# ALPCAH: Sample-wise Heteroscedastic PCA with Tail Singular Value Regularization

arXiv: https://arxiv.org/abs/2307.02745 · HTML: https://mlsanity.com/paper/2307.02745/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 5 of its 22 references that are in the map.
- First posted on arXiv 6 Jul 2023.

## Builds on (most cited first, 5 of 5)

- [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)
- [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)

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

- [ALPCAH: Subspace Learning for Sample-wise Heteroscedastic Data](https://mlsanity.com/paper/2505.07272/) (2025)
- [ALPCAHUS: Subspace Clustering for Heteroscedastic Data](https://mlsanity.com/paper/2505.18918/) (2025)

## Abstract (arXiv)

Principal component analysis (PCA) is a key tool in the field of data dimensionality reduction that is useful for various data science problems. However, many applications involve heterogeneous data that varies in quality due to noise characteristics associated with different sources of the data. Methods that deal with this mixed dataset are known as heteroscedastic methods. Current methods like HePPCAT make Gaussian assumptions of the basis coefficients that may not hold in practice. Other methods such as Weighted PCA (WPCA) assume the noise variances are known, which may be difficult to know in practice. This paper develops a PCA method that can estimate the sample-wise noise variances and use this information in the model to improve the estimate of the subspace basis associated with the low-rank structure of the data. This is done without distributional assumptions of the low-rank component and without assuming the noise variances are known. Simulations show the effectiveness of accounting for such heteroscedasticity in the data, the benefits of using such a method with all of the data versus retaining only good data, and comparisons are made against other PCA methods established in the literature like PCA, Robust PCA (RPCA), and HePPCAT. Code available at https://github.com/javiersc1/ALPCAH