论文标题
关于最低$ \ ell_2 $ interpolator的鲁棒性
On the robustness of the minimum $\ell_2$ interpolator
论文作者
论文摘要
我们用最小的$ \ ell_2 $ -norm $ \hatβ$在一般高维线性回归框架中分析插装器，其中$ \ mathbb y = \ mathbbxβ^*+ξ$其中$ \ mathbb x $是随机的$ n \ times p $ ntimes p $ onrix nondission $ ntime \ Mathbb r^n $。我们证明，以较高的可能性，该估计器的预测损失从上面有限为$（\ |β^*\ |^2_2r_ {cn}（cn}（σ）\ vee \ | eCC \ | eCC \ |^2）/n $，其中$ r_ {k}（k}（et s） $σ$的特征值。这些界限显示了速率的过渡。对于高信号与噪声比，$ \ |β^*\ |^2_2r_ {cn}（σ）/n $大致改善现有的速率。对于低信号与噪声比，我们还提供了较大概率的下限持有。根据$σ$的srectrum的假设，此下限是订单$ \ | ξ\ | _2^2/n $，与上限匹配。因此，在较大的噪声状态下，我们能够以很大的概率准确地跟踪预测误差。当插值在高维度上无害时，此结果可以提供新的见解。
We analyse the interpolator with minimal $\ell_2$-norm $\hatβ$ in a general high dimensional linear regression framework where $\mathbb Y=\mathbb Xβ^*+ξ$ where $\mathbb X$ is a random $n\times p$ matrix with independent $\mathcal N(0,Σ)$ rows and without assumption on the noise vector $ξ\in \mathbb R^n$. We prove that, with high probability, the prediction loss of this estimator is bounded from above by $(\|β^*\|^2_2r_{cn}(Σ)\vee \|ξ\|^2)/n$, where $r_{k}(Σ)=\sum_{i\geq k}λ_i(Σ)$ are the rests of the sum of eigenvalues of $Σ$. These bounds show a transition in the rates. For high signal to noise ratios, the rates $\|β^*\|^2_2r_{cn}(Σ)/n$ 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 $Σ$, this lower bound is of order $\| ξ\|_2^2/n$, 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.
