In many signal processing applications, one aims to reconstruct a signal that has a simple representation with respect to a certain basis or frame. Fundamental elements of the basis known as “atoms” allow us to define “atomic norms” that can be used to construct convex regularizers for the reconstruction problem. Efficient algorithms are available to solve the reconstruction problems in certain special cases, but an approach that works well for general atomic norms remains to be found. This paper describes an optimization algorithm called CoGEnT , which produces solutions with succinct atomic representations for reconstruction problems, generally formulated with atomic- norm constraints. CoGEnT combines a greedy selection scheme based on the conditional gradient approach with a backward (or “truncation”) step that exploits the quadratic nature of the objective to reduce the basis size. We establish convergence properties and validate the algorithm via extensive numerical experiments on a suite of signal processing applications. Our algorithm and analysis are also novel in that they allow for inexact forward steps. In practice, CoGEnT significantly outperforms the basic conditional gradient method, and indeed many methods that are tailored to specific applications, when the truncation steps are defined appropriately. We also introduce several novel applications that are enabled by the atomic-norm framework, including tensor completion, moment problems in signal processing, and graph deconvolution.