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A regularized eigenmatrix method for unstructured sparse recovery

Koung Hee Leem, Jun Liu, George Pelekanos

TL;DR

The paper tackles robust sparse recovery from unstructured samples by replacing the ill-conditioned pseudo-inverse step in the data-driven eigenmatrix method with a Tikhonov-regularized solution of $\widehat{G}\bm{v}=\widetilde{\bm u}$. This leads to constructing $A=[\widetilde{\bm u},\widehat{G}\Lambda\bm{v},\dots,\widehat{G}\Lambda^{l}\bm{v}]$ without forming the eigenmatrix $M$, enabling effective spike location and weight recovery even under substantial noise. Across five numerical examples (rational, spectral, Fourier, Laplace, sparse deconvolution), the regularized approach (using IMPC or L-curve for parameter selection) outperforms the original pseudo-inverse method, demonstrating improved robustness and accuracy. The work highlights a practical, scalable improvement for unstructured sparse recovery and suggests extensions to multidimensional data and sampling optimization.

Abstract

The recently developed data-driven eigenmatrix method shows very promising reconstruction accuracy in sparse recovery for a wide range of kernel functions and random sample locations. However, its current implementation can lead to numerical instability if the threshold tolerance is not appropriately chosen. To incorporate regularization techniques, we propose to regularize the eigenmatrix method by replacing the computation of an ill-conditioned pseudo-inverse by the solution of an ill-conditioned least square system, which can be efficiently treated by Tikhonov regularization. Extensive numerical examples confirmed the improved effectiveness of our proposed method, especially when the noise levels are relatively high.

A regularized eigenmatrix method for unstructured sparse recovery

TL;DR

The paper tackles robust sparse recovery from unstructured samples by replacing the ill-conditioned pseudo-inverse step in the data-driven eigenmatrix method with a Tikhonov-regularized solution of . This leads to constructing without forming the eigenmatrix , enabling effective spike location and weight recovery even under substantial noise. Across five numerical examples (rational, spectral, Fourier, Laplace, sparse deconvolution), the regularized approach (using IMPC or L-curve for parameter selection) outperforms the original pseudo-inverse method, demonstrating improved robustness and accuracy. The work highlights a practical, scalable improvement for unstructured sparse recovery and suggests extensions to multidimensional data and sampling optimization.

Abstract

The recently developed data-driven eigenmatrix method shows very promising reconstruction accuracy in sparse recovery for a wide range of kernel functions and random sample locations. However, its current implementation can lead to numerical instability if the threshold tolerance is not appropriately chosen. To incorporate regularization techniques, we propose to regularize the eigenmatrix method by replacing the computation of an ill-conditioned pseudo-inverse by the solution of an ill-conditioned least square system, which can be efficiently treated by Tikhonov regularization. Extensive numerical examples confirmed the improved effectiveness of our proposed method, especially when the noise levels are relatively high.
Paper Structure (11 sections, 17 equations, 5 figures)

This paper contains 11 sections, 17 equations, 5 figures.

Table of Contents

  1. Background
  2. Review of the eigenmatrix method
  3. A regularized eigenmatrix method
  4. Tikhonov regularization for solving (\ref{['Gv=u']})
  5. Numerical results
  6. Example 1 (Rational approximation)
  7. Example 2 (Spectral function approximation)
  8. Example 3 (Fourier inversion)
  9. Example 4 (Laplace inversion)
  10. Example 5 (Sparse deconvolution)
  11. Conclusion

Figures (5)

  • Figure 1: Rational approximation (Columns from left to right: $\sigma=10^{-1},10^{-2},10^{-3}$). The exact spikes in solid line and the recovered spikes in dashed line. The errors measure the 2-norm difference in spike locations and weights, respectively.
  • Figure 2: Spectral function approximation (Columns from left to right:$\sigma=10^{-1},10^{-2},10^{-3}$). The exact spikes in solid line and the recovered spikes in dashed line. The errors measure the 2-norm difference in spike locations and weights, respectively.
  • Figure 3: Fourier inversion (Columns from left to right:$\sigma=10^{-1},10^{-2},10^{-3}$). The exact spikes in solid line and the recovered spikes in dashed line. The errors measure the 2-norm difference in spike locations and weights, respectively.
  • Figure 4: Laplace inversion (Columns from left to right: $2\sigma=10^{-1}, 10^{-2}, 10^{-3}$). The exact spikes in solid line and the recovered spikes in dashed line. The errors measure the 2-norm difference in spike locations and weights, respectively.
  • Figure 5: Sparse deconvolution (Columns from left to right:$\sigma=10^{-1},10^{-2},10^{-3}$). The exact spikes in solid line and the recovered spikes in dashed line. The errors measure the 2-norm difference in spike locations and weights, respectively.