  We consider the problem of estimating a low-rank matrix from a noisy observed
matrix. Previous work has shown that the optimal method depends crucially on
the choice of loss function. In this paper, we use a family of weighted loss
functions, which arise naturally for problems such as submatrix denoising,
denoising with heteroscedastic noise, and denoising with missing data. However,
weighted loss functions are challenging to analyze because they are not
orthogonally-invariant. We derive optimal spectral denoisers for these weighted
loss functions. By combining different weights, we then use these optimal
denoisers to construct a new denoiser that exploits heterogeneity in the signal
matrix to boost estimation with unweighted loss.
