# On the Error-Propagation of Inexact Hotelling's Deflation for Principal Component Analysis

arXiv: https://arxiv.org/abs/2310.04283 · HTML: https://mlsanity.com/paper/2310.04283/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 11 of its 37 references that are in the map.
- First posted on arXiv 6 Oct 2023.

## Builds on (most cited first, 11 of 11)

- [Llama 2: Open Foundation and Fine-Tuned Chat Models](https://mlsanity.com/paper/2307.09288/) (2023)
- [Switch Transformers: Scaling to Trillion Parameter Models with Simple and Efficient Sparsity](https://mlsanity.com/paper/2101.03961/) (2021)
- [Traces of Class/Cross-Class Structure Pervade Deep Learning Spectra](https://mlsanity.com/paper/2008.11865/) (2020)
- [EigenGame: PCA as a Nash Equilibrium](https://mlsanity.com/paper/2010.00554/) (2020)
- [First Efficient Convergence for Streaming k-PCA: a Global, Gap-Free, and Near-Optimal Rate](https://mlsanity.com/paper/1607.07837/) (2016)
- [Streaming PCA: Matching Matrix Bernstein and Near-Optimal Finite Sample Guarantees for Oja's Algorithm](https://mlsanity.com/paper/1602.06929/) (2016)
- [Accelerated Stochastic Power Iteration](https://mlsanity.com/paper/1707.02670/) (2017)
- [Unperturbed: spectral analysis beyond Davis-Kahan](https://mlsanity.com/paper/1706.06516/) (2017)
- [Super-Acceleration with Cyclical Step-sizes](https://mlsanity.com/paper/2106.09687/) (2021)
- [Provable Deterministic Leverage Score Sampling](https://mlsanity.com/paper/1404.1530/) (2014)
- [Asymmetry Helps: Eigenvalue and Eigenvector Analyses of Asymmetrically Perturbed Low-Rank Matrices](https://mlsanity.com/paper/1811.12804/) (2018)

## Built on by (most cited first, 1 of 1)

- [AdaPaD: Adaptive Parallel Deflation for PEFT with Self-Correcting Rank Discovery](https://mlsanity.com/paper/2605.10741/) (2026)

## Abstract (arXiv)

Principal Component Analysis (PCA) aims to find subspaces spanned by the so-called principal components that best represent the variance in the dataset. The deflation method is a popular meta-algorithm that sequentially finds individual principal components, starting from the most important ones and working towards the less important ones. However, as deflation proceeds, numerical errors from the imprecise estimation of principal components propagate due to its sequential nature. This paper mathematically characterizes the error propagation of the inexact Hotelling's deflation method. We consider two scenarios: i) when the sub-routine for finding the leading eigenvector is abstract and can represent various algorithms; and ii) when power iteration is used as the sub-routine. In the latter case, the additional directional information from power iteration allows us to obtain a tighter error bound than the sub-routine agnostic case. For both scenarios, we explicitly characterize how the errors progress and affect subsequent principal component estimations.