The sampling process of Denoising Diffusion Probabilistic Models (DDPMs) can be accelerated by leveraging second-order information in the form of approximations to the denoising posterior covariance -- allowing samples of acceptable quality to be produced in fewer but larger sampling steps. Previous attempts at using such information have used drastic (e.g.\ diagonal) simplifications of the covariance. These do not do justice to the peculiar statistical structure of natural images, which exhibit strong non-diagonal correlations between pixels and color channels, and a slow-decaying power-law frequency spectrum. Here, we develop a novel covariance model that captures these features. Our Kronecker-DCT (K-DCT) model uses a Kronecker-factored decomposition of inter-color covariances and spatial covariances modeled in the frequency domain using the Discrete Cosine Transform (DCT). The use of the DCT reduces the computational complexity from quadratic to log-linear, resulting in negligible computational and memory overhead in each denoising step. By learning K-DCT-structured amortizations of the denoising posterior covariance using pre-trained score models on CIFAR-10, Celeb-A, ImageNet and LSUN datasets, we show improved performance compared to previous SOTA denoising samplers, both in terms of FID and likelihoods, especially in the regime of few denoising steps.
Training-free feature forecasting accelerates diffusion sampling by predicting features at skipped denoising steps. Recent work has mainly focused on designing stronger forecasters. Yet forecast error varies sharply across steps, and open-loop caches trust the forecast in full at every skipped step. This fixed trust is what breaks as acceleration turns aggressive. The missing question is not only how to forecast better, but when and how much to trust a forecast. We show that reliability can be observed from the cache itself. Two forecasts agree where the feature trajectory is smooth, and they diverge where prediction turns hard. Their disagreement is a cheap runtime signal, and it costs no extra denoiser evaluation. Based on this signal, we introduce RACER, a training-free closed-loop controller with two responses. It continuously shrinks uncertain forecasts toward the last computed feature. At the riskiest steps, RACER refreshes the feature and repays the added evaluation by skipping a later scheduled one. We derive a deterministic error bound for the shrinkage and empirically evaluate its validity and tightness across acceleration regimes. At the same number of denoiser evaluations, RACER improves the strongest open-loop baseline across SD3.5-Large, FLUX.1-dev, Wan2.1-14B, and HunyuanVideo on DrawBench, VBench, and COCO. On SD3.5, we further show that RACER samples faster at equal quality. RACER generalizes across forecasting designs as well. For example, it recovers much of the quality lost on a Taylor base. These results show that reliable diffusion acceleration also depends on how forecasts are used. Code is available at https://github.com/LiZaiyuan0619/RACER
We propose a novel diffusion model, Flicker-DDPM, which incorporates flicker (1/f) noise inspired by self-organized criticality (SOC), a widely observed phenomenon in natural systems. Unlike denoising diffusion probabilistic models (DDPMs), which employ isotropic white noise in the forward process, Flicker-DDPM adopts colored noise with power-law spectra to better match the spectral statistics of natural images, whose power spectra typically follow P(k) proportional to 1/k^α. To this end, we develop a colored-noise module based on a spatial correlation kernel, σ(d) = (d + 1)^{-η}, and theoretically establish that adjusting η controls the spectral exponent α of the generated 1/fα noise, enabling adaptation to datasets with diverse spectral characteristics. On CIFAR-10, Flicker DDPM matches or surpasses the generation quality of a standard DDPM baseline using 3.33 times fewer sampling steps, with negligible additional computational cost per step. We further develop a frequency-domain linear theory demonstrating that spectrally matched colored noise linearizes the reverse trajectory, theoretically explaining the observed sampling acceleration.