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

Marius F. R. Juston, Ramavarapu S. Sreenivas, Dustin Nottage, Ahmet Soylemezoglu

TL;DR

This work introduces a novel LDL$^\\top$-based Linear Matrix Inequality framework to design general $\\mathcal{L}$-Lipschitz deep residual networks, enabling end-to-end parameterization that preserves expressiveness while guaranteeing Lipschitz continuity. By reformulating ResNets as LMI constraints and exploiting block LDL$^\\top$ decomposition with positive diagonal blocks, the authors provide closed-form parameterizations and efficient computations (via Cholesky/Woodbury identities) that extend Lipschitz guarantees to inner layers and CNNs. Empirical results on 121 UCI datasets show competitive or superior certified robustness and accuracy compared to SDP-based layers, with LDLT variants achieving 3–13% gains in certified settings. The framework supports a broad range of architectures (including linear nets and U-Nets) and offers a scalable path toward robust, certifiable neural networks for adversarial defense and safety-critical control tasks.

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.

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

TL;DR

This work introduces a novel LDL-based Linear Matrix Inequality framework to design general -Lipschitz deep residual networks, enabling end-to-end parameterization that preserves expressiveness while guaranteeing Lipschitz continuity. By reformulating ResNets as LMI constraints and exploiting block LDL decomposition with positive diagonal blocks, the authors provide closed-form parameterizations and efficient computations (via Cholesky/Woodbury identities) that extend Lipschitz guarantees to inner layers and CNNs. Empirical results on 121 UCI datasets show competitive or superior certified robustness and accuracy compared to SDP-based layers, with LDLT variants achieving 3–13% gains in certified settings. The framework supports a broad range of architectures (including linear nets and U-Nets) and offers a scalable path toward robust, certifiable neural networks for adversarial defense and safety-critical control tasks.

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 -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 -Lipschitz continuity; however, using a new decomposition approach for certifying LMI feasibility, we extend the construction of -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 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.

Paper Structure

This paper contains 25 sections, 29 theorems, 120 equations, 2 figures, 12 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Contributions
  3. LMI Formulation
  4. LDL Decomposition
  5. Block Decompositions
  6. Constraints
  7. Variable Parameterization
  8. Diagonal Block Simplifications
  9. B Expansion
  10. Recursive Normalizer
  11. Computation Extensions
  12. Square Root Computation
  13. Convolution Layers
  14. Other Architectures
  15. Linear Network
  16. ...and 10 more sections

Key Result

Theorem 1

Given a real symmetric positive-definite matrix, the factorization may be written as where $\boldsymbol{L}$ is a lower unit triangular (unitrangular) matrix, and $\boldsymbol{D}$ is a (block) diagonal matrix. If $\boldsymbol{A}$ is positive definite, then $\boldsymbol{D}$ will also be positive definite Watkins2002. Where the following recursive relations apply for the entries of $\b

Figures (2)

  • Figure 1: Structure of the factors in the block $\mathrm{LDL}^\top$ decomposition used to certify positive semidefiniteness of the LMI. Visualizations are generated from a randomly initialized $\mathrm{LDL}^\top$ network with input/output dimension $4$ and hidden layer widths $[32,\,64,\,256,\,256]$.
  • Figure 2: $B$ matrix artifacting in network architecture with input and output size of $4$ with layer widths of $[32, 64, 128]$ respectively

Theorems & Definitions (29)

  • Theorem 1
  • Lemma 2
  • Lemma 3
  • Lemma 4
  • Lemma 5
  • Lemma 6
  • Lemma 7
  • Lemma 8
  • Lemma 9
  • Theorem 10
  • ...and 19 more