设计矩阵的零模式驱动过参数化回归中的多重下降
The Zero Pattern of a Design Matrix Drives Multiple Descent in Over-parameterized Regression
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中文总结 AI 辅助
研究过参数化线性回归，放宽协变量独立且协方差矩阵非退化的假设，推导预测风险的确定性等价物，证明协方差矩阵退化和依赖性会导致多重下降，用新图形表示及相关分解刻画峰值位置和方差奇异配置。
中文摘要 AI 辅助
在过去十年中，过参数化线性回归得到了广泛研究。然而，大多数现有工作假设协变量是独立的，且其协方差矩阵是非退化的。本文放宽了这两个假设，并在消失岭（vanishing-ridge） regime中推导了预测风险的确定性等价物。我们表明协方差矩阵的退化和依赖性会导致多重下降，并刻画了相应峰值可能出现的位置。我们的证明使用了方差轮廓的一种新颖图形表示。我们表明最大匹配和相关二分图的Dulmage-Mendelsohn分解确定了方差变得奇异的配置。
英文摘要
Over-parameterized linear regression has been widely studied over the last decade. However, most existing works assume that the covariates are independent and that their covariance matrices are non-degenerate. In this paper, we relax both assumptions and derive deterministic equivalents for the prediction risk in a vanishing-ridge regime. We show that degeneracy of the covariance matrices and dependence can lead to multiple descent, and characterize where the corresponding peaks can occur. Our proofs use a novel graph representation of the variance profile. We show that maximum matchings and the Dulmage--Mendelsohn decomposition of the associated bipartite graph identify the configurations at which the variance becomes singular.
发表机构
- University of Warwick（华威大学）
- Harvard University（哈佛大学）
机构由 AI 辅助整理，请以论文原文为准。
