用于平滑且可解释的归一化流的解析双射
Analytic Bijections for Smooth and Interpretable Normalizing Flows
AI总结：
提出三类全局光滑、解析可逆的双射函数，替代耦合流中的仿射变换或样条，并设计径向流架构，在径向结构目标上以千分之一参数达到耦合流质量。
摘要：
归一化流中的一个关键挑战是找到表达力强的可逆标量双射。现有方法面临权衡：仿射变换光滑且解析可逆但缺乏表达力；单调样条提供局部控制但仅分段光滑且作用于有界域；残差流实现光滑性但需要数值求逆。我们引入了三类解析双射，它们全局光滑（$C^\infty$），定义在整个$\mathbb{R}$上，且以闭式解析可逆，结合了先前方法的有利性质。除了作为耦合流中的即插即用替代品（其性能匹配或超越样条），我们还开发了径向流：一种使用直接参数化的新颖架构，在保持角度方向的同时变换径向坐标。径向流表现出卓越的训练稳定性，产生几何可解释的变换，并且在具有径向结构的目标上，能以$1000$倍更少的参数达到与耦合流相当的质量。我们在1D和2D基准测试上进行了全面评估，并通过$\phi^4$格点场论实验证明了其在更高维物理问题中的适用性，其中我们的双射优于仿射基线，并能够解决模式崩溃问题的特定设计。
英文摘要：
A key challenge in normalizing flows is finding expressive invertible scalar bijections. Existing approaches face trade-offs: affine transformations are smooth and analytically invertible but lack expressivity; monotonic splines offer local control but are only piecewise smooth and act on bounded domains; residual flows achieve smoothness but need numerical inversion. We introduce three families of analytic bijections that are globally smooth ($C^\infty$), defined on all of $\mathbb{R}$, and analytically invertible in closed form, combining the favorable properties of prior approaches. Beyond serving as drop-in replacements in coupling flows, where they match or exceed spline performance, we develop radial flows: a novel architecture using direct parametrization that transforms the radial coordinate while preserving angular direction. Radial flows exhibit exceptional training stability, produce geometrically interpretable transformations, and on targets with radial structure can achieve comparable quality to coupling flows with $1000\times$ fewer parameters. We provide comprehensive evaluation on 1D and 2D benchmarks, and demonstrate applicability to higher-dimensional physics problems through experiments on $ϕ^4$ lattice field theory, where our bijections outperform affine baselines and enable problem-specific designs that address mode collapse.
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