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Sparse free deconvolution under unknown noise level via eigenmatrix

Lexing Ying

TL;DR

The paper tackles blind free deconvolution of sparse spectral measures under unknown noise levels in both additive (deformed Wigner) and multiplicative (deformed MP) regimes. It integrates the eigenmatrix method for unstructured sparse recovery with a noise-detection mechanism that hinges on the singular-value structure of an intermediate matrix, enabling simultaneous estimation of the noise parameter and the sparse spectral locations/weights. The approach yields principled procedures for known and unknown noise (or dimension-to-sample ratio) in both settings, accompanied by numerical demonstrations showing accurate reconstruction of the sparse spectrum $\mu_A$ from observed $\mu_C$. This contributes a practical framework for spectral estimation and covariance-type problems where noise characteristics are inaccessible, with potential impact on statistics and data science applications requiring robust deconvolution under uncertainty.

Abstract

This note considers the spectral estimation problems of sparse spectral measures under unknown noise levels. The main technical tool is the eigenmatrix method for solving unstructured sparse recovery problems. When the noise level is determined, the free deconvolution reduces the problem to an unstructured sparse recovery problem to which the eigenmatrix method can be applied. To determine the unknown noise level, we propose an optimization problem based on the singular values of an intermediate matrix of the eigenmatrix method. Numerical results are provided for both the additive and multiplicative free deconvolutions.

Sparse free deconvolution under unknown noise level via eigenmatrix

TL;DR

The paper tackles blind free deconvolution of sparse spectral measures under unknown noise levels in both additive (deformed Wigner) and multiplicative (deformed MP) regimes. It integrates the eigenmatrix method for unstructured sparse recovery with a noise-detection mechanism that hinges on the singular-value structure of an intermediate matrix, enabling simultaneous estimation of the noise parameter and the sparse spectral locations/weights. The approach yields principled procedures for known and unknown noise (or dimension-to-sample ratio) in both settings, accompanied by numerical demonstrations showing accurate reconstruction of the sparse spectrum from observed . This contributes a practical framework for spectral estimation and covariance-type problems where noise characteristics are inaccessible, with potential impact on statistics and data science applications requiring robust deconvolution under uncertainty.

Abstract

This note considers the spectral estimation problems of sparse spectral measures under unknown noise levels. The main technical tool is the eigenmatrix method for solving unstructured sparse recovery problems. When the noise level is determined, the free deconvolution reduces the problem to an unstructured sparse recovery problem to which the eigenmatrix method can be applied. To determine the unknown noise level, we propose an optimization problem based on the singular values of an intermediate matrix of the eigenmatrix method. Numerical results are provided for both the additive and multiplicative free deconvolutions.
Paper Structure (11 sections, 28 equations, 8 figures)

This paper contains 11 sections, 28 equations, 8 figures.

Table of Contents

  1. Introduction
  2. Related work.
  3. Contributions.
  4. Eigenmatrix
  5. Additive deconvolution
  6. Known $\sigma$
  7. Unknown $\sigma$
  8. Multiplicative deconvolution
  9. Known $q$
  10. Unknown $q$
  11. Discussion

Figures (8)

  • Figure 3.1: Logarithm of singular values of $T(\sigma)$ as a function of $\sigma$. Left: large $N$ limit. Middle: $N=1024$. Right: loss landscape.
  • Figure 3.2: Left: the histogram of the eigenvalues of $C$. Middle: the loss landscape as a function of $\sigma$, and the global minimum is around the ground truth $\sigma$. Right: the exact (red) and reconstructed (blue) ${\mu_A}$.
  • Figure 3.3: Left: the histogram of the eigenvalues of $C$. Middle: the loss landscape as a function of $\sigma$, and the global minimum is around the ground truth $\sigma$. Right: the exact (red) and reconstructed (blue) ${\mu_A}$.
  • Figure 3.4: Left: the histogram of the eigenvalues of $C$. Middle: the loss landscape as a function of $\sigma$, and the global minimum is around the ground truth $\sigma$. Right: the exact (red) and reconstructed (blue) ${\mu_A}$.
  • Figure 4.1: Logarithm of singular values of $T(q)$ as a function of $q$. Left: large $N$ limit. Middle: $N=1024$. Right: loss landscape.
  • ...and 3 more figures

Theorems & Definitions (6)

  • Example 3.1
  • Example 3.2
  • Example 3.3
  • Example 4.1
  • Example 4.2
  • Example 4.3