I am reading the work Blessing of Dimensionality High dimensional feature and its efficient compression for the face verification CVPR 2013. One of the key contributes is the authors propose a new so called rotated sparse regression method for the dimension reduction. However, I don't know how to get the closed form solution for following optimization problem. The details are not given in the paper. Could someone give me a hint to  derive the following closed form in the following  
Solution to the optimization problem in Blessing of Dimensionality High ... the face verification
optimization
There's a well known problem called the orthogonal Procrustes problem,$ \min_{Q} \| AQ-B \|_{F} $subject to the constraint$ Q^{T}Q=I. $Note that the norm in the objective function is the Frobenius norm rather than the matrix 2-norm.  For this problem, the solution is $Q=UV^{T}$where $A^{T}B$ has the SVD$A^{T}B=U\Sigma V^{T}$.  This solution to the Procrustes problem is derived in lots of places.  See for example the Wikipedia page here:http://en.wikipedia.org/wiki/Orthogonal_Procrustes_problemIf your problem had the objective $\min \| R^{T}Y-B^{T}X \|_{F} $ rather than $\| R^{T}Y-B^{T}X \|_{2}^{2}$, then the problem could be rewritten as a Procrustes problem by transposing inside the norm,$\min \| Y^{T}R - X^{T}B \|_{F}$subject to $R^{T}R=I$.  The solution would be $R=UV^{T}$where $YX^{T}B$ has the singular value decomposition $YX^{T}B=U\Sigma V^{T}$.This matches up exactly with the solution given in the paper that you've quoted.  I don't believe that minimizing the 2-norm is equivalent to minimizing the Frobenius norm.  It seems to me that the authors of the paper you cite may have confused the Frobenius norm and the 2-norm.    Added after I finally got my hands on Watson's 1993 paper...  SeeG. A. Watson.  Solving Generalizations of Orthogonal Procrustes Problems. In Contributions in Numerical Mathematics, World Scientific Series in Applicable Analysis, World Scientifc Publishers, 1993.  The classical solution using the SVD is optimal with respect to the 2-norm only if the matrices are square ($m=n$), and the $A$ matrix is the identity- that is, we're minimizing $\min \| Q-B \|_{2}$.  Otherwise, Watson offers an iterative algorithm that requires the solution of a quadratic programming problem at each iteration.  