# On Minimum Trace Factor Analysis -- An Old Song Sung to a New Tune

arXiv: https://arxiv.org/abs/2402.02459 · HTML: https://mlsanity.com/paper/2402.02459/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 13 of its 51 references that are in the map.
- First posted on arXiv 4 Feb 2024.

## Builds on (most cited first, 13 of 13)

- [Convex optimization](https://mlsanity.com/paper/2106.01946/) (2021)
- [Spectral Methods for Data Science: A Statistical Perspective](https://mlsanity.com/paper/2012.08496/) (2020)
- [Noisy Matrix Completion: Understanding Statistical Guarantees for Convex Relaxation via Nonconvex Optimization](https://mlsanity.com/paper/1902.07698/) (2019)
- [Tensor SVD: Statistical and Computational Limits](https://mlsanity.com/paper/1703.02724/) (2017)
- [Rate-Optimal Perturbation Bounds for Singular Subspaces with Applications to High-Dimensional Statistics](https://mlsanity.com/paper/1605.00353/) (2016)
- [Heteroskedastic PCA: Algorithm, Optimality, and Applications](https://mlsanity.com/paper/1810.08316/) (2018)
- [Inference for Heteroskedastic PCA with Missing Data](https://mlsanity.com/paper/2107.12365/) (2021)
- [Entrywise Estimation of Singular Vectors of Low-Rank Matrices with Heteroskedasticity and Dependence](https://mlsanity.com/paper/2105.13346/) (2021)
- [Certifiably Optimal Low Rank Factor Analysis](https://mlsanity.com/paper/1604.06837/) (2016)
- [ePCA: High Dimensional Exponential Family PCA](https://mlsanity.com/paper/1611.05550/) (2016)
- [Subspace Estimation from Unbalanced and Incomplete Data Matrices: ℓ2,∞ Statistical Guarantees](https://mlsanity.com/paper/1910.04267/) (2019)
- [Selecting Regularization Parameters for nuclear norm type minimization problems](https://mlsanity.com/paper/2204.06206/) (2022)
- [Linear models based on noisy data and the Frisch scheme](https://mlsanity.com/paper/1304.3877/) (2013)

## Abstract (arXiv)

Dimensionality reduction methods, such as principal component analysis (PCA) and factor analysis, are central to many problems in data science. There are, however, serious and well-understood challenges to finding robust low dimensional approximations for data with significant heteroskedastic noise. This paper introduces a relaxed version of Minimum Trace Factor Analysis (MTFA), a convex optimization method with roots dating back to the work of Ledermann in 1940. This relaxation is particularly effective at not overfitting to heteroskedastic perturbations and addresses the commonly cited Heywood cases in factor analysis and the recently identified "curse of ill-conditioning" for existing spectral methods. We provide theoretical guarantees on the accuracy of the resulting low rank subspace and the convergence rate of the proposed algorithm to compute that matrix. We develop a number of interesting connections to existing methods, including HeteroPCA, Lasso, and Soft-Impute, to fill an important gap in the already large literature on low rank matrix estimation. Numerical experiments benchmark our results against several recent proposals for dealing with heteroskedastic noise.