# Semialgebraic Representation of Monotone Deep Equilibrium Models and Applications to Certification

arXiv: https://arxiv.org/abs/2106.01453 · HTML: https://mlsanity.com/paper/2106.01453/

## Key facts

- Cited by 16 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 2 Jun 2021.

## Built on by (most cited first, 16 of 16)

- [A Closer Look at the Adversarial Robustness of Deep Equilibrium Models](https://mlsanity.com/paper/2306.01429/) (2023)
- [Robustness Certificates for Implicit Neural Networks: A Mixed Monotone Contractive Approach](https://mlsanity.com/paper/2112.05310/) (2021)
- [Convolutional Neural Networks as 2-D systems](https://mlsanity.com/paper/2303.03042/) (2023)
- [FI-ODE: Certifiably Robust Forward Invariance in Neural ODEs](https://mlsanity.com/paper/2210.16940/) (2022)
- [Deep Equilibrium Models Meet Federated Learning](https://mlsanity.com/paper/2305.18646/) (2023)
- [Subhomogeneous Deep Equilibrium Models](https://mlsanity.com/paper/2403.00720/) (2024)
- [Comparative Analysis of Interval Reachability for Robust Implicit and Feedforward Neural Networks](https://mlsanity.com/paper/2204.00187/) (2022)
- [Improving Adversarial Robustness of DEQs with Explicit Regulations Along the Neural Dynamics](https://mlsanity.com/paper/2306.01435/) (2023)
- [Abstract Interpretation of Fixpoint Iterators with Applications to Neural Networks](https://mlsanity.com/paper/2110.08260/) (2021)
- [Verifying Properties of Binary Neural Networks Using Sparse Polynomial Optimization](https://mlsanity.com/paper/2405.17049/) (2024)
- [Stochastic Sample Approximations of (Local) Moduli of Continuity](https://mlsanity.com/paper/2509.15368/) (2025)
- [Lipschitz Multiscale Deep Equilibrium Models: A Theoretically Guaranteed and Accelerated Approach](https://mlsanity.com/paper/2602.03297/) (2026)
- [Representation Recycling for Streaming Video Analysis](https://mlsanity.com/paper/2204.13492/) (2022)
- [Robust Training and Verification of Implicit Neural Networks: A Non-Euclidean Contractive Approach](https://mlsanity.com/paper/2208.03889/) (2022)
- [Sparse Polynomial Optimization: Theory and Practice](https://mlsanity.com/paper/2208.11158/) (2022)
- [Tractable hierarchies of convex relaxations for polynomial optimization on the nonnegative orthant](https://mlsanity.com/paper/2209.06175/) (2022)

## Abstract (arXiv)

Deep equilibrium models are based on implicitly defined functional relations and have shown competitive performance compared with the traditional deep networks. Monotone operator equilibrium networks (monDEQ) retain interesting performance with additional theoretical guaranties. Existing certification tools for classical deep networks cannot directly be applied to monDEQs for which much fewer tools exist. We introduce a semialgebraic representation for ReLU based monDEQs which allows to approximate the corresponding input output relation by semidefinite programming (SDP). We present several applications to network certification and obtain SDP models for the following problems : robustness certification, Lipschitz constant estimation, ellipsoidal uncertainty propagation. We use these models to certify robustness of monDEQs w.r.t. a general Lq norm. Experimental results show that the proposed models outperform existing approaches for monDEQ certification. Furthermore, our investigations suggest that monDEQs are much more robust to L2 perturbations than L∞ perturbations.