用于可扩展局部生成建模的重整化群流匹配

中文摘要 AI 辅助

尽管生成模型在复杂数据建模方面取得了显著成功，但它们面临着一个基本权衡：全局方法可捕捉完整结构一致性，但计算成本高；而局部模型效率高，却常无法复现长程相关性和全局一致性。重整化群（RG）通过无缝连接不同长度尺度的空间结构，在每一步保留准局域描述的同时保留长程相关性，从而弥合了这一差距。我们提出了重整化群流匹配（RGFM），这是一种系统地构建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从长波长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：准局域性和尺度分离。我们严格证明，对于RG波数尺度Λ、线性系统尺寸L和规定的误差容限ε，RGFM概率流可通过作用于空间范围O(Λ⁻¹[lnL + ln(1/ε)])的局部速度场准确近似。这一特性使得尺寸为O(lnL)的patch（块）可用于局部生成建模，且计算成本随系统体积近线性缩放。我们在代表性一维分布中数值证明，局部RGFM可复现远超其感受野的长程相关性，而传统局部流匹配在长距离处表现出显著误差。在FFHQ图像上，RGFM在64×64和256×256分辨率下生成的样本比局部流匹配更具一致性和更高质量。我们的结果确立了RG引导的概率流作为仅使用局部计算就能捕捉长程结构的可扩展生成建模的有前景途径。

英文摘要

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.

发表机构

• The University of Tokyo（东京大学）
• Institute for Physics of Intelligence, The University of Tokyo（东京大学智能物理研究所）

机构由 AI 辅助整理，请以论文原文为准。

补充信息

备注

42 pages, 9 figures