# Quantization Robustness of Monotone Operator Equilibrium Networks

arXiv: https://arxiv.org/abs/2603.10562 · HTML: https://mlsanity.com/paper/2603.10562/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- Builds on 11 of its 25 references that are in the map.
- First posted on arXiv 11 Mar 2026.

## Builds on (most cited first, 11 of 11)

- [Quantization and Training of Neural Networks for Efficient Integer-Arithmetic-Only Inference](https://mlsanity.com/paper/1712.05877/) (2017)
- [Deep Equilibrium Models](https://mlsanity.com/paper/1909.01377/) (2019)
- [A White Paper on Neural Network Quantization](https://mlsanity.com/paper/2106.08295/) (2021)
- [Monotone operator equilibrium networks](https://mlsanity.com/paper/2006.08591/) (2020)
- [Lipschitz Bounded Equilibrium Networks](https://mlsanity.com/paper/2010.01732/) (2020)
- [Recurrent Equilibrium Networks: Flexible Dynamic Models with Guaranteed Stability and Robustness](https://mlsanity.com/paper/2104.05942/) (2021)
- [Deep Neural Network Structures Solving Variational Inequalities](https://mlsanity.com/paper/1808.07526/) (2018)
- [Learning over All Stabilizing Nonlinear Controllers for a Partially-Observed Linear System](https://mlsanity.com/paper/2112.04219/) (2021)
- [QEBVerif: Quantization Error Bound Verification of Neural Networks](https://mlsanity.com/paper/2212.02781/) (2022)
- [Synthesis of Stabilizing Recurrent Equilibrium Network Controllers](https://mlsanity.com/paper/2204.00122/) (2022)
- [Circuit realization and hardware linearization of monotone operator equilibrium networks](https://mlsanity.com/paper/2509.13793/) (2025)

## Built on by (most cited first, 3 of 3)

- [Quantizing Recursive Reasoning Models](https://mlsanity.com/paper/2607.16237/) (2026)
- [Quantizing Looped Transformers: Feedback Exposure and Calibration Blindness](https://mlsanity.com/paper/2609.30820/) (2026)
- [A Tilted Bowl Is Not a Slippery Slope: Compressing Looped Models](https://mlsanity.com/paper/2609.39277/) (2026)

## Abstract (arXiv)

Monotone operator equilibrium networks are implicit-layer models whose output is the unique equilibrium of a monotone operator, guaranteeing existence, uniqueness, and convergence. When deployed on low-precision hardware, weights are quantized, potentially destroying these guarantees. We analyze weight quantization as a spectral perturbation of the underlying monotone inclusion. Convergence of the quantized solver is guaranteed whenever the spectral-norm weight perturbation is smaller than the monotonicity margin; the displacement between quantized and full-precision equilibria is bounded in terms of the perturbation size and margin; and a condition number characterizing the ratio of the operator norm to the margin links quantization precision to forward error. MNIST experiments confirm a phase transition at the predicted threshold: three- and four-bit post-training quantization diverge, while five-bit and above converge. The backward-pass guarantee enables quantization-aware training, which recovers provable convergence at four bits.