I want to solve the following problem:

$$\arg\min_{u,v} \|W\odot(u v^\mathsf{T}-M)\|_\mathrm{F}$$

where $u$ and $v$ are $N\times 1$ vectors, $W$ and $M$ are $N\times N$ matrices, $\odot$ represents an elementwise multiplication, and $\|\cdot\|_\mathrm{F}$ is the Frobenius norm.

The matrix $W$ weights the contribution of each element of the error matrix to the overall cost. In the absence of this matrix, the problem could be solved straightforwardly using SVD. However, the weight matrix complicates things. Do you know of a solution?


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