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

TL;DR

This paper introduces Lie-LPP, a novel dimension reduction algorithm on SPD matrix Lie groups that preserves spatial structure, improving image recognition tasks like face and action recognition.

Contribution

The paper develops a new dimension reduction method directly on SPD matrix Lie groups, overcoming limitations of traditional vector-based manifold learning.

Findings

Lie-LPP effectively reduces SPD matrix dimensions in recognition tasks.

Experimental results show improved accuracy in face and human action recognition.

The method preserves the spatial structure of SPD matrices during reduction.

Abstract

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.