# SOFARI: High-Dimensional Manifold-Based Inference

arXiv: https://arxiv.org/abs/2309.15032 · HTML: https://mlsanity.com/paper/2309.15032/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 6 of its 29 references that are in the map.
- First posted on arXiv 26 Sep 2023.

## Builds on (most cited first, 6 of 6)

- [Confidence Intervals and Hypothesis Testing for High-Dimensional Regression](https://mlsanity.com/paper/1306.3171/) (2013)
- [SOFAR: large-scale association network learning](https://mlsanity.com/paper/1704.08349/) (2017)
- [High-Dimensional Knockoffs Inference for Time Series Data](https://mlsanity.com/paper/2112.09851/) (2021)
- [Scalable Interpretable Multi-Response Regression via SEED](https://mlsanity.com/paper/1608.03686/) (2016)
- [Adaptive estimation of the rank of the coefficient matrix in high dimensional multivariate response regression models](https://mlsanity.com/paper/1704.02381/) (2017)
- [Innovated scalable efficient estimation in ultra-large Gaussian graphical models](https://mlsanity.com/paper/1605.03313/) (2016)

## Built on by (most cited first, 2 of 2)

- [GeoERM: Geometry-Aware Multi-Task Representation Learning on Riemannian Manifolds](https://mlsanity.com/paper/2505.02972/) (2025)
- [LIDS: LLM Summary Inference Under the Layered Lens](https://mlsanity.com/paper/2603.00105/) (2026)

## Abstract (arXiv)

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.