chapter5_PCA_Procrustes.Rnw

\chapter{Procrustes Analysis for Data Fusion}
\label{chap:PoM}
\chaptermark{Procrustes Analysis}



\section{Procrustes Analysis}
Given two configurations of $n$ points in $d$-dimensional Euclidean space, Procrustean methods fit one configuration to the other so that the points align as well as possible in the $\ell_2$-sense. Let us denote the configurations by two $n \times d$ matrices: ${\X}_1$, ${\X}_2$. 
\begin{thm}
Let  $\mathbf{Q}=\argmin_{\mathbf{P^T}\mathbf{P}=\mathbf{P}\mathbf{P^T}=\mathbf{I}}\|{\mathbf{X}_1-\mathbf{X}_2}\mathbf{P}\|_{F}^2$ ,   $\mathbf{\tilde{X}}_2= \mathbf{X}_2\mathbf{Q}$, 
and let  
$\mathbf{X}=\left[\begin{array}{c}
\mathbf{X}_1\\
\mathbf{\tilde{X}}_2
\end{array}\right]$.

For $w>0$, let $\mathbf{Y}_{w} = \left[\begin{array}{c}
\mathbf{Y}_1\\
\mathbf{Y}_2
\end{array}\right]$  be  a $2n \times p$ configuration matrix obtained by the minimization of 
$ f(\mcY, M) =(1-w)\left({\sigma{(\mcY_1)}}+{\sigma{(\mcY_2)}}\right)+w\|{\mcY_1-\mcY_2}\|_{F}^2 $ with respect to  $\mcY=\left[\begin{array}{c}
\mcY_1\\
\mcY_2
\end{array}\right]$ with the constraint that $\mcY_1$ and $\mcY_2$ are two $n \times p$ configuration matrices with column means $\bm{0}$. Then, $$lim_{w\rightarrow0}\mathbf{Y}_{w}=\mathbf{X}\mathbf{R}$$ for a $p\times p$ orthogonal matrix $\mathbf{R}$. ($\mathbf{R}$ is a transformation matrix with a rotation and possibly a reflection component.)
\end{thm}