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
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[lnL+ln(1/ε)]) for RG wavenumber scale Λ, linear system size L, and prescribed error tolerance ε. This property enables local generative modeling with patches of size O(lnL) 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 ϕloc 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 Λ(t) is related to the logarithmic time t by Λ(t)=ΛUVe−t/τ with τ≪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 ϕ0∼pdata yields a noisy configuration ϕt and the corresponding conditional velocity field ut(ϕt∣ϕ0) . After transforming ϕt and ut according to the site-decimation step in panel (b), a local velocity field vloc,θ is trained by flow matching on local patches. (d) Representative 64×64 and 256×256 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 R3 -valued vector field ϕ=(ϕR,ϕG,ϕB) on the two-dimensional pixel lattice, where ϕ(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 Ktk in the Polchinski RG flow ( 24 ).
Figure 3: (a) Site-decimation transformation R and its probabilistic inverse R−1 . The coarse-grained field ϕ′ is obtained by applying the discrete cosine transform (DCT) to ϕ , discarding the high-wavenumber modes ϕ> with ∣k∣>Λ(t) , and transforming the retained modes ϕ< back onto the decimated lattice. Conversely, ϕ is reconstructed probabilistically by sampling the missing modes from pGS(ϕ>) and applying the inverse DCT to the combined modes. (b) Score-locality length ξt along the RG diffusion path pt . While ξt grows as cΛ(t)−1 , the transformation R , which reduces the linear lattice size from L to L/b , restores the locality length from ξt to ξt/b in lattice units. (c) Successive site-decimation transformations define the rescaled probability path p~t , whose score-locality length remains O(L0) 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 vloc,t .
Figure 4: (a) Decomposition of the spatial domain ΩL into local regions A and B and the remaining region C=A∪B . (b) Global and local velocity fields. In general, the velocity field v(ϕ,t) depends on the entire field configuration ϕ . The local velocity field on region A , denoted by vA(ϕAB,t) , depends only on the field configuration ϕAB on A∪B .
Figure 5: Flow chart of the main theorems and their proofs in Sec. IV .
Figure 6: Toy example of a conditionally local distribution p(ϕ)=∫dzp(ϕ∣z)p(z) , where p(ϕ∣z)=∏x=1LN(ϕx;z,σ2) and p(z)=Uniform{±1} . (a) After marginalizing over z , p(ϕ) exhibits nonlocal global structure. (b) For a fixed z , the conditional distribution p(ϕ∣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 lcorr=64 . (c) Correlation function C(r)=⟨ϕxϕx+r⟩ 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 lcorr=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)=⟨ϕxϕx+r⟩ computed from the generated samples, compared with the exact result in Eq. ( 118 ). The parameters are L=1024 and σ=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 64×64 , 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 64×64 images as a function of the receptive patch size lA+2lB (see Fig. 4 for the geometry). The target-patch size is fixed at lA=16 , while the receptive patch size is varied among lA+2lB=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 256×256 . The models are trained on a subset consisting of the first 5,000 images in the FFHQ dataset, resized to 256×256 , 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
Λ0
τ
βˉ0,kmax
αˉ1,kmin
64×64
π/40
87.42
0.1224
2.9×10−5
8.2×10−7
256×256
π/160
21.90
0.1224
4.5×10−4
8.7×10−7
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
6×105
1
256
128
32
6×105
4
256
64
32
5×105
16
256
32
32
3×105
64
256
16
16
3×105
64
64
Appendix
Table 2: Training and sampling settings for the 256×256 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.
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 χ2-divergence between the generated and true data distributions, leveraging the contraction property of the diffusion process. In practice, the stepwise structure of LFM is naturally amenable to model distillation, and various distillation techniques can be applied to accelerate generation. We empirically demonstrate that LFM achieves competitive generative performance compared to FM on unconditional generation of tabular and image datasets, and on conditional generation of robotic manipulation policies.
Chen Xu, Xiuyuan Cheng, Yao Xie
H. Milton Stewart School of Industrial and Systems Engineering, Georgia Institute of Technology. · Department of Mathematics, Duke University
In this paper, we present Laplacian multiscale flow matching (LapFlow), a novel framework that enhances flow matching by leveraging multi-scale representations for image generative modeling. Our approach decomposes images into Laplacian pyramid residuals and processes different scales in parallel through a mixture-of-transformers (MoT) architecture with causal attention mechanisms. Unlike previous cascaded approaches that require explicit renoising between scales, our model generates multi-scale representations in parallel, eliminating the need for bridging processes. The proposed multi-scale architecture not only improves generation quality but also accelerates the sampling process and promotes scaling flow matching methods. Through extensive experimentation on CelebA-HQ and ImageNet, we demonstrate that our method achieves superior sample quality with fewer GFLOPs and faster inference compared to single-scale and multi-scale flow matching baselines. The proposed model scales effectively to high-resolution generation (up to 1024×1024) while maintaining lower computational overhead.
Zelin Zhao, Petr Molodyk, Haotian Xue +1
Georgia Institute of Technology Atlanta, GA 30332, USA
Despite the remarkable empirical success of flow-matching models, their statistical generalization guarantees remain underdeveloped. Existing analyses often impose restrictive assumptions on the estimated velocity field and yield convergence rates that fail to reflect the intrinsic low-dimensional structure common in real data, such as natural images and molecular geometries. In this work, we study the statistical generalization of flow-matching models for learning an unknown distribution Pdata from finitely many samples. We derive finite-sample error bounds on the learned generative distribution, measured in the Wasserstein-p distance, for all p≥1. Specifically, given n i.i.d. samples from Pdata, we show that, for every d>dp∗(Pdata) and appropriately chosen network architectures and hyperparameters, the learned distribution PFM satisfies Wp(PFM,Pdata)≲n−1/d+n−1/(2p)(log(1/ξ))1/(2p) with probability at least 1−ξ, where dp∗(Pdata) denotes the Wasserstein-p dimension of the target measure. Our results demonstrate that flow matching naturally adapts to the intrinsic geometry of data and mitigates the curse of dimensionality, as the convergence exponent depends on the intrinsic rather than ambient dimension. These guarantees remain meaningful in high-dimensional regimes and provide a theoretical explanation for the empirical success of flow matching on structured data distributions under substantially more relaxed assumptions than those in existing analyses.
Saptarshi Chakraborty, Quentin Berthet, Peter L. Bartlett
Department of Statistics, University of Michigan · Google DeepMind · Department of Statistics, University of California, Berkeley +1