凸正则化的谱依赖：右旋转不变设计下的基本极限
Spectral Dependence of Convex Regularization: Fundamental Limits under Right-Rotationally Invariant Designs
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中文总结 AI 辅助
该研究针对右旋转不变设计的高维线性回归，证明凸惩罚最小二乘估计器的渐近风险以Bayes VAMP算法的风险为下界，揭示设计矩阵完整奇异值分布对凸正则化局限性的决定作用。
中文摘要 AI 辅助
我们研究了具有右旋转不变设计矩阵的高维线性回归中凸正则化估计的基本极限。我们证明了凸惩罚最小二乘估计器的渐近风险以一种名为Bayes VAMP的近似消息传递算法的风险为下界，并进一步刻画了该下界可达的条件。作为证明下界定理的技术组成部分，我们刻画了对于每个固定扰动强度λ>0，经ℓ₂扰动的凸估计器的渐近性能，这填补了文献中的空白——此前文献要求λ足够大或对凸估计器的类别施加限制。我们方法的优势在于可在高维极限中直接分析谱对下界的影响，尤其可证明下界在设计矩阵的极限谱分布的排序上是单调的。这些结果明确了设计矩阵的完整奇异值分布（而非仅纵横比或平均测量强度）如何决定凸正则化的局限性。
英文摘要
We study the fundamental limits of convex-regularized estimation in high-dimensional linear regression with right-rotationally invariant design matrices. We show that the asymptotic risks of convex-penalized least squares estimators are lower bounded by the risk of an approximate message passing algorithm known as Bayes VAMP, and we further characterize when the lower bound is attainable. As a technical ingredient in the proof of our lower bound theorem, we characterize the asymptotic performance of the $\ell_2$-perturbed convex estimator for every fixed perturbation strength $λ>0$. This closes a gap in the literature, in which $λ$ was required to be sufficiently large or restrictions were imposed on the class of convex estimators. The benefit of our approach is that we can conduct a direct analysis of the spectrum's impact on the lower bound in the high-dimensional limit. In particular, we can show that the lower bound is monotone in an ordering on the limiting spectral distributions of the design matrix. These results isolate how the full singular-value distribution of the design---not merely the aspect ratio or average measurement strength---governs the limitations of convex regularization.
