How is Procrustes Distance defined between two given datasets? Assume that the datasets each of them having "k" points in "n" dimensions.
I couldn't find a proper source anywhere with a mathematical expression. I found some leads, but they are in research papers, etc.
In wikipedia there is a statement which says - "minimum sum of squared distances between corresponding landmarks" but is pointwise or corresponding points?, I'm not sure. Please help. Thanks
Procrustes Distance is defined as follows:
As you have already mentioned, assume that we have two datasets M and N of size [k x n], where k = number of datapoints, and n = no. of dimensions.
Procrustes method "transforms" M and N into A and B respectively.
And the method of the above transformation is governed (by translate, rotate and scaling rules) in order to minimise the Procrustes Distance between A and B.
$$ Procrustes\_distance(A, B) = \sqrt{\sum_{j=0}^n\sum_{i=0}^k(x^{A}_{ij} - x^{B}_{ij})^{2}} $$
In other words, consider $X = A - B$; Then, $$ Procrustes\_distance(A,B)=\operatorname{trace}\left(X^{T} X\right)^{\frac{1}{2}} $$