面向统计回归的用户友好逼近理论
User-friendly approximation theory for statistical regression
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中文总结 AI 辅助
本文融合逼近论与统计学，为广义最小二乘回归建立统一误差界，涵盖加权范数与相关噪声，并改进傅里叶和切比雪夫回归的勒贝格常数至对数阶，提供可复用的显式回归保证。
中文摘要 AI 辅助
使用带噪声函数值进行回归的误差自然分解为来自模型空间选择的逼近误差和来自观测噪声的随机误差。统计理论主要关注随机误差，而逼近理论则侧重于逼近误差，且通常假设观测无噪声。我们将逼近理论和统计学的工具结合成一个框架，用于在广义最小二乘回归中同时界定这两种误差。结果涵盖加权上确界范数和$L^2(\mu)$-范数、实值和复值函数，以及在各种尾部和矩假设下的相关噪声。加权勒贝格函数和常数控制最佳逼近误差的放大，而浓度不等式则提供随机误差的界限。我们将这些结果组织成具有显式常数的可重用界限，并在可能的情况下建立锐度。对于常见的随机回归策略，我们证明了勒贝格函数和常数围绕相应$L^2(\mu)$-投影的浓度。对于标准采样网格上的傅里叶和切比雪夫回归，我们获得了勒贝格常数的显式、渐近紧的$\mathcal O(\log m)$界限，其中$m$是模型维度。这改进了先前在回归设置中获得的$\mathcal O(\sqrt{m})$界限，并匹配了插值已知的界限。一个逼近界限目录和实例说明了如何将这些结果转化为显式的回归保证。
英文摘要
The error of regression using noisy function values naturally decomposes into an approximation error from the choice of model space and stochastic error from the noise on the observations. Statistical theory is mostly concerned with the stochastic error while approximation theory focuses on the approximation error, often subject to the assumption of noiseless observations. We combine the tools from approximation theory and statistics into a framework for bounding both errors in generalized least squares regression. The results cover weighted sup-norms and $L^2(μ)$-norms, real- and complex-valued functions, and correlated noise under a range of tail and moment assumptions. Weighted Lebesgue functions and constants control the amplification of the best approximation error, while concentration inequalities provide bounds on the stochastic error. We organize these results into reusable bounds with explicit constants and establish sharpness where possible. For the common strategy of randomized regression we prove concentration of the Lebesgue function and constant around those of the corresponding $L^2(μ)$-projection. For Fourier and Chebyshev regression on standard sampling grids, we obtain explicit, asymptotically tight $\mathcal O(\log m)$ bounds on the Lebesgue constants where $m$ is the model dimension. This improves upon the previous $\mathcal O(\sqrt{m})$ bound obtained for the regression setting and matches the bounds known for interpolation. A catalogue of approximation bounds and worked examples illustrates how to turn these results into explicit regression guarantees.
