An efficient kernel product for automatic differentiation libraries, with applications to measure transport

This paper presents a memory-efficient implementation of the kernel matrix-vector product (sparse convolution) and the way to link it with automatic differentiation libraries such as PyTorch. This piece of software alleviates the major bottleneck of autodiff libraries as far as diffeomorphic shape registration is concerned: memory consumption. As a result, symbolic python code can now scale up to large point clouds and shapes (> 100, 000 vertices). To showcase the value of automatic differentiation to the LDDMM community, we introduce the normalized Hamiltonian setting and show that it corresponds to a spatially regularized optimal transport of mass distributions: made tractable by autodiff libraries, the kernel normalization trick turns an extrinsic image deformation routine into an intrinsic measure transportation program.