Wasserstein Centroid‐Based Binary Classification for Distributional Data

We introduce a binary classification method for analyzing random objects in a non‐linear space. Unlike traditional classification approaches that maximize the mean difference between groups and minimize within‐group variance based on Euclidean distance, we consider the distance that accounts for dissimilarities between two random objects under intrinsic conditions. Accordingly, we need a distinct method for calculating the mean difference between groups, and this requirement poses a challenge in computing group variances because they do not exist in the same space. To address these challenges, we focus on the Wasserstein distance between two random objects measured locally in a tangent space. Additionally, we employ logarithmic mapping and a parallel transport operator to adequately compute the mean difference and variance. Consequently, we can effectively incorporate the central and dispersion characteristics of objects with intrinsic conditions and accurately calculate the distance for classification. Through repeated simulations under various scenarios, we demonstrate the advantages of our proposed approach in terms of classification performance relative to other approaches. Furthermore, we explore its potential in practical applications, such as the classification of glioblastoma multiforme pixel intensity data.