几何感知扩散引导：基于曲率自适应管状校正
Geometry-Aware Diffusion Guidance via Curvature-Adaptive Tubular Correction
- Shanghai Jiao Tong University（上海交通大学）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
针对扩散模型强引导导致的采样轨迹偏离问题，提出曲率自适应管状校正（CAT），通过分解法向与切向位移并优化几何预算，在七个逆问题及黑洞重建中稳定引导并提升感知指标与FID。
AI中文摘要：
梯度引导的扩散采样器为逆问题和条件生成提供了灵活的先验，但强引导可能将采样轨迹移入学习分数支持不佳的区域。现有的切平面投影策略限制了偏离等密度面的一阶运动，却丢弃了可能有用的法向运动，并忽视了曲面上切向运动引起的二阶偏离。我们引入了曲率自适应管状校正（CAT），这是一种无需训练的即插即用插件，在共享的、依赖于噪声的几何预算内调节这两种效应。CAT将引导梯度分解为法向和切向分量，法向位移按一阶计费，切向位移按方向曲率计费，并通过一维对偶方程获得它们联合最优的幅值。Armijo回溯根据实际引导目标校准所得的有限步长，而矩阵无关的方向导数避免了构建完整的分数雅可比矩阵。我们为管状近似、校正的唯一性以及目标函数的充分下降建立了局部保证。在FFHQ和ImageNet上的七个逆问题中，CAT改进了所评估的像素空间和潜在空间宿主采样器，在感知指标上尤其获得一致的提升。它还在InverseBench上改善了黑洞重建，并在所有测试的无分类器引导尺度下取得了相比方法最低的FID，同时保持了稳定的饱和度和对比度。这些结果支持曲率感知的管状控制作为稳定扩散引导的可复用机制。
英文摘要：
Gradient-guided diffusion samplers provide flexible priors for inverse problems and conditional generation, but strong guidance can move the sampling trajectory into regions where the learned score is poorly supported. Existing tangent-projection strategies limit first-order departure from an iso-density surface, yet discard potentially useful normal motion and overlook the second-order departure induced by tangent motion on a curved surface. We introduce curvature-adaptive tubular correction (CAT), a training-free plugin that regulates both effects within a shared, noise-dependent geometric budget. CAT decomposes the guidance gradient into normal and tangent components, charges normal displacement at first order and tangent displacement according to directional curvature, and obtains their jointly optimal magnitudes from a one-dimensional dual equation. Armijo backtracking calibrates the resulting finite step against the actual guidance objective, while matrix-free directional derivatives avoid constructing the full score Jacobian. We establish local guarantees for the tubular approximation, uniqueness of the correction, and sufficient objective decrease. Across seven inverse problems on FFHQ and ImageNet, CAT improves the evaluated pixel- and latent-space host samplers, with particularly consistent gains in perceptual metrics. It also improves black hole reconstruction on InverseBench and yields the lowest FID among the compared methods at every tested classifier-free guidance scale, while maintaining stable saturation and contrast. These results support curvature-aware tubular control as a reusable mechanism for stabilizing diffusion guidance.
