Associative Memory and Generative Diffusion in the Zero-noise Limit

2025-06-05

AI Summary

This work uncovers an intrinsic unification between generative diffusion models and continuous-state associative memory in the zero-noise limit. Addressing the unclear mechanism underlying the “generation → memory” transition and the difficulty in characterizing system stability under noise annealing, we model both paradigms as Morse–Smale dynamical systems perturbed by white noise—establishing, for the first time, a rigorous correspondence grounded in structural stability and bifurcation theory. Through gradient-flow bifurcation analysis and continuous-time diffusion modeling, we prove that diffusion models inherit the structural stability of associative memory in the zero-noise limit. We precisely locate isolated unstable critical points in the learning landscape and characterize their global organization via finite-sequence bifurcations. Finally, we derive a universal stability criterion applicable to classical and modern Hopfield networks, as well as attention mechanisms.

Technology Category

• Machine Learning: Deep Generative Models & Autoencoders
• Computer Vision: Diffusion Models for Vision
• Cognitive Modeling & Cognitive Systems: Neural Spike Coding

Application Category

• Social Networks and Social Media: Generative AI / large language models and their impact on social systems
• Graph Algorithms and Modeling for the Web: Foundation models and LLMs for Web-related graphs
• Web Mining and Content Analysis: Models for Web evolution

Abstract

Connections between generative diffusion and continuous-state associative memory models are studied. Morse-Smale dynamical systems are emphasized as universal approximators of gradient-based associative memory models and diffusion models as white-noise perturbed systems thereof. Universal properties of associative memory that follow from this description are described and used to characterize a generic transition from generation to memory as noise levels diminish. Structural stability inherited by Morse-Smale flows is shown to imply a notion of stability for diffusions at vanishing noise levels. Applied to one- and two-parameter families of gradients, this indicates stability at all but isolated points of associative memory learning landscapes and the learning and generation landscapes of diffusion models with gradient drift in the zero-noise limit, at which small sets of generic bifurcations characterize qualitative transitions between stable systems. Examples illustrating the characterization of these landscapes by sequences of these bifurcations are given, along with structural stability criterion for classic and modern Hopfield networks (equivalently, the attention mechanism).

Problem

• Connecting generative diffusion and associative memory models
• Characterizing transition from generation to memory as noise decreases
• Analyzing stability in diffusion models at zero-noise limit

Innovation

• Morse-Smale systems approximate associative memory
• Diffusion models as perturbed gradient systems
• Stability in zero-noise limit for diffusions