[Paper Review] Analytic Bijections for Smooth and Interpretable Normalizing Flows

Mathis Gerdes, Miranda C. N. Cheng|arXiv (Cornell University)|Jan 15, 2026

Model Reduction and Neural Networks0 citations

TL;DR

The paper introduces three families of analytic, globally smooth, and analytically invertible bijections for normalizing flows, including a novel radial flow architecture with interpretable geometry and high training stability.

ABSTRACT

A key challenge in designing normalizing flows is finding expressive scalar bijections that remain invertible with tractable Jacobians. 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 -- cubic rational, sinh, and cubic polynomial -- that are globally smooth ($C^\infty$), defined on all of $\mathbb{R}$, and analytically invertible in closed form, combining the favorable properties of all prior approaches. These bijections serve as drop-in replacements in coupling flows, matching or exceeding spline performance. Beyond coupling layers, 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 $1000 imes$ 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.

Motivation & Objective

• Address limitations of existing normalizing flow bijections (affine, splines, residuals) in expressivity, smoothness, and invertibility.
• Develop three analytic bijection families that are globally defined on R and analytically invertible in closed form.
• Introduce radial flows that transform the radial coordinate while preserving angular structure for interpretability and efficiency.
• Demonstrate improved performance and stability on 1D/2D benchmarks and physics-inspired high-dimensional problems.

Proposed method

• Propose three families of analytic bijections that are globally smooth (C^^) on R and analytically invertible in closed form.
• Develop radial flows with direct parametrization that transform the radial coordinate while preserving angular direction.
• Evaluate bijections as drop-in replacements in coupling flows and compare to spline-based approaches across 1D, 2D, and higher-dimensional tasks.
• Demonstrate training stability and interpretability of radial flows, with performance on targets exhibiting radial structure.

Experimental results

Research questions

• RQ1 Can analytic, globally smooth bijections be constructed that are analytically invertible in closed form and defined on the entire real line?
• RQ2 Do radial flow architectures provide stability, interpretability, and parameter efficiency on radially-structured targets?
• RQ3 How do the proposed bijections perform relative to affine and spline-based flows on 1D/2D benchmarks and physics-inspired problems?

Key findings

• Three families of analytic bijections that are globally smooth, defined on R, and analytically invertible in closed form.
• Radial flows offer exceptional training stability and geometrically interpretable transformations.
• Radial flows can achieve comparable quality to coupling flows with up to 1000 fewer parameters on radially structured targets.
• Experiments on 1D/2D benchmarks and phi^4 lattice field theory demonstrate improvements over affine baselines and enable problem-specific designs to address mode collapse.
• New CIFAR-10 and tabular-data results illustrate broad applicability and readability of the approach.