LDLT $\mathcal{L}$-Lipschitz Network: Generalized Deep End-To-End Lipschitz Network Construction

• URL: http://arxiv.org/abs/2512.05915v1
• Date: Fri, 05 Dec 2025 17:51:08 GMT
• Title: LDLT $\mathcal{L}$-Lipschitz Network: Generalized Deep End-To-End Lipschitz Network Construction
• Authors: Marius F. R. Juston, Ramavarapu S. Sreenivas, Dustin Nottage, Ahmet Soylemezoglu,
• Abstract summary: Controlling the Lipschitz constant in neural networks has emerged as an essential area of research to enhance adversarial robustness and network certifiability.<n>This paper presents a rigorous approach to the general design of $mathcalL$-Lipschitz deep residual networks using a Linear Matrix Inequality (LMI) framework.
• Score: 3.744861320984297
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: 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 $\mathcal{L}$-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 $\mathcal{L}$-Lipschitz continuity; however, using a new $LDL^\top$ decomposition approach for certifying LMI feasibility, we extend the construction of $\mathcal{L}$-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.

Related papers

• Hierarchical Zero-Order Optimization for Deep Neural Networks [33.991611257471114]
  We propose Hierarchical Zeroth-Order (HZO) optimization, a novel divide-and-conquer strategy that decomposes the depth dimension of the network.<n>We prove that HZO reduces the query complexity from $O(ML2)$ to $O(ML log L)$ for a network of width $M$ and depth $L$, representing a significant leap over existing ZO methodologies.
  arXiv  Detail & Related papers  (2026-02-11T07:56:07Z)
• A Scalable Approach for Safe and Robust Learning via Lipschitz-Constrained Networks [2.8960888722909566]
  Lipschitz-constrained global training constraints for neural networks (NNs) are proposed.<n>We show that the proposed formulation of Lipschitz-constrained NNs can be significantly improved.
  arXiv  Detail & Related papers  (2025-06-30T15:42:23Z)
• L-Lipschitz Gershgorin ResNet Network [0.0]
  This paper uses a rigorous approach to design $mathcalL$-Lipschitz deep residual networks.<n>The ResNet architecture was reformulated as a pseudo-tri-diagonal LMI with off-diagonal elements.<n>To address the lack of explicit eigenvalue computations for such matrix structures, the Gershgorin circle theorem was employed.
  arXiv  Detail & Related papers  (2025-02-28T17:57:57Z)
• Optimization Guarantees of Unfolded ISTA and ADMM Networks With Smooth Soft-Thresholding [57.71603937699949]
  We study optimization guarantees, i.e., achieving near-zero training loss with the increase in the number of learning epochs. We show that the threshold on the number of training samples increases with the increase in the network width.
  arXiv  Detail & Related papers  (2023-09-12T13:03:47Z)
• A Unified Algebraic Perspective on Lipschitz Neural Networks [88.14073994459586]
  This paper introduces a novel perspective unifying various types of 1-Lipschitz neural networks. We show that many existing techniques can be derived and generalized via finding analytical solutions of a common semidefinite programming (SDP) condition. Our approach, called SDP-based Lipschitz Layers (SLL), allows us to design non-trivial yet efficient generalization of convex potential layers.
  arXiv  Detail & Related papers  (2023-03-06T14:31:09Z)
• Rethinking Lipschitz Neural Networks for Certified L-infinity Robustness [33.72713778392896]
  We study certified $ell_infty$ from a novel perspective of representing Boolean functions. We develop a unified Lipschitz network that generalizes prior works, and design a practical version that can be efficiently trained.
  arXiv  Detail & Related papers  (2022-10-04T17:55:27Z)
• Robust Training and Verification of Implicit Neural Networks: A Non-Euclidean Contractive Approach [64.23331120621118]
  This paper proposes a theoretical and computational framework for training and robustness verification of implicit neural networks. We introduce a related embedded network and show that the embedded network can be used to provide an $ell_infty$-norm box over-approximation of the reachable sets of the original network. We apply our algorithms to train implicit neural networks on the MNIST dataset and compare the robustness of our models with the models trained via existing approaches in the literature.
  arXiv  Detail & Related papers  (2022-08-08T03:13:24Z)
• Singular Value Perturbation and Deep Network Optimization [29.204852309828006]
  We develop new theoretical results on matrix perturbation to shed light on the impact of architecture on the performance of a deep network. In particular, we explain what deep learning practitioners have long observed empirically: the parameters of some deep architectures are easier to optimize than others. A direct application of our perturbation results explains analytically why a ResNet is easier to optimize than a ConvNet.
  arXiv  Detail & Related papers  (2022-03-07T02:09:39Z)
• Training Certifiably Robust Neural Networks with Efficient Local Lipschitz Bounds [99.23098204458336]
  Certified robustness is a desirable property for deep neural networks in safety-critical applications. We show that our method consistently outperforms state-of-the-art methods on MNIST and TinyNet datasets.
  arXiv  Detail & Related papers  (2021-11-02T06:44:10Z)
• Scalable Lipschitz Residual Networks with Convex Potential Flows [120.27516256281359]
  We show that using convex potentials in a residual network gradient flow provides a built-in $1$-Lipschitz transformation. A comprehensive set of experiments on CIFAR-10 demonstrates the scalability of our architecture and the benefit of our approach for $ell$ provable defenses.
  arXiv  Detail & Related papers  (2021-10-25T07:12:53Z)
• Automatic Perturbation Analysis for Scalable Certified Robustness and Beyond [171.07853346630057]
  Linear relaxation based perturbation analysis (LiRPA) for neural networks has become a core component in robustness verification and certified defense. We develop an automatic framework to enable perturbation analysis on any neural network structures. We demonstrate LiRPA based certified defense on Tiny ImageNet and Downscaled ImageNet.
  arXiv  Detail & Related papers  (2020-02-28T18:47:43Z)

This list is automatically generated from the titles and abstracts of the papers in this site.

This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.