# Matrix Denoising with Partial Noise Statistics: Optimal Singular Value Shrinkage of Spiked F-Matrices

arXiv: https://arxiv.org/abs/2211.00986 · HTML: https://mlsanity.com/paper/2211.00986/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 2 Nov 2022.

## Built on by (most cited first, 3 of 3)

- [Matrix Denoising with Doubly Heteroscedastic Noise: Fundamental Limits and Optimal Spectral Methods](https://mlsanity.com/paper/2405.13912/) (2024)
- [Design a Metric Robust to Complicated High Dimensional Noise for Efficient Manifold Denoising](https://mlsanity.com/paper/2401.03921/) (2024)
- [On the Noise Sensitivity of the Randomized SVD](https://mlsanity.com/paper/2305.17435/) (2023)

## Abstract (arXiv)

We study the problem of estimating a large, low-rank matrix corrupted by additive noise of unknown covariance, assuming one has access to additional side information in the form of noise-only measurements. We study the Whiten-Shrink-reColor (WSC) workflow, where a "noise covariance whitening" transformation is applied to the observations, followed by appropriate singular value shrinkage and a "noise covariance re-coloring" transformation. We show that under the mean square error loss, a unique, asymptotically optimal shrinkage nonlinearity exists for the WSC denoising workflow, and calculate it in closed form. To this end, we calculate the asymptotic eigenvector rotation of the random spiked F-matrix ensemble, a result which may be of independent interest. With sufficiently many pure-noise measurements, our optimally-tuned WSC denoising workflow outperforms, in mean square error, matrix denoising algorithms based on optimal singular value shrinkage which do not make similar use of noise-only side information; numerical experiments show that our procedure's relative performance is particularly strong in challenging statistical settings with high dimensionality and large degree of heteroscedasticity.