I have the following minimization problem of:
$$ \min_W \sum_{n=1}^{N} \| l_n - P_n W x_n \|^2_2$$
where $l_n$ is a $C$ dimensional vector, $P_n$ is a $C \times L$ dimensional matrix, $W$ is a $L \times D$ dimensional matrix and $x_n$ is a $D$ dimensional vector. We also know that $D > C > L$ specifically. This is a convex problem and encouraged by its similarity with ordinary least squares $||b-Ax||_2^2$, I first tried to find a closed solution for it, without opting for numerical approaches. I take the derivative with respect to $W$ and find:
$$\dfrac{d}{dW}\sum_{n=1}^{N} \| l_n - P_nWx_n \|^2_2 = \sum_{n=1}^{N}-2P_n^T(l_n - P_nWx_n)x_n^T$$
But after this point, setting the derivative to zero and solving for the matrix $W$ doesn't seem to be doable to me. I just wanted to be sure and ask if we can solve the expression $$\sum_{n=1}^{N}-2P_n^T(l_n - P_nWx_n)x_n^T = 0$$ for $W$ analytically, without making use of tools of numerical optimization.