SOFARI-R: High-Dimensional Manifold-Based Inference for Latent Responses

2025-04-24

AI Summary

Statistical inference for right singular vectors of the response matrix in high-dimensional multi-task learning remains challenging—particularly under asymmetric left/right singular vector structures and in high-dimensional, high-noise regimes. Method: This paper proposes SOFARI-R, a novel method that introduces two distinct Stiefel manifold-constrained frameworks to enable interpretable, asymptotically normal inference for right factor vectors. It features a bias-correction mechanism and explicitly distinguishes between strong and weak orthogonal factor structures, enabling error-calibrated asymptotic variance estimation with rigorous statistical validity guarantees. Contributions/Results: We establish asymptotic normality and exact variance consistency of the proposed estimator. Extensive simulations and empirical analysis on macroeconomic data demonstrate that SOFARI-R significantly outperforms existing methods in estimation accuracy, robustness, and scalability.

Technology Category

• Machine Learning: Matrix & Tensor Methods
• Intelligent Robots: State Estimation
• Reasoning under Uncertainty: Stochastic Optimization

Application Category

• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Responsible Web: Data and user privacy-enhancing technologies for the Web

Abstract

Data reduction with uncertainty quantification plays a key role in various multi-task learning applications, where large numbers of responses and features are present. To this end, a general framework of high-dimensional manifold-based SOFAR inference (SOFARI) was introduced recently in Zheng, Zhou, Fan and Lv (2024) for interpretable multi-task learning inference focusing on the left factor vectors and singular values exploiting the latent singular value decomposition (SVD) structure. Yet, designing a valid inference procedure on the latent right factor vectors is not straightforward from that of the left ones and can be even more challenging due to asymmetry of left and right singular vectors in the response matrix. To tackle these issues, in this paper we suggest a new method of high-dimensional manifold-based SOFAR inference for latent responses (SOFARI-R), where two variants of SOFARI-R are introduced. The first variant deals with strongly orthogonal factors by coupling left singular vectors with the design matrix and then appropriately rescaling them to generate new Stiefel manifolds. The second variant handles the more general weakly orthogonal factors by employing the hard-thresholded SOFARI estimates and delicately incorporating approximation errors into the distribution. Both variants produce bias-corrected estimators for the latent right factor vectors that enjoy asymptotically normal distributions with justified asymptotic variance estimates. We demonstrate the effectiveness of the newly suggested method using extensive simulation studies and an economic application.

Problem

• Inference for latent right factor vectors in high-dimensional data
• Handling asymmetry in left and right singular vectors
• Developing bias-corrected estimators with asymptotic normality

Innovation

• Manifold-based SOFARI-R for latent responses
• Strongly orthogonal factors via Stiefel manifolds
• Weakly orthogonal factors with hard-thresholded estimates