Analyzing the structure and function of proteins is a key part of understanding biology at the molecular and cellular level. In addition, a major engineering challenge is to design new proteins in a principled and methodical way. Current computational modeling methods for protein design are slow and often require human oversight and intervention. Here, we apply Generative Adversarial Networks (GANs) to the task of generating protein structures, toward application in fast de novo protein design. We encode protein structures in terms of pairwise distances between alpha-carbons on the protein backbone, which eliminates the need for the generative model to learn translational and rotational symmetries. We then introduce a convex formulation of corruption-robust 3D structure recovery to fold the protein structures from generated pairwise distance maps, and solve these problems using the Alternating Direction Method of Multipliers. We test the effectiveness of our models by predicting completions of corrupted protein structures and show that the method is capable of quickly producing structurally plausible solutions.