Renormalization Group Flow Matching for Scalable Local Generative Modeling

重整化群流匹配：用于可扩展局部生成建模

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.

摘要： 尽管生成模型在建模复杂数据方面取得了显著成功，但它们面临着一个根本性的权衡。全局方法能够捕捉完整的结构连贯性，但计算成本高昂；而局部模型虽然高效，却往往无法重现长程相关性和全局连贯性。重整化群（RG）通过无缝连接不同长度尺度下的空间结构，弥补了这一差距，在每一步保持准局部描述的同时，保留了长程相关性。

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.

我们引入了重整化群流匹配（RGFM），这是一个系统化构建跨不同空间尺度数据生成的生成框架。通过使用精确的 RG 流作为概率路径，RGFM 能够从长波长结构到短波长结构逐步生成数据。为了协调可扩展性与全局结构，我们利用了 RG 的两个关键特性：准局部性和尺度分离。

We rigorously show that the RGFM probability flow can be accurately approximated by local velocity fields acting over a spatial range $O(\Lambda^{-1}[\ln L+\ln(1/\varepsilon)])$ for RG wavenumber scale $\Lambda$, 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.

我们严格证明了 RGFM 概率流可以由局部速度场精确近似，该速度场作用于空间范围 $O(\Lambda^{-1}[\ln L+\ln(1/\varepsilon)])$，其中 $\Lambda$ 为 RG 波数尺度，$L$ 为线性系统尺寸，$\varepsilon$ 为预设的误差容限。这一特性使得使用 $O(\ln L)$ 大小的补丁进行局部生成建模成为可能，且计算成本随系统体积呈近线性增长。

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.

我们通过数值实验证明，在代表性的一维分布中，局部 RGFM 能够重现远超其感受野的长程相关性，而传统的局部流匹配在长距离上表现出显著误差。在 FFHQ 图像上，RGFM 在 64x64 和 256x256 分辨率下生成的样本比局部流匹配更具连贯性且质量更高。我们的研究结果表明，RG 引导的概率流是实现可扩展生成建模的一条有前途的途径，它仅利用局部计算即可捕捉长程结构。