# SOC-ICNN: From Polyhedral to Conic Geometry for Learning Convex Surrogate Functions

arXiv: https://arxiv.org/abs/2604.22355 · HTML: https://mlsanity.com/paper/2604.22355/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 20 of its 21 references that are in the map.
- First posted on arXiv 24 Apr 2026.

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

- [OptNet: Differentiable Optimization as a Layer in Neural Networks](https://mlsanity.com/paper/1703.00443/) (2017)
- [Differentiable Convex Optimization Layers](https://mlsanity.com/paper/1910.12430/) (2019)
- [Input Convex Neural Networks](https://mlsanity.com/paper/1609.07152/) (2016)
- [Understanding Deep Neural Networks with Rectified Linear Units](https://mlsanity.com/paper/1611.01491/) (2016)
- [Optimal transport mapping via input convex neural networks](https://mlsanity.com/paper/1908.10962/) (2019)
- [Optimal Control Via Neural Networks: A Convex Approach](https://mlsanity.com/paper/1805.11835/) (2018)
- [Flexible Differentiable Optimization via Model Transformations](https://mlsanity.com/paper/2206.06135/) (2022)
- [Principled Weight Initialisation for Input-Convex Neural Networks](https://mlsanity.com/paper/2312.12474/) (2023)
- [On the Depth of Monotone ReLU Neural Networks and ICNNs](https://mlsanity.com/paper/2505.06169/) (2025)
- [BPQP: A Differentiable Convex Optimization Framework for Efficient End-to-End Learning](https://mlsanity.com/paper/2411.19285/) (2024)
- [Learning Optimal Power Flow Value Functions with Input-Convex Neural Networks](https://mlsanity.com/paper/2310.04605/) (2023)
- [Convexity in ReLU Neural Networks: beyond ICNNs?](https://mlsanity.com/paper/2501.03017/) (2025)
- [Input Convex Kolmogorov Arnold Networks](https://mlsanity.com/paper/2505.21208/) (2025)
- [Real-Time Machine-Learning-Based Optimization Using Input Convex Long Short-Term Memory Network](https://mlsanity.com/paper/2311.07202/) (2023)
- [Fast and Reliable N-k Contingency Screening with Input-Convex Neural Networks](https://mlsanity.com/paper/2410.00796/) (2024)
- [ICNN-enhanced 2SP: Leveraging input convex neural networks for solving two-stage stochastic programming](https://mlsanity.com/paper/2505.05261/) (2025)
- [Differentiable Convex Optimization Layers in Neural Architectures: Foundations and Perspectives](https://mlsanity.com/paper/2412.20679/) (2024)
- [Learning Parametric Convex Functions](https://mlsanity.com/paper/2506.04183/) (2025)
- [Input Convex Lipschitz Recurrent Neural Networks for Robust and Efficient Process Modeling and Optimization](https://mlsanity.com/paper/2401.07494/) (2024)
- [Parametric Nonconvex Optimization via Convex Surrogates](https://mlsanity.com/paper/2604.05640/) (2026)

## Built on by (most cited first, 1 of 1)

- [Dual Certified White-Box Inference for Input Convex Neural Networks](https://mlsanity.com/paper/2605.04722/) (2026)

## Abstract (arXiv)

Classical ReLU-based Input Convex Neural Networks (ICNNs) are equivalent to the optimal value functions of Linear Programming (LP). This intrinsic structural equivalence restricts their representational capacity to piecewise-linear polyhedral functions. To overcome this representational bottleneck, we propose the SOC-ICNN, an architecture that generalizes the underlying optimization class from LP to Second-Order Cone Programming (SOCP). By explicitly injecting positive semi-definite curvature and Euclidean norm-based conic primitives, our formulation introduces native smooth curvature into the representation while preserving a rigorous optimization-theoretic interpretation. We formally prove that SOC-ICNNs strictly expand the representational space of ReLU-ICNNs without increasing the asymptotic order of forward-pass complexity. Extensive experiments demonstrate that SOC-ICNN substantially improves function approximation, while delivering competitive downstream decision quality. The code is available at https://anonymous.4open.science/r/SOC-ICNN-4B18/.