SVD Approach to Data Unfolding

TL;DR

This paper addresses the ill-posed problem of unfolding detector-distorted distributions in high-energy physics by reframing it as a linear inverse problem and applying a regularized SVD-based solution. It develops a concise, fully error-propagating algorithm that uses the response matrix, a Monte Carlo–driven normalization, and a curvature-based regularization term with a data-driven rule for selecting the regularization strength. The method produces smooth, stable unfolded spectra with complete covariance information and is validated on two illustrative examples. The approach offers a practical, widely applicable unfolding framework that avoids ad-hoc binning or covariance simplifications while maintaining tractable computation.

Abstract

Distributions measured in high energy physics experiments are usually distorted and/or transformed by various detector effects. A regularization method for unfolding these distributions is re-formulated in terms of the Singular Value Decomposition (SVD) of the response matrix. A relatively simple, yet quite efficient unfolding procedure is explained in detail. The concise linear algorithm results in a straightforward implementation with full error propagation, including the complete covariance matrix and its inverse. Several improvements upon widely used procedures are proposed, and recommendations are given how to simplify the task by the proper choice of the matrix. Ways of determining the optimal value of the regularization parameter are suggested and discussed, and several examples illustrating the use of the method are presented.

Figures (2)

• Figure 1: a). The probability matrix $\hat{A}$ corresponding to the response function (\ref{['blex1']}). b). The true distribution (\ref{['blex2']}) (solid curve) compared to the measured histogram $b$ and the unfolded distribution $x^{(\tau)}$ for $\tau=s_{10}^2$. c). The absolute values of $d_i$ (solid line) compared to the regularized r.h.s. (dashed line) and the one unaffected by the statistical fluctuations (dotted line). The horizontal line shows statistical errors in $d_i$, while the arrow indicates the boundary between the significant and non-significant equations. d). The deviation of the unfolded distribution from the true exact one (see text for details).
• Figure 2: a). The simulated number-of-events response matrix $A$. b). The true test distribution $x^{\mathrm{test}}$ (solid line) compared to the unfolded one (data points). The dashed histogram corresponds to the initial distribution $x^{\mathrm{ini}}$ according to which the response matrix was generated. c). The absolute values of $d_i$ (solid line) compared to the regularized r.h.s. (dashed line) and the one unaffected by the statistical fluctuations (dotted line). The horizontal line shows statistical errors in $d_i$, while the arrow indicates the boundary between the significant and non-significant equations. d). The deviation of the unfolded distribution from the true exact one (see text for details).