algorithm on SPD matrix space to transformhigh-dimensional SPD matrices into low-dimensional SPD matrices. Our work isbased on the fact that the set of all SPD matrices with the same size has a Liegroup structure, and we aim to transform the manifold learning to the SPDmatrix Lie group. We use the basic idea of the manifold learning algorithmcalled locality preserving projection (LPP) to construct the correspondingLaplacian matrix on the SPD matrix Lie group. Thus, we call our approachLie-LPP to emphasize its Lie group character. We present a detailed algorithmanalysis and show through experiments that Lie-LPP achieves effective resultson human action recognition and human face recognition.