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.
Generative modeling within constrained sets is essential for scientific and engineering applications involving physical, geometric, or safety requirements (e.g., molecular generation, robotics). We present a unified framework for constrained diffusion models on generic nonconvex feasible sets Σ that simultaneously enforces equality and inequality constraints throughout the diffusion process. Our framework incorporates both overdamped and underdamped dynamics for forward and backward sampling. A key algorithmic innovation is a computationally efficient landing mechanism that replaces costly and often ill-defined projections onto Σ, ensuring feasibility without iterative Newton solves or projection failures. By leveraging underdamped dynamics, we accelerate mixing toward the prior distribution, effectively alleviating the high simulation costs typically associated with constrained diffusion. Empirically, this approach reduces function evaluations and memory usage during both training and inference while preserving sample quality. On benchmarks featuring equality and mixed constraints, our method achieves comparable sample quality to state-of-the-art baselines while significantly reducing computational cost, providing a practical and scalable solution for diffusion on nonconvex feasible sets.
Constrained generative models aim to produce samples that satisfy complex feasibility constraints while remaining faithful to the data distribution. Existing constrained generation methods typically enforce constraints either through training-time optimization or sampling-time correction. Training-time optimization approaches optimize on states induced by the training distribution, which can differ substantially from those encountered during sampling. Sampling-time correction methods instead modify the sampling process at inference, introducing distribution shift and requiring expensive tuning, particularly for few-step sampling. We propose a fine-tuning framework that incorporates constraint guidance obtained through online rollout into the training process, which aligns training with sampling by differentiating through the fixed noise schedule used to numerically integrate the denoising process. This exposes the model to violations that arise along the denoising trajectory and aligns diffusion learning with the sampling process. Experiments across multiple tasks show that our method improves constraint satisfaction while maintaining competitive sampling quality compared to prior methods.
Recently, diffusion models have been widely adopted in generative modeling and have served as foundational models for many image generation tasks. To control the generation without costly re-training or fine-tuning, many works seek inference-time guidance methods to steer the latent via a differentiable objective at inference time. However, these methods cannot effectively preserve the original Gaussian distribution because they introduce distributional drift, thereby degrading the sample quality. To address this gap, we propose DiffRGD, a distribution-aware guidance framework that explicitly preserves the latent Gaussian structure. DiffRGD formulates each sampling step as a constrained optimization problem on a spherical manifold induced by the latent Gaussian distribution, and solves it efficiently via Riemannian Gradient Descent (RGD). DiffRGD is a plug-and-play method that can be seamlessly integrated into any pre-trained diffusion model. Extensive experiments demonstrate that DiffRGD outperforms previous methods in most image restoration and conditional generation tasks. Our project page is available at https://diffrgd.github.io/.