High-dimensional principal component analysis with heterogeneous missingness

Zhu, Ziwei, Wang, Tengyao ORCID: 0000-0003-2072-6645 and Samworth, Richard J. (2022) High-dimensional principal component analysis with heterogeneous missingness. Journal of the Royal Statistical Society. Series B: Statistical Methodology, 84 (5). 2000 - 2031. ISSN 1369-7412

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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.

Item Type:	Article
Official URL:	https://rss.onlinelibrary.wiley.com/journal/146798...
Additional Information:	© 2022 The Authors
Divisions:	Statistics
Subjects:	H Social Sciences > HA Statistics
Date Deposited:	16 Dec 2022 17:15
Last Modified:	12 Dec 2024 03:29
URI:	http://eprints.lse.ac.uk/id/eprint/117647

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