Geometric Characteristics of the Wasserstein Metric on SPD(n) and Its Applications on Data Processing

Abstract

The Wasserstein distance, especially among symmetric positive-definite matrices, has broad and deep influences on the development of artificial intelligence (AI) and other branches of computer science. In this paper, by involving the Wasserstein metric on SPD(n), we obtain computationally feasible expressions for some geometric quantities, including geodesics, exponential maps, the Riemannian connection, Jacobi fields and curvatures, particularly the scalar curvature. Furthermore, we discuss the behavior of geodesics and prove that the manifold is globally geodesic convex. Finally, we design algorithms for point cloud denoising and edge detecting of a polluted image based on the Wasserstein curvature on SPD(n). The experimental results show the efficiency and robustness of our curvature-based methods.

ECG Classification Based on Wasserstein Scalar Curvature

“… becomes a Riemannian manifold after being endowed with the natural Riemannian metric induced by Wasserstein distance [ 28 ]. We reveal the local differences of point clouds on [ 29 ] by Wasserstein scalar curvature (WSC). By analyzing the Wasserstein scalar curvature histogram (WSCH) of the converted point clouds, we originally defined a new data characteristic, called Wasserstein scalar curvature dispersion (WSCD) which can express the pathological changes of heart.…”