Outlier Robust and Sparse Estimation of Linear Regression Coefficients

We consider outlier robust and sparse estimation of linear regression coefficients when covariates and noise are sampled, respectively, from an $\mathfrak{L}$-subGaussian distribution and a heavy-tailed distribution, and additionally, the covariates and noise are contaminated by adversarial outliers. We deal with two cases: known or unknown covariance of the covariates. Particularly, in the former case, our estimator attains nearly information theoretical optimal error bound, and our error bound is sharper than that of earlier studies dealing with similar situations. Our estimator analysis relies heavily on Generic Chaining to derive sharp error bounds.