# A Bayesian sparse factor model with adaptive posterior concentration

arXiv: https://arxiv.org/abs/2305.18488 · HTML: https://mlsanity.com/paper/2305.18488/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 8 of its 46 references that are in the map.
- First posted on arXiv 29 May 2023.

## Builds on (most cited first, 8 of 8)

- [Estimating Number of Factors by Adjusted Eigenvalues Thresholding](https://mlsanity.com/paper/1909.10710/) (2019)
- [Sufficient Forecasting Using Factor Models](https://mlsanity.com/paper/1505.07414/) (2015)
- [FarmTest: Factor-Adjusted Robust Multiple Testing with Approximate False Discovery Control](https://mlsanity.com/paper/1711.05386/) (2017)
- [Large Covariance Estimation through Elliptical Factor Models](https://mlsanity.com/paper/1507.08377/) (2015)
- [On the identifiability of Bayesian factor analytic models](https://mlsanity.com/paper/2004.05105/) (2020)
- [Spike and slab Bayesian sparse principal component analysis](https://mlsanity.com/paper/2102.00305/) (2021)
- [Expandable Factor Analysis](https://mlsanity.com/paper/1407.1158/) (2014)
- [Order-invariant prior specification in Bayesian factor analysis](https://mlsanity.com/paper/1409.7672/) (2014)

## Abstract (arXiv)

In this paper, we propose a new Bayesian inference method for a high-dimensional sparse factor model that allows both the factor dimensionality and the sparse structure of the loading matrix to be inferred. The novelty is to introduce a certain dependence between the sparsity level and the factor dimensionality, which leads to adaptive posterior concentration while keeping computational tractability. We show that the posterior distribution asymptotically concentrates on the true factor dimensionality, and more importantly, this posterior consistency is adaptive to the sparsity level of the true loading matrix and the noise variance. We also prove that the proposed Bayesian model attains the optimal detection rate of the factor dimensionality in a more general situation than those found in the literature. Moreover, we obtain a near-optimal posterior concentration rate of the covariance matrix. Numerical studies are conducted and show the superiority of the proposed method compared with other competitors.