Locality preserving projection on SPD matrix Lie group: algorithm and analysis
This work addresses a domain-specific limitation in computer vision by enabling direct dimensionality reduction on SPD matrices, which is incremental as it adapts an existing manifold learning method to a new mathematical structure.
The authors tackled the problem of reducing the dimensionality of symmetric positive definite (SPD) matrices used in image recognition without breaking their spatial structure, by proposing Lie-LPP, a dimension reduction algorithm on the SPD matrix Lie group, which achieved effective results in human action and face recognition experiments.
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.