去噪即投影：带梯度引导扩散的约束优化
Denoising as Projection: Constrained Optimization with Gradient-Guided Diffusion
- MIT（麻省理工学院）
- University of Wisconsin–Madison（威斯康星大学麦迪逊分校）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
本文提出一种将目标梯度融入去噪步骤的投影梯度引导扩散更新方法，用于结构化可行集上的约束优化，在三类设置中保证收敛，实验验证其平衡目标下降与几何保留的有效性。
AI中文摘要：
扩散模型不仅越来越多地用于从学习到的数据分布中采样，还用于生成优化特定任务目标的样本。一种常见方法是利用外部目标的梯度引导反向扩散过程。然而，当数据分布支撑在结构化可行集（如流形或约束集）上时，梯度引导可能会使样本偏离学习到的数据几何结构。在本文中，我们基于Stein去噪算子可作为数据几何结构近似投影的观察，研究了一种简单的投影梯度引导扩散更新方法。所提出的更新方法将目标梯度融入去噪步骤，产生一种仅使用预训练去噪器和梯度评估的推理时方法。我们将该更新分析为针对学习到的可行几何结构上约束优化的不精确投影梯度方法。我们的理论涵盖三种设置：线性流形、紧致凸可行集和紧致黎曼子流形。在所有这些设置中，我们证明了下降和有限时间收敛保证。数值实验支持该理论解释，并说明了所提出的更新如何平衡目标下降与学习几何结构的保留。
英文摘要：
Diffusion models are increasingly used not only for sampling from learned data distributions, but also for generating samples that optimize task-specific objectives. A common approach is to guide the reverse diffusion process using gradients of an external objective. However, when the data distribution is supported on a structured feasible set, such as a manifold or a constraint set, gradient guidance can move samples away from the learned data geometry. In this paper, we study a simple projected-gradient-guided diffusion update based on the observation that the Stein denoising operator can act as an approximate projection onto the data geometry. The proposed update incorporates the objective gradient inside the denoising step, yielding an inference-time method that uses only a pretrained denoiser and gradient evaluations. We analyze this update as an inexact projected-gradient method for constrained optimization over learned feasible geometries. Our theory covers three settings: linear manifolds, compact convex feasible sets, and compact Riemannian submanifolds. In all these settings, we prove descent and finite-time convergence guarantees. Numerical experiments support the theoretical interpretation and illustrate how the proposed update balances objective descent with preservation of the learned geometry.
