Factorizable Normalizing Flows for parameter-dependent density morphing
2026-06-29 • Machine Learning
Machine Learning
AI summaryⓘ
The authors address a challenge where models need to understand how data distributions change with continuous parameters, which is hard to do if modeled separately for every parameter combination. They propose Factorizable Normalizing Flows (FNFs), which start with a baseline model and learn simple, separate changes for each parameter. This approach avoids huge computational costs by combining effects additively instead of looking at all parameter mixes together. Their method works well on test problems, is easier to interpret, and scales efficiently with more parameters, useful for scientific fields like high energy physics.
Normalizing FlowsDensity EstimationContinuous ParametersFactorizationLikelihoodHigh Energy PhysicsParameter DeformationInferenceUnbinned Likelihood FitsModel Scalability
Authors
Davide Valsecchi, Mauro Donegà, Rainer Wallny
Abstract
Normalizing Flows excel at modeling a single fixed density, yet many problems across the sciences, such as high energy physics, instead require modeling how that density deforms as a function of continuous parameters: the strength of a physical effect, a calibration constant, or a source of systematic uncertainty. Learning a separate flow for every parameter configuration quickly becomes intractable, since the number of joint settings grows exponentially with the number of parameters. We introduce Factorizable Normalizing Flows (FNFs), which represent the parameter-dependent density as a fixed, high-fidelity flow for a reference configuration composed with a learnable transformation that is polynomial in the parameters and factorized over them. This structure has a practical consequence: each parameter's effect is learned in isolation, from samples in which that parameter alone is varied. The combined response of many parameters is then recovered by summation at inference, without ever sampling their combinatorially large joint space. On a controlled problem with two interpretable deformations applied jointly to the data, the learned transformation reproduces the true deformations and matches the optimal likelihood, while optional interaction terms capture residual correlations when several parameters vary strongly at once. The resulting model is interpretable, scales linearly with the number of parameters, and keeps the likelihood tractable. This provides a general tool for any inference workflow requiring continuous density morphing, and directly enables the next generation of unbinned likelihood fits in high energy physics.