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The Value of Covariance Matching in Gaussian DDPMs and the Lanczos Sampler

NeurIPS 2026 · Probabilistic and scientific modeling › Variational and generative sampling · approach: Posterior sampling with diffusion guidance · 0 citations (OpenAlex)

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Shows full posterior covariance matching in Gaussian DDPMs reduces path KL to O(1/T^2) and proposes the training-free, matrix-free Lanczos Gaussian Sampler (LGS) that samples the full covariance using Jacobian-vector products, improving FID over OCM-DDPM on CIFAR-10, CelebA and ImageNet 64.

Problem

Gaussian DDPM reverse kernels with isotropic or diagonal covariance incur Omega(1/T) path-space KL error, which matters for procedures like classifier guidance that perturb the whole reverse trajectory; full posterior covariance is impractical to store or learn at image scale.

Approach

Matching the full posterior covariance yields O(1/T^2) path KL; the covariance-vector product is available via a Jacobian-vector product of the posterior mean (Tweedie), so a Lanczos/Krylov approximation of Sigma^{1/2} z with m matvecs samples from the full Gaussian without storing covariance or training an auxiliary model.

Key results

• LGS-DDPM (m=3, l=2) reaches FID 3.85 on CIFAR-10 at T=50 vs 5.83 for OCM-DDPM at T=50.
• On CelebA 64x64 at T=100, LGS-DDPM (m=3, l=2) achieves FID 4.09 vs 7.09 for OCM-DDPM.
• Under matched wall-clock budget, LGSb-DDPM at T=50 (FID 5.40 CIFAR-10, 6.69 CelebA) beats OCM-DDPM at T=67 (6.10, 8.10); on ImageNet OCM-DDPM remains better (19.23 vs 21.19 at matched budget).

Surprising findings

• Batching covariance computations across steps (l=2) gave the best FID in 5 of 9 settings, despite using staler covariance, which the authors conjecture is implicit averaging.
• LGS on ImageNet 64x64 does not consistently beat OCM-DDPM; at T=100 OCM-DDPM (18.09) is competitive with or better than LGS variants under matched budget.

Limitations

• On ImageNet 64x64, LGS-based variants are competitive but not uniformly better than OCM-DDPM under matched wall-clock budget.
• Experiments use single pretrained score checkpoints, so variability from retraining is not measured.
• Theory uses bounded data assumption ||x0|| <= D and a Gaussian-channel argument that yields polynomial dependence on D.

Experimental setup

Datasets: CIFAR-10 CelebA 64x64 ImageNet 64x64

Benchmarks: CIFAR-10 CelebA 64x64 ImageNet 64x64

Metrics: FID wall-clock time per sample

Baselines: DDPM (beta_t) DDPM (tilde beta_t) OCM-DDPM

Builds on: OCM-DDPM Improved DDPM Krylov/Lanczos matrix function approximation (MMS18) I-MMSE identity

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