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2020

On the robustness of the minimum $\ell_2$ interpolator

Chinot, Geoffrey, Lerasle, Matthieu

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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.

On the robustness of the minimum $\ell_2$ interpolator · Around

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