CAT-Flow: Curvature-Adaptive sTeps for Flow Matching

Flow Matching has emerged as a leading framework for generative modeling, powering state-of-the-art systems such as FLUX and Stable Diffusion 3.5. However, the iterative nature of its ODE-based sampling process creates a fundamental efficiency bottleneck: the quality of generated samples is highly sensitive to the choice of step-sizes, and current models typically require 20 to 30 steps for good quality. In this work, we propose two lightweight, training-free algorithms, CAT-OV and CAT-OT that adapt step-sizes at inference time based on a novel connection between Flow Matching sampling and gradient flow. Our algorithms are computed efficiently by not requiring additional neural function evaluations. Specifically, CAT-OT estimates curvature over time via a finite-difference approximation of the time-derivative of the vector field, while CAT-OV approximates curvature over the state space via a gradient of the vector field. Under suitable conditions, both methods have truncation error bounds of constant order. Empirically, CAT-OV and CAT-OT outperform existing step-size heuristics in image quality metrics across four text- to-image Flow Matching models, reducing the number of generation steps required to reach comparable quality by up to 40%.

Magnitude-Direction Decoupling for Fast Video Generation with Flow Matching Models

Flow matching models for video generation achieve impressive performance but suffer from high computational overhead due to iterative denoising. In fact, the original model is not necessary for all denoising steps, allowing some steps to use lightweight alternatives for faster sampling. However, directly using caching or lightweight models can deviate from the original denoising trajectory, resulting in suboptimal performance. Through empirical analysis, we find that lightweight models can robustly capture the magnitude components of the original model's output, while caching provides reliable directional guidance. Building on this insight, we propose the Magnitude-Direction Decoupling (MDD) method, which adaptively employs a direction-calibrated lightweight model as a substitute for the original model to accelerate inference and effectively correct deviations in the denoising trajectory. Moreover, MDD further reduces inference costs by reusing magnitude information under classifier-free guidance (CFG). As a result, MDD offers a more reliable and lightweight solution to accelerate sampling. Experiments show that MDD outperforms existing acceleration methods, delivering promising speedups (e.g., up to 2.95x on Wan2.1) while preserving high visual fidelity and content richness.

x-Prediction Is All You Need:Training-Free Accelerated Generation via Endpoint Decodability

Diffusion and flow matching models generate high-quality samples, but their ODE samplers often need tens to hundreds of neural function evaluations (NFEs). This remains a practical challenge for released checkpoints, since many accelerators require additional design choices and training cost through retraining, distillation, or trajectory redesign. We investigate a different route based on $x$-prediction. During sampling, standard affine probability paths already expose $x_0$ information: an intermediate state and its path velocity determine a principled estimate of the clean sample. We formalize this property as \textbf{endpoint decodability} and show that the decoder is the minimum-MSE estimator $\mathbb{E}[x_0\mid x_t]$ under the usual $\ell_2$ objective. This yields \textbf{Truncated Jump Sampling} (TJS): stop the ODE at an early-exit time $t^*$ and return the decoded $x_0$. TJS requires no retraining, distillation, or architecture change. Across SDXL, SD3.5M, Z-Image-Turbo, and three class-conditional benchmarks, it reduces NFEs by 20--70\% with near-matched quality. The analysis also shows why endpoint prediction can work without straightening the trajectory, providing inference acceleration without trajectory redesign.

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

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.

A Tutorial on Diffusion Theory: From Differential Equations to Diffusion Models

This tutorial develops diffusion models from the viewpoint of differential equations. We begin with the conditional Gaussian forward process and show that this path admits both an ordinary differential equation (ODE) representation and a stochastic differential equation (SDE) representation. Averaging the conditional process over the data distribution then yields marginalized forward ODE and SDE formulations that transport the data distribution $p_0=p_{\mathrm{data}}$ to a Gaussian prior $p_1=\mathcal{N}(0,I)$. We next derive the corresponding reverse-time dynamics, namely the reverse SDE and the reverse probability-flow ODE, both of which are governed by the marginal score $\grad\log p_t(x)$. This leads to a training objective for score estimation and shows that the standard noise-prediction objective is equivalent to score matching up to an additive constant independent of the model parameters. We then discuss sampling methods for the learned reverse dynamics, including DPM-Solver, as well as guided sampling through classifier guidance and classifier-free guidance. Finally, we compare DDPM and DDIM with the reverse SDE/ODE framework and show that they share the same training objective, while DDPM sampling corresponds to discrete reverse-SDE sampling and DDIM sampling corresponds to reverse-ODE sampling.