The Interval Denoiser, a theoretically rigorous framework for latent-free generation, derived directly from the flow matching ODE, establishes an exact analytical mapping for intermediate trajectory states and is shown to reside on a low-dimensional manifold across any time interval.
Abstract
Modern diffusion and flow-based models are increasingly moving toward few-step, latent-free generation to bypass the computational overhead of multi-step sampling and the reconstruction bottlenecks of external autoencoders. We propose the Interval Denoiser, a theoretically rigorous framework for latent-free generation. Derived directly from the flow matching ODE, it establishes an exact analytical mapping for intermediate trajectory states. Unlike prior formulations, our prediction is shown to reside on a low-dimensional manifold across any time interval, making the regression tractable for a network operating directly on pixels. Furthermore, by avoiding empirical algebraic substitutions, our formulation correctly isolates the pure time derivative to prevent biased gradient evaluations and ensure exact first-order optimization. By analyzing this objective, we equip our framework with residual clipping and a time-sampling curriculum, enabling effective long-interval training and improving few-step performance. Trained from scratch on ImageNet 256x256, our model achieves an FID of 4.55 in one step (1-NFE) and 3.98 in two steps (2-NFE) without perceptual losses.
Energy-Guided Flow Matching is introduced that explicitly models a coarse-to-fine generative trajectory by moving endpoint that evolves smoothly from low-frequency image to clean image and requires no adaptation of the backbone and training data.
Diffusion models achieve impressive image-generation quality but remain expensive at inference time. Diffusion distillation reduces sampling steps, yet many distilled models, including SDXL-Lightning and distribution matching distillation methods, suffer from degraded Fr\'echet Inception Distance (FID). We analyze this phenomenon through a PAC-style generalization bound. Our analysis suggests that aggressive early-step redirection of the velocity field makes the distillation target harder to learn, enlarging the train-test gap. As a result, early-step output distributions differ between training and inference, causing distribution mismatch in the intermediate noisy latent used as next-step inputs. We empirically validate this mechanism by showing reduced diversity in both intermediate features and final outputs. To address this issue, we propose EMPURPLE, a simple training-free method that recycles intermediate latents sampled from the original model. EMPURPLE is model-agnostic and improves FID by 7\% to 20\% across DMD2, Hyper-SD, FlashSD, and SDXL-Lightning. The repo is: https://github.com/TheLovesOfLadyPurple/Empurple-Training-Free-Algorithm-To-enhance-Diversity-of-The-Diffusion-Distillation-Model
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