Spectrum-Aware Debiasing: A Modern Inference Framework with Applications to Principal Components Regression
谱感知去偏：高维主成分回归的现代推断框架AI 解读 ↓
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谱感知去偏：高维主成分回归的现代推断框架
- 研究问题
- 摘要称，高维线性回归中的自由度调整去偏仅适用于独立同分布样本和次高斯协变量，本文要解决在存在结构化相依、重尾和低秩结构时如何进行高维去偏与推断的问题。
- 方法与思路
- 作者提出谱感知去偏方法，通过一个重新缩放的梯度下降步实现去偏，缩放因子由样本协方差矩阵的谱信息导出；在特征数与样本量成比例、协变量右旋转不变的设定下，建立估计量的渐近正态性并给出渐近方差的相合估计。
- 材料陈述的贡献
- 摘要称该方法将去偏推广到更广的相依与重尾情形，并给出两个副产品：高维下首个去偏主成分回归估计量，以及检验信号与样本协方差矩阵特征向量对齐性的原则性检验；技术上连接近似消息传递与去偏，并给出向量近似消息传递 Cauchy 性质的首个证明。
- 阅读关注点
- 应核对右旋转不变等设计假设的具体表述、渐近正态性所依赖的收敛概念与中心化缩放方式、渐近方差估计的相合性条件，以及去偏 PCR 与对齐检验的构造细节和模拟、实证数据的适用范围。
- 阅读边界
- 仅凭摘要无法判断定理的完整假设、有限样本表现与实证结论的稳健性；该文为 arXiv 预印本，未经同行评审验证，不能据此声称已被期刊接收。
原文摘要
Debiasing is a fundamental concept in high-dimensional statistics. While degrees-of-freedom adjustment is the state-of-the-art technique in high-dimensional linear regression, it is limited to i.i.d. samples and sub-Gaussian covariates. These constraints hinder its broader practical use. Here, we introduce Spectrum-Aware Debiasing--a novel method for high-dimensional regression. Our approach applies to problems with structured dependencies, heavy tails, and low-rank structures. Our method achieves debiasing through a rescaled gradient descent step, deriving the rescaling factor using spectral information of the sample covariance matrix. The spectrum-based approach enables accurate debiasing in much broader contexts. We study the common modern regime where the number of features and samples scale proportionally. We establish asymptotic normality of our proposed estimator (suitably centered and scaled) under various convergence notions when the covariates are right-rotationally invariant. Such designs have garnered recent attention due to their crucial role in compressed sensing. Furthermore, we devise a consistent estimator for its asymptotic variance. Our work has two notable by-products: first, we use Spectrum-Aware Debiasing to correct bias in principal components regression (PCR), providing the first debiased PCR estimator in high dimensions. Second, we introduce a principled test for checking alignment between the signal and the eigenvectors of the sample covariance matrix. This test is independently valuable for statistical methods developed using approximate message passing, leave-one-out, or convex Gaussian min-max theorems. We demonstrate our method through simulated and real data experiments. Technically, we connect approximate message passing algorithms with debiasing and provide the first proof of the Cauchy property of vector approximate message passing (V-AMP).