  We study the problem of estimating a $p$-dimensional $s$-sparse vector in a
linear model with Gaussian design and additive noise. In the case where the
labels are contaminated by at most $o$ adversarial outliers, we prove that the
$\ell_1$-penalized Huber's $M$-estimator based on $n$ samples attains the
optimal rate of convergence $(s/n)^{1/2} + (o/n)$, up to a logarithmic factor.
For more general design matrices, our results highlight the importance of two
properties: the transfer principle and the incoherence property. These
properties with suitable constants are shown to yield the optimal rates, up to
log-factors, of robust estimation with adversarial contamination.
