Before revision:  This framework builds upon a formulation of PD metrics as optimal transport (OT) problems , for which recent computational advances , in particular entropic regularization and its convolutional formulation on regular grids, can all be leveraged to provide efficient and (GPU) scalable computations .
After revision:  This framework builds upon a formulation of PD metrics as optimal transport (OT) problems , for which recent computational advances : the OT problem on a planar grid, when regularized with entropy, is convex can be solved in linear time using the Sinkhorn algorithm and convolutions. This results in scalable computations that can stream on GPUs .