提出三类全局光滑可解析逆的双射，提升生成模型表达力与可解释性。
Analytic Bijections for Smooth and Interpretable Normalizing Flows
- 设计三类在实数域上光滑可解析逆的双射函数，兼具表达力与数学优雅性。
- 径向流架构仅用千分之一参数实现与耦合流相当的生成质量，且训练更稳定。
- 适合需几何可解释性的高维物理建模，如ϕ⁴格点场论中的模式崩溃问题解决。
正则化流的核心挑战在于寻找表达力强的可逆标量双射。现有方法存在权衡：仿射变换虽光滑且可解析逆但表达力弱；单调样条具有局部控制但仅分段光滑且定义域有界；残差流虽光滑但需数值求逆。本文提出三类全局光滑（C∞）、定义于整个实数域、且可闭式解析逆的双射，融合了前序方法的优点。除作为耦合流的即插即用替代品外，还提出径向流：一种直接参数化径向坐标而保持角度方向的新架构。径向流展现卓越训练稳定性，生成几何可解释变换，在具径向结构的目标上，以1000倍少的参数达到与耦合流相当的生成质量。我们在一维和二维基准上进行全面评估，并通过ϕ⁴格点场论的高维物理问题实验展示了适用性，其双射优于仿射基线，支持针对问题设计，有效缓解模式崩溃。
原文摘要 · Abstract (English)
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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