8 papers · 1 filter
Strong law of large numbers for the -Karcher mean
Yongdo Lim, Miklós Pálfia
Sturm's strong law of large numbers in spaces has been an influential tool to study the geometric mean or also called Karcher barycenter of positive definite matr…
Operator means of probability measures
Fumio Hiai, Yongdo Lim
Let be the complete metric space consisting of positive invertible operators on an infinite-dimensional Hilbert space with the Thompson metric. We introduce the notion…
Strong Convexity of Sandwiched Entropies and Related Optimization Problems
Rajendra Bhatia, Tanvi Jain, Yongdo Lim
We present several theorems on strict and strong convexity, and higher order differential formulae for sandwiched quasi-relative entropy (a parametrised version of the classical fi…
Inequalities for the Wasserstein mean of positive definite matrices
Rajendra Bhatia, Tanvi Jain, Yongdo Lim
We prove majorization inequalities for different means of positive definite matrices. These include the Cartan mean (the Karcher mean), the log Euclidean mean, the Wasserstein mean…
On the Bures-Wasserstein distance between positive definite matrices
Rajendra Bhatia, Tanvi Jain, Yongdo Lim
The metric $d(A,B)=\left[ \tr\, A+\tr\, B-2\tr(A^{1/2}BA^{1/2})^{1/2}\right]^{1/2}$ on the manifold of positive definite matrices arises in various optimisation problem…
Log-majorizations for the (symplectic) eigenvalues of the Cartan barycenter
Fumio Hiai, Yongdo Lim
In this paper we show that the eigenvalue map and the symplectic eigenvalue map of positive definite matrices are Lipschitz for the Cartan-Hadamard Riemannian metric, and establish…