A New Riemannian Structure in SPD(n)

Symmetric positive definite matrix has a widely application in many branches of science and engineering, a key issue is about how to find the `nice' distance between two elements on SPD(n) consisting of symmetric positive definite matrices. By using the fiber bundle and constructing a Riemannian submersion from the general linear group GL(n) to SPD(n), we develop the closed-form expressions of a new Riemannian metric and associated Riemannian distance in SPD(n). This metric and distance can be used to obtain a more precise result compared with the widely used Euclidean distance, which is usually used in signal classification.