三角多项式模型期望重构风险的非渐近分析

中文摘要 AI 辅助

我们研究了不同采样方案下三角多项式模型的期望重构风险。通过数值实验，我们观察到当采样节点 $\{t_l\}_{l=1}^m$ 是独立同分布且在 $[0,1)$ 上均匀分布的随机变量时，相关的结构化随机矩阵 $\pmb{A} \in \mathbb{C}^{m \times N}$（其中 $A_{l,k} = e^{2\pi \mathrm{i} kt_l}, k \in \Gamma = \{-q, \dots, q\}, N = 2q+1$）经常变得近乎奇异或严重病态。因此，期望重构风险表现出发散行为。相反，当采样节点 $t_l$ 是等距点或等距网格的小随机扰动时，期望重构风险在插值阈值 $m=N$ 处经历急剧的相变。为了更好地理解这些不同现象背后的机制，我们通过谱量 $\sum_{i=1}^{r} \frac{1}{\sigma_i^2(\pmb{A})}$ 来刻画期望重构风险，其中 $\sigma_i(\pmb{A})$ 表示采样矩阵的奇异值。基于该谱表示，我们从理论上证明了在均匀分布随机采样下期望重构风险发散。此外，我们推导了等距采样情形下期望重构风险的显式公式，并建立了抖动采样下期望重构风险的上下界。

英文摘要

We investigate the expected reconstruction risk of trigonometric polynomial models under different sampling schemes. Through numerical experiments, we observe that when the sampling nodes $\{t_l\}_{l=1}^m$ are i.i.d. random variables uniformly distributed over $[0,1)$, the associated structured random matrix $\pmb{A} \in \mathbb{C}^{m \times N}$ with $A_{l,k} = e^{2π\mathrm{i} kt_l}, k \in Γ= \{-q, \dots, q\}, N = 2q+1$ frequently becomes nearly singular or severely ill-conditioned. As a consequence, the expected reconstruction risk exhibits divergent behavior. In contrast, when the sampling nodes $t_l$ are either equidistant points or small random perturbations of an equidistant grid, the expected reconstruction risk undergoes a sharp phase transition at the interpolation threshold $m=N$. To better understand the underlying mechanisms behind these different phenomena, we characterize the expected reconstruction risk through the spectral quantity $\sum_{i=1}^{r} \frac{1}{σ_i^2(\pmb{A})}$, where $σ_i(\pmb{A})$ denotes the singular values of the sampling matrix. Based on this spectral representation, we theoretically prove that the expected reconstruction risk diverges under uniformly distributed random sampling. Furthermore, we derive an explicit formula for the expected reconstruction risk in the equidistant sampling case and establish upper and lower bounds for the expected reconstruction risk under jittered sampling.

发表机构

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

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