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Abstract When the dimension of data is comparable to or larger than the number of data samples, principal components analysis (PCA) may exhibit problematic high-dimensional noise. In this work, we propose an empirical Bayes PCA method that reduces this noise by estimating a joint prior distribution for the principal components. EB-PCA is based on the classical Kiefer–Wolfowitz non-parametric maximum likelihood estimator for empirical Bayes estimation, distributional results derived from random matrix theory for the sample PCs and iterative refinement using an approximate message passing (AMP) algorithm. In theoretical ‘spiked’ models, EB-PCA achieves Bayes-optimal estimation accuracy in the same settings as an oracle Bayes AMP procedure that knows the true priors. Empirically, EB-PCA significantly improves over PCA when there is strong prior structure, both in simulation and on quantitative benchmarks constructed from the 1000 Genomes Project and the International HapMap Project. An illustration is presented for analysis of gene expression data obtained by single-cell RNA-seq.
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Independent Nonlinear Component Analysis
The idea of summarizing the information contained in a large number of variables by a small number of “factors” or “principal components” has been broadly adopted in statistics. This article introduces a generalization of the widely used principal component analysis (PCA) to nonlinear settings, thus providing a new tool for dimension reduction and exploratory data analysis or representation. The distinguishing features of the method include (i) the ability to always deliver truly independent (instead of merely uncorrelated) factors; (ii) the use of optimal transport theory and Brenier maps to obtain a robust and efficient computational algorithm; (iii) the use of a new multivariate additive entropy decomposition to determine the most informative principal nonlinear components, and (iv) formally nesting PCA as a special case for linear Gaussian factor models. We illustrate the method’s effectiveness in an application to excess bond returns prediction from a large number of macro factors. Supplementary materials for this article are available online.
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- PAR ID:
- 10416285
- Date Published:
- Journal Name:
- Journal of the American Statistical Association
- ISSN:
- 0162-1459
- Page Range / eLocation ID:
- 1 to 14
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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- Free Publicly Accessible Full Text
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Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.1080/01621459.2021.1990768
