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CCS: Controllable and Constrained Sampling with Diffusion Models via Initial Noise Perturbation
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Shows that DDIM output change is near-linear in initial-noise perturbation scale and proposes CCS, a spherical-interpolation noise perturbation with a binary-search controller, to sample diffusion outputs centered on a target mean at a specified diversity level.
Problem
Controlling the distribution of samples from a diffusion model so that they center on a target mean with a specified statistical spread (diversity), rather than constraining each sample individually, while preserving sample quality.
Approach
Empirically and theoretically, DDIM/ODE output change is approximately linear in the initial noise perturbation scale. CCS spherically interpolates the DDIM-inverted noise of a target image with fresh Gaussian noise, which keeps the noise norm near sqrt(d), and a controller binary-searches the interpolation angle C0 to hit a target rMSE.
Key results
- On FFHQ pixel diffusion at rMSE 0.12, CCS-C reaches PSNR 25.13 and SD 0.104 versus 23.52 and 0.088 for the best baseline CCDF-C.
- On CIFAR-10 at rMSE 0.11, CCS-C achieves PSNR 26.05 and IS 8.09, above CCDF-C (24.63, 7.91).
- Linearity R-square between sin(C0) and normalized residual norm is 0.995 (FFHQ), 0.988 (CIFAR-10), 0.959 (CelebA-HQ), 0.947 (fMoW).
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Use CCS spherical initial-noise perturbation instead of CCDF-C | Pixel diffusion sampling · FFHQ, rMSE target 0.12 | PSNR: 23.52 -> 25.13 | Helped |
| Use CCS spherical initial-noise perturbation instead of CCDF-C | Pixel diffusion sampling · FFHQ, rMSE target 0.12 | SD: 0.088 -> 0.104 | Helped |
| Use CCS spherical initial-noise perturbation instead of CCDF-C | Pixel diffusion sampling · FFHQ, rMSE target 0.12 | CLIP-IQA: 0.746 -> 0.750 | Helped |
| Use CCS spherical initial-noise perturbation instead of CCDF-C | Pixel diffusion sampling · FFHQ, rMSE target 0.12 | MUSIQ: 66.15 -> 66.79 | Helped |
| Use CCS spherical initial-noise perturbation instead of GP-C | Pixel diffusion sampling · CIFAR-10, rMSE target 0.11 | PSNR: 24.66 -> 26.05 | Helped |
| Use CCS spherical initial-noise perturbation instead of GP-C | Pixel diffusion sampling · CIFAR-10, rMSE target 0.11 | SD: 0.100 -> 0.107 | Helped |
| Use CCS spherical initial-noise perturbation instead of CCDF-C | Pixel diffusion sampling · CIFAR-10, rMSE target 0.11 | IS: 7.91 -> 8.09 | Helped |
| Use CCS spherical initial-noise perturbation instead of CCDF-C | Stable Diffusion 1.5 sampling · CelebA-HQ, rMSE target 0.07 | PSNR: 27.66 -> 30.29 | Helped |
| Use CCS spherical initial-noise perturbation instead of CCDF-C | Stable Diffusion 1.5 sampling · CelebA-HQ, rMSE target 0.07 | SD: 0.051 -> 0.053 | Helped |
| Use CCS spherical initial-noise perturbation instead of CCDF-C | Stable Diffusion 1.5 sampling · CelebA-HQ, rMSE target 0.07 | CLIP-IQA: 0.735 -> 0.732 | Hurt |
| Use CCS spherical initial-noise perturbation instead of CCDF-C | Stable Diffusion 1.5 sampling · CelebA-HQ, rMSE target 0.07 | MUSIQ: 49.29 -> 49.66 | Helped |
| Increase spherical perturbation scale C0 | CCS · FFHQ | R-square | Helped |
Surprising findings
- Output semantic changes also appear to vary linearly with the perturbation scale, not only pixel values.
- Directly adding Gaussian noise to the initial latent (GP) yields noisy samples and sample means, because it pushes the noise off the typical norm sphere.
Limitations
- Partial inversion is needed for latent diffusion with CFG because exact DDIM inversion is inaccurate, forcing an intermediate t0 choice tuned on one image.
- Theory relies on smoothness assumptions and high-dimensional concentration, and the mean-preservation guarantee holds only approximately for small C0.
- Evaluation uses a single target mean per setting and lacks a formal distribution-level metric.
Experimental setup
Datasets: FFHQ CIFAR-10 CelebA-HQ fMoW
Benchmarks: FFHQ CIFAR-10 CelebA-HQ
Metrics: PSNR SD CLIP-IQA MUSIQ Inception Score R-square
Baselines: GP-C ILVR-C DPS-C CCDF-C LDPS-C
Builds on: DDIM Diffusion posterior sampling ILVR Come-closer-diffuse-faster
Closest papers by meaning
- Align Your Steps: Optimizing Sampling Schedules in Diffusion Models (ICML 2024 · similarity 0.91)
- Constrained Diffusion with Trust Sampling (NeurIPS 2024 · similarity 0.91)
- CADS: Unleashing the Diversity of Diffusion Models through Condition-Annealed Sampling (ICLR 2024 · similarity 0.91)
- A Simple Early Exiting Framework for Accelerated Sampling in Diffusion Models (ICML 2024 · similarity 0.90)
- Fast constrained sampling in pre-trained diffusion models (NeurIPS 2025 · similarity 0.90)
- Simple and Fast Distillation of Diffusion Models (NeurIPS 2024 · similarity 0.90)
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