# Lipschitz Thinking: Ten Years of Certifiable-by-Design Robust Neural Networks

arXiv: https://arxiv.org/abs/2610.06252 · HTML: https://mlsanity.com/paper/2610.06252/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 1 of its 60 references that are in the map.
- First posted on arXiv 5 Oct 2026.

## Builds on (most cited first, 1 of 1)

- [Distilling the Knowledge in a Neural Network](https://mlsanity.com/paper/1503.02531/) (2015)

## Abstract (arXiv)

The Lipschitz property of a deep neural network provides a direct measure of its sensitivity to input perturbations and, when explicitly controlled, offers a principled way to limit the propagation of errors and improve robustness. Over the past decade, Lipschitz-bounded layers have been incorporated into increasingly expressive and high-performing deep models, narrowing the gap between empirical robustness and formal, by-design guarantees of stability. This article introduces the fundamental concepts underlying Lipschitz-bounded neural networks, explaining the principles behind Lipschitz-constrained layers, the mechanisms used to enforce their bounds, and how they yield robustness certificates at the cost of a single forward pass. The tutorial concludes by discussing emerging and open directions, highlighting Lipschitz control as a general framework for offering guaranteed, by-design stability.