# LDLT L-Lipschitz Network: Generalized Deep End-To-End Lipschitz Network Construction

arXiv: https://arxiv.org/abs/2512.05915 · HTML: https://mlsanity.com/paper/2512.05915/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 39 of its 59 references that are in the map.
- First posted on arXiv 5 Dec 2025.

## Builds on (most cited first, 30 of 39)

- [Deep Residual Learning for Image Recognition](https://mlsanity.com/paper/1512.03385/) (2015)
- [U-Net: Convolutional Networks for Biomedical Image Segmentation](https://mlsanity.com/paper/1505.04597/) (2015)
- [High-Resolution Image Synthesis with Latent Diffusion Models](https://mlsanity.com/paper/2112.10752/) (2021)
- [Explaining and Harnessing Adversarial Examples](https://mlsanity.com/paper/1412.6572/) (2014)
- [Squeeze-and-Excitation Networks](https://mlsanity.com/paper/1709.01507/) (2017)
- [MobileNets: Efficient Convolutional Neural Networks for Mobile Vision Applications](https://mlsanity.com/paper/1704.04861/) (2017)
- [Delving Deep into Rectifiers: Surpassing Human-Level Performance on ImageNet Classification](https://mlsanity.com/paper/1502.01852/) (2015)
- [Gaussian Error Linear Units (GELUs)](https://mlsanity.com/paper/1606.08415/) (2016)
- [Wide Residual Networks](https://mlsanity.com/paper/1605.07146/) (2016)
- [Aggregated Residual Transformations for Deep Neural Networks](https://mlsanity.com/paper/1611.05431/) (2016)
- [Searching for MobileNetV3](https://mlsanity.com/paper/1905.02244/) (2019)
- [Spectral Normalization for Generative Adversarial Networks](https://mlsanity.com/paper/1802.05957/) (2018)
- [Inception-v4, Inception-ResNet and the Impact of Residual Connections on Learning](https://mlsanity.com/paper/1602.07261/) (2016)
- [Fast and Accurate Deep Network Learning by Exponential Linear Units (ELUs)](https://mlsanity.com/paper/1511.07289/) (2015)
- [An Introduction to Variational Autoencoders](https://mlsanity.com/paper/1906.02691/) (2019)
- [Sigmoid-Weighted Linear Units for Neural Network Function Approximation in Reinforcement Learning](https://mlsanity.com/paper/1702.03118/) (2017)
- [Spectrally-normalized margin bounds for neural networks](https://mlsanity.com/paper/1706.08498/) (2017)
- [Self-Normalizing Neural Networks](https://mlsanity.com/paper/1706.02515/) (2017)
- [BinaryConnect: Training Deep Neural Networks with binary weights during propagations](https://mlsanity.com/paper/1511.00363/) (2015)
- [Empirical Evaluation of Rectified Activations in Convolutional Network](https://mlsanity.com/paper/1505.00853/) (2015)
- [Regularisation of Neural Networks by Enforcing Lipschitz Continuity](https://mlsanity.com/paper/1804.04368/) (2018)
- [Efficient and Accurate Estimation of Lipschitz Constants for Deep Neural Networks](https://mlsanity.com/paper/1906.04893/) (2019)
- [Lipschitz-Margin Training: Scalable Certification of Perturbation Invariance for Deep Neural Networks](https://mlsanity.com/paper/1802.04034/) (2018)
- [Long Short-Term Memory Based Recurrent Neural Network Architectures for Large Vocabulary Speech Recognition](https://mlsanity.com/paper/1402.1128/) (2014)
- [Analysis and Design of Optimization Algorithms via Integral Quadratic Constraints](https://mlsanity.com/paper/1408.3595/) (2014)
- [Monotone operator equilibrium networks](https://mlsanity.com/paper/2006.08591/) (2020)
- [Continuously Differentiable Exponential Linear Units](https://mlsanity.com/paper/1704.07483/) (2017)
- [A Dynamical System Perspective for Lipschitz Neural Networks](https://mlsanity.com/paper/2110.12690/) (2021)
- [A Unified Algebraic Perspective on Lipschitz Neural Networks](https://mlsanity.com/paper/2303.03169/) (2023)
- [Direct Parameterization of Lipschitz-Bounded Deep Networks](https://mlsanity.com/paper/2301.11526/) (2023)

## Abstract (arXiv)

Deep residual networks (ResNets) have demonstrated outstanding success in computer vision tasks, attributed to their ability to maintain gradient flow through deep architectures. Simultaneously, controlling the Lipschitz constant in neural networks has emerged as an essential area of research to enhance adversarial robustness and network certifiability. This paper presents a rigorous approach to the general design of L-Lipschitz deep residual networks using a Linear Matrix Inequality (LMI) framework. Initially, the ResNet architecture was reformulated as a cyclic tridiagonal LMI, and closed-form constraints on network parameters were derived to ensure L-Lipschitz continuity; however, using a new LDL^top decomposition approach for certifying LMI feasibility, we extend the construction of L-Lipchitz networks to any other nonlinear architecture. Our contributions include a provable parameterization methodology for constructing Lipschitz-constrained residual networks and other hierarchical architectures. Cholesky decomposition is also used for efficient parameterization. These findings enable robust network designs applicable to adversarial robustness, certified training, and control systems. The LDL^top formulation is shown to be a tight relaxation of the SDP-based network, maintaining full expressiveness and achieving 3%-13% accuracy gains over SLL Layers on 121 UCI data sets.