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A Phase Transition in Diffusion Models Reveals the Hierarchical Nature of Data (2402.16991v2)

Published 26 Feb 2024 in stat.ML, cond-mat.dis-nn, cs.CV, and cs.LG

Abstract: Understanding the structure of real data is paramount in advancing modern deep-learning methodologies. Natural data such as images are believed to be composed of features organised in a hierarchical and combinatorial manner, which neural networks capture during learning. Recent advancements show that diffusion models can generate high-quality images, hinting at their ability to capture this underlying structure. We study this phenomenon in a hierarchical generative model of data. We find that the backward diffusion process acting after a time $t$ is governed by a phase transition at some threshold time, where the probability of reconstructing high-level features, like the class of an image, suddenly drops. Instead, the reconstruction of low-level features, such as specific details of an image, evolves smoothly across the whole diffusion process. This result implies that at times beyond the transition, the class has changed but the generated sample may still be composed of low-level elements of the initial image. We validate these theoretical insights through numerical experiments on class-unconditional ImageNet diffusion models. Our analysis characterises the relationship between time and scale in diffusion models and puts forward generative models as powerful tools to model combinatorial data properties.

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Authors (3)
  1. Antonio Sclocchi (15 papers)
  2. Alessandro Favero (13 papers)
  3. Matthieu Wyart (89 papers)
Citations (17)
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Summary

Exploring the Hierarchical Structure of Data Through Diffusion Models

Introduction

In the quest to understand the nuances of data generation and the manifestation of hierarchical structures within, recent research pivots around the efficiency and revelation capacities of denoising diffusion probabilistic models (DDPMs). This paper explores how DDPMs, when applied to image data, illustrate not only the compositional nature of the data but also the underlying hierarchical genealogy of features that constitute the final perceptible forms. By rewind-examining the image generation process over time through the lens of class-unconditional ImageNet diffusion models, insights into the mechanics of feature composition and evolution emerged.

Investigation into Hierarchical Generativity

The research theorized that images, as a specimen of natural data, bear features that are not randomly scattered but are methodically hierarchical and combinatorial. The initial proposal posited that the diffusion process associated with DDPMs could serve as an investigative tool to unfold this hierarchical structuring of features. It was observed that during the backward diffusion process (akin to reverse time-evolution of image generation), a phase transition is encountered. This transition delineates a threshold beyond which the probability to reconstruct high-level features, such as the class identity of an image, experiences a dramatic decline, whereas the reconstruction of low-level features proceeds with relative smoothness across the diffusion timeline.

Numerical Experiments and Theoretical Insights

Utilizing class-unconditional ImageNet diffusion models for empirical scrutiny, it was confirmed that beyond the phase transition threshold, the generative models maintain the ability to assemble new samples by weaving together low-level features from the initial image. Concretely, this entails that attributes like color or shape could persist through the diffusion process, incorporating themselves into newly generated images that might belong to entirely different classes.

This duality - the sharp phase transition for class features and the smooth evolution of low-level details - underscores the sophisticated interplay between various levels of feature hierarchies in the generative process. The findings were bolstered by numerical analyses, which demonstrated a stark correspondence between the theoretical predictions and observable dynamics in state-of-the-art convolutional neural network (CNN) architectures when tasked with supervised classification on the generated data.

Implications and FuBrtIu7S2Ezture Outlook

The implications of this work are manifold. Theoretically, this paper enriches the comprehension of data compositionality and its implications for neural network training. Practically, it lays down a framework to further investigate the nature of generative models, especially in how they discern, preserve, and innovate upon the hierarchical structure within data.

Going forward, this line of inquiry could extend into textual data sets using diffusion LLMs. The exploration into the compositional and hierarchical nature of data through diffusion models not only provides a window into their generative mechanics but also potentially elucidates why these models exhibit remarkable generalization capabilities, and how they navigate between memorization and genuine data generation.

The advancement in understanding provided by this research could echo through future developments in machine learning methodologies, particularly in enhancing generative models' efficiency and in understanding the computational underpinnings of data structure recognition and replication.

Conclusion

This research presents a pioneering exploration into the compositional and hierarchical complexities of generative data models through the adaptive lens of diffusion models. It underscores a novel method of contemplating and studying the interconnectedness of data features and their evolution over time. As this inquiry broadens, it beckons a deeper understanding of the theoretical underpinnings of machine learning and the practical capabilities and limitations of current models.

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