  Existing Optimal Transport (OT) methods mainly derive the optimal transport
plan/matching under the criterion of transport cost/distance minimization,
which may cause incorrect matching in some cases. In many applications,
annotating a few matched keypoints across domains is reasonable or even
effortless in annotation burden. It is valuable to investigate how to leverage
the annotated keypoints to guide the correct matching in OT. In this paper, we
propose a novel KeyPoint-Guided model by ReLation preservation (KPG-RL) that
searches for the optimal matching (i.e., transport plan) guided by the
keypoints in OT. To impose the keypoints in OT, first, we propose a mask-based
constraint of the transport plan that preserves the matching of keypoint pairs.
Second, we propose to preserve the relation of each data point to the keypoints
to guide the matching. The proposed KPG-RL model can be solved by Sinkhorn's
algorithm and is applicable even when distributions are supported in different
spaces. We further utilize the relation preservation constraint in the
Kantorovich Problem and Gromov-Wasserstein model to impose the guidance of
keypoints in them. Meanwhile, the proposed KPG-RL model is extended to the
partial OT setting. Moreover, we deduce the dual formulation of the KPG-RL
model, which is solved using deep learning techniques. Based on the learned
transport plan from dual KPG-RL, we propose a novel manifold barycentric
projection to transport source data to the target domain. As applications, we
apply the proposed KPG-RL model to the heterogeneous domain adaptation and
image-to-image translation. Experiments verified the effectiveness of the
proposed approach.
