关于最小$\ell_2$阶的鲁棒性
On the robustness of the minimum $\ell_2$ interpolator
摘要：
我们分析了最小$\ell_2$ -范数$\hatβ$在一般的高维线性回归框架，其中$\mathbb Y=\mathbb Xβ^*+ξ$，其中$\mathbb X$是一个随机的$n\times p$矩阵与独立的$\mathcal N(0,Σ)$行，没有假设的噪声向量$ξ\in \mathbb R^n$。我们证明了，以高概率，该估计的预测损失是有界的从上由$(\|β^*\|^2_2r_{cn}(Σ)\vee \|ξ\|^2)/n$，其中$r_{k}(Σ)=\sum_{i\geq k}λ_i(Σ)$是其余的$Σ$的特征值之和。这些界限显示了速率的转变。对于高信噪比，速率$\|β^*\|^2_2r_{cn}(Σ)/n$广泛地改进了现有的速率。对于低信噪比，我们还提供了大概率的下限保持。在对$Σ$的谱的假设下，该下界是$\| ξ\|_2^2/n$阶的，与上界匹配。因此，在大噪声状态下，我们能够以很大的概率精确地跟踪预测误差。这一结果提供了新的见解时，插值可以是无害的高维。
英文摘要：
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.
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