Expected reconstruction risk varies by sampling method in trigonometric models
Non-asymptotic Analysis of Expected Reconstruction Risk for Trigonometric Polynomial Models
Information Theory
Summary
This work looks at how well trigonometric polynomial models can be reconstructed from sampled data under different sampling types. The authors found that when sampling points are randomly chosen across an interval, the reconstruction tends to be unstable and error-prone. However, when samples are taken evenly or with small variations from an even spacing, the reconstruction error shows a clear tipping point related to the number of samples used. They explain these differences by analyzing the matrix properties that arise from the sampled points, proving why random sampling leads to high errors and providing exact formulas for regular sampling setups.
What this means in practice
- •For signal processing engineers: Optimize sensor placement strategies to improve stability and accuracy in reconstructing periodic signals from sampled data.
- •For medical imaging developers: Design sampling protocols that minimize reconstruction errors for periodic or oscillatory imaging data by avoiding purely random sample distributions.
Authors
Hang Xu, Yi Shen
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.