# Resampling Sensitivity of High-Dimensional PCA

arXiv: https://arxiv.org/abs/2212.14531 · HTML: https://mlsanity.com/paper/2212.14531/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 10 of its 65 references that are in the map.
- First posted on arXiv 30 Dec 2022.

## Builds on (most cited first, 10 of 10)

- [Train faster, generalize better: Stability of stochastic gradient descent](https://mlsanity.com/paper/1509.01240/) (2015)
- [Predictive inference with the jackknife+](https://mlsanity.com/paper/1905.02928/) (2019)
- [Settling the Sharp Reconstruction Thresholds of Random Graph Matching](https://mlsanity.com/paper/2102.00082/) (2021)
- [Toward Better Generalization Bounds with Locally Elastic Stability](https://mlsanity.com/paper/2010.13988/) (2020)
- [Spectral Graph Matching and Regularized Quadratic Relaxations II: ErdHos-R'enyi Graphs and Universality](https://mlsanity.com/paper/1907.08883/) (2019)
- [Random Graph Matching in Geometric Models: the Case of Complete Graphs](https://mlsanity.com/paper/2202.10662/) (2022)
- [Database Alignment with Gaussian Features](https://mlsanity.com/paper/1903.01422/) (2019)
- [Black-box tests for algorithmic stability](https://mlsanity.com/paper/2111.15546/) (2021)
- [Spectral Alignment of Correlated Gaussian matrices](https://mlsanity.com/paper/1912.00231/) (2019)
- [Optimal Smoothed Analysis and Quantitative Universality for the Smallest Singular Value of Random Matrices](https://mlsanity.com/paper/2211.03975/) (2022)

## Abstract (arXiv)

The study of stability and sensitivity of statistical methods or algorithms with respect to their data is an important problem in machine learning and statistics. The performance of the algorithm under resampling of the data is a fundamental way to measure its stability and is closely related to generalization or privacy of the algorithm. In this paper, we study the resampling sensitivity for the principal component analysis (PCA). Given an  n × p  random matrix  X , let  X[k]  be the matrix obtained from  X  by resampling  k  randomly chosen entries of  X . Let  v  and  v[k]  denote the principal components of  X  and  X[k] . In the proportional growth regime  p/n → xi ∈ (0,1] , we establish the sharp threshold for the sensitivity/stability transition of PCA. When  k gg n5/3 , the principal components  v  and  v[k]  are asymptotically orthogonal. On the other hand, when  k ll n5/3 , the principal components  v  and  v[k]  are asymptotically colinear. In words, we show that PCA is sensitive to the input data in the sense that resampling even a negligible portion of the input may completely change the output.