Singular Value Decomposition, Hessian Errors, and Linear Algebra of Non-parametric Extraction of Partons from DIS

TL;DR

This paper applies singular value decomposition to analyze the Hessian matrix and linear systems in the non-parametric extraction of parton distributions from deep inelastic scattering data, revealing eigenvalue structures and error properties.

Contribution

It introduces a novel application of SVD to understand the null space and eigenvalue structure in parton extraction, providing insights into error propagation and data redundancy.

Findings

Identification of boundary between range and null space eigenvalues

Eigenvalue structure of the null space reveals information nullity

Analysis of error smallness linked to null space properties

Abstract

By singular value decomposition (SVD) of a numerically singular Hessian matrix and a numerically singular system of linear equations for the experimental data (accumulated in the respective ${\chi ^2}$ function) and constraints, least square solutions and their propagated errors for the non-parametric extraction of Partons from $F_2$ are obtained. SVD and its physical application is phenomenologically described in the two cases. Among the subjects covered are: identification and properties of the boundary between the two subsets of ordered eigenvalues corresponding to range and null space, and the eigenvalue structure of the null space of the singular matrix, including a second boundary separating the smallest eigenvalues of essentially no information, in a particular case. The eigenvector-eigenvalue structure of "redundancy and smallness" of the errors of two pdf sets, in our simplified Hessian model, is described by a secondary manifestation of deeper null space, in the context of SVD.