Traditional diffusion-based image restoration methods utilize degraded images as conditional input to effectively guide the reverse generation process, without modifying the original denoising diffusion process.

We address these challenges by investigating the primary contributors to error accumulation in DDIM inversion and identify the singularity problem in traditional noise schedules as a key issue.

A.1 Proof of Proposition 4.1 Proposition 4.1 For any bounded measurable function τ(t): [0, T ] R, the following Reverse SDEs [ (1 + τ Eq. (20) is a reverse-time SDE running[ from T to 0, thus (there)are two additional minus ] signs in Eq. (21) before term A.2 Two Reparameterizations and Exact Solution under Exponential Integrator In this subsection, we will show the exact solution of SDE in both data prediction reparameterization and noise prediction reparameterization. The noise term in data prediction has smaller variance than noise prediction ones, implying the necessity of adopting data prediction reparameterization for the SDE sampler. The computation of variance uses the Itô Isometry, which is a crucial fact of Itô integral. Similar with Proposition 4.2, Eq. (37) can be solved analytically, which is shown in the following propositon: Following the derivation in Proposition 4.2, the mean of the Itô integral term is: [ A.2.4 Comparison between Data and Noise Reparameterizations In Table 1 we perform an ablation study on data and noise reparameterizations, the experiment results show that under the same magnitude of stochasticity, the proposed SA-Solver in data reparameterization has a better convergence which leads to better FID results under the same NFEs. In this subsection, we provide a theoretical view of this phenomenon.

Diffusion models have gained traction as powerful algorithms for synthesizing high-quality images. Central to these algorithms is the diffusion process, a set of equations which maps data to noise in a way that can significantly affect performance. In this paper, we explore whether the diffusion process can be learned from data.

To achieve the highest perceptual quality, state-of-the-art diffusion models are optimized with objectives that typically look very different from the maximum likelihood andtheEvidence LowerBound (ELBO) objectives.

We provide more visualization results in Figure 1, Figure 1, Figure 1, and Figure 1. As mentioned in the limitation section of the main text, our method can generate realistic textures in most regions. However, it may restore incorrect small characters as shown in Figure 1, which is highly ill-posed. Compared with the Uformer, it shows consistent improvements in perceptual quality. Learning to see in the dark. We compare the PSNR-oriented methods and our method.

While the performance of DDGMs is impressive, not all of their aspects are fully understood.

Based onourformulation, weproposeDPM-Solver,afastdedicated high-order solver for diffusion ODEs with the convergence order guarantee.