Taming Nonconvexity with Epigraph Lifting

Abstract

We consider inverse problems regularized by a trained neural network with convex activations and unrestricted weights. The resulting optimization problem is generally nonconvex, and gradient-based methods may converge to poor local minima. We introduce a penalized epigraph lifting of the network and show that the lifted problem admits a differenceof-convex (DC) decomposition. For Lipschitz continuous activations, we establish finite exactness: the original objective is recovered exactly at a finite and explicitly computable penalty threshold ρ. Building on this structure, we propose H-DC (Homotopic DC), a homotopic scheme that solves a sequence of DC subproblems with increasing ρ, warm-starting each level from the previous solution. Numerical experiments on synthetic problems, a manifold denoiser regularizer, and image inpainting with a learned Mumford-Shah regularizer show that H-DC consistently escapes poor local minima and reaches solutions with lower objective values than gradient-based methods from the same initialization.

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BibTeX RIS

BibTeX

@misc{lage2026taming,
  title = {{Taming Nonconvexity with Epigraph Lifting}},
  author = {Clara Lage and Shing Wong and Matthias Ehrhardt Hok},
  howpublished = {HAL},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.16429},
  eprint = {2610.16429},
  archivePrefix = {arXiv},
  url = {https://hal.science/hal-05789819v1},
  note = {CC BY 4.0},
}

RIS

TY  - UNPB
TI  - Taming Nonconvexity with Epigraph Lifting
AU  - Clara Lage
AU  - Shing Wong
AU  - Matthias Ehrhardt Hok
PY  - 2026
DA  - 2026/10/09
PB  - HAL
DO  - 10.48550/arxiv.2610.16429
UR  - https://hal.science/hal-05789819v1
AB  - We consider inverse problems regularized by a trained neural network with convex activations and unrestricted weights. The resulting optimization problem is generally nonconvex, and gradient-based methods may converge to poor local minima. We introduce a penalized epigraph lifting of the network and show that the lifted problem admits a differenceof-convex (DC) decomposition. For Lipschitz continuous activations, we establish finite exactness: the original objective is recovered exactly at a finite and explicitly computable penalty threshold ρ. Building on this structure, we propose H-DC (Homotopic DC), a homotopic scheme that solves a sequence of DC subproblems with increasing ρ, warm-starting each level from the previous solution. Numerical experiments on synthetic problems, a manifold denoiser regularizer, and image inpainting with a learned Mumford-Shah regularizer show that H-DC consistently escapes poor local minima and reaches solutions with lower objective values than gradient-based methods from the same initialization.
ER  -

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