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Lipschitz Singularities in Diffusion Models
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Shows that noise- and v-prediction diffusion models have infinite Lipschitz constants in time near t=0 and proposes E-TSDM, which shares timestep conditions over sub-intervals near zero to reduce these singularities and improve FID.
Problem
Noise-prediction and v-prediction diffusion models exhibit unbounded Lipschitz constants with respect to the timestep near t=0 (because d sigma_t/dt diverges), threatening training stability and the accuracy of integration-based samplers.
Approach
Divide the near-zero timestep interval [0, t~) into n sub-intervals and feed the left endpoint of each sub-interval as the timestep condition during both training and sampling, so the network is constant within each sub-interval and its Lipschitz constant there is zero, with no change to architecture, loss, or forward process.
Key results
- On FFHQ 256x256 with the linear schedule, E-TSDM reduces FID-10k from 9.50 (DDPM baseline) to 6.62.
- With DPM-Solver-3 at 20 NFE, FID drops from 21.91 (DDPM) to 16.97 (E-TSDM), a reduction of over 22%; the abstract reports over 33% reduction for fast samplers.
- Super-resolution 64->256 on FFHQ improves PSNR from 24.64 to 25.61.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Share timestep conditions near zero using E-TSDM | DDPM, discrete-time diffusion · FFHQ 256×256, linear schedule | FID-10k: 9.50 -> $\bf6.62$ | Helped |
| Share timestep conditions near zero using E-TSDM | DDPM, discrete-time diffusion · FFHQ 256×256, quadratic schedule | FID-10k: 13.79 -> $\bf9.69$ | Helped |
| Share timestep conditions near zero using E-TSDM | DDPM, discrete-time diffusion · FFHQ 256×256, cosine schedule | FID-10k: 27.17 -> $\bf26.08$ | Helped |
| Share timestep conditions near zero using E-TSDM | DDPM, continuous-time diffusion · FFHQ 256×256, cosine schedule | FID-10k: 25.65 -> $\bf16.80$ | Helped |
| Share timestep conditions near zero | DDPM with DPM-Solver-3 · FFHQ 256×256, NFE 50 | FID-10k: 24.48 -> $\bf13.52$ | Helped |
| Share timestep conditions near zero | DDPM with DDIM · FFHQ 256×256, NFE 200 | FID-10k: 23.16 -> $\bf13.71$ | Helped |
| Share timestep conditions near zero in v-prediction | DDPM · FFHQ 256×256 | FID-10k: 10.85 -> $\bf9.00$ | Helped |
| Share timestep conditions near zero in v-prediction | DDPM · Lsun-Cat 256×256 | FID-10k: 18.40 -> $\bf13.86$ | Helped |
| Apply regularization to DDPM | DDPM · FFHQ 256×256 | FID-10k: 9.50 -> 9.18 | Helped |
| Apply regularization to DDPM | DDPM · CelebAHQ 256×256 | FID-10k: 8.05 -> 7.97 | Helped |
| Modify noise schedule | DDPM · FFHQ 256×256, linear schedule | FID-10k: 9.50 -> 8.67 | Helped |
| Modify noise schedule | DDPM · FFHQ 256×256, quadratic schedule | FID-10k: 13.79 -> 17.48 | Hurt |
Surprising findings
- The baseline's FID for DDIM and DPM-Solver worsens as NFE increases, a phenomenon not observed with E-TSDM.
- Modifying noise schedules so that d alpha/dt=0 at t=0 gives unstable results (e.g., quadratic FID 17.48 vs 13.79 baseline).
Limitations
- Not verified on large-scale text-to-image diffusion models.
- Reducing Lipschitz constants to zero within sub-intervals may introduce unknown harmful effects.
- Hyperparameters t~ and n require search for best results.
Experimental setup
Datasets: FFHQ 256x256 CelebAHQ 256x256 AFHQ-Cat 256x256 AFHQ-Wild 256x256 LSUN-Church 256x256 LSUN-Cat 256x256
Benchmarks: FFHQ 256x256 CelebAHQ 256x256 AFHQ-Cat AFHQ-Wild LSUN-Church LSUN-Cat
Metrics: FID-10k PSNR Lipschitz constant K(t,t')
Baselines: DDPM DDPM with regularization Modified noise schedules Remap (1/t, inverse-sigmoid) DDIM DPM-Solver
Builds on: DDPM DDIM DPM-Solver score SDE
Closest papers by meaning
- Tackling the Singularities at the Endpoints of Time Intervals in Diffusion Models (CVPR 2024 · similarity 0.89)
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- Diffusion Bridge Implicit Models (ICLR 2025 · similarity 0.88)
- Fast Solvers for Discrete Diffusion Models: Theory and Applications of High-Order Algorithms (NeurIPS 2025 · similarity 0.88)
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