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The studies of large-scale, high-dimensional data in fields such as genomicsand neuroscience have injected new insights into science. Yet, despiteadvances, they are confronting several challenges, often simultaneously: lackof interpretability, nonlinearity, slow computation, inconsistency anduncertain convergence, and small sample sizes compared to high featuredimensions. Here, we propose a relatively simple, scalable, and consistentnonlinear dimension reduction method that can potentially address these issuesin unsupervised settings. We call this method Statistical Quantile Learning(SQL) because, methodologically, it leverages on a quantile approximation ofthe latent variables together with standard nonparametric techniques (sieve orpenalyzed methods). We show that estimating the model simplifies into a convexassignment matching problem; we derive its asymptotic properties; we show thatthe model is identifiable under few conditions. Compared to its linearcompetitors, SQL explains more variance, yields better separation andexplanation, and delivers more accurate outcome prediction. Compared to itsnonlinear competitors, SQL shows considerable advantage in interpretability,ease of use and computations in large-dimensional settings. Finally, we applySQL to high-dimensional gene expression data (consisting of 20,263 genes from801 subjects), where the proposed method identified latent factors predictiveof five cancer types. The SQL package is available athttps://github.com/jbodelet/SQL.

Statistical Quantile Learning for Large, Nonlinear, and Additive Latent Variable Models