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

• URL: http://arxiv.org/abs/2605.22723v1
• Date: Thu, 21 May 2026 16:57:27 GMT
• Title: The Value of Covariance Matching in Gaussian DDPMs and the Lanczos Sampler
• Abstract summary: A central error measure in Gaussian DDPMs is the path-space KL divergence between the exact reverse chain and the learned Gaussian reverse process.<n>We show that matching the full posterior covariance breaks this barrier, yielding an order-wise improvement that reduces the path KL to $O(1/T2)$.<n>We prove that LGS approximation error decays exponentially in the number of Lanczos steps, where each Lanczos step requires a single Jacobian-vector product.
• Score: 4.538165276831438
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: 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 $Ω(1/T)$ 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 $O(1/T^2)$. 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.

Related papers

• Provably adaptive sampling with uniform and remasking discrete diffusion models [9.180350432640912]
  We study a first-order sampler based on the leave-one-out denoiser for uniform and remasking processes.<n>Our main result establishes an adaptive sampling guarantee.<n> Numerical experiments on structured synthetic distributions illustrate the predicted dimension-adaptive behavior.
  arXiv  Detail & Related papers  (2026-08-24T17:54:51Z)
• Causal Discovery in Equal Variance Linear Gaussian DAGs via SURE-Tuned Ridge Regression [18.884190660925444]
  SURE-Ridge is a non-iterative, closed-form estimator for equal variance linear Gaussian SEM.<n>It achieves the lowest structural Hamming distance in the small-sample regime and the lowest run time across all sample sizes tested.
  arXiv  Detail & Related papers  (2026-08-17T21:07:06Z)
• Closed-Form Spectral Regularization for Multi-Task Model Merging [96.82449201305234]
  Model merging combines several independently fine-tuned experts into a single multi-task model without any training data.<n>State-of-the-art merging methods formulate merging as a layer-wise interference problem.<n>We propose SWUDI, a closed-form method that combines a soft exponential filter, which matches the gradient-flow trajectory of iterative descent.
  arXiv  Detail & Related papers  (2026-06-05T14:00:47Z)
• Covariance-aware sampling for Diffusion Models [18.719332889440675]
  We present a co-aware sampler that improves the quality of pixelspace Diffusion Model (DM) sampling in the few-step regime.<n>We demonstrate that for pixel-based DMs, our method consistently produces superior samples compared to state-of-the-art second order samplers.
  arXiv  Detail & Related papers  (2026-05-13T07:46:06Z)
• Distributionally Robust Multi-Objective Optimization [47.16280600850848]
  We introduce distributionally robust multi-objective optimization (DRMOO)<n>We propose a double-loop MGDA that uses an inner loop to estimate dual variables and achieves a total sample complexity $mathcalO(-12)$ for reaching an $$-Pareto-stationary point.<n>To further improve efficiency, we handle generalized-smooth biased estimates, removing the need for double sampling.
  arXiv  Detail & Related papers  (2026-05-07T04:24:17Z)
• Closing the Approximation Gap of Partial AUC Optimization: A Tale of Two Formulations [121.39938773554523]
  The Area Under the ROC Curve (AUC) is a pivotal evaluation metric in real-world scenarios with both class imbalance and decision constraints.<n>We present two simple instance-wise minimax reformulations to close the approximation gap of PAUC optimization.<n>The resulting algorithms enjoy a linear per-iteration computational complexity w.r.t. the sample size and a convergence rate of $O(-2/3)$ for typical one-way and two-way PAUCs.
  arXiv  Detail & Related papers  (2025-12-01T02:52:33Z)
• Feynman-Kac Correctors in Diffusion: Annealing, Guidance, and Product of Experts [64.34482582690927]
  We provide an efficient and principled method for sampling from a sequence of annealed, geometric-averaged, or product distributions derived from pretrained score-based models.<n>We propose Sequential Monte Carlo (SMC) resampling algorithms that leverage inference-time scaling to improve sampling quality.
  arXiv  Detail & Related papers  (2025-03-04T17:46:51Z)
• Online Covariance Estimation in Nonsmooth Stochastic Approximation [14.818683408659764]
  We consider applying approximation (SA) methods to solve nonsmooth variational inclusion problems.<n>Our convergence construction establish the best-known for statistical estimation methods.
  arXiv  Detail & Related papers  (2025-02-07T20:16:51Z)
• Double Variance Reduction: A Smoothing Trick for Composite Optimization Problems without First-Order Gradient [40.22217106270146]
  Variance reduction techniques are designed to decrease the sampling variance, thereby accelerating convergence rates of first-order (FO) and zeroth-order (ZO) optimization methods. In composite optimization problems, ZO methods encounter an additional variance called the coordinate-wise variance, which stems from the random estimation. This paper proposes the Zeroth-order Proximal Double Variance Reduction (ZPDVR) method, which utilizes the averaging trick to reduce both sampling and coordinate-wise variances.
  arXiv  Detail & Related papers  (2024-05-28T02:27:53Z)
• Towards Efficient and Optimal Covariance-Adaptive Algorithms for Combinatorial Semi-Bandits [12.674929126684528]
  We address the problem of semi-bandits, where a player selects among P actions from the power set of a set containing d base items. We show that our approach efficiently leverages the semi-bandit feedback and outperforms bandit feedback approaches.
  arXiv  Detail & Related papers  (2024-02-23T08:07:54Z)
• Improving Diffusion Models for Inverse Problems Using Optimal Posterior Covariance [52.093434664236014]
  Recent diffusion models provide a promising zero-shot solution to noisy linear inverse problems without retraining for specific inverse problems. Inspired by this finding, we propose to improve recent methods by using more principled covariance determined by maximum likelihood estimation.
  arXiv  Detail & Related papers  (2024-02-03T13:35:39Z)
• Sampling and estimation on manifolds using the Langevin diffusion [45.57801520690309]
  Two estimators of linear functionals of $mu_phi $ based on the discretized Markov process are considered.<n>Error bounds are derived for sampling and estimation using a discretization of an intrinsically defined Langevin diffusion.
  arXiv  Detail & Related papers  (2023-12-22T18:01:11Z)
• Adaptive Annealed Importance Sampling with Constant Rate Progress [68.8204255655161]
  Annealed Importance Sampling (AIS) synthesizes weighted samples from an intractable distribution. We propose the Constant Rate AIS algorithm and its efficient implementation for $alpha$-divergences.
  arXiv  Detail & Related papers  (2023-06-27T08:15:28Z)
• Stochastic Approximation Approaches to Group Distributionally Robust Optimization and Beyond [89.72693227960274]
  This paper investigates group distributionally robust optimization (GDRO) with the goal of learning a model that performs well over $m$ different distributions. To reduce the number of samples in each round from $m$ to 1, we cast GDRO as a two-player game, where one player conducts and the other executes an online algorithm for non-oblivious multi-armed bandits. In the second scenario, we propose to optimize the average top-$k$ risk instead of the maximum risk, thereby mitigating the impact of distributions.
  arXiv  Detail & Related papers  (2023-02-18T09:24:15Z)
• Langevin Monte Carlo: random coordinate descent and variance reduction [7.464874233755718]
  Langevin Monte Carlo (LMC) is a popular Bayesian sampling method. We investigate how to enhance computational efficiency through the application of RCD (random coordinate descent) on LMC.
  arXiv  Detail & Related papers  (2020-07-26T18:14:36Z)

This list is automatically generated from the titles and abstracts of the papers in this site.

This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.