Renormalization Group Flow Matching for Scalable Local Generative Modeling

• URL: http://arxiv.org/abs/2608.23696v1
• Date: Mon, 24 Aug 2026 18:00:04 GMT
• Title: Renormalization Group Flow Matching for Scalable Local Generative Modeling
• Abstract summary: We introduce renormalization group flow matching (RGFM), a generative framework that systematically structures data generation across different spatial scales.<n>By using an exact RG flow as the probability path, RGFM progressively generates data from long- to short-wavelength structures.<n>We numerically demonstrate that local RGFM reproduces long-range correlations far beyond its receptive field in representative one-dimensional distributions.
• Score: 0.0
• License: http://creativecommons.org/licenses/by/4.0/
• 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.

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