Deep learning with missing data

Tianyi Wu, Tengyao Wang and Richard J. Samworth

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

Abstract: In the context of multivariate nonparametric regression with missing covariates, we propose Pattern Embedded Neural Networks (PENNs), which can be applied in conjunction with any existing imputation technique. In addition to a neural net work trained on the imputed data, PENNs pass the vectors of observation indicators through a second neural network to provide a compact representation. The outputs are then combined in a third neural network to produce final predictions. Our main theoretical result exploits an assumption that the observation patterns can be partitioned into cells on which the Bayes regression function behaves similarly, and belongs to a compositional H¨older class. It provides a finite-sample excess risk bound that holds for an arbitrary missingness mechanism, and in combination with a complementary minimax lower bound, demonstrates that our PENN estimator attains in typical cases the minimax rate of convergence as if the cells of the par tition were known in advance, up to a poly-logarithmic factor in the sample size. Numerical experiments on simulated, semi-synthetic and real data confirm that the PENN estimator consistently improves, often dramatically, on standard neural net works without pattern embedding. Code to reproduce our experiments, as well as a tutorial on how to apply our method, is publicly available.

Keywords: deep learning; missing data; nonparametric regression (search for similar items in EconPapers)
JEL-codes: C1 (search for similar items in EconPapers)
Date: 2026-07-29
References: Add references at CitEc
Citations:

Published in Journal of the Royal Statistical Society. Series B: Statistical Methodology, 29, July, 2026. ISSN: 1369-7412

Downloads: (external link)
https://researchonline.lse.ac.uk/id/eprint/139005/ 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:139005

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 ().