Point Cloud-based Variational Autoencoder Inverse Mappers (PC-VAIM) - An Application on Quantum Chromodynamics Global Analysis.

Correctly mapping the experimental data to quantum probability distributions is a critical step to characterize nucleon structure and the emergence of hadrons, in terms of quark and gluon degrees of freedom. Since the actual parameters of interest are not directly measurable, but instead are inferred from experimental observables, this is fundamentally an inverse problem of recovering parameters from observables. In addition to the well-known challenges such as ill-posedness in general inverse problems, an application specific issue here is that the experimental data are observed on kinematics bins which are usually irregular and varying. In this paper, to address this ill-defined, varying observable space problem, we represent the observables together with their kinematics bins as an unstructured, high-dimensional point cloud. We incorporate a permutation invariant neural network framework to handle the observables in unstructured and unordered point cloud representations. We incorporate the point cloud representation into the Variational Autoencoder Inverse Mapper (VAIM) framework. The point cloud-based VAIM (PC-VAIM) enables the underlying deep neural networks to learn how the observables are distributed across kinematics. We demonstrate the effectiveness of PC-VAIM on a toy inverse problem, and then on constructing the inverse function mapping Quantum Correlation Functions (QCF) to observables in a Quantum Chromodynamics (QCD) analysis of nucleon structure.