For a given $m n$-length vector $a$, the problem of finding an $m$-length vector $x$ and an $n$-length vector $y$ that minimize
$$ \lVert a - x \otimes y \rVert^2 $$
is known as the Nearest Kronecker Product problem, and there is a beautiful method by Van Loan and Pitsianis (1993) that solves it using the singular value decomposition. In short, if we denote by $\text{vec}^{-1}(\cdot)$ the inverse of the $\text{vec}$ operator for $n \times m$ matrices, we get
$$ \lVert a - x \otimes y \rVert^2 = \lVert a - \text{vec}(yx') \rVert^2 = \lVert \text{vec}^{-1}(a) - yx' \rVert_F^2 $$
Where $\lVert \cdot \rVert_F$ denotes the Frobenius norm. This reduces to the problem of finding a rank-one approximation to $A = \text{vec}^{-1}(a)$, and the solution can be easily obtained from the SVD of $A$.
For my own research, I am interested in the related problem of minimizing
$$ \lVert a - x \otimes y \rVert_W^2 = (a - x \otimes y)'W(a - x \otimes y) $$
Where $W$ is postive definite. Note that the Van Loan method does not generalize easily since
$$ \lVert a - x \otimes y \rVert_W^2 = \left\lVert W^{\frac 1 2}a - W^{\frac 1 2} (x \otimes y) \right\rVert^2 $$
And the vector $W^{\frac 1 2} (x \otimes y)$ cannot be easily written as the $\text{vec}$ of some matrix that depends nicely on $x$ and $y$.