Lipschitz Continuity in Deep Learning: A Systematic Review of Theoretical Foundations, Estimation Methods, Regularization Approaches, and Certifiable Robustness

• URL: http://arxiv.org/abs/2607.16329v2
• Date: Tue, 21 Jul 2026 08:39:11 GMT
• Title: Lipschitz Continuity in Deep Learning: A Systematic Review of Theoretical Foundations, Estimation Methods, Regularization Approaches, and Certifiable Robustness
• Abstract summary: Lipschitz continuity is a fundamental property of neural networks that characterizes their sensitivity to input perturbations.<n>Despite its importance, research on Lipschitz continuity is scattered across various domains, lacking a unified perspective.<n>This paper addresses this gap by providing a systematic review of Lipschitz continuity in deep learning.
• Score: 1.5484595752241122
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: Lipschitz continuity is a fundamental property of neural networks that characterizes their sensitivity to input perturbations. It plays a pivotal role in deep learning, governing \textbf{robustness}, \textbf{generalization} and \textbf{optimization dynamics}. Despite its importance, research on Lipschitz continuity is scattered across various domains, lacking a unified perspective. This paper addresses this gap by providing a systematic review of Lipschitz continuity in deep learning. We explore its \textbf{theoretical foundations}, \textbf{estimation methods}, \textbf{regularization approaches}, and \textbf{certifiable robustness}. By reviewing existing research through the lens of Lipschitz continuity, this survey serves as a comprehensive reference for researchers and practitioners seeking a deeper understanding of Lipschitz continuity and its implications in deep learning.

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