SOFARI: High-Dimensional Manifold-Based Inference

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 article, 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. Supplementary materials for this article are available online, including a standardized description of the materials available for reproducing the work.

多任务学习（Multi-task learning）是一种广泛应用于从各类任务中获取信息的技术。近期，基于系数矩阵内稀疏奇异值分解（sparse singular value decomposition, SVD）的稀疏正交因子回归（sparse orthogonal factor regression, SOFAR）框架被提出，用于可解释多任务学习，能够跨不同层发现有意义的潜在特征-响应关联网络。然而，由于稀疏SVD约束所继承的正交性限制，对潜在因子矩阵进行精确推断仍具有挑战性。本文提出一种名为高维流形基SOFAR推断（high-dimensional manifold-based SOFAR inference, SOFARI）的新方法，该方法借鉴了Neyman近正交推断，并融入了SVD约束所施加的Stiefel流形结构。通过利用对推断至关重要的底层Stiefel流形结构，SOFARI为潜在左因子向量和奇异值提供了易用的偏差校正估计量，我们证明这些估计量具有渐近零均值正态分布且方差可估计。我们引入两种SOFARI变体以处理强正交和弱正交的潜在因子，其中后者覆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，验证了SOFARI的有效性并证实了我们的理论结果。