# Neural network training under semidefinite constraints

arXiv: https://arxiv.org/abs/2201.00632 · HTML: https://mlsanity.com/paper/2201.00632/

## Key facts

- Cited by 11 papers in the map (data through 14 Oct 2026).
- Builds on 15 of its 27 references that are in the map.
- First posted on arXiv 3 Jan 2022.

## Builds on (most cited first, 15 of 15)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [Intriguing properties of neural networks](https://mlsanity.com/paper/1312.6199/) (2013)
- [Convex optimization](https://mlsanity.com/paper/2106.01946/) (2021)
- [Improved Training of Wasserstein GANs](https://mlsanity.com/paper/1704.00028/) (2017)
- [Efficient and Accurate Estimation of Lipschitz Constants for Deep Neural Networks](https://mlsanity.com/paper/1906.04893/) (2019)
- [Safety Verification and Robustness Analysis of Neural Networks via Quadratic Constraints and Semidefinite Programming](https://mlsanity.com/paper/1903.01287/) (2019)
- [Training robust neural networks using Lipschitz bounds](https://mlsanity.com/paper/2005.02929/) (2020)
- [The Singular Values of Convolutional Layers](https://mlsanity.com/paper/1805.10408/) (2018)
- [Label-Free Supervision of Neural Networks with Physics and Domain Knowledge](https://mlsanity.com/paper/1609.05566/) (2016)
- [Stability Analysis using Quadratic Constraints for Systems with Neural Network Controllers](https://mlsanity.com/paper/2006.07579/) (2020)
- [Imposing Hard Constraints on Deep Networks: Promises and Limitations](https://mlsanity.com/paper/1706.02025/) (2017)
- [Imitation Learning with Stability and Safety Guarantees](https://mlsanity.com/paper/2012.09293/) (2020)
- [Certifying Incremental Quadratic Constraints for Neural Networks via Convex Optimization](https://mlsanity.com/paper/2012.05981/) (2020)
- [Offset-free setpoint tracking using neural network controllers](https://mlsanity.com/paper/2011.14006/) (2020)
- [Linear systems with neural network nonlinearities: Improved stability analysis via acausal Zames-Falb multipliers](https://mlsanity.com/paper/2103.17106/) (2021)

## Built on by (most cited first, 11 of 11)

- [RAYEN: Imposition of Hard Convex Constraints on Neural Networks](https://mlsanity.com/paper/2307.08336/) (2023)
- [Lipschitz constant estimation for 1D convolutional neural networks](https://mlsanity.com/paper/2211.15253/) (2022)
- [Lipschitz-bounded 1D convolutional neural networks using the Cayley transform and the controllability Gramian](https://mlsanity.com/paper/2303.11835/) (2023)
- [Synthesis of Stabilizing Recurrent Equilibrium Network Controllers](https://mlsanity.com/paper/2204.00122/) (2022)
- [Synthesizing Neural Network Controllers with Closed-Loop Dissipativity Guarantees](https://mlsanity.com/paper/2404.07373/) (2024)
- [State space representations of the Roesser type for convolutional layers](https://mlsanity.com/paper/2403.11938/) (2024)
- [LipKernel: Lipschitz-Bounded Convolutional Neural Networks via Dissipative Layers](https://mlsanity.com/paper/2410.22258/) (2024)
- [Identification For Control Based on Neural Networks: Approximately Linearizable Models](https://mlsanity.com/paper/2409.15858/) (2024)
- [A Scalable Approach for Safe and Robust Learning via Lipschitz-Constrained Networks](https://mlsanity.com/paper/2506.23977/) (2025)
- [Scalable Incremental Robustness Analysis of Neural Network Feedback Systems](https://mlsanity.com/paper/2609.22576/) (2026)
- [Learning Provable Neural Network Observer for Uncertain Dynamical Systems](https://mlsanity.com/paper/2609.30819/) (2026)

## Abstract (arXiv)

This paper is concerned with the training of neural networks (NNs) under semidefinite constraints, which allows for NN training with robustness and stability guarantees. In particular, we focus on Lipschitz bounds for NNs. Exploiting the banded structure of the underlying matrix constraint, we set up an efficient and scalable training scheme for NN training problems of this kind based on interior point methods. Our implementation allows to enforce Lipschitz constraints in the training of large-scale deep NNs such as Wasserstein generative adversarial networks (WGANs) via semidefinite constraints. In numerical examples, we show the superiority of our method and its applicability to WGAN training.