Renormalization Group Flow Matching for Scalable Local Generative Modeling

Kanta Masuki; Yuto Ashida

cs.LGAug 24, 2026

Renormalization Group Flow Matching for Scalable Local Generative Modeling

Authors: Kanta Masuki, Yuto Ashida

Organizations: Department of Physics, The University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo 113-0033, Japan · Institute for Physics of Intelligence, The University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo 113-0033, Japan

Abstract

Despite their remarkable success in modeling complex data, generative models face a fundamental tradeoff. Global approaches can capture full structural coherence but suffer from high computational costs, while local models are efficient but often fail to reproduce long-range correlations and global coherence. The renormalization group (RG) bridges this gap by seamlessly connecting spatial structures across different length scales, retaining quasi-local descriptions at each step while preserving long-range correlations. We introduce renormalization group flow matching (RGFM), a generative framework that systematically structures data generation across different spatial scales. By using an exact RG flow as the probability path, RGFM progressively generates data from long- to short-wavelength structures. To reconcile scalability with global structure, we exploit two key properties of the RG: quasi-locality and scale separation. We rigorously show that the RGFM probability flow can be accurately approximated by local velocity fields acting over a spatial range $O(Λ^{-1}[\ln L+\ln(1/\varepsilon)])$ for RG wavenumber scale $Λ$ , linear system size $L$ , and prescribed error tolerance $\varepsilon$ . This property enables local generative modeling with patches of size $O(\ln L)$ and a computational cost that scales nearly linearly with the system volume. We numerically demonstrate that local RGFM reproduces long-range correlations far beyond its receptive field in representative one-dimensional distributions, while conventional local flow matching exhibits substantial errors at long distances. On FFHQ images, RGFM yields far more coherent and higher-quality samples than local flow matching at 64x64 and 256x256. Our results establish RG-guided probability flows as a promising route toward scalable generative modeling that captures long-range structure using only local computation.

Figures & tables

Figure 1: (a) Local generative modeling. Exploiting the underlying spatial structure of the data, the velocity field in a target region $A$ (shaded in blue) is approximated using only the configuration $\phi_{\rm loc}$ in its surrounding neighborhood. Data are generated by integrating the ODE flow defined by this locally approximated velocity field, which is learned using a local neural network (NN). (b) Schematic of the probability path used for local generative modeling with renormalization group flow matching (RGFM). The RG wavenumber scale $\Lambda(t)$ is related to the logarithmic time $t$ by $\Lambda(t)=\Lambda_{\rm UV}e^{-t/\tau}$ with $\tau\ll 1$ being a characteristic time scale specified later. Along the forward path from $t=0$ to $t=1$ , the coarse-graining diffusion process, which we call the RG diffusion, is alternated with site-decimation transformations. The RG diffusion can be reversed by solving the ODE generated by a local velocity field, while each site-decimation step can be reversed at the level of probability distributions by resampling the discarded Gaussian modes. The entire path can therefore be reversed from $t=1$ to $t=0$ using only local patches. (c) Training scheme for local RGFM. Applying the RG diffusion to a data configuration $\phi_{0}\sim p_{\rm data}$ yields a noisy configuration $\phi_{t}$ and the corresponding conditional velocity field $u_{t}(\phi_{t}|\phi_{0})$ . After transforming $\phi_{t}$ and $u_{t}$ according to the site-decimation step in panel (b), a local velocity field $v_{{\rm loc},\theta}$ is trained by flow matching on local patches. (d) Representative $64\times 64$ and $256\times 256$ image samples generated by local RGFM and the standard local FM. While the local model based on the standard FM fails to generate globally consistent samples, the local RGFM produces substantially more globally coherent samples.

Figure 2: (a) Correspondence between image data and an $\mathbb{R}^{3}$ -valued vector field $\vec{\phi}=(\phi_{R},\phi_{G},\phi_{B})$ on the two-dimensional pixel lattice, where $\vec{\phi}(x)$ represents the RGB value of the pixel at position $x$ . (b) Data generation in the standard flow matching (FM) and the renormalization group flow matching (RGFM). (c) Behavior of the RG cutoff function $K_{tk}$ in the Polchinski RG flow ( 24 ).

Figure 3: (a) Site-decimation transformation $\mathcal{R}$ and its probabilistic inverse $\mathcal{R}^{-1}$ . The coarse-grained field $\phi^{\prime}$ is obtained by applying the discrete cosine transform (DCT) to $\phi$ , discarding the high-wavenumber modes $\phi_{>}$ with $|k|>\Lambda(t)$ , and transforming the retained modes $\phi_{<}$ back onto the decimated lattice. Conversely, $\phi$ is reconstructed probabilistically by sampling the missing modes from $p_{\rm GS}(\phi_{>})$ and applying the inverse DCT to the combined modes. (b) Score-locality length $\xi_{t}$ along the RG diffusion path $p_{t}$ . While $\xi_{t}$ grows as $c\Lambda(t)^{-1}$ , the transformation $\mathcal{R}$ , which reduces the linear lattice size from $L$ to $L/b$ , restores the locality length from $\xi_{t}$ to $\xi_{t}/b$ in lattice units. (c) Successive site-decimation transformations define the rescaled probability path $\tilde{p}_{t}$ , whose score-locality length remains $O(L^{0})$ throughout the probability path from $t=0$ to $t=1$ . Each segment can thus be approximately tracked by the ODE flow ( 3 ) generated by a local velocity field $v_{{\rm loc},t}$ .

Figure 4: (a) Decomposition of the spatial domain $\Omega_{L}$ into local regions $A$ and $B$ and the remaining region $C=\overline{A\cup B}$ . (b) Global and local velocity fields. In general, the velocity field $v(\phi,t)$ depends on the entire field configuration $\phi$ . The local velocity field on region $A$ , denoted by $v^{A}(\phi_{AB},t)$ , depends only on the field configuration $\phi_{AB}$ on $A\cup B$ .

Panel of Figure 5 — Figure 5: Flow chart of the main theorems and their proofs in Sec. IV .

Figure 6: Toy example of a conditionally local distribution $p(\phi)=\int dz\,p(\phi|z)p(z)$ , where $p(\phi|z)=\prod_{x=1}^{L}\mathcal{N}(\phi_{x};z,\sigma^{2})$ and $p(z)={\rm Uniform}\{\pm 1\}$ . (a) After marginalizing over $z$ , $p(\phi)$ exhibits nonlocal global structure. (b) For a fixed $z$ , the conditional distribution $p(\phi|z)$ is a local Gaussian distribution.

Figure 7: (a) Architecture of the convolutional neural network (CNN) used to learn the local velocity field in Secs. V.1 and V.2 . The finite receptive field of the network is indicated schematically. (b) Representative configurations sampled from the one-dimensional Ising model ( 112 ) and generated by local-CNN RGFM and FM for the correlation length

l_{\rm corr}=64

. (c) Correlation function

C(r)=\langle\phi_{x}\phi_{x+r}\rangle

computed from the generated samples, compared with the exact result in Eq. ( 114 ). We set the system size

L=1024

and the Ising coupling

K

such that the correlation length takes the values

l_{\rm corr}=32

64

128

, and

256

Figure 8: (a) Representative configurations sampled from the conditionally local distribution in Eq. ( 115 ) and generated by local-CNN RGFM and FM. (b) Correlation function

C(r)=\langle\phi_{x}\phi_{x+r}\rangle

computed from the generated samples, compared with the exact result in Eq. ( 118 ). The parameters are

L=1024

and

\sigma=0.05

, and we use the same CNN as in Fig. 7 (a).

Figure 9: (a) Typical image samples generated by local generative modeling with the RGFM and the standard FM. The models are trained on the FFHQ dataset resized to a resolution of $64\times 64$ , using target patches of linear size $16$ and receptive patches of linear size $32$ . (b) Fréchet inception distance (FID) scores 36 ; 86 of the generated $64\times 64$ images as a function of the receptive patch size $l_{A}+2l_{B}$ (see Fig. 4 for the geometry). The target-patch size is fixed at $l_{A}=16$ , while the receptive patch size is varied among $l_{A}+2l_{B}=32,48$ , and $64$ . We generate $50{,}000$ images with each trained model. (c) Typical image samples generated by local generative modeling with the RGFM and the standard FM at a resolution of $256\times 256$ . The models are trained on a subset consisting of the first $5{,}000$ images in the FFHQ dataset, resized to $256\times 256$ , using local patches with receptive and target patch sizes of $32$ and $16$ , respectively.

Appendix figures & tables2 assets

Supplementary material from the paper’s appendix.

Appendix

Resolution	$m$	$\Lambda_{0}$	$\tau$	$\bar{\beta}_{0,k_{\rm max}}$	$\bar{\alpha}_{1,k_{\rm min}}$
$64\times 64$	$\pi/40$	$87.42$	$0.1224$	$2.9\times 10^{-5}$	$8.2\times 10^{-7}$
$256\times 256$	$\pi/160$	$21.90$	$0.1224$	$4.5\times 10^{-4}$	$8.7\times 10^{-7}$

Appendix

Table 1: Parameters of the RG schedule used in the image-generation experiments.

$L$	Receptive size	Training steps	Batch size	Local patches
$256$	$32$	$6\times 10^{5}$	$1$	$256$
$128$	$32$	$6\times 10^{5}$	$4$	$256$
$64$	$32$	$5\times 10^{5}$	$16$	$256$
$32$	$32$	$3\times 10^{5}$	$64$	$256$
$16$	$16$	$3\times 10^{5}$	$64$	$64$

Appendix

Table 2: Training and sampling settings for the

256\times 256

RGFM experiment. The batch size denotes the number of full images, while the number of local patches counts the inputs supplied to the U-Net in one optimization batch.

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Flow Matching (FM) is a simulation-free method for learning a continuous, invertible flow that interpolates between two distributions, and in particular generates data from noise. Inspired by the variational nature of the diffusion process as a gradient flow, we introduce a stepwise FM model, Local Flow Matching (LFM), which sequentially learns a sequence of FM submodels, each matching a diffusion process up to the time-step size in the data-to-noise direction. In each step, the two distributions to be interpolated by the sub-flow model are closer than those in the full-flow matching model, which interpolates data to noise distributions, enabling smaller models with more efficient training. This variational perspective also allows us to prove a theoretical generation guarantee for the proposed flow model in terms of the

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