# Adapting to Unknown Noise Distribution in Matrix Denoising

arXiv: https://arxiv.org/abs/1810.02954 · HTML: https://mlsanity.com/paper/1810.02954/

## Key facts

- Cited by 5 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 6 Oct 2018.

## Built on by (most cited first, 5 of 5)

- [Inference for Heteroskedastic PCA with Missing Data](https://mlsanity.com/paper/2107.12365/) (2021)
- [Matrix Denoising with Doubly Heteroscedastic Noise: Fundamental Limits and Optimal Spectral Methods](https://mlsanity.com/paper/2405.13912/) (2024)
- [Detection of Signal in the Spiked Rectangular Models](https://mlsanity.com/paper/2104.13517/) (2021)
- [Low coordinate degree algorithms I: Universality of computational thresholds for hypothesis testing](https://mlsanity.com/paper/2403.07862/) (2024)
- [Bayesian reconstruction of memories stored in neural networks from their connectivity](https://mlsanity.com/paper/2105.07416/) (2021)

## Abstract (arXiv)

We consider the problem of estimating an unknown matrix boldsymbolX∈ mathbb Rm× n, from observations boldsymbolY = boldsymbolX+boldsymbolW where boldsymbolW is a noise matrix with independent and identically distributed entries, as to minimize estimation error measured in operator norm. Assuming that the underlying signal boldsymbolX is low-rank and incoherent with respect to the canonical basis, we prove that minimax risk is equivalent to (√mvee√n)/√IW in the high-dimensional limit m,n→∞, where IW is the Fisher information of the noise. Crucially, we develop an efficient procedure that achieves this risk, adaptively over the noise distribution (under certain regularity assumptions). Letting boldsymbolX = boldsymbolUboldsymbolSigmaboldsymbolV^sf T --where boldsymbolU∈ mathbb Rm× r, boldsymbolV∈mathbb Rn× r are orthogonal, and r is kept fixed as m,n→∞-- we use our method to estimate boldsymbolU, boldsymbolV. Standard spectral methods provide non-trivial estimates of the factors boldsymbolU,boldsymbolV (weak recovery) only if the singular values of boldsymbolX are larger than (mn)1/4rm Var(W11)1/2. We prove that the new approach achieves weak recovery down to the the information-theoretically optimal threshold (mn)1/4IW1/2.