REMARKS ON THE PROPERTIES OF WASSERSTEIN MEDIANS ON SYMMETRIC CONES

Abstract

The author studied a one-parameter family of divergences and the related medianminimization problem of finite points over these divergences in a symmetric cone [14].The unique solution of the minimization problem with a weight ω is called the ω-weighted Wasserstein median. Recently, in the special symmetric cone of the positive denite matrices, Hwang and Kim [8] explored several properties of the Wasserstein mean and found bounds for the Wasserstein mean with respect to Löwner order. Also Kim and Lee [10] presented some relations between the Wasserstein mean and other well-known matrix meanssuch as the power mean, harmonic mean and Karcher mean. Motivated by these results, asan application of the previous work [14], we investigate several properties of the ω-weightedWasserstein median which mainly extend the corresponding ones in [8, 10] into a general symmetric cone Ω with a purely Jordan-algebraic technique.