  The studies of large-scale, high-dimensional data in fields such as genomics
and neuroscience have injected new insights into science. Yet, despite
advances, they are confronting several challenges, often simultaneously: lack
of interpretability, nonlinearity, slow computation, inconsistency and
uncertain convergence, and small sample sizes compared to high feature
dimensions. Here, we propose a relatively simple, scalable, and consistent
nonlinear dimension reduction method that can potentially address these issues
in unsupervised settings. We call this method Statistical Quantile Learning
(SQL) because, methodologically, it leverages on a quantile approximation of
the latent variables together with standard nonparametric techniques (sieve or
penalyzed methods). We show that estimating the model simplifies into a convex
assignment matching problem; we derive its asymptotic properties; we show that
the model is identifiable under few conditions. Compared to its linear
competitors, SQL explains more variance, yields better separation and
explanation, and delivers more accurate outcome prediction. Compared to its
nonlinear competitors, SQL shows considerable advantage in interpretability,
ease of use and computations in large-dimensional settings. Finally, we apply
SQL to high-dimensional gene expression data (consisting of 20,263 genes from
801 subjects), where the proposed method identified latent factors predictive
of five cancer types. The SQL package is available at
https://github.com/jbodelet/SQL.
