Unsupervised manifold learning with polynomial mapping on symmetric positive definite matrices

Abstract

In this paper, an unsupervised manifold learning algorithm with polynomial mapping on the symmetric positive-definite (SPD) matrix manifold is introduced by matrix information geometry method for data dimensional reduction. Firstly, the mathematical knowledge about the SPD matrix manifold is presented including the metric, geodesic and submanifold. And then, the high dimensional information coordinates are given by different SPD matrix data for constructing the polynomial kernel matrix, weight matrix and sparsity preserving matrix. Next, the manifold learning algorithm on the SPD matrix manifold is proposed by polynomial mapping with geodesic distance. Finally, comparing with some conventional methods in terms of accuracy rate and time cost, the preliminary analysis results indicate that the proposed approach is able to offer a consistent and comprehensive method to realize the SPD matrix data dimensional reduction.