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On Exact Inversion of DPM-Solvers
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Proposes exact inversion of DDIM and high-order DPM-Solver++(2M) via implicit backward Euler (gradient descent or forward step) with high-order term approximation, improving image/noise reconstruction, watermark classification, and background-preserving editing.
Problem
Finding the exact initial noise (inversion) for images generated by fast DPM-solvers, especially high-order multistep solvers, and remaining robust under large classifier-free guidance.
Approach
Treat each explicit DDIM inversion step as an implicit backward Euler problem solved by gradient descent or a forward step method instead of fixed-point iteration, and for multistep DPM-Solver++(2M) approximate the unavailable past-state high-order terms using a fine-grained naive DDIM inversion.
Key results
- Algorithm 1 reduces DDIM 50-step inversion image/noise reconstruction NMSE to below 1e-6 in pixel-space DPM, versus substantially higher errors for naive DDIM inversion.
- Algorithm 2 gives the best Tree-ring watermark classification, with WM2 detection accuracy of 58% for Alg. 2 versus 36% for naive DDIM inversion with D† and 16% naive inversion... (confusion matrix diagonal values 98/32/58/10/23/77).
- Background-preserving editing NMSE on background is 12.8e-3 for Alg. 1 vs 18.4e-3 for naive DDIM with decoder inversion and 30.4e-3 for naive DDIM, close to oracle 11.0e-3.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Use backward Euler inversion instead of naïve DDIM inversion | DDIM image editing · Background-preserving editing (60 × 5 images) | NMSE (×10−3): 30.4±4.8 -> 12.8±2.0 | Helped |
| Add decoder inversion D† to naïve DDIM inversion (ablation) | LDM image editing · Background-preserving editing (60 × 5 images) | NMSE (×10−3): 30.4±4.8 -> 18.4±2.0 | Helped |
| Use backward Euler with high-order term approximation instead of naïve DDIM inversion | DPM-Solver++(2M) inversion · Tree-ring watermark reconstruction, WM 1 | NMAE: 0.24 -> 0.14 | Helped |
| Use backward Euler with high-order term approximation instead of naïve DDIM inversion | DPM-Solver++(2M) inversion · Tree-ring watermark reconstruction, WM 2 | NMAE: 0.28 -> 0.17 | Helped |
| Use backward Euler with high-order term approximation instead of naïve DDIM inversion | DPM-Solver++(2M) inversion · Tree-ring watermark reconstruction, WM 3 | NMAE: 0.26 -> 0.12 | Helped |
| Add decoder inversion D† to naïve DDIM inversion (ablation) | DPM-Solver++(2M) inversion · Tree-ring watermark reconstruction, WM 1 | NMAE: 0.24 -> 0.21 | Helped |
| Add decoder inversion D† to naïve DDIM inversion (ablation) | DPM-Solver++(2M) inversion · Tree-ring watermark reconstruction, WM 2 | NMAE: 0.28 -> 0.25 | Helped |
| Add decoder inversion D† to naïve DDIM inversion (ablation) | DPM-Solver++(2M) inversion · Tree-ring watermark reconstruction, WM 3 | NMAE: 0.26 -> 0.24 | Helped |
Surprising findings
- Naive DDIM inversion with finer step sizes saturates and cannot further reduce reconstruction error, indicating that the error comes from the discretized solver trajectory rather than inversion step size.
Limitations
- Substantially higher computational cost than naive DDIM inversion.
- Requires prior knowledge of the prompt for LDMs.
- Not perfectly exact for accelerated schedulers and LDMs; described as near-exact.
Experimental setup
Datasets: ImageNet64
Benchmarks: ImageNet64 (pixel-space DPM reconstruction) LDM reconstruction Tree-ring watermark classification Patashnik et al. background-preserving editing
Metrics: NMSE LPIPS SSIM NMAE confusion matrix accuracy runtime
Baselines: naive DDIM inversion Pan et al. fixed point iteration (AIDI) Wallace et al. EDICT Zhang et al. bi-directional integration
Builds on: DDIM DPM-Solver++ Pan et al. AIDI fixed-point inversion Tree-ring watermark Prompt-to-prompt background-preserving editing (Patashnik et al.)
Closest papers by meaning
- Accelerating Diffusion Sampling with Optimized Time Steps (CVPR 2024 · similarity 0.89)
- UDPM: Upsampling Diffusion Probabilistic Models (NeurIPS 2024 · similarity 0.89)
- A Modular Conditional Diffusion Framework for Image Reconstruction (NeurIPS 2024 · similarity 0.89)
- On Inference Stability for Diffusion Models (AAAI 2024 · similarity 0.89)
- An Edit Friendly DDPM Noise Space: Inversion and Manipulations (CVPR 2024 · similarity 0.89)
- D^2-DPM: Dual Denoising for Quantized Diffusion Probabilistic Models (AAAI 2025 · similarity 0.88)
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