## Normalizing Flows: A New Frontier Normalizing flows are a powerful method for modeling complex probability distributions, but their effectiveness depends on the choice of efficient and invertible transformations. A new study published on arXiv presents three families of analytic functions that promise to significantly improve the performance and interpretability of these models. ## The New Analytic Functions The research introduces three new classes of analytic bijections: cubic rational, hyperbolic sine (sinh), and cubic polynomial. These functions offer several advantages over existing techniques: * **Smoothness:** They are globally smooth ($C^\infty$). * **Domain:** They are defined on all of $\mathbb{R}$. * **Invertibility:** They are analytically invertible in closed form. These characteristics make them ideal for replacing existing transformations in coupling flows, with performance equal to or better than that of splines. ## Radial Flows and Applications In addition to coupling layers, the researchers developed radial flows, a novel architecture that transforms the radial coordinate while preserving the angular direction. Radial flows exhibit exceptional training stability and produce geometrically interpretable transformations. On targets with radial structure, they can achieve comparable quality to coupling flows with $1000\times$ fewer parameters. The research includes a comprehensive evaluation on 1D and 2D benchmarks, and demonstrates applicability to higher-dimensional physics problems through experiments on $\phi^4$ lattice field theory, where the new bijections outperform affine baselines and enable problem-specific designs that address mode collapse.