Geodesics And Geodesic Flow Kernel On Grassmann Manifold

Grassmann manifold is formed by arbitrarily taking the same low dimensional linear subspace from the high-dimensional vector space and endowing it with topological structure and differential structure.This is an abstract but classical differentiable manifold,which is a natural generalization of real projective space.It has important significance both in theory and in practical application.For example,in manifold learning,high-dimensional data is compressed into low-dimensional data,and the geodesic on Grassmann manifold will be used when measuring distance.This paper focuses on the Grassmann manifold as the research object,aiming to reveal the concept and theory of the manifold,as well as its closely related applications.The explicit description of geodesic equation on Grassmann manifold is derived from three different angles,and the theoretical analysis of geodesic flow kernel method is given.Firstly,considering the real projective space is a special case of Grassmann manifold.The quotient space structure of real projective space is expounded.By constructing equivalence relation,equivalence class is obtained,and then quotient set is obtained.Thus,quotient topological structure and differential structure of Grassmann manifold are described further.By exploring the geometric definition of the Grassmann manifold and related properties,it’s deduced that there is the locally unique geodesics on Grassmann manifold in view of the standard inner product,and its explicit parametric representation and geodesic distance are given.Secondly,it should be noted that every point on the Grassmann manifold is an equivalence class.This provides a foundation upon which we can further describe the quotient space structure of the Grassmann manifold and define a Riemannian metric on it.The equations of parallel transport satisfied on the particular Grassmann manifold is given,and the geodesic equations on the Grassmann manifold based on horizontal lifting is discussed by combining the singular value decomposition.Thridly,the abstract definition of the rolling map is used to construct a rolling map on the Grassmann manifold.The kinematic equation of the rolling on the affine space corresponding to a certain point is derived.Rolling along a straight line,it is obtained that a geodesic on the Grassmann manifold can develop into a geodesic on the corresponding tangent space.Finally,the basic idea of the geodesic flow kernel method is used to treat source domain data and target domain data as two points on the Grassmann manifold.The geodesic flow deduction process of geodesic subspace sampling is analyzed.The steps of geodesic flow kernel method are given,and the geodesic flow characterized between two points can form a path between two subspaces.