Abstract

Co-authors: Garvesh Raskutti (University of Wisconsin), Yuan Li (University of Wisconsin)

Sparse high-dimensional linear regression and inverse problems have received substantial attention over the past two decades. Much of this work assumes that explanatory variables are only mildly correlated. However, in modern applications ranging from functional MRI to genome-wide association studies, we observe highly correlated explanatory variables and associated design matrices 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 explanatory variables, 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 correlated explanatory variables. I will show how the graph structure encapsulated by this regularizer interacts with correlated design matrices to yield provably a ccurate estimates. The proposed approach outperforms standard methods in a variety of experiments on simulated and real fMRI data.

This is joint work with Yuan Li and Garvesh Raskutti.