Decomposition of the Outer Product

Question

I have an $m \times n$ matrix that is statistical data. I have a theoretical model about how that data is constructed in which the matrix is the outer product of two vectors. Let's call those vectors $u$ and $v$. Their size is $m \times 1$ and $n\times 1$ respectively such that the outer product is $uv^{\ \prime}$. The hitch is that there is also missing data, i.e. not all cells in the matrix are in the data. In other words, it is an outer product that is then multiplied cell by cell by a matrix of the same size with zeros and ones. Any idea how I can estimate the vectors $\hat{u}$ and $\hat{v}$ that yield an outer product matrix that is the closest to the one in the data? I do know where the missing data is (i.e. I know the matrix of zeros and ones that is multiplied by the outer product matrix). Thank you!