Title: Provable Defenses against Spatially Transformed Adversarial Inputs: Impossibility and Possibility Results. Abstract: One intriguing property of neural networks is their inherent vulnerability to adversarial inputs, which are maliciously crafted samples to trigger target networks to misbehave. The state-of-the-art attacks generate adversarial inputs using either pixel perturbation or spatial transformation. Thus far, several provable defenses have been proposed against pixel perturbation-based attacks; yet, little is known about whether such solutions exist for spatial transformation-based attacks. This paper bridges this striking gap by conducting the first systematic study on provable defenses against spatially transformed adversarial inputs. Our findings convey mixed messages. On the impossibility side, we show that such defenses may not exist in practice: for any given networks, it is possible to find legitimate inputs and imperceptible transformations to generate adversarial inputs that force arbitrarily large errors. On the possibility side, we show that it is still feasible to construct adversarial training methods to significantly improve the resilience of networks against adversarial inputs over empirical datasets. We believe our findings provide insights for designing more effective defenses against spatially transformed adversarial inputs.