PCA consistency in high dimension, low sample size context
Open accessAbstract
Principal Component Analysis (PCA) is an important tool of dimension reduction especially when the dimension (or the number of variables) is very high. Asymptotic studies where the sample size is fixed, and the dimension grows [i.e., High Dimension, Low Sample Size (HDLSS)] are becoming increasingly relevant. We investigate the asymptotic behavior of the Principal Component (PC) directions. HDLSS asymptotics are used to study consistency, strong inconsistency and subspace consistency. We show that if the first few eigenvalues of a population covariance matrix are large enough compared to the others, then the corresponding estimated PC directions are consistent or converge to the appropriate subspace (subspace consistency) and most other PC directions are strongly inconsistent. Broad sets of sufficient conditions for each of these cases are specified and the main theorem gives a catalogue of possible combinations. In preparation for these results, we show that the geometric representation of HDLSS data holds under general conditions, which includes a ρ-mixing condition and a broad range of sphericity measures of the covariance matrix.
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The increase in E-distance with falling cell numbers is due to the inability of PCA to adequately represent data in the low sample regime.
“This is due to the inability of PCA to adequately represent data in the low sample regime51.”
From scPerturb: harmonized single-cell perturbation data (2024) · cites this paper as [51]
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References 21 · Cited by 303
- Authors
- Jung, S., & Marron, J. S.
- Affiliations
- University of North Carolina at Chapel Hill (US)
- Venue
- The Annals of Statistics
- Publisher
- Institute of Mathematical Statistics
- Published
- 2009-12-01
- Volume
- 37
- Issue
- 6B
- Pages
- 4104-4130
- ISSN
- 0090-5364 (ISSN-L)
- 2168-8966
- OpenAlex
- W2078957532
- Cited by
- 303
- References
- 21
- Citations by year
- 2026: 4 · 2025: 10 · 2024: 16 · 2023: 17 · 2022: 13 · 2021: 24 · 2020: 19 · 2019: 39 · 2018: 23 · 2017: 11 · 2016: 20 · 2015: 29 · 2014: 21 · 2013: 22 · 2012: 16
- FWCI
- 9.43
- Type
- journal article
- Language
- en
- Open access
- bronze
- doi.org
- Locations
- The Annals of Statistics (publishedVersion)
- arXiv (Cornell University) (submittedVersion)
- Carolina Digital Repository (University of North Carolina at Chapel Hill) (submittedVersion)
- Project Euclid (Cornell University) (submittedVersion)
- arXiv (Cornell University)
- UNC Libraries
- Funders
- University of Leeds
- Topics
- Physical Sciences › Mathematics › Statistics and Probability › Statistical Methods and Applications
- Physical Sciences › Mathematics › Statistics and Probability › Statistical Methods and Inference
- Physical Sciences › Mathematics › Statistics and Probability › Advanced Statistical Methods and Models
- Keywords
- principal component analysis, high dimension low sample size, asymptotic consistency, dimensionality reduction
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