Matrix denoising for weighted loss functions and heterogeneous signals

We consider the problem of recovering a low-rank matrix from a noisy observed matrix. Previous work has shown that the optimal method for recovery depends crucially on the choice of loss function. We use a family of weighted loss functions, which arise naturally in many settings such as heteroscedastic noise and missing data. 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 for more accurate recovery with unweighted loss.