Diffusion Model for Manifold Data: Score Decomposition, Curvature, and Statistical Complexity

Abstract

Diffusion models have become a leading framework in generative modeling, yet their theoretical understanding -- especially for high-dimensional data concentrated on low-dimensional structures -- remains incomplete. This paper investigates how diffusion models learn such structured data, focusing on two key aspects: statistical complexity and influence of data geometric properties. By modeling data as samples from a smooth Riemannian manifold, our analysis reveals crucial decompositions of score functions in diffusion models under different levels of injected noise. We also highlight the interplay of manifold curvature with the structures in the score function. These analyses enable an efficient neural network approximation to the score function, built upon which we further provide statistical rates for score estimation and distribution learning. Remarkably, the obtained statistical rates are governed by the intrinsic dimension of data and the manifold curvature. These results advance the statistical foundations of diffusion models, bridging theory and practice for generative modeling on manifolds.

Figures (3)

• Figure 1: Demonstration of tangent space $T_x\mathcal{M}$, geodesic, and exponential map based on $x \in \mathcal{M}$ and $v \in T^d_{x}\mathcal{M}$.
• Figure 2: Score decomposition for linear subspace and general manifold. For linear subspace data, the projection $\Pi_A$ is globally defined for all $t$ and noisy state $x$. The score function decomposes into two orthogonal components. For a smooth manifold, the score function behaves distinctly according to the magnitude of the added noise. In the large noise regime, score function decomposes corresponding to tangent spaces on the manifold. In the small noise regime, local decomposition exists and is centered around the projection point $\Pi_{\mathcal{M}}(x_{t_2}, t_2)$ of a noisy state $x_{t_2}$.
• Figure 3: Illustration of the neural network architecture. A time switching network aggregates sub-networks for small noise and large noise respectively. Both regimes involve Taylor approximation of on-support scores, and projection approximation as part of orthogonal scores. These approximators are implemented by neural networks.

Theorems & Definitions (106)

• Theorem 1.1: Informal
• Definition 2.1: Chart
• Definition 2.2: $C^\infty$ Atlas
• Definition 2.3: Smooth Manifold
• Definition 2.4: Exponential map
• Definition 2.5: Injectivity radius
• Definition 2.6: Reach Federer
• Proposition 2.7: Proof in AL
• Remark 2.8: Atlas using exponential map
• Definition 2.9: $C^s$ Functions on $\mathcal{M}$