Generalized Bures-Wasserstein Geometry for Positive Definite Matrices Andi Han * Bamdev Mishra † Pratik Jawanpuria † Junbin Gao * Abstract This paper proposes a generalized Bures-Wasserstein (BW) Riemannian geometry for the manifold of symmetric positive definite matrices. We explore the generalization of the BW geometry in three different ways: 1) by generalizing the Lyapunov operator in the metric, 2) by generalizing the orthogonal Procrustes distance, and 3) by generalizing the Wasserstein distance between the Gaussians. We show that they all lead to the same geometry. The proposed generalization is parameterized by a symmetric positive definite matrix M such that when M = I, we recover the BW geometry. We derive expressions for the distance, geodesic, exponential/logarithm maps, Levi-Civita connection, and sectional curvature under the generalized BW geometry. We also present applications and experiments that illustrate the efficacy of the proposed geometry. 1 Introduction Symmetric positive definite (SPD) matrices are fundamental in various fields of applications, such as machine learning, signal processing, computer vision, and medical imaging, where the object of interest is primarily modelled as a covariance/kernel matrix or a diffusion tensor [14, 19, 35, 41, 51]. The set of SPD matrices, denoted as S n ++ , is a convex subset of the Euclidean space R n(n+1)/2 . To measure the (dis)similarity between SPD matrices, one needs to assign a metric (a smooth inner product structure) on S n ++ , which yields a Riemannian manifold. Consequently, there exist a number of Riemannian metrics such as the Affine-Invariant [9, 41, 49], Log-Euclidean [5, 6], and Log-Cholesky [33] metrics, to name a few. Existing works also explore metrics induced from symmetric divergences [45, 46, 47]. Recently, the Bures-Wasserstein (BW) distance has gained popularity, especially in machine learning applications [10, 20, 36, 52]. It is defined as d bw (X, Y)=  tr(X) + tr(Y) − 2tr(X 1/2 YX 1/2 ) 1/2  1/2 , (1) where X and Y are SPD matrices and tr(X) 1/2 denotes the trace of the matrix square root. It has been shown in [10, 36] that the BW distance (1) induces a Riemannian metric and geometry on the manifold of SPD matrices. The BW metric between symmetric matrices U, V on T X S n ++ is defined as g bw (U, V)= 1 2 tr(L X [U]V)= 1 2 vec(U) ⊤ (X ⊗ I + I ⊗ X) −1 vec(V), (2) * University of Sydney (andi.han@sydney.edu.au, junbin.gao@sydney.edu.au). † Microsoft India. (bamdevm@microsoft.com, pratik.jawanpuria@microsoft.com). 1 arXiv:2110.10464v1 [math.FA] 20 Oct 2021