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A Geometric Perspective on Diffusion Models
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Recent years have witnessed significant progress in developing effective training and fast sampling techniques for diffusion models. A remarkable advancement is the use of stochastic differential equations (SDEs) and their marginal-preserving ordinary differential equations (ODEs) to describe data perturbation and generative modeling in a unified framework. In this paper, we carefully inspect the ODE-based sampling of a popular variance-exploding SDE and reveal several intriguing structures of its sampling dynamics. We discover that the data distribution and the noise distribution are smoothly connected with a quasi-linear sampling trajectory and another implicit denoising trajectory that even converges faster. Meanwhile, the denoising trajectory governs the curvature of the corresponding sampling trajectory and its finite differences yield various second-order samplers used in practice. Furthermore, we establish a theoretical relationship between the optimal ODE-based sampling and the classic mean-shift (mode-seeking) algorithm, with which we can characterize the asymptotic behavior of diffusion models and identify the empirical score deviation. Code is available at \url{https://github.com/zju-pi/diff-sampler}.
Forward citations
Cited by 3 Pith papers
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x-Prediction Is All You Need:Training-Free Accelerated Generation via Endpoint Decodability
Affine probability paths make the clean endpoint algebraically recoverable from any intermediate state and velocity, so early-exit decoding yields training-free 20–70% NFE savings at near-matched quality.
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x-Prediction Is All You Need:Training-Free Accelerated Generation via Endpoint Decodability
Truncated Jump Sampling stops the ODE early and outputs the algebraically decoded x0 estimate, reducing NFEs by 20-70% across six model families with no retraining.
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x-Prediction Is All You Need:Training-Free Accelerated Generation via Endpoint Decodability
A pretrained diffusion or flow-matching model's endpoint estimate E[x0|xt], decoded from an intermediate state and velocity, is already good enough that truncating the ODE early saves 20-70% of NFEs with near-matched quality.
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