Statistical Quantile Learning for Large Additive Latent Variable Models

Large and complex datasets, emerging from technological advancements in fields such as genomics and brain imaging, hold ample promise for gaining new scientific insights. Yet, their inherent nonlinearity and high dimensionality present considerable theoretical, methodological, and application challenges to the statistics and machine learning community. This article introduces Statistical Quantile Learning (SQL), a new nonparametric method for estimating large additive latent variable models. The distinguishing features of the SQL framework include the following. (i) A nonparametric approach relying on penalization and sieves: it offers a scalable and computationally simple, yet potent, alternative to deep learning methods. (ii) Rooted in statistical theory: SQL is consistent and achieves optimal rates of convergence in the large-dimensional case under mild assumptions. (iii) Adapted to large and high dimensional settings: we show that, numerically and theoretically, SQL’s performance improves as the data dimensionality increases. (iv) Interpretability through an identifiable additive model structure. After presenting the theoretical properties, we show that SQL can outperform variational autoencoders (VAE) in simulation studies. Finally, we apply SQL to high-dimensional gene expression data (consisting of 20,263 genes from 801 subjects), where the proposed method identifies latent factors predictive of five cancer types. Supplementary materials for this article are available online, including a standardized description of the materials available for reproducing the work.