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 orthogonality constraints inherited from the sparse SVD constraint. In this paper, we suggest a novel approach called 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, SOFARI provides 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.

翻译：多任务学习是一种广泛用于利用不同任务信息的技术。近年来，基于系数矩阵内稀疏奇异值分解（SVD）的稀疏正交因子回归（SOFAR）框架被引入可解释的多任务学习，能够发现跨不同层的有意义的潜在特征-响应关联网络。然而，由于稀疏SVD约束带来的正交性限制，对潜在因子矩阵进行精确推断仍然具有挑战性。在本文中，我们提出一种称为高维流形SOFAR推断（SOFARI）的新方法，该方法借鉴Neyman近正交推断，同时结合SVD约束所施加的Stiefel流形结构。通过利用潜在的Stiefel流形结构，SOFARI为潜在左因子向量和奇异值提供偏差校正估计量，我们证明这些估计量具有均值为零、方差可估的渐近正态分布。我们引入两种SOFARI变体来处理强正交和弱正交潜在因子，其中后者涵盖更广泛的应用场景。我们通过模拟示例和经济预测中的真实数据应用，展示SOFARI的有效性并验证我们的理论结果。