  We analyse the interpolator with minimal $\ell_2$-norm $\hat{\beta}$ in a
general high dimensional linear regression framework where $\mathbb Y=\mathbb
X\beta^*+\xi$ where $\mathbb X$ is a random $n\times p$ matrix with independent
$\mathcal N(0,\Sigma)$ rows and without assumption on the noise vector $\xi\in
\mathbb R^n$. We prove that, with high probability, the prediction loss of this
estimator is bounded from above by $(\|\beta^*\|^2_2r_{cn}(\Sigma)\vee
\|\xi\|^2)/n$, where $r_{k}(\Sigma)=\sum_{i\geq k}\lambda_i(\Sigma)$ are the
rests of the sum of eigenvalues of $\Sigma$. These bounds show a transition in
the rates. For high signal to noise ratios, the rates
$\|\beta^*\|^2_2r_{cn}(\Sigma)/n$ broadly improve the existing ones. For low
signal to noise ratio, we also provide lower bound holding with large
probability. Under assumptions on the sprectrum of $\Sigma$, this lower bound
is of order $\| \xi\|_2^2/n$, matching the upper bound. Consequently, in the
large noise regime, we are able to precisely track the prediction error with
large probability. This results give new insight when the interpolation can be
harmless in high dimensions.
