Principles of Lipschitz continuity in neural networks

• URL: http://arxiv.org/abs/2602.04078v1
• Date: Tue, 03 Feb 2026 23:30:08 GMT
• Title: Principles of Lipschitz continuity in neural networks
• Authors: Róisín Luo,
• Abstract summary: This thesis seeks to advance a principled understanding of the principles of Lipschitz continuity in neural networks.<n>It examines how Lipschitz continuity modulates the behavior of neural networks with respect to features in the input data.
• Score: 0.304585143845864
• License: http://creativecommons.org/licenses/by-nc-nd/4.0/
• Abstract: Deep learning has achieved remarkable success across a wide range of domains, significantly expanding the frontiers of what is achievable in artificial intelligence. Yet, despite these advances, critical challenges remain -- most notably, ensuring robustness to small input perturbations and generalization to out-of-distribution data. These critical challenges underscore the need to understand the underlying fundamental principles that govern robustness and generalization. Among the theoretical tools available, Lipschitz continuity plays a pivotal role in governing the fundamental properties of neural networks related to robustness and generalization. It quantifies the worst-case sensitivity of network's outputs to small input perturbations. While its importance is widely acknowledged, prior research has predominantly focused on empirical regularization approaches based on Lipschitz constraints, leaving the underlying principles less explored. This thesis seeks to advance a principled understanding of the principles of Lipschitz continuity in neural networks within the paradigm of machine learning, examined from two complementary perspectives: an internal perspective -- focusing on the temporal evolution of Lipschitz continuity in neural networks during training (i.e., training dynamics); and an external perspective -- investigating how Lipschitz continuity modulates the behavior of neural networks with respect to features in the input data, particularly its role in governing frequency signal propagation (i.e., modulation of frequency signal propagation).

Related papers

• On the Stability of Neural Networks in Deep Learning [3.843574434245427]
  This thesis examines how neural networks respond to perturbations at both the input and parameter levels.<n>We study Lipschitz networks as a principled way to constrain sensitivity to perturbations, thereby improving generalization, adversarial robustness, and training stability.
  arXiv  Detail & Related papers  (2025-10-29T08:38:43Z)
• Estimating Neural Network Robustness via Lipschitz Constant and Architecture Sensitivity [6.468625143772815]
  This paper investigates the robustness of neural networks in perception systems. We identify the Lipschitz constant as a key metric for quantifying and enhancing network robustness.
  arXiv  Detail & Related papers  (2024-10-30T18:38:42Z)
• Feature contamination: Neural networks learn uncorrelated features and fail to generalize [5.642322814965062]
  Learning representations that generalize under distribution shifts is critical for building robust machine learning models.<n>We show that even allowing a neural network to explicitly fit the representations obtained from a teacher network that can generalize out-of-distribution is insufficient for the generalization of the student network.
  arXiv  Detail & Related papers  (2024-06-05T15:04:27Z)
• ECLipsE: Efficient Compositional Lipschitz Constant Estimation for Deep Neural Networks [0.8993153817914281]
  Lipschitz constant plays a crucial role in certifying the robustness of neural networks to input perturbations. Efforts have been made to obtain tight upper bounds on the Lipschitz constant. We provide a compositional approach to estimate Lipschitz constants for deep feed-forward neural networks.
  arXiv  Detail & Related papers  (2024-04-05T19:36:26Z)
• Quantum-Inspired Analysis of Neural Network Vulnerabilities: The Role of Conjugate Variables in System Attacks [54.565579874913816]
  Neural networks demonstrate inherent vulnerability to small, non-random perturbations, emerging as adversarial attacks. A mathematical congruence manifests between this mechanism and the quantum physics' uncertainty principle, casting light on a hitherto unanticipated interdisciplinarity.
  arXiv  Detail & Related papers  (2024-02-16T02:11:27Z)
• Universal Scaling Laws of Absorbing Phase Transitions in Artificial Deep Neural Networks [0.8932296777085644]
  Conventional artificial deep neural networks operating near the phase boundary of the signal propagation dynamics, also known as the edge of chaos, exhibit universal scaling laws of absorbing phase transitions.<n>We exploit the fully deterministic nature of the propagation dynamics to elucidate an analogy between a signal collapse in the neural networks and an absorbing state.
  arXiv  Detail & Related papers  (2023-07-05T13:39:02Z)
• Some Fundamental Aspects about Lipschitz Continuity of Neural Networks [6.576051895863941]
  Lipschitz continuity is a crucial functional property of any predictive model. We examine and characterise the Lipschitz behaviour of Neural Networks. We show a remarkable fidelity of the lower Lipschitz bound, identify a striking Double Descent trend in both upper and lower bounds to the Lipschitz and explain the intriguing effects of label noise on function smoothness and generalisation.
  arXiv  Detail & Related papers  (2023-02-21T18:59:40Z)
• Gradient Descent in Neural Networks as Sequential Learning in RKBS [63.011641517977644]
  We construct an exact power-series representation of the neural network in a finite neighborhood of the initial weights. We prove that, regardless of width, the training sequence produced by gradient descent can be exactly replicated by regularized sequential learning.
  arXiv  Detail & Related papers  (2023-02-01T03:18:07Z)
• Critical Learning Periods for Multisensory Integration in Deep Networks [112.40005682521638]
  We show that the ability of a neural network to integrate information from diverse sources hinges critically on being exposed to properly correlated signals during the early phases of training. We show that critical periods arise from the complex and unstable early transient dynamics, which are decisive of final performance of the trained system and their learned representations.
  arXiv  Detail & Related papers  (2022-10-06T23:50:38Z)
• How You Start Matters for Generalization [26.74340246715699]
  We show that the generalization of neural networks is heavily tied to their initializes. We make a case against the controversial flat-minima conjecture.
  arXiv  Detail & Related papers  (2022-06-17T05:30:56Z)
• Rank Diminishing in Deep Neural Networks [71.03777954670323]
  Rank of neural networks measures information flowing across layers. It is an instance of a key structural condition that applies across broad domains of machine learning. For neural networks, however, the intrinsic mechanism that yields low-rank structures remains vague and unclear.
  arXiv  Detail & Related papers  (2022-06-13T12:03:32Z)
• Formalizing Generalization and Robustness of Neural Networks to Weight Perturbations [58.731070632586594]
  We provide the first formal analysis for feed-forward neural networks with non-negative monotone activation functions against weight perturbations. We also design a new theory-driven loss function for training generalizable and robust neural networks against weight perturbations.
  arXiv  Detail & Related papers  (2021-03-03T06:17:03Z)
• Provably Training Neural Network Classifiers under Fairness Constraints [70.64045590577318]
  We show that overparametrized neural networks could meet the constraints. Key ingredient of building a fair neural network classifier is establishing no-regret analysis for neural networks.
  arXiv  Detail & Related papers  (2020-12-30T18:46:50Z)

This list is automatically generated from the titles and abstracts of the papers in this site.

This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.