Matrix Linear Least Squares Problem with Diagonal Matrix Constraint How could one solve the following least-squares problem with Frobenius Norm and diagonal matrix constraint?
$$\hat{S} := \arg \min_{S} \left\| Y - XUSV^T \right\|_{F}^{2}$$
where the $S$ is a diagonal matrix and $U,V$ are column-orthogonal matrix. Is there any fast algorithm?

Vectorize $S$, express matrix multiplication by $X,U$ from left and $V^T$ from right with Kronecker products. Now you have a linear least squares problem in which you can add the extra term $\left\| D \text{vec}(S) \right\|_F^2$, where you make $D$ to be a diagonal weight matrix with large positive values corresponding to  off-diagonal elements in $S$. This will regularize away any non-diagonal solution $\hat S$.

EDIT:
From wikipedia, you can use the following relation to build your objective term:
$${\bf AXB = C} \Leftrightarrow ( {\bf B^T \otimes A} )\text{vec}({\bf X}) =\text{vec}({\bf C})$$
Then to build the regularizing term - enforcing a diagonal matrix, just add $+\lambda\|{\bf D}\text{vec}({\bf X})\|_F^2$, where $$\cases{D_{ii} = \cases{0,&index $i$ corresponds to diagonal element in X\\1,&index $i$ corresponds to non-diagonal element in X}\\D_{ij} = 0, i\neq j}$$
