Generalised Diffusion Probabilistic Scale-Spaces

TL;DR

This work develops a generalised scale-space theory for diffusion probabilistic models (DPMs), treating the evolution of probability distributions under forward diffusion and the learned reverse denoising as a stochastic scale-space. It establishes forward and reverse scale-space properties, including semigroup structure, Lyapunov (entropy) measures, and invariances for variance-preserving/exploding and blurring diffusion, and connects these dynamics to osmosis filtering and inverse heat dissipation. The theory is complemented by empirical comparisons showing how DPM trajectories relate to deterministic osmosis and classical diffusion in terms of entropy evolution, variance, and Fréchet-Inception distance. The results unify generative diffusion with scale-space concepts and offer design guidance for forward processes that leverage classical filtering insights, potentially benefiting both theory and practical diffusion-based generation.

Abstract

Diffusion probabilistic models excel at sampling new images from learned distributions. Originally motivated by drift-diffusion concepts from physics, they apply image perturbations such as noise and blur in a forward process that results in a tractable probability distribution. A corresponding learned reverse process generates images and can be conditioned on side information, which leads to a wide variety of practical applications. Most of the research focus currently lies on practice-oriented extensions. In contrast, the theoretical background remains largely unexplored, in particular the relations to drift-diffusion. In order to shed light on these connections to classical image filtering, we propose a generalised scale-space theory for diffusion probabilistic models. Moreover, we show conceptual and empirical connections to diffusion and osmosis filters.

Figures (4)

• Figure 1: Forward DPM Trajectory. Starting from each sample of the initial distribution $p(\bm u_0)$, infinitely many trajectories exist. In each step of the trajectory, noise is added according to the transition probability $p(\bm u_i | \bm u_{i-1})$.
• Figure 2: Semigroup Property for Forward DPM. Due to the Markov property, each intermediate scale $i$ can be reached either from the training distribution $p(\bm u_0)$ in $i$ steps, or from $p(\bm u_{i-k})$ in $k$ steps. Note that this property does not apply to individual images as in classical scale-spaces. Instead, it refers to probability distributions which are visualised by samples from four different trajectories.
• Figure 3: Visual comparison of trajectories. The diffusion probabilistic model (a) behaves distinctively different compared to the other approaches since it does not perform blurring in the image domain. Due to the minimal amounts of added noise, inverse heat dissipation (b) closely resembles homogeneous diffusion (c). With a suitable noise schedule, blurring diffusion (d) closely resembles osmosis filtering (e).
• Figure 4: Quantitative Comparison of Diffusion Probabilistic Models and Model-based Filters. Both the variance evolution over time in (a) and the FID w.r.t. the osmosis distributions in (b) suggest that DPM differs significantly from the classical diffusion and osmosis filters. Heat dissipation approximates diffusion, while blurring diffusion approximates osmosis.

Theorems & Definitions (17)

• Proposition 1: Transition Probability from the Initial Distribution
• proof
• Proposition 2: Semigroup Property
• proof
• Proposition 3: Increasing Conditional Entropy
• proof
• Proposition 4: Permutation Invariant Trajectories
• proof
• Corollary 5: Permutation Invariant Distributions
• Proposition 6: Convergences to a Normal Distribution