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L-Lipschitz Gershgorin ResNet Network

Marius F. R. Juston, William R. Norris, Dustin Nottage, Ahmet Soylemezoglu

TL;DR

The paper tackles enforcing Lipschitz continuity in deep residual networks to enhance adversarial robustness and certifiability. It reformulates ResNet as a Linear Matrix Inequality (LMI) with a pseudo-tri-diagonal structure and uses the Gershgorin circle theorem to bound eigenvalues, yielding closed-form parameter constraints that guarantee $\mathcal{L}$-Lipschitz continuity. Key contributions include a parameterization framework for Lipschitz-constrained deep residual networks, a compositional approach for recursive inner-layer systems, and an analysis of Gershgorin-based limitations that can over-constrain nonlinearity. The work highlights foundational theory for robust Lipschitz design and points to future directions in alternative eigenvalue approximations to recover expressiveness in constrained networks.

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 bound in neural networks has emerged as an essential area of research for enhancing adversarial robustness and network certifiability. This paper uses a rigorous approach to design $\mathcal{L}$-Lipschitz deep residual networks using a Linear Matrix Inequality (LMI) framework. The ResNet architecture was reformulated as a pseudo-tri-diagonal LMI with off-diagonal elements and derived closed-form constraints on network parameters to ensure $\mathcal{L}$-Lipschitz continuity. To address the lack of explicit eigenvalue computations for such matrix structures, the Gershgorin circle theorem was employed to approximate eigenvalue locations, guaranteeing the LMI's negative semi-definiteness. Our contributions include a provable parameterization methodology for constructing Lipschitz-constrained networks and a compositional framework for managing recursive systems within hierarchical architectures. These findings enable robust network designs applicable to adversarial robustness, certified training, and control systems. However, a limitation was identified in the Gershgorin-based approximations, which over-constrain the system, suppressing non-linear dynamics and diminishing the network's expressive capacity.

L-Lipschitz Gershgorin ResNet Network

TL;DR

The paper tackles enforcing Lipschitz continuity in deep residual networks to enhance adversarial robustness and certifiability. It reformulates ResNet as a Linear Matrix Inequality (LMI) with a pseudo-tri-diagonal structure and uses the Gershgorin circle theorem to bound eigenvalues, yielding closed-form parameter constraints that guarantee -Lipschitz continuity. Key contributions include a parameterization framework for Lipschitz-constrained deep residual networks, a compositional approach for recursive inner-layer systems, and an analysis of Gershgorin-based limitations that can over-constrain nonlinearity. The work highlights foundational theory for robust Lipschitz design and points to future directions in alternative eigenvalue approximations to recover expressiveness in constrained networks.

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 bound in neural networks has emerged as an essential area of research for enhancing adversarial robustness and network certifiability. This paper uses a rigorous approach to design -Lipschitz deep residual networks using a Linear Matrix Inequality (LMI) framework. The ResNet architecture was reformulated as a pseudo-tri-diagonal LMI with off-diagonal elements and derived closed-form constraints on network parameters to ensure -Lipschitz continuity. To address the lack of explicit eigenvalue computations for such matrix structures, the Gershgorin circle theorem was employed to approximate eigenvalue locations, guaranteeing the LMI's negative semi-definiteness. Our contributions include a provable parameterization methodology for constructing Lipschitz-constrained networks and a compositional framework for managing recursive systems within hierarchical architectures. These findings enable robust network designs applicable to adversarial robustness, certified training, and control systems. However, a limitation was identified in the Gershgorin-based approximations, which over-constrain the system, suppressing non-linear dynamics and diminishing the network's expressive capacity.

Paper Structure

This paper contains 14 sections, 5 theorems, 44 equations, 6 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Contributions
  3. LMI Formulation
  4. General LMI Solution
  5. Last block
  6. Middle blocks
  7. First layer
  8. Weighted norm constraint
  9. Elementwise vs. row constraint bound switching
  10. $C_l$ constraints
  11. $C_1$ constraints
  12. LMI parameterization
  13. Experiment
  14. Conclusion

Key Result

Theorem 2.1

Let $A$ be a complex matrix $n \times n$ matrix, with entries $a_{ij}$. For $i \in \{1, \cdots,n\}$ let $R_i$ be the sum of the absolute values of the non-diagonal entries of the $i$-th row: Let $D(a_{ii}, R_{i}) \subseteq \mathbb{C}$ be a closed disc centered at $a_{ii}$ with radius $R_i$, every eigenvalue of $A$ lies within at least one of the Gershgorin discs $D(a_{ii}, R_{i})$.

Figures (6)

  • Figure 1: Eigenvalue distribution
  • Figure 2: LMI Gershgorin Circles
  • Figure 3: Backwards pass
  • Figure 4: Forward pass
  • Figure 5: Trained L-Lipschitz network output over multiple optimizers
  • ...and 1 more figures

Theorems & Definitions (8)

  • Theorem 2.1
  • Corollary 1
  • Theorem 3.1
  • proof
  • Theorem 3.2
  • proof
  • Theorem 3.3
  • proof