I have the following problem
$$ W^* := \arg\min_W \| WX - Y \|_F $$
where $W^TW = I$ and $I$ is the identity matrix.
Instead of making yet another regression problem we can find optimal orthogonal transformation using singular value decomposition. It turns out that optimal transformation $W^*$ can be expressed via SVD components: $$X^TY=U\Sigma V^T\text{, singular value decompostion}$$ $$W^*=UV^T$$
I solved this task like a linear regression and get weight in next form $$W^*=(X^{.T}X)^{-1}(X^{.T}Y)$$ but I can't solve it clearly using svd, I always have a variable $\Sigma$. I am doing something wrong and I have an error somewhere but I can't understand where ...