Comprehensive neural network methods for heteroscedastic partial linear quantile regression in ultra-high dimensions: Deep and shallow architectures

We study heteroscedastic partial linear quantile regression in ultra-high dimensions allowing an exponentially growing dimension of the high-dimensional linear covariates. Recent advances show that deep neural networks (DNNs) can estimate the unknown nonlinear component beyond the limits of classical nonparametric methods. However, two fundamental challenges remain: the additional sparsity tuning commonly required by sparse neural networks substantially increases computational cost, and the trade-off among predictive accuracy, variable-selection consistency, and model complexity between deep and shallow architectures is not well understood. To address these challenges, we develop a unified neural framework with complementary theoretical guarantees. When the true non-linear component lies in a hierarchically compositional function space, we establish theory for fully connected deep network estimators; when it lies in a variation space, we provide parallel theory for fully connected shallow network estimators. Our results establish sharp-rate estimation consistency, asymptotic normality, asymptotically valid inference for the key linear covariates, and consistent variable selection for the sparse linear coefficient vector at the target quantile level. Building on this theoretical characterization, we propose a data-adaptive depth-and-width selection procedure that achieves substantial computational savings while preserving statistical guarantees. Extensive simulations demonstrate state-of-the-art predictive performance, stable recovery of the active set, and marked computational gains of shallow architectures when the underlying structure favors low-order variation. An application to high-dimensional metabolomic data demonstrates the practical advantages of the proposed method in identifying biologically meaningful metabolites while maintaining strong predictive performance.