Multidimensional unstructured sparse recovery via eigenmatrix

TL;DR

This work extends the eigenmatrix framework to multidimensional unstructured sparse recovery, where observations $u(s)=\sum_k w_k G(s,x_k)$ are contaminated by noise. By constructing dimension-wise eigenmatrices $M^t$ that satisfy $M^t\mathbf{g}(x)\approx x^t\mathbf{g}(x)$ on product grids, the method converts spike recovery into an algebraic problem via ESPRIT-like joint eigenanalysis, followed by least-squares recovery of weights. The approach handles unstructured sampling and noisy data, with 2D and 3D numerical demonstrations on sparse deconvolution and Fourier inversion that show robust spike localization under moderate noise. The framework unifies several multidimensional sparse recovery problems under a data-driven, algebraic procedure and suggests directions for error analysis and extensions to other spectral methods.

Abstract

This note considers the multidimensional unstructured sparse recovery problems. Examples include Fourier inversion and sparse deconvolution. The eigenmatrix is a data-driven construction with desired approximate eigenvalues and eigenvectors proposed for the one-dimensional problems. This note extends the eigenmatrix approach to multidimensional problems. Numerical results are provided to demonstrate the performance of the proposed method.

Figures (4)

• Figure 1: Sparse deconvolution. $G(s,x) = \frac{1}{\|s-x\|}$. $X=[-1,1]^2$. $\{s_j\}$ are $J=1024$ random points in $[-2,2]^2$ outside $X$. Columns: $\sigma$ equals to $10^{-2}$, $10^{-3}$, and $10^{-4}$, respectively. Rows: the easy case (top) and the hard case (bottom).
• Figure 2: Fourier inversion. $G(s,x) = \exp(\pi i s \cdot x)$. $X=[-1,1]^2$. $\{s_j\}$ are $J=1024$ randomly chosen points in $[-8,8]^2$. Columns: $\sigma$ equals to $10^{-2}$, $10^{-3}$, and $10^{-4}$, respectively. Rows: the easy case (top) and the hard case (bottom).
• Figure 3: Sparse deconvolution. $G(s,x) = \frac{1}{\|s-x\|^{1/2}}$. $X=[-1,1]^3$. $\{s_j\}$ are $J=8192$ random points in $[-2,2]^3$ outside $X$. Columns: $\sigma$ equals to $10^{-4}$, $10^{-5}$, and $10^{-6}$, respectively. Rows: the easy case (top) and the hard case (bottom).
• Figure 4: Fourier inversion. $G(s,x) = \exp(\pi i s \cdot x)$. $X=[-1,1]^3$. $\{s_j\}$ are $J=8192$ randomly chosen points in $[-4,4]^3$. Columns: $\sigma$ equals to $10^{-3}$, $10^{-4}$, and $10^{-5}$, respectively. Rows: the easy case (top) and the hard case (bottom).

Theorems & Definitions (8)

• Remark 1
• Remark 2
• Remark 3
• Remark 4
• Example 1: Sparse deconvolution
• Example 2: Fourier inversion
• Example 3: Sparse deconvolution
• Example 4: Fourier inversion