# Inference for Low-Rank Models

arXiv: https://arxiv.org/abs/2107.02602 · HTML: https://mlsanity.com/paper/2107.02602/

## Key facts

- Cited by 7 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 6 Jul 2021.

## Built on by (most cited first, 7 of 7)

- [Inference for Heteroskedastic PCA with Missing Data](https://mlsanity.com/paper/2107.12365/) (2021)
- [Inference of Dependency Knowledge Graph for Electronic Health Records](https://mlsanity.com/paper/2312.15611/) (2023)
- [Online Policy Learning and Inference by Matrix Completion](https://mlsanity.com/paper/2404.17398/) (2024)
- [A Generalized Latent Factor Model Approach to Mixed-data Matrix Completion with Entrywise Consistency](https://mlsanity.com/paper/2211.09272/) (2022)
- [Statistical Inference in Tensor Completion: Optimal Uncertainty Quantification and Statistical-to-Computational Gaps](https://mlsanity.com/paper/2410.11225/) (2024)
- [A Unified Framework for Estimation of High-dimensional Conditional Factor Models](https://mlsanity.com/paper/2209.00391/) (2022)
- [Addressing Dynamic and Sparse Qualitative Data: A Hilbert Space Embedding of Categorical Variables](https://mlsanity.com/paper/2308.11781/) (2023)

## Abstract (arXiv)

This paper studies inference in linear models with a high-dimensional parameter matrix that can be well-approximated by a ``spiked low-rank matrix.'' A spiked low-rank matrix has rank that grows slowly compared to its dimensions and nonzero singular values that diverge to infinity. We show that this framework covers a broad class of models of latent-variables which can accommodate matrix completion problems, factor models, varying coefficient models, and heterogeneous treatment effects. For inference, we apply a procedure that relies on an initial nuclear-norm penalized estimation step followed by two ordinary least squares regressions. We consider the framework of estimating incoherent eigenvectors and use a rotation argument to argue that the eigenspace estimation is asymptotically unbiased. Using this framework we show that our procedure provides asymptotically normal inference and achieves the semiparametric efficiency bound. We illustrate our framework by providing low-level conditions for its application in a treatment effects context where treatment assignment might be strongly dependent.