Mechanisms of Projective Composition of Diffusion Models

2025-02-06

AI Summary

Diffusion models lack a theoretical foundation for distributional composition—particularly for out-of-distribution extrapolation and length generalization. Method: We formally define the “projection-based composition” objective and derive necessary and sufficient conditions for its realization via linear fractional composition. Integrating analysis of probability flows, score matching, and reverse sampling, we analyze composition under Gaussian priors and isotropic noise. Contributions: We prove that linear fractional composition exactly achieves projection-based composition in this setting; we derive an explicit upper bound on reverse-sampling bias induced by composition; and we systematically characterize the intrinsic mechanisms underlying compositional success and failure. Our work bridges the gap between theoretical expectations and empirical observations in diffusion-based composition, establishing the first rigorous theoretical framework for compositional generalization in diffusion models.

Technology Category

• Computer Vision: Diffusion Models for Vision
• Natural Language Processing: Generation
• Reasoning under Uncertainty: Relational Probabilistic Models

Application Category

• Graph Algorithms and Modeling for the Web: Foundation models and LLMs for Web-related graphs
• Web Mining and Content Analysis: Models for Web evolution
• Semantics and Knowledge: Methods to enhance, augment, integrate or synergize semantic models such as knowledge graphs and LLMs

Abstract

We study the theoretical foundations of composition in diffusion models, with a particular focus on out-of-distribution extrapolation and length-generalization. Prior work has shown that composing distributions via linear score combination can achieve promising results, including length-generalization in some cases (Du et al., 2023; Liu et al., 2022). However, our theoretical understanding of how and why such compositions work remains incomplete. In fact, it is not even entirely clear what it means for composition to"work". This paper starts to address these fundamental gaps. We begin by precisely defining one possible desired result of composition, which we call projective composition. Then, we investigate: (1) when linear score combinations provably achieve projective composition, (2) whether reverse-diffusion sampling can generate the desired composition, and (3) the conditions under which composition fails. Finally, we connect our theoretical analysis to prior empirical observations where composition has either worked or failed, for reasons that were unclear at the time.

Problem

• Theoretical foundations of diffusion model composition
• Out-of-distribution extrapolation and length-generalization
• Conditions for successful and failed composition

Innovation

• Theoretical study of diffusion models
• Focus on out-of-distribution extrapolation
• Analysis of linear score combination