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A note on the convergence of RED algorithms under minimal hypotheses and open questions

Yann Traonmilin, J. -F Aujol

Abstract

In this note, we give a convergence result for a modified ''regularization-by-denoising''(RED) algorithm under a restricted isometry condition on measurements and a restricted Lipschitz condition on the considered deep projective prior. This study leads to open questions about the convergence of RED algorithms.

A note on the convergence of RED algorithms under minimal hypotheses and open questions

Abstract

In this note, we give a convergence result for a modified ''regularization-by-denoising''(RED) algorithm under a restricted isometry condition on measurements and a restricted Lipschitz condition on the considered deep projective prior. This study leads to open questions about the convergence of RED algorithms.
Paper Structure (4 sections, 1 theorem, 20 equations)

This paper contains 4 sections, 1 theorem, 20 equations.

Table of Contents

  1. Introduction
  2. Notations
  3. Convergence of a modified RED algorithm
  4. On the convergence of classical RED

Key Result

Theorem 3.1

Suppose $\Sigma$ is a proximinal set. Consider modified RED iterations def:mod_RED with $\|P_\Sigma - P_\Sigma^\perp\|_2 \leq \eta$. Let $\beta = \beta_\Sigma(P_\Sigma)$ and $\delta = \delta_\Sigma(\mu A^TA)$. Suppose $r = (\delta \beta + |1-\lambda|\| I - \mu A^TA\|_{\mathrm{op}}) < 1$, then

Theorems & Definitions (6)

  • Definition 2.1
  • Definition 2.2: Generalized projection
  • Definition 2.3: Proximinal sets and orthogonal projections
  • Definition 2.4: Restricted Lipschitz property
  • Theorem 3.1
  • proof