While reading the following paper on an optimization problem, there was a variant of an orthogonal Procrustes problem, where the solution is an element of the Stiefel manifold. The authors provided a closed-form solution, but I couldn't understand how they derived it!
- T. Neumann, K. Varanasi, C. Theobalt, M. Magnor, M. Wacker, Compressed manifold modes for mesh processing, Eurographics Symposium on Geometry Processing 2014.
Given $Y$ is an $n \times k$ matrix and $D$ is an $n \times n$ diagonal matrix of strictly positive values on its main diagonal, the following optimization problem in $n \times k$ matrix $\Psi$
$$ {\bf \Psi}^{*} = \arg\min_{{\bf \Psi}, {\bf \Psi}^{\top}{\bf \Psi} = {\bf I}} \left\Vert {\bf D}^{-\frac12} {\bf \Psi} - {\bf Y} \right\Vert_2^2 $$
The $\left\Vert\cdot\right\Vert_2$ refers to the Frobenius norm, and the solution they used is as follows:
$$ {\bf \Psi}^{*} = \bf{Y}\bf{V}\bf{W}^{-\frac12}\bf{V}^{\top} $$
where $\bf{V}$, $\bf{W}$ and $\bf{V}^{\top}$ are obtained via the SVD decomposition
$$ (\bf{D}^{\frac{1}{2}}\bf{Y})^{\top}(\bf{D}^{\frac{1}{2}}\bf{Y}) = \bf{V}\bf{W}\bf{V}^{\top} $$
I attached the screenshot of the problem from the paper (we want to solve equation 10, the equations of interest are 14, 15 and 16), any help is appreciated!
