Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Selecting Optimal Sampling Rate for Stable Super-Resolution

Nuha Diab

TL;DR

The paper tackles stable super-resolution of spike trains from noisy Fourier data, focusing on ill-conditioning when node separation is below the Rayleigh limit. It develops a spectral-based preprocessing approach that identifies an admissible decimation rate $\rho$ by analyzing the Toeplitz spectrum derived from decimated samples, thereby improving conditioning before applying SR algorithms. The authors establish that for $M$-cluster configurations, the $(M+1)$-th singular value of the Toeplitz matrix $T_{\rho}$ scales as $\sigma_{M+1}(T_{\rho}) \asymp \Delta_{\rho}^2$, and connect this to the Vandermonde structure to guide rate selection. Building on this, they introduce Enhanced Decimated Prony (EDP) and Decimated Matrix Pencil (DMP) methods, with an algorithm to pick the optimal decimation rate and a de-aliasing step using co-prime shifts; EDP, in particular, attains the min-max error bounds in multi-cluster SR and offers substantial speedups over prior approaches. The results provide a principled preprocessing step that robustly improves SR performance in applications such as optics, imaging, and spectroscopy.

Abstract

We investigate the recovery of nodes and amplitudes from noisy frequency samples in spike train signals, also known as the super-resolution (SR) problem. When the node separation falls below the Rayleigh limit, the problem becomes ill-conditioned. Admissible sampling rates, or decimation parameters, improve the conditioning of the SR problem, enabling more accurate recovery. We propose an efficient preprocessing method to identify the optimal sampling rate, significantly enhancing the performance of SR techniques.

Selecting Optimal Sampling Rate for Stable Super-Resolution

TL;DR

The paper tackles stable super-resolution of spike trains from noisy Fourier data, focusing on ill-conditioning when node separation is below the Rayleigh limit. It develops a spectral-based preprocessing approach that identifies an admissible decimation rate by analyzing the Toeplitz spectrum derived from decimated samples, thereby improving conditioning before applying SR algorithms. The authors establish that for -cluster configurations, the -th singular value of the Toeplitz matrix scales as , and connect this to the Vandermonde structure to guide rate selection. Building on this, they introduce Enhanced Decimated Prony (EDP) and Decimated Matrix Pencil (DMP) methods, with an algorithm to pick the optimal decimation rate and a de-aliasing step using co-prime shifts; EDP, in particular, attains the min-max error bounds in multi-cluster SR and offers substantial speedups over prior approaches. The results provide a principled preprocessing step that robustly improves SR performance in applications such as optics, imaging, and spectroscopy.

Abstract

We investigate the recovery of nodes and amplitudes from noisy frequency samples in spike train signals, also known as the super-resolution (SR) problem. When the node separation falls below the Rayleigh limit, the problem becomes ill-conditioned. Admissible sampling rates, or decimation parameters, improve the conditioning of the SR problem, enabling more accurate recovery. We propose an efficient preprocessing method to identify the optimal sampling rate, significantly enhancing the performance of SR techniques.

Paper Structure

This paper contains 8 sections, 6 theorems, 31 equations, 3 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Identification of an Optimal Rate
  4. Main result and proofs
  5. Numerical Validation
  6. Algorithm
  7. Selecting an optimal sampling rate
  8. Optimality of EDP

Key Result

Theorem 1

Let $Q \in \mathbb{C}^{n\times n}$ be a matrix of the form $Q = V^*DV$, where $D$ is a diagonal complex matrix and $V \in \mathbb{C}^{n\times n}$. Then we have: where $\lambda_n((DD^*)^{1\over 2}) \leq \theta_i \leq \lambda_1((DD^*)^{1\over 2})$.

Figures (3)

  • Figure 1: (right) $X$ has one cluster of size $\ell_1$ with $SRF = 6$. (left) $X$ has two clusters of sizes $\ell_1$ and $\ell_2$ with $SRF = 3$. The plotted $\sqrt{\sigma_{k+1}(T_{\rho})}$ values are scaled by $C := \frac{n}{\Omega}$ for better visualization.
  • Figure 2: (right) single cluster configuration. (left) multi-cluster configuration. For both experiments the noise level is $10^{-6}$, $N_{\rho} = 900$ and $N_b = 3\Delta^{-1}$.
  • Figure 3: EDP - asymptotic optimality. For cluster node $x_1$, $\mathcal{K}_{x_1}$ (left) scales like $SRF^{2\ell_1 - 2}$, while the $\mathcal{K}_{a_1}$ (right) scales like $SRF^{2\ell_1 - 1}$. For the non-cluster node $x_4$, both $\mathcal{K}_{x_4}$ and $\mathcal{K}_{a_4}$ are lower bounded by a constant. These scaling rates are optimal.

Theorems & Definitions (19)

  • Definition 1
  • Definition 2
  • Definition 3: Single cluster configuration
  • Definition 4: Multi-cluster configuration
  • Definition 5
  • Theorem 1
  • proof
  • Theorem 2
  • proof
  • Theorem 3: Proposition 7.1 in batenkov2021single
  • ...and 9 more