# Phase Transition in the Recovery of Rank One Matrices Corrupted by Gaussian Noise

## TL;DR

This paper investigates the phase transition phenomenon in recovering rank-one matrices from noisy data, revealing critical thresholds where PCA succeeds or fails in high-dimensional settings.

## Contribution

It characterizes the phase transition in eigen-structure for rank-one matrix recovery under Gaussian noise, extending understanding of PCA limitations in high-dimensional regimes.

## Key findings

- Identifies critical noise thresholds for successful recovery
- Demonstrates phase transition behavior in eigenvalues
- Provides theoretical insights into PCA performance under noise

## Abstract

In datasets where the number of parameters is fixed and the number of samples is large, principal component analysis (PCA) is a powerful dimension reduction tool. However, in many contemporary datasets, when the number of parameters is comparable to the sample size, PCA can be misleading. A closely related problem is the following: is it possible to recover a rank-one matrix in the presence of a large amount of noise? In both situations, there is a phase transition in the eigen-structure of the matrix.