# Taming Nonconvexity with Epigraph Lifting

arXiv: https://arxiv.org/abs/2610.16429 · HTML: https://mlsanity.com/paper/2610.16429/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 20 of its 30 references that are in the map.
- First posted on arXiv 12 Oct 2026.

## Builds on (most cited first, 20 of 20)

- [Towards Evaluating the Robustness of Neural Networks](https://mlsanity.com/paper/1608.04644/) (2016)
- [Efficient Neural Network Robustness Certification with General Activation Functions](https://mlsanity.com/paper/1811.00866/) (2018)
- [Input Convex Neural Networks](https://mlsanity.com/paper/1609.07152/) (2016)
- [GAN Inversion: A Survey](https://mlsanity.com/paper/2101.05278/) (2021)
- [The Little Engine that Could: Regularization by Denoising (RED)](https://mlsanity.com/paper/1611.02862/) (2016)
- [A Convex Relaxation Barrier to Tight Robustness Verification of Neural Networks](https://mlsanity.com/paper/1902.08722/) (2019)
- [Adversarial Regularizers in Inverse Problems](https://mlsanity.com/paper/1805.11572/) (2018)
- [A Neural-Network-Based Convex Regularizer for Inverse Problems](https://mlsanity.com/paper/2211.12461/) (2022)
- [Learning Weakly Convex Regularizers for Convergent Image-Reconstruction Algorithms](https://mlsanity.com/paper/2308.10542/) (2023)
- [Lifted Neural Networks](https://mlsanity.com/paper/1805.01532/) (2018)
- [Fenchel Lifted Networks: A Lagrange Relaxation of Neural Network Training](https://mlsanity.com/paper/1811.08039/) (2018)
- [Single Loop Gaussian Homotopy Method for Non-convex Optimization](https://mlsanity.com/paper/2203.05717/) (2022)
- [Deep Equilibrium Learning of Explicit Regularizers for Imaging Inverse Problems](https://mlsanity.com/paper/2303.05386/) (2023)
- [Learning Regularization Functionals for Inverse Problems: A Comparative Study](https://mlsanity.com/paper/2510.01755/) (2025)
- [Weakly Convex Regularisers for Inverse Problems: Convergence of Critical Points and Primal-Dual Optimisation](https://mlsanity.com/paper/2402.01052/) (2024)
- [A primal-dual algorithm for image reconstruction with input-convex neural network regularizers](https://mlsanity.com/paper/2410.12441/) (2024)
- [Using positive spanning sets to achieve d-stationarity with the Boosted DC Algorithm](https://mlsanity.com/paper/1907.11471/) (2019)
- [Exact continuous relaxations of l0-regularized criteria with non-quadratic data terms](https://mlsanity.com/paper/2402.06483/) (2024)
- [Explicit and Implicit Graduated Optimization in Deep Neural Networks](https://mlsanity.com/paper/2412.11501/) (2024)
- [A Preconditioned Difference of Convex Algorithm for Truncated Quadratic Regularization with Application to Imaging](https://mlsanity.com/paper/2001.01268/) (2020)

## Abstract (arXiv)

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 difference-of-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 barρ. 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.