Non-asymptotic Analysis of Expected Reconstruction Risk for 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 {tl}l=1m are i.i.d. random variables uniformly distributed over [0,1), the associated structured random matrix pmbA ∈ ℂm × N with Al,k = e2πi ktl, k ∈ Γ= {−q, …, 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 tl 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 ∑i=1r 1/σi2(pmbA), where σi(pmbA) 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.
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BibTeX
@misc{xu2026nonasymptotic,
title = {{Non-asymptotic Analysis of Expected Reconstruction Risk for Trigonometric Polynomial Models}},
author = {Hang Xu and Yi Shen},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.29286},
eprint = {2609.29286},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.29286},
}
RIS
TY - UNPB
TI - Non-asymptotic Analysis of Expected Reconstruction Risk for Trigonometric Polynomial Models
AU - Hang Xu
AU - Yi Shen
PY - 2026
DA - 2026/09/24
PB - arXiv
DO - 10.48550/arxiv.2609.29286
UR - https://arxiv.org/abs/2609.29286
AB - 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 pmbA ∈ ℂ^m × N with A_l,k = e^2πi kt_l, k ∈ Γ= {−q, …, 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 ∑_i=1^r 1/σ_i^2(pmbA), where σ_i(pmbA) 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.
ER -