Matrix denoising for weighted loss functions and heterogeneous signals
This work addresses matrix denoising for applications like submatrix denoising and heteroscedastic noise, offering a method to handle weighted losses and improve estimation, though it appears incremental as it builds on prior work on loss function dependence.
The paper tackles the problem of estimating a low-rank matrix from noisy observations by deriving optimal spectral denoisers for weighted loss functions, which are challenging due to lack of orthogonal invariance, and constructs a new denoiser that exploits signal heterogeneity to boost estimation with unweighted loss.
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.