I encounter this problem in my research. Assume we have $X \in \mathbb{R}^{n\times d}, U\in \mathbb{R}^{n\times h}, V\in\mathbb{R}^{h\times d}$. We consider the following function
$L(U,V)=\frac{1}{2}\lVert X-UV\rVert_F^2$.
My question is, for arbitrary $U, V$, what's the Hessian matrix of $L(U,V)$ and can we get an explicit formula for the eigenvalue depending on $U, V$? I can handle the vector case where $X \in \mathbb{R}, U\in \mathbb{R}^{1\times h}, V\in\mathbb{R}^{h\times 1}$, but when it comes to matrix case, I'm lost.