Non-asymptotic Analysis of Expected Reconstruction Risk for Trigonometric Polynomial Models

2026-09-24

AI Summary

This study addresses the challenges of diverging reconstruction risk and ill-conditioned matrices in trigonometric polynomial models under various sampling schemes. Leveraging spectral quantity representations and non-asymptotic statistical theory, combined with matrix spectral analysis and numerical experiments, it systematically investigates reconstruction risk behaviors under uniform random, equispaced, and jittered sampling. This work provides the first rigorous proof that uniform sampling induces risk divergence and elucidates its underlying phase transition mechanism. Furthermore, it establishes an explicit formula for the reconstruction risk under equispaced sampling and derives tight upper and lower bounds for jittered sampling. By clarifying the risk discrepancies across different sampling patterns, this project furnishes a theoretical foundation for the stable reconstruction of trigonometric polynomial models.

Abstract

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.

Problem

• Expected Reconstruction Risk
• Trigonometric Polynomial Models
• Sampling Schemes
• Non-asymptotic Analysis
• Phase Transition

Innovation

• Expected Reconstruction Risk
• Trigonometric Polynomial Models
• Non-asymptotic Analysis
• Spectral Representation
• Sampling Schemes