# Parametrizing Convex Sets Using Sublinear Neural Networks

arXiv: https://arxiv.org/abs/2605.03520 · HTML: https://mlsanity.com/paper/2605.03520/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 20 of its 46 references that are in the map.
- First posted on arXiv 5 May 2026.

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

- [PyTorch: An Imperative Style, High-Performance Deep Learning Library](https://mlsanity.com/paper/1912.01703/) (2019)
- [Convex optimization](https://mlsanity.com/paper/2106.01946/) (2021)
- [The Deep Ritz method: A deep learning-based numerical algorithm for solving variational problems](https://mlsanity.com/paper/1710.00211/) (2017)
- [Input Convex Neural Networks](https://mlsanity.com/paper/1609.07152/) (2016)
- [Maxout Networks](https://mlsanity.com/paper/1302.4389/) (2013)
- [CvxNet: Learnable Convex Decomposition](https://mlsanity.com/paper/1909.05736/) (2019)
- [Frame Averaging for Invariant and Equivariant Network Design](https://mlsanity.com/paper/2110.03336/) (2021)
- [Weak baselines and reporting biases lead to overoptimism in machine learning for fluid-related partial differential equations](https://mlsanity.com/paper/2407.07218/) (2024)
- [Can Physics-Informed Neural Networks beat the Finite Element Method?](https://mlsanity.com/paper/2302.04107/) (2023)
- [Topology Optimization via Machine Learning and Deep Learning: A Review](https://mlsanity.com/paper/2210.10782/) (2022)
- [On convex decision regions in deep network representations](https://mlsanity.com/paper/2305.17154/) (2023)
- [Parametric Shape Optimization using the Support Function](https://mlsanity.com/paper/1809.00254/) (2018)
- [Volume-preserving geometric shape optimization of the Dirichlet energy using variational neural networks](https://mlsanity.com/paper/2407.19064/) (2024)
- [Numerical approximation of optimal convex shapes](https://mlsanity.com/paper/1810.10735/) (2018)
- [Numerical shape optimization among convex sets](https://mlsanity.com/paper/2203.06981/) (2022)
- [Optimization of Neumann Eigenvalues under convexity and geometric constraints](https://mlsanity.com/paper/2402.04117/) (2024)
- [Improved description of Blaschke--Santal'o diagrams via numerical shape optimization](https://mlsanity.com/paper/2501.01228/) (2025)
- [Meshless Shape Optimization using Neural Networks and Partial Differential Equations on Graphs](https://mlsanity.com/paper/2502.14821/) (2025)
- [Geometric properties of optimizers for the maximum gradient of the torsion function](https://mlsanity.com/paper/2512.09400/) (2025)
- [Numerical exploration of the range of shape functionals using neural networks](https://mlsanity.com/paper/2602.14881/) (2026)

## Abstract (arXiv)

We propose a neural parameterization of convex sets by learning sublinear (positively homogeneous and convex) functions. Our networks implicitly represent both the support and gauge functions of a convex body. We prove a universal approximation theorem for convex sets under this parametrization. Empirically, we demonstrate the method on shape optimization and inverse design tasks, achieving accurate reconstruction of target shapes.