Abstract
Modern generative models can produce high-quality samples, but independent samples under the same condition often yields highly similar outputs. Particle-based methods increase diversity by letting the samples in a batch interact, typically through a repulsive force. These forces, however, also push each sample away from the data distribution and produce visible artifacts. We introduce EDDY, a training-free particle guidance method designed to avoid this failure mode. Instead of a repulsive gradient, EDDY couples particles through anti-symmetric matrix fields passed through a Stein operator, a family of drift perturbations that leaves the Fokker--Planck equation invariant. Unlike repulsive guidance, these interactions do not alter a particle's distribution while the batch is independent. For the same reason they cannot create diversity on their own, so EDDY pairs them with a negatively correlated initialization that keeps each particle's prior exact. To use EDDY with perceptual kernels such as DINOv2, we approximate its second-order terms with finite differences and Hutchinson estimates. Across FLUX.1-dev, FLUX.2-klein and SDXL, EDDY achieves higher image quality and prompt alignment than existing particle guidance methods at matched diversity.
EDDY's Building Block
Let \(k(x,y)\) be a similarity kernel (e.g. RBF). Naively, driving particles away from \(x_{\mathrm{star}}\) can be achieved using the negative kernel gradient:
Unfortunately, this distorts the marginal distribution. Instead, EDDY constructs an anti-symmetric matrix field using direction \( v_\textrm{star} \) and applies Stein's operator, producing a vortex-like (eddy-like) motion that preserves the marginal exactly:
Show the math
Naive repulsion uses the negative kernel gradient as a repulsive direction \(r := -\nabla_x k(x,\,x_{\mathrm{star}})\).
EDDY instead builds an anti-symmetric matrix from repulsive vector \(r\) and an arbitrary transport vector \(v_\textrm{star}\): \[ A \;:=\; r \otimes v_\textrm{star} \;-\; v_\textrm{star} \otimes r, \] and applies Stein's operator with respect to the underlying distribution \(p\): \[ \mathcal{A}_p(A) :\;=\; A\,\nabla\log p(x_{\mathrm{star}}) \;+\; \operatorname{div} A. \] Interestingly, \(\text{div}\,A\) is the divergence-free kernel \[ \bigl(\nabla^2_x k(x,x_\textrm{star}) \;-\; \Delta_x k(x,x_\textrm{star}) \bigr) \; v_\textrm{star}, \] which dominates \(\mathcal{A}_p(A)\) as the dimension grows.
EDDY Promotes Diversity
When the direction \(v_1,\ldots,v_n\) is chosen as the velocity each particle \(x_1,\ldots,x_n\) follows in a diffusion or flow matching process, EDDY's building block naturally promotes diversity — steering particles toward different modes without altering any particle's marginal distribution:
EDDY in Text-to-Image Generation
EDDY applied to FLUX.1‑dev and Stable Diffusion XL on COCO captions. Each panel shows four generated variants for the same prompt and random seed — I.I.D samples independently while EDDY steers all four particles apart without changing any individual marginal distribution.
BibTeX
@article{vinograd2026eddy,
title = {Diverse Sampling in Diffusion Models with Divergence-Free Particle Guidance},
author = {Vinograd, Gal and Achituve, Idan and Fetaya, Ethan},
journal = {arXiv preprint arXiv:2605.06553},
year = {2026}
}