Hidden Block Regression: A General Framework for Multi-Response Models with Group Structures and Hidden Variables

Multi-response models with group structures and hidden variables are prevalent in complex data analysis. However, existing methods often fall short in capturing the intricate interactions between observed and unobserved components. In this paper, we introduce Hidden Block Regression (HBR), a novel method that provides a unified framework for modeling grouped structures influenced by unobservable factors. By integrating hidden block detection with group structure preservation, HBR offers a flexible strategy that generalizes to various forms of hidden factor involvement, enabling accurate prediction and variable selection. We derive theoretical deviation bounds for the HBR estimator, demonstrating its capacity to identify hidden variables while maintaining essential group structure information. Extensive simulations reveal that HBR consistently outperforms existing methods, particularly in scenarios involving complex interactions between observed and latent factors. A case study on a yeast dataset underscores its practical utility in identifying key predictors in such contexts.