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Dimensionality Reduction, Regularization, and Variable Selection

Forward Stepwise Deep Autoencoder-Based Monotone Nonlinear Dimensionality Reduction Methods

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Pages 519-529 | Received 02 Oct 2019, Accepted 13 Nov 2020, Published online: 27 Jan 2021
 

Abstract

Dimensionality reduction is an unsupervised learning task aimed at creating a low-dimensional summary and/or extracting the most salient features of a dataset. Principal component analysis is a linear dimensionality reduction method in the sense that each principal component is a linear combination of the input variables. To allow features that are nonlinear functions of the input variables, many nonlinear dimensionality reduction (NLDR) methods have been proposed. In this article, we propose novel NLDR methods based on bottleneck deep autoencoders. Our contributions are 2-fold: (1) We introduce a monotonicity constraint into bottleneck deep autoencoders for estimating a single nonlinear component and propose two methods for fitting the model. (2) We propose a new, forward stepwise deep learning architecture for estimating multiple nonlinear components. The former helps extract interpretable, monotone components when the assumption of monotonicity holds, and the latter helps evaluate reconstruction errors in the original data space for a range of components. We conduct numerical studies to compare different model fitting methods and use two real data examples from the studies of human immune responses to HIV to illustrate the proposed methods. Supplementary materials for this article are available online.

Supplementary Materials

Additional figures and tables and derivation of the gradients.

Acknowledgments

The authors thank the editor, the AE, and two anonymous referees for their highly constructive comments. The authors are also indebted to the investigators of the immune correlates study of mother-to-child transmission of HIV-1, in particular Sallie Permar, and the participants and investigators of HVTN 505, in particular Georgia Tomaras and Julie McElrath, for providing the biomarker data for the examples. The authors thank Lindsay N. Carpp for help with editing.

Additional information

Funding

This work was supported by the National Institutes of Health (R01-AI122991; UM1-AI068635; S10OD028685).

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