Before revision:  To achieve this goal, we introduce the Ghost Point Diffusion Maps (GPDM) estimator on an extended manifold, identified by the set of point clouds on the unknown original manifold together with a set of ghost points  specified along the  tangential direction at the boundary of the original manifold .
After revision:  To achieve this goal, we introduce the Ghost Point Diffusion Maps (GPDM) estimator on an extended manifold, identified by the set of point clouds on the unknown original manifold together with a set of ghost points specified along the tangential direction at the sampled points at the boundary .