# Matrix denoising for weighted loss functions and heterogeneous signals

## TL;DR

This paper develops optimal spectral denoisers for low-rank matrix estimation under weighted loss functions, addressing challenges of heterogeneity, missing data, and heteroscedastic noise, to improve denoising performance.

## Contribution

It introduces a framework for deriving optimal spectral denoisers for weighted loss functions and combines them to exploit heterogeneity in signals for enhanced estimation.

## Key findings

- Derived optimal spectral denoisers for weighted loss functions.
- Constructed a new denoiser leveraging heterogeneity to improve accuracy.
- Addressed analysis challenges of non-orthogonally-invariant weighted losses.

## Abstract

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.