Authors: Xianfeng GU (Stony Brook University)*; Yalin Wang (Arizona State University); Dongsheng An (Stony Brook University); Min Zhang (Brigham and Women’s Hospital, Harvard Medical School); Na Lei (Dalian University of Technology); Tong Zhao (inria); Jianfeng Wu (Arizona State University); Xiaoyin Xu (Brigham and Women’s Hospital, Harvard Medical School)
Abstract: Biomarkers play an important role in preclinical/early detection and intervention in Alzheimer’s disease (AD). However, obtaining effective biomarkers for AD is still a big challenge. In this work, we propose to use the worst transportation cost as the biomarker for the A beta brain surfaces.
The worst transportation (WT) aims to find the least economical way to transport one measure to the other, which contrasts to the optimal transportation (OT) that find the most economical way between measures. In contrast, maps find the The transportation cost of an OT map is the Wasserstein distance between the measures, which has been broadly applied for shape analysis.
To compute the WT cost, we generalize the Brenier theorem for the OT map to the WT map, and show that the WT map is the gradient of a concave function satisfying the Monge-Ampere equation. We also develop an algorithm to compute the WT map based on computational geometry. Finally, we successfully use the WT cost to study the group difference between the A beta positive AD patients and A beta negative cognitively unimpaired subjects.