Considering the following matrix-vector multiplication: \begin{align} (A\otimes B)x \end{align} where $A$ is $n\times n$ matrix and $B$ is $m \times m$ matrix. In naive way, it can be computed with complexity $O(n^2m+nm^2)$ rather than the complexity of schoolbook multiplication $O(n^2m^2)$, considering the decomposition \begin{align} (A\otimes I_m)(I_n\otimes B)x. \end{align} Is there any clever way to reduce the complexity?
(it is a duplicated question but there is no answer in the link.)