Learning functional relationships from noisy data is a central problem in scientific inference. Spectral methods approximate unknown functions by expanding them in a basis and estimating the corresponding coefficients from data, but the stability of these coefficients under noise remains poorly understood. Here we study supervised regression with additive label noise using sparse spectral representations across multiple bases and dimensions. We show that noise induces a predictable drift in the learned coefficient vector whose magnitude depends on the effective number of active spectral modes. After whitening the empirical feature geometry, we derive a closed-form expression for the overlap between noisy and noiseless coefficient vectors, revealing a universal degradation curve governed by a single intrinsic noise scale. Numerical experiments across Fourier, Legendre, Bessel, and Haar bases confirm the theoretical prediction. The results demonstrate that spectral learning exhibits a fundamental noise threshold beyond which coefficient estimates become unstable, placing intrinsic limits on recovering functional structure from noisy data.

翻译：从含噪数据中学习函数关系是科学推理中的核心问题。谱方法通过将未知函数在基函数上展开并从数据中估计相应系数来逼近函数，然而噪声对这些系数稳定性的影响尚不明确。本文研究采用稀疏谱表示在多基函数与多维度条件下处理含有附加标签噪声的监督回归问题。研究表明，噪声会导致学习得到的系数向量产生可预测的漂移，其幅度取决于有效活跃谱模式的数量。在对经验特征几何结构进行白化处理后，我们推导出含噪与无噪系数向量之间重叠度的闭式表达式，揭示了由单一固有噪声尺度控制的通用退化曲线。基于傅里叶基、勒让德基、贝塞尔基和哈尔基的数值实验验证了这一理论预测。结果表明，谱学习存在基本噪声阈值，超过该阈值后系数估计将变得不稳定，从而对从含噪数据中恢复函数结构施加了固有限制。