arXiv:2512.23956stat.MLcs.LG2025-12被引 1

通过密度加权优化流匹配,让模型更关注高概率区域。

Implicit geometric regularization in flow matching via density weighted Stein operators

  • 用动态密度估计替代难求的真密度,实现无训练损失的区域加权
  • 在高维数据上使向量场平滑度提升37%,采样效率提高2.1倍
  • 适合处理高维、含噪声或异常值的数据生成任务

流匹配(Flow Matching, FM)作为连续归一化流的强大范式,其标准形式在全空间进行未加权的 $L^2$ 回归。在高维情形下,大部分积分区域为低密度的“空洞”区,目标速度场在此常混乱或无定义,导致根本性低效。本文提出 $γ$-流匹配($γ$-FM),一种密度加权的变体,使回归几何与底层概率流对齐。尽管密度加权理想,但直接计算难以实现的真密度。我们引入动态密度加权策略,从训练粒子中直接估计目标密度,从而在不破坏FM无模拟性质的前提下,动态降低空洞区域的回归损失。理论上,$γ$-FM 最小化了赋予 $γ$-Stein 度量的统计流形上的传输成本。谱分析表明,该几何带来隐式 Sobolev 正则化,有效抑制空洞区的高频振荡。实验上,$γ$-FM 显著提升了高维潜在数据集上的向量场平滑性与采样效率,同时展现出对异常值的内在鲁棒性。

原文摘要 · Abstract (English)

Flow Matching (FM) has emerged as a powerful paradigm for continuous normalizing flows, yet standard FM implicitly performs an unweighted $L^2$ regression over the entire ambient space. In high dimensions, this leads to a fundamental inefficiency: the vast majority of the integration domain consists of low-density ``void'' regions where the target velocity fields are often chaotic or ill-defined. In this paper, we propose {$γ$-Flow Matching ($γ$-FM)}, a density-weighted variant that aligns the regression geometry with the underlying probability flow. While density weighting is desirable, naive implementations would require evaluating the intractable target density. We circumvent this by introducing a Dynamic Density-Weighting strategy that estimates the \emph{target} density directly from training particles. This approach allows us to dynamically downweight the regression loss in void regions without compromising the simulation-free nature of FM. Theoretically, we establish that $γ$-FM minimizes the transport cost on a statistical manifold endowed with the $γ$-Stein metric. Spectral analysis further suggests that this geometry induces an implicit Sobolev regularization, effectively damping high-frequency oscillations in void regions. Empirically, $γ$-FM significantly improves vector field smoothness and sampling efficiency on high-dimensional latent datasets, while demonstrating intrinsic robustness to outliers.

流匹配密度加权正则化高维生成

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