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PreprintCC BY 4.0HAL

From proximal alternating minimization techniques to composite proximal gradient

  • Clara Lage
  • Nelly Pustelnik
  • Audrey Repetti
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Abstract

The discrete Mumford-Shah formulation leads to a challenging optimization problem involving the joint minimization of two variables. This class of problems is commonly addressed through alternating minimization strategies, notably the PALM algorithm and its variants. In this work, we establish a connection between the discrete Mumford-Shah formulation and a class of composite minimization problems for which majorization-minimization algorithms have recently been developed. Under assumptions satisfied by the main applications of interest, we prove the equivalence between these two formulations, thereby providing an alternative perspective on Mumford-Shah-type optimization problems. We then numerically compare the resulting majorization-minimization approach with PALM-based methods in two distinct applications: Modulo Sampling, where the formulation arises in a nonlinear signal reconstruction problem, and the classical problem of image contour from noisy data.

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Cite

BibTeXRIS
BibTeX
@misc{lage2026from,
  title = {{From proximal alternating minimization techniques to composite proximal gradient}},
  author = {Clara Lage and Nelly Pustelnik and Audrey Repetti},
  howpublished = {HAL},
  year = {2026},
  month = sep,
  url = {https://hal.science/hal-05767932v1},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - From proximal alternating minimization techniques to composite proximal gradient
AU  - Clara Lage
AU  - Nelly Pustelnik
AU  - Audrey Repetti
PY  - 2026
DA  - 2026/09/28
PB  - HAL
UR  - https://hal.science/hal-05767932v1
AB  - The discrete Mumford-Shah formulation leads to a challenging optimization problem involving the joint minimization of two variables. This class of problems is commonly addressed through alternating minimization strategies, notably the PALM algorithm and its variants. In this work, we establish a connection between the discrete Mumford-Shah formulation and a class of composite minimization problems for which majorization-minimization algorithms have recently been developed. Under assumptions satisfied by the main applications of interest, we prove the equivalence between these two formulations, thereby providing an alternative perspective on Mumford-Shah-type optimization problems. We then numerically compare the resulting majorization-minimization approach with PALM-based methods in two distinct applications: Modulo Sampling, where the formulation arises in a nonlinear signal reconstruction problem, and the classical problem of image contour from noisy data.
ER  -