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We consider the problem of estimating a low-rank matrix from a noisy observedmatrix. Previous work has shown that the optimal method depends crucially onthe choice of loss function. In this paper, we use a family of weighted lossfunctions, 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 notorthogonally-invariant. We derive optimal spectral denoisers for these weightedloss functions. By combining different weights, we then use these optimaldenoisers to construct a new denoiser that exploits heterogeneity in the signalmatrix to boost estimation with unweighted loss.

Matrix denoising for weighted loss functions and heterogeneous signals