Sparse models for machine learning have received substantial attention over the past two decades. Model selection, or determining which features are the best explanatory variables, is critical to the interpretability of a learned model. Much of this work assumes that features are only mildly correlated. However, in modern applications ranging from functional MRI to genome-wide association studies, we observe highly correlated features that do not exhibit key properties (such as the restricted eigenvalue condition). In this talk, I will describe novel methods for robust sparse linear regression in these settings. Using side information about the strength of correlations among features, we form a graph with edge weights corresponding to pairwise correlations. This graph is used to define a graph total variation regularizer that promotes similar weights for highly correlated features. I will show how the graph structure encapsulated by this regularizer helps precondition correlated features to yield provably accurate estimates. The proposed approach outperforms several previous approaches in a variety of experiments on simulated and real fMRI data. This is joint work with Yuan Li, Ben Mark, and Garvesh Raskutti.