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Spectral Properties of Infinitely Smooth Kernel Matrices in the Single Cluster Limit, with Applications to Multivariate Super-Resolution

Nuha Diab, Dmitry Batenkov

TL;DR

This work addresses the stability of recovering sparse measures from bandlimited Fourier data in high dimensions by studying the spectral properties of infinitely smooth kernel matrices in the flat limit when nodes form a single cluster. It develops a general eigenvalue-scaling framework: for a kernel matrix $K_{\epsilon}(\mathcal{X})$ with discrete moment order $m=\mu(\mathcal{X})$, eigenvalues cluster into $m+1$ groups with $\lambda_{0,0}=O(1)$ and $\lambda_{k,j}=O(\epsilon^{2k})$, where the group sizes $t_k$ are determined by ranks of multivariate Vandermonde submatrices. The paper specializes to the Dirichlet kernel, proves nondegeneracy and geometric-principled conditions (GC/poisedness) under which the scaling is exact, and connects these results to the conditioning of multidimensional Vandermonde matrices in the super-resolution regime. Numerical experiments on Dirichlet kernels and multivariate SR tasks (NLS and 2D-ESPRIT) validate the theory and reveal geometry-dependent stability, guiding sampling choices and algorithm design for high-dimensional SR problems.

Abstract

We study the spectral properties of infinitely smooth multivariate kernel matrices when the nodes form a single cluster. We show that the geometry of the nodes plays an important role in the scaling of the eigenvalues of these kernel matrices. For the multivariate Dirichlet kernel matrix, we establish a criterion for the sampling set ensuring precise scaling of eigenvalues. Additionally, we identify specific sampling sets that satisfy this criterion. Finally, we discuss the implications of these results for the problem of super-resolution, i.e. stable recovery of sparse measures from bandlimited Fourier measurements.

Spectral Properties of Infinitely Smooth Kernel Matrices in the Single Cluster Limit, with Applications to Multivariate Super-Resolution

TL;DR

This work addresses the stability of recovering sparse measures from bandlimited Fourier data in high dimensions by studying the spectral properties of infinitely smooth kernel matrices in the flat limit when nodes form a single cluster. It develops a general eigenvalue-scaling framework: for a kernel matrix with discrete moment order , eigenvalues cluster into groups with and , where the group sizes are determined by ranks of multivariate Vandermonde submatrices. The paper specializes to the Dirichlet kernel, proves nondegeneracy and geometric-principled conditions (GC/poisedness) under which the scaling is exact, and connects these results to the conditioning of multidimensional Vandermonde matrices in the super-resolution regime. Numerical experiments on Dirichlet kernels and multivariate SR tasks (NLS and 2D-ESPRIT) validate the theory and reveal geometry-dependent stability, guiding sampling choices and algorithm design for high-dimensional SR problems.

Abstract

We study the spectral properties of infinitely smooth multivariate kernel matrices when the nodes form a single cluster. We show that the geometry of the nodes plays an important role in the scaling of the eigenvalues of these kernel matrices. For the multivariate Dirichlet kernel matrix, we establish a criterion for the sampling set ensuring precise scaling of eigenvalues. Additionally, we identify specific sampling sets that satisfy this criterion. Finally, we discuss the implications of these results for the problem of super-resolution, i.e. stable recovery of sparse measures from bandlimited Fourier measurements.
Paper Structure (14 sections, 21 theorems, 74 equations, 5 figures)

This paper contains 14 sections, 21 theorems, 74 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Eigenvalues of smooth kernel matrices in the flat limit
  4. Eigenvalues of the multidimensional Dirichlet kernel
  5. A nondegeneracy condition
  6. Geometric characterization condition and the uniform grid
  7. Vandermonde matrices in the super-resolution regime
  8. Numerical Experiments
  9. Dirichlet kernel eigenvalue scaling
  10. Super-resolution
  11. Local Stability
  12. 2D-ESPRIT
  13. Taylor Expansion
  14. Auxiliary Lemmas

Key Result

Theorem 3.1

Let $m=\mu({\cal X})$ be the discrete moment order of the set ${\cal X}{\color{black}\subset\mathbb{R}^d}$. For any infinitely smooth kernel $\mathcal{K}\in \mathcal{C^{(\infty,\infty)}}(\Omega)$ with $\mathbf{0}\in\Omega$, and small enough $\epsilon$, the eigenvalues of $K_{\epsilon}({\cal X})$ spl

Figures (5)

  • Figure 1: Demonstration of the proof of Lemma \ref{['lem:rescaled-lattice']}.
  • Figure 2: Eigenvalues of Dirichlet kernel with samples on the grid for the different node geometries. The $x$-axis is $\epsilon$ denoting the smallest distance between the nodes.
  • Figure 3: NLS reconstruction, general position/line/parabola, $d=2$.
  • Figure 4: NLS reconstruction, general position/line/parabola, $d=3$.
  • Figure 5: Accuracy of 2D ESPRIT for different geometries.

Theorems & Definitions (60)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3: Multivariate Vandermonde matrix
  • Definition 2.4
  • Definition 2.5: Discrete moment order, schabak2005
  • Theorem 3.1
  • Remark 3.2
  • proof
  • Definition 3.3
  • Lemma 3.4
  • ...and 50 more