= Gaussian score event for the Lasso
{c}
{title2=$\Omega=\{\|X^T\varepsilon\|_\infty/n\leq c\lambda\}$}
Let $0<c<1$, $A>0$, $\sigma>0$ and $p\geq2$. For <design matrix> columns of norm $\sqrt n$ and $\varepsilon\sim N_n(0,\sigma^2I)$, the event $\|X^T\varepsilon\|_\infty/n\leq c\lambda$ has probability at least $1-p^{1-A^2c^2/2}$ when $\lambda=A\sigma\sqrt{\log(p)/n}$. Apply the <sharp two-sided Gaussian tail bound> to each standardized score and then the <union bound>; independence of the scores is not required.
Back to article page
