Matrix size of the SVD

I need to reach the svd matrices of J for a control law customization, and I have two questions:
1. I am surprised of the size of U, which is not square in svd_boost_HH.
2. There are 3 different svd decompositions: svd_HH, svd_boost_HH and svd_macie_HH. What are the features of each one?

Thank you very much,
Sébastien Rubrecht.