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

arXiv: https://arxiv.org/abs/2605.22723 · HTML: https://mlsanity.com/paper/2605.22723/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Placed in Diffusion foundations and sampling, one of 14 directions in Diffusion models.
- Builds on 19 of its 26 references that are in the map.
- First posted on arXiv 21 May 2026.

## Placed in

- Diffusion models / [Diffusion foundations and sampling](https://mlsanity.com/diffusion/diffusion-foundations-and-sampling/)

## Builds on (most cited first, 19 of 19)

- [Denoising Diffusion Probabilistic Models](https://mlsanity.com/paper/2006.11239/) (2020)
- [Denoising Diffusion Implicit Models](https://mlsanity.com/paper/2010.02502/) (2020)
- [Diffusion Models Beat GANs on Image Synthesis](https://mlsanity.com/paper/2105.05233/) (2021)
- [Deep Unsupervised Learning using Nonequilibrium Thermodynamics](https://mlsanity.com/paper/1503.03585/) (2015)
- [Scalable Diffusion Models with Transformers](https://mlsanity.com/paper/2212.09748/) (2022)
- [Improved Denoising Diffusion Probabilistic Models](https://mlsanity.com/paper/2102.09672/) (2021)
- [Deep Learning Face Attributes in the Wild](https://mlsanity.com/paper/1411.7766/) (2014)
- [Sampling is as easy as learning the score: theory for diffusion models with minimal data assumptions](https://mlsanity.com/paper/2209.11215/) (2022)
- [Analytic-DPM: an Analytic Estimate of the Optimal Reverse Variance in Diffusion Probabilistic Models](https://mlsanity.com/paper/2201.06503/) (2022)
- [Nearly d-Linear Convergence Bounds for Diffusion Models via Stochastic Localization](https://mlsanity.com/paper/2308.03686/) (2023)
- [Convergence for score-based generative modeling with polynomial complexity](https://mlsanity.com/paper/2206.06227/) (2022)
- [Towards Faster Non-Asymptotic Convergence for Diffusion-Based Generative Models](https://mlsanity.com/paper/2306.09251/) (2023)
- [Estimating the Optimal Covariance with Imperfect Mean in Diffusion Probabilistic Models](https://mlsanity.com/paper/2206.07309/) (2022)
- [KL Convergence Guarantees for Score diffusion models under minimal data assumptions](https://mlsanity.com/paper/2308.12240/) (2023)
- [On the Posterior Distribution in Denoising: Application to Uncertainty Quantification](https://mlsanity.com/paper/2309.13598/) (2023)
- [Stability of the Lanczos Method for Matrix Function Approximation](https://mlsanity.com/paper/1708.07788/) (2017)
- [Optimal Convergence Analysis of DDPM for General Distributions](https://mlsanity.com/paper/2510.27562/) (2025)
- [Improving Probabilistic Diffusion Models With Optimal Diagonal Covariance Matching](https://mlsanity.com/paper/2406.10808/) (2024)
- [A Sharp KL-Convergence Analysis for Diffusion Models under Minimal Assumptions](https://mlsanity.com/paper/2508.16306/) (2025)

## Abstract (arXiv)

A central error measure in Gaussian DDPMs is the path-space KL divergence between the exact reverse chain and the learned Gaussian reverse process. This quantity is especially relevant for procedures such as classifier guidance, which perturb the entire reverse trajectory rather than only the terminal sample. Prior analyses show that standard isotropic reverse covariances suffer an unavoidable Omega(1/T) path-KL error as the number of denoising steps T grows. We show that matching the full posterior covariance breaks this barrier, yielding an order-wise improvement that reduces the path KL to O(1/T2). To make full covariance matching practical, we introduce the Lanczos Gaussian sampler (LGS), a training-free, matrix-free method for sampling from the optimal reverse covariance using only covariance-vector products, which are available through Jacobian-vector products of the posterior mean. LGS avoids dense covariance storage and auxiliary covariance models. We prove that LGS approximation error decays exponentially in the number of Lanczos steps, where each Lanczos step requires a single Jacobian-vector product. Empirically, using only just three such steps improves sample quality over strong diagonal-covariance baselines, including OCM-DDPM, across standard image benchmarks. This identifies full covariance matching as both theoretically valuable and practically accessible for fast DDPM sampling.