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Reconstruction with prior support information and non-Gaussian constraints

Xiaotong Liu, Yiyu Liang

TL;DR

A novel model, termed the Weighted Basis Pursuit Dequantization, which incorporates prior support information by assigning weights on the $\ell_1$ norm in the $\ell_1$ minimization process and replaces the $\ell_2$ norm with the $\ell_p$ norm in the constraint is introduced.

Abstract

In this study, we introduce a novel model, termed the Weighted Basis Pursuit Dequantization ($ω$-BPDQ$_p$), which incorporates prior support information by assigning weights on the $\ell_1$ norm in the $\ell_1$ minimization process and replaces the $\ell_2$ norm with the $\ell_p$ norm in the constraint. This adjustment addresses cases where noise deviates from a Gaussian distribution, such as quantized errors, which are common in practice. We demonstrate that Restricted Isometry Property (RIP$_{p,q}$) and Weighted Robust Null Space Property ($ω$-RNSP$_{p,q}$) ensure stable and robust reconstruction within $ω$-BPDQ$_p$, with the added observation that standard Gaussian random matrices satisfy these properties with high probability. Moreover, we establish a relationship between RIP$_{p,q}$ and $ω$-RNSP$_{p,q}$ that RIP$_{p,q}$ implies $ω$-RNSP$_{p,q}$. Additionally, numerical experiments confirm that the incorporation of weights and the non-Gaussian constraint results in improved reconstruction quality.

Reconstruction with prior support information and non-Gaussian constraints

TL;DR

A novel model, termed the Weighted Basis Pursuit Dequantization, which incorporates prior support information by assigning weights on the norm in the minimization process and replaces the norm with the norm in the constraint is introduced.

Abstract

In this study, we introduce a novel model, termed the Weighted Basis Pursuit Dequantization (-BPDQ), which incorporates prior support information by assigning weights on the norm in the minimization process and replaces the norm with the norm in the constraint. This adjustment addresses cases where noise deviates from a Gaussian distribution, such as quantized errors, which are common in practice. We demonstrate that Restricted Isometry Property (RIP) and Weighted Robust Null Space Property (-RNSP) ensure stable and robust reconstruction within -BPDQ, with the added observation that standard Gaussian random matrices satisfy these properties with high probability. Moreover, we establish a relationship between RIP and -RNSP that RIP implies -RNSP. Additionally, numerical experiments confirm that the incorporation of weights and the non-Gaussian constraint results in improved reconstruction quality.

Paper Structure

This paper contains 8 sections, 12 theorems, 116 equations, 6 figures.

Table of Contents

  1. Introduction
  2. $\omega$-BPDQ$_p$ and Its Reconstruction Guarantee
  3. RIP$_{p,q}$
  4. $\omega$-RNSP$_{p,q}$
  5. With Gaussian matrices
  6. Numerical Implementation
  7. Proximal Optimization
  8. Experiments

Key Result

Theorem 2.2

For any given weight $\omega=[\omega_1,\ldots,\omega_N]^T\in [\theta,1]^N$, $\theta\in(0,1)$, which is defined as w, if the matrix $\Phi$ satisfies RIP$_{p,q}$ of order $2s$ with $\delta_{2s}$, $s\geqslant2$ and $p,q\in[2,+\infty)$, then the vector $x\in\mathbb{C}^N$ and the solution of wbpdqp satis where $\epsilon$ is defined as wbpdqp, $\sigma_s(\boldsymbol{x})_{\omega,1}$ is defined as sigmax,

Figures (6)

  • Figure 1: the original signal, the recover signals and their SNRs for Basis Pursuit and weighted Basis Pursuit with $\gamma=0.5$
  • Figure 2: the error of reconstruction for Basis Pursuit and weighted Basis Pursuit with $\gamma=0.5$
  • Figure 3: the original signal, the recover signals and their SNRs for $\omega$-BPDQ$_p$ with $p=10,2,\infty$
  • Figure 4: DR convergence for $\omega$-BPDQ$_p$ with $p=10,2,\infty$
  • Figure 5: the error of reconstruction for $\omega$-BPDQ$_p$ with $p=10,2,\infty$
  • ...and 1 more figures

Theorems & Definitions (24)

  • Definition 2.1: RIP$_{p,q}$(see Definition 1 in jac2011)
  • Theorem 2.2
  • Lemma 2.3: see Lemma 2 in jac2011
  • proof
  • proof : proof of \ref{['thmripp2']}
  • Remark 2.4
  • Definition 2.5: $\omega$-RNSP$_{p,q}$
  • Theorem 2.6
  • proof
  • Proposition 2.7
  • ...and 14 more