🤖 AI Summary
This work addresses the exponential computational cost of modeling probability densities under continuously varying parameters by introducing Factorized Normalizing Flows (FNF). FNF represents parameter-dependent densities as a composition of a fixed, high-fidelity normalizing flow defined at a reference configuration and a factorizable polynomial transformation of the parameters. By leveraging an additive structure—with or without interaction terms—the approach enables independent learning of individual parameter effects and linearly combines multi-parameter responses, thereby circumventing the combinatorial explosion in the joint parameter space. FNF offers interpretability, linear scalability with respect to the number of parameters, and exact likelihood evaluation. In biaxial deformation experiments, it accurately reproduces true deformations and achieves state-of-the-art likelihood scores, demonstrating direct applicability to binning-free continuous density estimation tasks in fields such as high-energy physics.
📝 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.