Procrustes Analysis

Definition

Procrustes Transformation without Scaling

Consider two $n\times p$ matrices ${\mathbf{X}}_1$ and a target ${\mathbf{X}}_2$. What we want is to find a Procrustes transformation of $X_1$ whose result is closest to $X_2$,

The optimization problem is defined as $\underset{\mu ,\mathbf{R}}{\min }||{\mathbf{X}}_2-({\mathbf{X}}_1\mathbf{R}+1_n\mathbf{\mu })|{|}_F$ where $\mathbf{R}$ is a $p\times p$ Orthonormal Matrix (perform rotation and reflection), ${\mathbf{\mu }}_{1\times p}$ act as location parameter, and $||\cdot |{|}_F$ is a Frobenius Norm.

Let ${\tilde{x}}_1$ and ${\tilde{x}}_2$ be the columnwise mean vectors of the matrices, and ${\tilde{\mathbf{X}}}_1$ and ${\tilde{\mathbf{X}}}_2$ be the demeaned matrices. Consider the SVD ${\tilde{\mathbf{X}}}_1^{⊺}{\tilde{\mathbf{X}}}_2=\mathbf{UD}{\mathbf{V}}^{⊺}$. Then, the solution of the optimization problem is given by $\hat{\mathbf{R}}=\mathbf{U}{\mathbf{V}}^{⊺},\hat{\mu }={\tilde{x}}_2-\hat{\mathbf{R}}{\tilde{x}}_1$ and the minimal distance is referred to as the Procrustes distance.

Procrustes Transformation with Scaling

Consider demeaned matrices ${\mathbf{X}}_1$ and ${\mathbf{X}}_2$. The Procrustes distance with scaling is obtained from more general optimization problem. $\underset{\beta ,\mathbf{R}}{\min }||{\mathbf{X}}_2-\beta {\mathbf{X}}_1\mathbf{R}|{|}_F$ where $\beta >0$ is a positive scalar.

The solution for $\mathbf{R}$ is as before ($\hat{\mathbf{R}}=\mathbf{U}{\mathbf{V}}^{⊺}$), with $\hat{\beta }=\mathrm{tr}(D)/||{\mathbf{X}}_1|{|}_F^2$

Procrustes Average

Consider demeaned and scaled matrices $\{{\mathbf{X}}_l|l=1,2,\dots ,L\}$. Procrustes average problem finds the shape $\mathbf{M}$ closest in average squared Procrustes distance to all the given shapes. $\underset{\{{\mathbf{R}}_l{\}}_1^L,\mathbf{M}}{\min }||{\mathbf{X}}_l{\mathbf{R}}_l-\mathbf{M}|{|}_F^2$ This is solved by a simple algorithm

1. Initialize $\mathbf{M}={\mathbf{X}}_1$
2. Solve the $L$ Procrustes rotation problems with $\mathbf{M}$ fixed, yielding ${\mathbf{X}}_l^{\prime }\leftarrow \mathbf{X}{\hat{\mathbf{R}}}_l$
3. Let $\mathbf{M}\leftarrow \frac{1}{L}\sum _{l=1}^L{\mathbf{X}}_l^{\prime }$
4. Iterate step 1 and 2 until it converges.