  Learning maps between data samples is fundamental. Applications range from
representation learning, image translation and generative modeling, to the
estimation of spatial deformations. Such maps relate feature vectors, or map
between feature spaces. Well-behaved maps should be regular, which can be
imposed explicitly or may emanate from the data itself. We explore what induces
regularity for spatial transformations, e.g., when computing image
registrations. Classical optimization-based models compute maps between pairs
of samples and rely on an appropriate regularizer for well-posedness. Recent
deep learning approaches have attempted to avoid using such regularizers
altogether by relying on the sample population instead. We explore if it is
possible to obtain spatial regularity using an inverse consistency loss only
and elucidate what explains map regularity in such a context. We find that deep
networks combined with an inverse consistency loss and randomized off-grid
interpolation yield well behaved, approximately diffeomorphic, spatial
transformations. Despite the simplicity of this approach, our experiments
present compelling evidence, on both synthetic and real data, that regular maps
can be obtained without carefully tuned explicit regularizers, while achieving
competitive registration performance.
