With recent advances in computing hardware and surges of deep-learning architectures, learning-based deep image registration methods have surpassed their traditional counterparts, in terms of metric performance and inference time. However, these methods focus on improving performance measurements such as Dice, resulting in less attention given to model behaviors that are equally desirable for registrations, especially for medical imaging. This paper investigates these behaviors for popular learning-based deep registrations under a sanity-checking microscope. We find that most existing registrations suffer from low inverse consistency and nondiscrimination of identical pairs due to overly optimized image similarities. To rectify these behaviors, we propose a novel regularization-based sanity-enforcer method that imposes two sanity checks on the deep model to reduce its inverse consistency errors and increase its discriminative power simultaneously. Moreover, we derive a set of theoretical guarantees for our sanity-checked image registration method, with experimental results supporting our theoretical findings and their effectiveness in increasing the sanity of models without sacrificing any performance.

Interacting agent and particle systems are extensively used to model complex phenomena in science and engineering. We consider the problem of learning interaction kernels in these dynamical systems constrained to evolve on Riemannian manifolds from given trajectory data. The models we consider are based on interaction kernels depending on pairwise Riemannian distances between agents, with agents interacting locally along the direction of the shortest geodesic connecting them. We show that our estimators converge at a rate that is independent of the dimension of the state space, and derive bounds on the trajectory estimation error, on the manifold, between the observed and estimated dynamics. We demonstrate the performance of our estimator on two classical first order interacting systems: Opinion Dynamics and a Predator-Swarm system, with each system constrained on two prototypical manifolds, the $2$-dimensional sphere and the Poincaré disk model of hyperbolic space.