Analytic Bijections for Smooth and Interpretable Normalizing Flows

2026-01-15

AI Summary

This work addresses the longstanding trade-off among expressiveness, smoothness, and computational efficiency in normalizing flow methods. We propose three families of analytically invertible bijections defined over the entire real line—rational cubic, sinh-based, and cubic polynomial—that simultaneously achieve global $C^\infty$ smoothness, closed-form inverses, strong expressivity, and tractable Jacobian determinants for the first time. Building upon these, we introduce a novel radial flow architecture that directly parameterizes radial coordinate transformations while preserving angular coordinates, yielding high numerical stability and geometric interpretability. Experiments demonstrate that our approach matches or exceeds spline flows on 1D/2D benchmarks and outperforms affine coupling flows in high-dimensional $\phi^4$ lattice field theory tasks with only one-thousandth the number of parameters, effectively mitigating mode collapse and enabling physics-informed, customizable modeling.

Technology Category

• Machine Learning: Learning with Manifolds
• Reasoning under Uncertainty: Relational Probabilistic Models
• Computer Vision: Diffusion Models for Vision

Application Category

• Graph Algorithms and Modeling for the Web: Graph embeddings and representation learning for Web-related graphs
• User Modeling, Personalization and Recommendation: Fairness-aware retrieval and ranking
• Responsible Web: Measurement, analysis, and circumvention of Web censorship

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 $1000\times$ fewer parameters. We provide comprehensive evaluation on 1D and 2D benchmarks, and demonstrate applicability to higher-dimensional physics problems through experiments on $\phi^4$ lattice field theory, where our bijections outperform affine baselines and enable problem-specific designs that address mode collapse.

Problem

• normalizing flows
• analytic bijections
• invertible functions
• smoothness
• tractable Jacobians

Innovation

• analytic bijections
• normalizing flows
• radial flows
• closed-form invertibility
• smooth transformations