# ALPCAH: Subspace Learning for Sample-wise Heteroscedastic Data

arXiv: https://arxiv.org/abs/2505.07272 · HTML: https://mlsanity.com/paper/2505.07272/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 11 of its 57 references that are in the map.
- First posted on arXiv 12 May 2025.

## Builds on (most cited first, 11 of 11)

- [Nonconvex Optimization Meets Low-Rank Matrix Factorization: An Overview](https://mlsanity.com/paper/1809.09573/) (2018)
- [Low-rank Solutions of Linear Matrix Equations via Procrustes Flow](https://mlsanity.com/paper/1507.03566/) (2015)
- [Heteroskedastic PCA: Algorithm, Optimality, and Applications](https://mlsanity.com/paper/1810.08316/) (2018)
- [Robust Subspace Learning: Robust PCA, Robust Subspace Tracking, and Robust Subspace Recovery](https://mlsanity.com/paper/1711.09492/) (2017)
- [Partial Sum Minimization of Singular Values in Robust PCA: Algorithm and Applications](https://mlsanity.com/paper/1503.01444/) (2015)
- [Permutation methods for factor analysis and PCA](https://mlsanity.com/paper/1710.00479/) (2017)
- [HePPCAT: Probabilistic PCA for Data with Heteroscedastic Noise](https://mlsanity.com/paper/2101.03468/) (2021)
- [Selecting the number of components in PCA via random signflips](https://mlsanity.com/paper/2012.02985/) (2020)
- [Weighted principal component analysis: a weighted covariance eigendecomposition approach](https://mlsanity.com/paper/1412.4533/) (2014)
- [ALPCAH: Sample-wise Heteroscedastic PCA with Tail Singular Value Regularization](https://mlsanity.com/paper/2307.02745/) (2023)
- [Streaming Heteroscedastic Probabilistic PCA with Missing Data](https://mlsanity.com/paper/2310.06277/) (2023)

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

- [ALPCAHUS: Subspace Clustering for Heteroscedastic Data](https://mlsanity.com/paper/2505.18918/) (2025)
- [PET-TURTLE: Deep Unsupervised Support Vector Machines for Imbalanced Data Clusters](https://mlsanity.com/paper/2601.03237/) (2026)

## Abstract (arXiv)

Principal component analysis (PCA) is a key tool in the field of data dimensionality reduction. However, some applications involve heterogeneous data that vary in quality due to noise characteristics associated with each data sample. Heteroscedastic methods aim to deal with such mixed data quality. This paper develops a subspace learning method, named ALPCAH, that can estimate the sample-wise noise variances and use this information to improve the estimate of the subspace basis associated with the low-rank structure of the data. Our method makes no distributional assumptions of the low-rank component and does not assume that the noise variances are known. Further, this method uses a soft rank constraint that does not require subspace dimension to be known. Additionally, this paper develops a matrix factorized version of ALPCAH, named LR-ALPCAH, that is much faster and more memory efficient at the cost of requiring subspace dimension to be known or estimated. Simulations and real data experiments show the effectiveness of accounting for data heteroscedasticity compared to existing algorithms. Code available at https://github.com/javiersc1/ALPCAH.