Organizations: Institute for Particle Physics and Astrophysics, D-PHYS, ETH Zurich, Zurich, Switzerland
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.
We investigate a normalizing-flow approach for reconstructing parton distribution functions (PDFs) from synthetic matrix-element data. Our framework combines Gaussian Process priors with invertible neural networks to learn a posterior distribution over PDFs consistent with limited Ioffe-time data. We demonstrate that the architecture preserves physical constraints and extrapolation properties.
Modern reinforcement learning (RL) algorithms have found success by using powerful probabilistic models, such as transformers, energy-based models, and diffusion/flow-based models. To this end, RL researchers often choose to pay the price of accommodating these models into their algorithms -- diffusion models are expressive, but are computationally intensive due to their reliance on solving differential equations, while autoregressive transformer models are scalable but typically require learning discrete representations. Normalizing flows (NFs), by contrast, seem to provide an appealing alternative, as they enable likelihoods and sampling without solving differential equations or autoregressive architectures. However, their potential in RL has received limited attention, partly due to the prevailing belief that normalizing flows lack sufficient expressivity. We show that this is not the case. Building on recent work in NFs, we propose a single NF architecture which integrates seamlessly into RL algorithms, serving as a policy, Q-function, and occupancy measure. Our approach leads to much simpler algorithms, and achieves higher performance in imitation learning, offline, goal conditioned RL and unsupervised RL.
Normalizing flows are powerful generative models that learn an invertible mapping between complex data distributions and simple latent distributions, typically a standard normal density. However, this choice of latent density can impose unnecessary complexity on the learned flow transformation due to the topological mismatch between the latent and data densities, leading to slower training and suboptimal performance. In this work, we propose using mixtures of probabilistic principal component analyzers (MPPCA) as the latent density for normalizing flows. We simplify the learned flow transformation by learning a latent distribution that more closely aligns with the data distribution in terms of KL divergence, thus enabling faster convergence and improved generative performance. Critically, MPPCA models can be fit quickly and cheaply using the expectation-maximization algorithm, making them a practical choice for initializing latent distributions even in high-dimensional generative tasks. We validate our method on both tabular and image datasets, demonstrating consistent gains in training efficiency and generation quality compared to baselines.
Liam A. Kruse, Houjun Liu, Alexandros E. Tzikas +2