随机傅里叶级数模型的双重下降

中文摘要 AI 辅助

我们研究了使用随机部分离散傅里叶变换（DFT）矩阵的最小二乘线性回归问题，对该模型的泛化误差进行了严格分析。通过利用随机矩阵理论的工具，我们推导了Moore-Penrose估计器风险的精确非渐近界，这些界适用于有限维问题，并揭示了其与样本量、维数和噪声方差等关键参数的精确依赖关系。随后，我们在线性回归背景下获得了双重下降现象的表征，展示了当参数数量$p$和样本数量$n$趋于无穷大且$p/n$固定时，风险如何演变。该分析依赖于随机傅里叶矩阵的Stieltjes变换的应用，从而能够精确描述这些矩阵的谱性质及其对回归性能的影响。为了验证我们的理论发现，我们提供了几个数值示例来说明双重下降曲线。这些模拟与我们的推导界紧密吻合，证实了它们在欠参数化和过参数化情况下的预测能力。

英文摘要

We investigate the least squares linear regression problem with random partial Discrete Fourier Transform (DFT) matrices, providing a rigorous analysis of the model's generalization error. By leveraging tools from random matrix theory, we derive exact non-asymptotic bounds for the risk of the Moore-Penrose estimator, which hold for finite-dimensional problems and reveal the precise dependence on key parameters such as the sample size, dimension, and noise variance. Then we obtain a characterization of the double descent phenomenon in the linear regression context, demonstrating how the risk evolves when the number of parameters $p$ and the number of samples $n$ tend to infinity, with $p/n$ fixed. The analysis relies on applications of the Stieltjes transform for random Fourier matrices, enabling a precise description of the spectral properties of these matrices and their impact on regression performance. To validate our theoretical findings, we present several numerical examples that illustrate the double descent curves. These simulations align closely with our derived bounds, confirming their predictive power in both under-parameterized and over-parameterized regimes.

发表机构

• Zhejiang Sci-Tech University（浙江理工大学）

机构由 AI 辅助整理，请以论文原文为准。