Abstract
Recent advances in diffusion models have demonstrated their remarkable ability to capture complex image distributions, but the geometric properties of the learned data manifold remain poorly understood. We address this gap by introducing a score-based Riemannian metric that leverages the Stein score function from diffusion models to characterize the intrinsic geometry of the data manifold without requiring explicit parameterization. Our approach defines a metric tensor in the ambient space that stretches distances perpendicular to the manifold while preserving them along tangential directions, effectively creating a geometry where geodesics naturally follow the manifold's contours. We develop efficient algorithms for computing these geodesics and demonstrate their utility for both interpolation between data points and extrapolation beyond the observed data distribution. Through experiments on synthetic data with known geometry, Rotated MNIST, and complex natural images via Stable Diffusion, we show that our score-based geodesics capture meaningful transformations that respect the underlying data distribution. Our method consistently outperforms baseline approaches on perceptual metrics (LPIPS) and distribution-level metrics (FID, KID), producing smoother, more realistic image transitions. These results reveal the implicit geometric structure learned by diffusion models and provide a principled way to navigate the manifold of natural images through the lens of Riemannian geometry.
Abstract (translated)
最近在扩散模型方面的进展已经展示了它们捕捉复杂图像分布的非凡能力，但学习到的数据流形的几何属性仍然鲜为人知。为了解决这一问题，我们引入了一种基于评分函数的黎曼度量方法，该方法利用扩散模型中的Stein评分函数来表征数据流形的内在几何性质，并且不需要显式的参数化方式。 我们的方法在包围空间中定义了一个度量张量，在这个度量张量中，沿着与流形垂直的方向拉伸距离，而在沿切线方向则保持不变。这有效地创建了一种几何结构，在这种结构下测地线自然地遵循流形的轮廓。我们开发了用于计算这些测地线的有效算法，并展示了它们在数据点之间插值以及超出观察到的数据分布外推中的实用性。 通过使用具有已知几何特征的人工数据、旋转后的MNIST数据集和通过Stable Diffusion生成的复杂自然图像进行实验，我们表明我们的基于评分函数的测地线捕捉到了有意义的变化过程，这些变化过程尊重底层数据分布。在感知度量（LPIPS）和分布级度量（FID, KID）上，我们的方法始终优于基线方法，生成了更平滑、更逼真的图像转换。 这些结果揭示了扩散模型所学到的隐含几何结构，并通过黎曼几何提供了一种导航自然图象流形的合理途径。
URL
https://arxiv.org/abs/2505.11128
