SOFARI: High-Dimensional Manifold-Based Inference

TL;DR

SOFARI introduces a novel manifold-based inference method for high-dimensional multi-task learning models, enabling accurate, bias-corrected inference on latent factors within the sparse SVD framework.

Contribution

It develops a new inference approach leveraging Stiefel manifold structure, addressing challenges in orthogonality constraints for latent factor inference in SOFAR models.

Findings

SOFARI provides asymptotically normal estimators for latent factors.

Simulation studies confirm the accuracy of SOFARI's inference.

Real data application demonstrates practical utility in economic forecasting.

Abstract

Multi-task learning is a widely used technique for harnessing information from various tasks. Recently, the sparse orthogonal factor regression (SOFAR) framework, based on the sparse singular value decomposition (SVD) within the coefficient matrix, was introduced for interpretable multi-task learning, enabling the discovery of meaningful latent feature-response association networks across different layers. However, conducting precise inference on the latent factor matrices has remained challenging due to the orthogonality constraints inherited from the sparse SVD constraints. In this paper, we suggest a novel approach called the high-dimensional manifold-based SOFAR inference (SOFARI), drawing on the Neyman near-orthogonality inference while incorporating the Stiefel manifold structure imposed by the SVD constraints. By leveraging the underlying Stiefel manifold structure that is crucial to enabling inference, SOFARI provides easy-to-use bias-corrected estimators for both latent left factor vectors and singular values, for which we show to enjoy the asymptotic mean-zero normal distributions with estimable variances. We introduce two SOFARI variants to handle strongly and weakly orthogonal latent factors, where the latter covers a broader range of applications. We illustrate the effectiveness of SOFARI and justify our theoretical results through simulation examples and a real data application in economic forecasting.