# Matrices with Gaussian noise: optimal estimates for singular subspace perturbation

arXiv: https://arxiv.org/abs/1803.00679 · HTML: https://mlsanity.com/paper/1803.00679/

## Key facts

- Cited by 6 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 2 Mar 2018.

## Built on by (most cited first, 6 of 6)

- [Enhancing Accuracy in Deep Learning Using Random Matrix Theory](https://mlsanity.com/paper/2310.03165/) (2023)
- [Spectral Perturbation Bounds for Low-Rank Approximation with Applications to Privacy](https://mlsanity.com/paper/2510.25670/) (2025)
- [S3LoRA: Safe Spectral Sharpness-Guided Pruning in Adaptation of Agent Planner](https://mlsanity.com/paper/2508.15068/) (2025)
- [Perturbation Bounds for Low-Rank Inverse Approximations under Noise](https://mlsanity.com/paper/2510.25571/) (2025)
- [Countering Multi-modal Representation Collapse through Rank-targeted Fusion](https://mlsanity.com/paper/2511.06450/) (2025)
- [Improved Guarantees for Heterogeneous Treatment-Effect Estimation via Matrix Completion](https://mlsanity.com/paper/2605.30319/) (2026)

## Abstract (arXiv)

The Davis-Kahan-Wedin sin Theta theorem describes how the singular subspaces of a matrix change when subjected to a small perturbation. This classic result is sharp in the worst case scenario. In this paper, we prove a stochastic version of the Davis-Kahan-Wedin sin Theta theorem when the perturbation is a Gaussian random matrix. Under certain structural assumptions, we obtain an optimal bound that significantly improves upon the classic Davis-Kahan-Wedin sin Theta theorem. One of our key tools is a new perturbation bound for the singular values, which may be of independent interest.