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Multidimensional extrapolated global proximal gradient and applications for image processing

Abdeslem Hafid Bentbib, Khalide Jbilou, Ridwane Tahiri

TL;DR

This paper tackles constrained high-dimensional tensor minimization with a general non-smooth regularizer by extending proximal gradient techniques to the tensor setting. It introduces the Global Tensorial Double Proximal Gradient framework, combining an unconstrained proximal step with Tseng’s projection to a convex set Ω, and derives efficient proximal solves via dual formulations. To overcome slow convergence, two tensor extrapolation schemes, GT-TET and HOSVD-MPE, are integrated in a restarted fashion, yielding accelerated variants of TISTA and TDPG that outperform their non-extrapolated counterparts in color image completion tasks. The results demonstrate that the proposed approach robustly handles a broad class of regularizers (including l1, TV, and nuclear norm) and achieves faster convergence with competitive reconstruction quality, highlighting practical impact for multidimensional image and data completion problems.

Abstract

The proximal gradient method is a generic technique introduced to tackle the non-smoothness in optimization problems, wherein the objective function is expressed as the sum of a differentiable convex part and a non-differentiable regularization term. Such problems with tensor format are of interest in many fields of applied mathematics such as image and video processing. Our goal in this paper is to address the solution of such problems with a more general form of the regularization term. An adapted iterative proximal gradient method is introduced for this purpose. Due to the slowness of the proposed algorithm, we use new tensor extrapolation methods to enhance its convergence. Numerical experiments on color image deblurring are conducted to illustrate the efficiency of our approach.

Multidimensional extrapolated global proximal gradient and applications for image processing

TL;DR

This paper tackles constrained high-dimensional tensor minimization with a general non-smooth regularizer by extending proximal gradient techniques to the tensor setting. It introduces the Global Tensorial Double Proximal Gradient framework, combining an unconstrained proximal step with Tseng’s projection to a convex set Ω, and derives efficient proximal solves via dual formulations. To overcome slow convergence, two tensor extrapolation schemes, GT-TET and HOSVD-MPE, are integrated in a restarted fashion, yielding accelerated variants of TISTA and TDPG that outperform their non-extrapolated counterparts in color image completion tasks. The results demonstrate that the proposed approach robustly handles a broad class of regularizers (including l1, TV, and nuclear norm) and achieves faster convergence with competitive reconstruction quality, highlighting practical impact for multidimensional image and data completion problems.

Abstract

The proximal gradient method is a generic technique introduced to tackle the non-smoothness in optimization problems, wherein the objective function is expressed as the sum of a differentiable convex part and a non-differentiable regularization term. Such problems with tensor format are of interest in many fields of applied mathematics such as image and video processing. Our goal in this paper is to address the solution of such problems with a more general form of the regularization term. An adapted iterative proximal gradient method is introduced for this purpose. Due to the slowness of the proposed algorithm, we use new tensor extrapolation methods to enhance its convergence. Numerical experiments on color image deblurring are conducted to illustrate the efficiency of our approach.
Paper Structure (18 sections, 6 theorems, 60 equations, 22 figures, 2 tables, 6 algorithms)

This paper contains 18 sections, 6 theorems, 60 equations, 22 figures, 2 tables, 6 algorithms.

Table of Contents

  1. Introduction
  2. Preliminaries and notation
  3. Global tensorial double proximal gradient method
  4. Global tensorial double proximal gradient algorithm
  5. Approximate unconstrained solution via Tseng's splitting algorithm
  6. Selection of parameters $\alpha_{k}$ and $\beta_{l}$
  7. Special cases
  8. Case $1$. $\mathcal{L}=id_\mathbb{T}$ and $\phi =\|.\|_{1}$
  9. Case $2$ $\mathcal{L}=\nabla$ and $\phi =\||.\||_{1}$
  10. Case $3$. $\mathcal{L}=\mathcal{\nabla}$ and $\varphi_{\mu} =\mu\||\mathcal{L}( .)\||_{1}+\|| . \||_{*}$
  11. Convergence acceleration using Extrapolation
  12. GT-TET
  13. HOSVD-MPE
  14. Accelerated version of GTPG algorithm
  15. Numerical experiment
  16. ...and 3 more sections

Key Result

Proposition 2.5

parikh40 Let $\mathcal{Y} \in \mathbb{T}$, then there exists a unique minimizer $\mathcal{X}$ for the problem

Figures (22)

  • Figure 6.1: The original images (1st row), the (randomly) $55\%$ incompleted images (2nd row),
  • Figure 6.3: The curves of relative errorr.
  • Figure 6.4: The curves of PSNR.
  • Figure 6.5: The curves of the accelertion rate.
  • Figure 6.6: The original images (1st row) and the uniformly incomplete images (2nd row).
  • ...and 17 more figures

Theorems & Definitions (14)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Proposition 2.5
  • Definition 2.6
  • Proposition 2.7
  • Proposition 3.1
  • proof
  • Proposition 3.2
  • ...and 4 more