Information Criteria for Selecting Parton Distribution Function Solutions

TL;DR

This work tackles the non-uniqueness of parton distribution function (PDF) solutions arising from ill-posed inverse problems in global QCD analyses. It introduces an absolute shape criterion based on Rényi entropy $H_2(X)$ to classify and select diverse PDF shapes, and complements it with relative criteria via Rényi divergence $D_2$ and Wasserstein distance $W_1$, using Pareto fronts to identify representative metamorphs. The methods are demonstrated on Fantômas pion PDFs, showing that Rényi-entropy-driven selection yields results qualitatively consistent with relative metrics and also supports clustering with UMAP. The approach provides a computationally efficient, minimalistic framework for capturing model diversity and guiding robust PDF ensembles, with clear paths for extension to proton PDFs and higher flavor-dimensional spaces.

Abstract

In data-driven determination of Parton Distribution Functions (PDFs) in global QCD analyses, uncovering the true underlying distributions is complicated by a highly convoluted inverse problem. The determination of PDFs can be understood as the inference of a function supported on $[0,1]$, a problem that admits multiple acceptable solutions. An ensemble of solutions exists that pass all standard goodness-of-fit criteria. In this paper, we propose algorithms for the classification, clustering, and selection of solutions to the determination of PDFs, or any functions on $[0,1]$, based on the characterization of their shape. We explore information-theoretic based (Rényi entropy and divergence) and optimal-transport based (Wasserstein distance) criteria. In particular, we advocate for the use of the Rényi entropy as an {\it absolute} estimator per solution, as opposed to {\it relative} estimators that compare solutions pairwise. We show that the Rényi entropy can characterize the space of solutions {\it w.r.t.} the PDF shapes. Paired with the identification of the optimal combination of solutions via Pareto fronts, it provides a plausible and minimalist selection algorithm. Moreover, Rényi entropy proves versatile for use in clustering applications.

Figures (6)

• Figure 1: Illustration of values for $H_2(X)$ for various shapes. (Left) ideal case of shapes ranging from (in decreasing order) steep (dot-dashed red), spread-out (3-dot-dashed cyan) and highly-concentrated (dashed green), for tweaked Gaussian point (blue dots). (Right) similar color code and characteristics, with the addition of the short-dashed blue-grey curve whose $H_2(X)$ is similar to the steep curve. Both show a similar pattern in variations, not magnitude, at the evaluation points.
• Figure 2: Rényi 3-dimensional space. Left panel: The light gray points represent all acceptable solutions. Larger red points represent the vertices of the Pareto hypersurfaces, i.e. floor and roof. Right panel: The four clusters identified through the UMAP nearest-neighbor algorithm (orange, green, cyan, and dark blue), along with the vertices of the Pareto fronts (low opacity red circles). Both are illustrated for $X_{0.25, 20}$.
• Figure 3: Selected shapes (color) of the pool of acceptable metamorph solutions generated for the Fantômas pion PDFs (light grey). Rényi entropy-based coordinates are filtered as Pareto vertices. The black dots represent the $x$ values at which the entropy was evaluated.
• Figure 4: Color matrix based on the value of the estimator $H_2^{\{V, g, S\}}(X)$ for UMAP-identified clusters displayed in the right panel of Fig. \ref{['fig:abs_renyi_vertices']}. The values of the average Rényi entropy are rescaled between $\left[0,1\right]$ (from dark blue to dark red) for improved visualization. To each cluster corresponds a different pattern in projected entropies.
• Figure 5: Comparison of the selection obtained by the absolute criterion (dotted-dashed red), and two relative criteria -- Rényi divergence (long dashed mustard) and Wasserstein distance (3 dots-dashed cyan).