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Unfolding Algorithms

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\section{Unfolding Algorithms}\label{sec:UnfAlg}
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The main software package for cross-section extraction (unfolding) in T2K is called the xsTool~\cite{xsTool}. xsTool integrates different unfolding algorithms from RooUnfold~\cite{RooUnfAlg} and is responsible for the calculation of the error propagation.

Generally, the unfolding algorithms can be categorized into three main groups:
\begin{itemize}
\item bin-by-bin, \item matrix inversion, and \item Bayesian methods.
\end{itemize}
The RooUnfold package supports bin-by-bin, unregularized matrix inversion, Singular Value Decomposition (SVD), TUnfold, and iterative Bayes (based on D'Agostini 1995 …show more content…

\nonumber
\end{align}
Where, T denotes the matrix transpose, and the vectors u and v are the left-singular and right-singular vectors for $\sigma$, respectively~\cite{SVDRef}.
}, cf.~\cite[section 4]{SVD} to obtain a pseudo-inverse\footnote{A pseudo-inverse matrix $A^+$ fulfills
Moore and Penrose, cf.\cite[section 2]{Pinv} conditions:
\begin{itemize}
\item $AA^+A = A$ \item $A^+AA^+ = A^+$ \item $(AA^+)^* = AA^+ $ \item $(A^+A)^* = A^+A $
\end{itemize}
i.e. $AA^+$ and $A^+A$ map the columns of A and $A^+$ to themselves, besides, $AA^+$ and $A^+A$ are Hermitian. } for the detector response.
This method is ill-posed as described in \cref{fig:Unf_IllPosed}, hence is not recommended, cf.~\cite[Section 3.4]{RooUnfAlg}.
\subsection{Regularized Singular Value Decomposition}
Singular Value Decomposition (SVD) technique provides a pseudo-inverse for the detector response matrix ($A_{m\times n}$) by factoring it as
\begin{equation}
A = USV^T,
\end{equation}
where, U and V are orthogonal $m \times m$ and $n \times n$ matrices, respectively, $U^T = U^{-1}$ and $V^T = V^{-1}$; S is $m \times n$ diagonal matrix of the singular values (non negative); and the columns of U and V represent the left- and right-singular vectors, respectively.

This process could be visualized as a rotation by an orthogonal matrix, then stretching by a diagonal one and finally another rotation to bring the truth vector as close as possible to the measured vector.

The problem is

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