Geometric Asymptotics of Score Mixing and Guidance in Diffusion Models

Liu K, Zuazua E (2026)

Publication Language: English

Publication Status: Submitted

Publication Type: Unpublished / Preprint

Future Publication Type: Journal article

Publication year: 2026

Open Access Link: https://dcn.nat.fau.eu/wp-content/uploads/2605.12231.pdf

Abstract

Diffusion models are routinely guided in practice by combining multiple score fields, yet the mathematical structure of score mixing is still poorly understood. We study the small-time generation dynamics driven by mixed scores

s=λ∇logu1+(1−λ)∇logu2,λ≥0,

in the heat-flow framework, where u1,u2 are heat evolutions of two compactly supported probability measures. This single formulation covers both the mixture-of-experts regime (0≤λ≤1) and the classifier-free guidance regime (λ>1). Exploiting a Laplace-Varadhan principle under a similarity-time rescaling, we show that the small-time generation dynamics is governed by the explicit geometric potential

Φλ=λd21+(1−λ)d22,

which depends only on the supports of the initial measures and on the mixing parameter. This gives a rigorous reduction from a singular, non-autonomous score-driven dynamics to autonomous Clarke-type subgradient inclusions. In the empirical setting of finite Dirac mixtures, the limiting potential is piecewise quadratic with a Voronoi-type structure; this rigidity yields convergence of all autonomous limiting trajectories to critical points and a conditional convergence criterion for the original generation flow toward local minimizers of the potential, with rate (t√) in the smooth stable case.

Authors with CRIS profile

Enrique Zuazua Iriondo Lehrstuhl für Dynamics, Control, Machine Learning and Numerics (Alexander von Humboldt-Professur)

Involved external institutions

Université Bourgogne Europe FR France (FR)

How to cite

APA:

Liu, K., & Zuazua, E. (2026). Geometric Asymptotics of Score Mixing and Guidance in Diffusion Models. (Unpublished, Submitted).

MLA:

Liu, Kang, and Enrique Zuazua. Geometric Asymptotics of Score Mixing and Guidance in Diffusion Models. Unpublished, Submitted. 2026.

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