High-dimensional principal component analysis with heterogeneous missingness

Ziwei Zhu, Tengyao Wang and Richard J. Samworth

LSE Research Online Documents on Economics from London School of Economics and Political Science, LSE Library

Abstract: We study the problem of high-dimensional Principal Component Analysis (PCA) with missing observations. In a simple, homogeneous observation model, we show that an existing observed-proportion weighted (OPW) estimator of the leading principal components can (nearly) attain the minimax optimal rate of convergence, which exhibits an interesting phase transition. However, deeper investigation reveals that, particularly in more realistic settings where the observation probabilities are heterogeneous, the empirical performance of the OPW estimator can be unsatisfactory; moreover, in the noiseless case, it fails to provide exact recovery of the principal components. Our main contribution, then, is to introduce a new method, which we call primePCA, that is designed to cope with situations where observations may be missing in a heterogeneous manner. Starting from the OPW estimator, primePCA iteratively projects the observed entries of the data matrix onto the column space of our current estimate to impute the missing entries, and then updates our estimate by computing the leading right singular space of the imputed data matrix. We prove that the error of primePCA converges to zero at a geometric rate in the noiseless case, and when the signal strength is not too small. An important feature of our theoretical guarantees is that they depend on average, as opposed to worst-case, properties of the missingness mechanism. Our numerical studies on both simulated and real data reveal that primePCA exhibits very encouraging performance across a wide range of scenarios, including settings where the data are not Missing Completely At Random.

Keywords: heterogeneous missingness; high-dimensional statistics; iterative projections; missing data; principal component analysis (search for similar items in EconPapers)
JEL-codes: C1 (search for similar items in EconPapers)
Pages: 32 pages
Date: 2022-11-20
New Economics Papers: this item is included in nep-ecm
References: View references in EconPapers View complete reference list from CitEc
Citations: View citations in EconPapers (2)

Published in Journal of the Royal Statistical Society. Series B: Statistical Methodology, 20, November, 2022, 84(5), pp. 2000 - 2031. ISSN: 1369-7412

Downloads: (external link)
http://eprints.lse.ac.uk/117647/ Open access version. (application/pdf)

Related works:
This item may be available elsewhere in EconPapers: Search for items with the same title.

Export reference: BibTeX RIS (EndNote, ProCite, RefMan) HTML/Text

Persistent link: https://EconPapers.repec.org/RePEc:ehl:lserod:117647

Access Statistics for this paper

More papers in LSE Research Online Documents on Economics from London School of Economics and Political Science, LSE Library LSE Library Portugal Street London, WC2A 2HD, U.K.. Contact information at EDIRC.
Bibliographic data for series maintained by LSERO Manager ().