Robust Sparse Regression under Adversarial Corruption

We consider high dimensional sparse regression with arbitrary – possibly, severe or coordinated – errors in the covariates matrix. We are interested in understanding how many corruptions we can tolerate, while identifying the correct support. To the best of our knowledge, neither standard outlier rejection techniques, nor recently developed robust regression algorithms (that focus only on corrupted response variables), nor recent algorithms for dealing with stochastic noise or erasures, can provide guarantees on support recovery. As we show, neither can the natural brute force algorithm that takes exponential time to find the subset of data and support columns, that yields the smallest regression error. We explore the power of a simple idea: replace the essential linear algebraic calculation – the inner product – with a robust counterpart that cannot be greatly affected by a controlled number of arbitrarily corrupted points: the trimmed inner product. We consider three popular algorithms in the uncorrupted setting: Thresholding Regression, Lasso, and the Dantzig selector, and show that the counterparts obtained using the trimmed inner product are provably robust.