Geometry Structure On Positive-definite Matrix Manifold And Algorithms Research

The main purpose of this paper is to study the geometric structures and algorithms on positive-definite matrix manifolds.First,with the help of the geometric structure on the manifold of positive definite matrix,the Riemann gradient descent algorithm and the logarithmic gradient descent algorithm are respectively proposed to calculate the numerical solutions of the Stein matrix equation and a class of linear matrix equations.Secondly,the iterative algorithm of the arithmetic mean matrix derived from the Euclidean metric on the positive-definite matrix manifold and the Riemann mean matrix derived from the affine invariant metric is introduced,and the geometric algorithm of the Riemannian mean matrix based on the log-Euclidean metric is given.A non-iterative method for calculating the geodesic distance between a positive-definite matrix and a mean matrix is proposed.Finally,a point cloud denoising algorithm based on the geometric structure of the positive-definite matrix manifold is proposed.Finally,a point cloud denoising algorithm based on the geometric structure of the positive definite matrix manifold is proposed,and the denoising effect of the algorithm is described by the precision,recall and accuracy.According to the research content and method,the full text is divided into four chapters.The first chapter summarizes the classical contents and applications of information geometry,and introduces related concepts such as positive-definite matrix manifold geometry and gradient algorithm,as well as the research content and main results of this paper.Chapter 2 presents the Riemannian gradient descent algorithm and the logarithmic gradient descent algorithm for computing the numerical solutions of the Stein equation and a class of linear matrix equations.Based on the Riemannian geometry of positive-definite matrix manifolds,the affine invariant metrics on the Riemannian manifold and the geodesic distance induced by the log-Euclidean metric are used as the objective function,and the Riemann gradient of the two objective functions are calculated.Finally,numerical examples show that the logarithmic gradient descent algorithm is more computationally efficient than the Riemannian gradient descent algorithm.Chapter 3 summarizes the geometric algorithms of the mean matrix derived from Euclidean metric,affine invariant metric and log-Euclidean metric for a given number of symmetric positive-definite matrices,and proposes a method to calculate the difference between the positive-definite matrix and the mean matrix.Finally,numerical simulation is used to compare the difference of the mean matrix under different metrics and the effectiveness of the non-iterative method for the difference of means.Chapter 4 presents a point cloud denoising algorithm based on positive-definite matrix manifold geometry.Considering the difference of local statistical properties between noise and valid data,the original data point cloud is mapped into the normal distribution family manifold,which becomes the parametric point cloud.The Euclidean metric and the logarithmic Euclidean metric are respectively assigned to the normal distribution family manifold,and the K-means clustering algorithm is applied to cluster the parameter point cloud,so as to classify the original point cloud corresponding to the parameter point cloud.Finally,denoising experiments are carried out on the Teapot point cloud data with high density noise.