Robust Estimation of a Sparse Linear Model: Provable Guarantees with Non-convexity

In this paper, we address the problem of sparse regression vector estimation in the presence of corrupted samples, with a particular focus on accurately identifying the support. Traditional methods, such as the Least Absolute Shrinkage and Selection Operator (LASSO), often fail in such scenarios, exhibiting inconsistency. To tackle this challenge, we propose a combinatorial, non-convex, and robust variant of LASSO framework, designed to enhance estimation accuracy under corruption. Our approach is supported by theoretical guarantees, which establish its reliability and robustness. Our method also handles corruption from heavy-tailed distributions, with only a few bounded moments. We validate our theoretical results through extensive experiments, comparing the performance of our method against the LASSO and its other robust variants. These comparisons highlight the efficacy of our framework, demonstrating its practical applicability in sparse regression tasks involving corrupted data.