Eigenmatrix for unstructured sparse recovery

TL;DR

The paper addresses unstructured sparse recovery with a general kernel $G(s,x)$ and noisy observations by introducing the eigenmatrix $M$, a data-driven operator designed so that $M \mathbf{g}(x) \approx x \mathbf{g}(x)$ for $\mathbf{g}(x)=[G(s_j,x)]$. This enables a unified Prony/ESPRIT–style recovery that does not require structured sampling, by constructing $M$ on complex or real analytic domains (via unit-disk boundary grids or Chebyshev grids) and applying postprocessing to estimate spike locations $\{x_k\}$ and weights $\{w_k\}$. The method is demonstrated across rational approximation, spectral function estimation, Fourier inversion, Laplace inversion, and sparse deconvolution, showing robustness to noise and providing accurate initial guesses for subsequent LS refinement; however, tasks like Laplace inversion remain more noise-sensitive. Overall, the eigenmatrix framework offers a principled, data-driven, unified approach to several unstructured sparse recovery problems with potential for extensions to other spectral-type methods.

Abstract

This note considers the unstructured sparse recovery problems in a general form. Examples include rational approximation, spectral function estimation, Fourier inversion, Laplace inversion, and sparse deconvolution. The main challenges are the noise in the sample values and the unstructured nature of the sample locations. This note proposes the eigenmatrix, a data-driven construction with desired approximate eigenvalues and eigenvectors. The eigenmatrix offers a new way for these sparse recovery problems. Numerical results are provided to demonstrate the efficiency of the proposed method.

Figures (6)

• Figure 1: The eigenmatrix for the example in Section \ref{['sec:pe']}. $X=\mathbb{S}\equiv\{z: |z|=1\}\subset\mathbb{C}$, $S=\mathbb{R}$, $G(s,x) = x^s$ (with the branch cut at $x=-1$). $n_s=32$. Left: $M$ when $s_j=j$ for $0\le j <n_s$. Middle: $M$ when $s_j$ is a random perturbation of the integer lattice. Right: $M$ when $\{s_j\}$ are chosen uniformly in $[0,n_s]$.
• Figure 2: Rational approximation. $G(s,x) = \frac{1}{s-x}$. $X=\mathbb{D}$. $\{s_j\}$ are random points outside the unit disk, each with a modulus between $1.2$ and $2.2$. $n_s=40$. Columns: $\sigma$ equals to $10^{-2}$, $10^{-3}$, and $10^{-4}$. Rows: the easy test (well-separated) and the hard test (with two nearby spikes).
• Figure 3: Spectral function approximation. $G(s,x) = \frac{1}{s-x}$. $X=[-1,1]$. $\{s_j\}$ is the Matsubara grid from $-\frac{(2N-1)\pi}{\beta}i$ to $\frac{(2N-1)\pi}{\beta}i$ with $\beta=100$ and $N=128$. Columns: $\sigma$ equals to $10^{-2}$, $10^{-3}$, and $10^{-4}$. Rows: the easy test (well-separated) and the hard test (with two nearby spikes).
• Figure 4: Fourier inversion. $G(s,x) = \exp(\pi i s x)$. $X=[-1,1]$. $\{s_j\}$ are randomly chosen points in $[-5,5]$. $n_s=128$. Columns: $\sigma$ equals to $10^{-2}$, $10^{-3}$, and $10^{-4}$, respectively. Rows: the easy test (well-separated) and the hard test (with two nearby spikes).
• Figure 5: Laplace inversion. $G(s,x) = \exp(-sx)$. $X=[0.1,2.1]$. $\{s_j\}$ are random samples in $[0,10]$. $n_s=100$. Columns: $\sigma$ equals to $10^{-5}$, $10^{-6}$, and $10^{-7}$, respectively. Rows: the easy test (well-separated) and the hard test (with two nearby spikes).

Theorems & Definitions (13)

• Remark 1
• Remark 2
• Remark 3
• Remark 4
• Remark 5
• Remark 6
• Remark 7
• Example 1: Rational approximation
• Example 2: Spectral function approximation
• Example 3: Fourier inversion