Riemannian Manifold Metric Learning Based On High-order Statistical Features

With the rapid development of Internet technology and deep learning,data with high-order statistical features plays an important role in more tasks.Compared to the first order statistical features such as mean vector features,high-order statistical feature of the data set,such as the covariance matrix,contains more information about the relations between data samples and is distributed on Riemannian manifold.Traditional metric learning methods mainly focus on the low-dimensional low-order vector features of data samples,and uses the similarity and dissimilarity between sample pairs to perform discriminative independent metrics for different types of samples.However,in recent years,many works have found that compared to traditional metric methods based on vector features of the sample,methods based Riemannian manifold can make better use of the structure information of the manifold.Moreover,the experimental results also show that using the Riemannian manifold can perform metric learning effectively.At present,the widely used Riemannian manifold are mainly distributed in high-dimensional manifold spaces with different expressions.However,most of existing manifold-based metric learning methods are designed for only one specific manifold,and cannot be widely applied.In addition,Support Vector Machine and other methods based on kernel mapping also lack an effective means of directly applying to manifolds.Aiming at the metric learning problem on Riemannian manifolds,two metric learning methods are proposed in this paper.The methods mainly includes towards generalized and efficient metric learning on Riemannian manifold,support vector metric learning on symmetric positive definite manifold.The main contributions and innovative work of this paper are shown as follows:1.A generalized efficient Riemannian manifold metric learning method is proposed.This method can be adopted to some manifolds,like symmetric positive definite manifold,Gaussian manifold and Grassmann manifold.Based on the metric of similar sample pairs,our proposed method measures the metric between the dissimilar sample pairs by introducing the inverse of the metric matrix.In addition,this method has the global closed-form solution by computing matrix geometric mean between inverse of similarity matrix and dissimilarity matrix.2.Aiming at lack of samples per class,based on the kernel-based Support Vector Machine model,a new kind of positive definite kernel for point pairs on symmetric positive definite manifold is defined to transform the multi-classification problem to a point pair classification problem,which can be efficiently solved by standard kernelbased support vector machines.