证明流匹配的KL散度有确定上界，提升其统计效率可信度。
On Flow Matching KL Divergence
- 基于L2损失有界推导出KL散度的确定上界
- 在总变差距离下实现近似极小极大最优效率
- 适用于追求理论保证的生成模型研究者
我们推导出流匹配分布近似中Kullback-Leibler（KL）散度的确定性、非渐近上界。若L2流匹配损失不超过ε² > 0，则真实数据分布与估计分布之间的KL散度被限制在A₁ε + A₂ε²以内，其中常数A₁和A₂仅依赖于数据和速度场的正则性。该上界表明，在总变差（TV）距离下，流匹配变换器具有统计收敛率。我们证明，流匹配在估计光滑分布时接近极小极大最优效率。结果使流匹配在TV距离下的统计效率可与扩散模型相媲美。合成数据和学习速度场的数值实验验证了理论结论。
原文摘要 · Abstract (English)
We derive a deterministic, non-asymptotic upper bound on the Kullback-Leibler (KL) divergence of the flow-matching distribution approximation. In particular, if the $L_2$ flow-matching loss is bounded by $ε^2 > 0$, then the KL divergence between the true data distribution and the estimated distribution is bounded by $A_1 ε+ A_2 ε^2$. Here, the constants $A_1$ and $A_2$ depend only on the regularities of the data and velocity fields. Consequently, this bound implies statistical convergence rates of Flow Matching Transformers under the Total Variation (TV) distance. We show that, flow matching achieves nearly minimax-optimal efficiency in estimating smooth distributions. Our results make the statistical efficiency of flow matching comparable to that of diffusion models under the TV distance. Numerical studies on synthetic and learned velocities corroborate our theory.
Thank you to arXiv for use of its open access interoperability. PaperDance 不是 arXiv 官方产品；中文卡片由大模型生成，请以原文为准。