Variational Trajectory Optimization of Anisotropic Diffusion Schedules

We develop a principled framework for analyzing and designing noise schedules in diffusion models. We show that one can recast this design problem as an optimal control problem, whose state is the Fisher information of the diffusion process, which evolves according to an ODE, and whose control input is the noise schedule. The objective of the optimal control problem is a functional involving the Fisher information, which is shown to be an upper bound on the Kullback-Leibler sampling error. By solving this optimal control problem, we obtain sufficient conditions on noise schedules under which $\tilde{\mathcal{O}}(d/n)$ sampling error is achievable, where $d$ is the data dimension and $n$ is the number of discretization steps. While existing theoretical works also prove that $\tilde{\mathcal{O}}(d/n)$ sampling error bounds are achievable, these results hold for specific noise schedules that differ from those used in practice. Under additional assumptions on the data distribution, we derive a family of noise schedules that recovers the functional forms of schedules widely used in practice. We show that these schedules achieve the $\tilde{\mathcal{O}}(d/n)$ error bound when their parameters are chosen according to our analysis. Our theory offers a principled approach to noise schedule design, yielding explicit functional forms and guidance for selecting their parameters.

Score-based diffusion models achieve state-of-the-art performance for inverse problems, but their practical deployment is hindered by long inference times and cumbersome hyperparameter tuning. While pretrained diffusion models can be reused across tasks without retraining, inference-time hyperparameters such as the noise schedule and posterior sampling weights typically require ad-hoc adjustment for each problem setup. We propose principled reparameterizations that induce invariances, allowing the same hyperparameters to be reused across multiple problems without re-tuning. In addition, building on the RED-diff framework, which reformulates posterior sampling as an optimization problem, we further develop the OptDiff pipeline. OptDiff provides a simplified tuning framework that facilitates the integration of convex optimization tools to accelerate inference. Experiments on image reconstruction, deblurring, and super-resolution show substantial speedups and improved image quality.

Multi-representation diffusion models can improve visual synthesis by denoising complementary views of an image, but their performance depends critically on the asynchronous schedule that determines when each representation is denoised. We propose to learn this schedule. Our method formulates asynchronous flow matching over multiple representation spaces and uses a schedule-corrected objective that keeps each representation's local noising-time weights fixed as the schedule changes. We instantiate the schedule with a flexible parametric class that is convex and monotone by construction, and learn it using a fast joint probe with less than 1% additional training compute. On ImageNet 256x256, the learned schedule substantially improves both convergence speed and final quality under a matched 675M-parameter XL backbone. With AutoGuidance, our 200-epoch model reaches FID 1.05, matching the 800-epoch SFD-XL baseline with 4x less training. Training to 600 epochs further improves to FID 1.02, outperforming the 1B-parameter SFD-XXL result of FID 1.04 while using a smaller model. In the unguided setting, our 200-epoch model reaches FID 2.37, already below the best 800-epoch SFD-XL result (2.54) at 4x less training, and improves to FID 2.14 at 600 epochs. Code is available at https://github.com/bsq532087/LWD