# HePPCAT: Probabilistic PCA for Data with Heteroscedastic Noise

arXiv: https://arxiv.org/abs/2101.03468 · HTML: https://mlsanity.com/paper/2101.03468/

## Key facts

- Cited by 8 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 10 Jan 2021.

## Built on by (most cited first, 8 of 8)

- [The Fisher-Rao geometry of CES distributions](https://mlsanity.com/paper/2310.01032/) (2023)
- [ALPCAH: Sample-wise Heteroscedastic PCA with Tail Singular Value Regularization](https://mlsanity.com/paper/2307.02745/) (2023)
- [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)
- [HeMPPCAT: Mixtures of Probabilistic Principal Component Analysers for Data with Heteroscedastic Noise](https://mlsanity.com/paper/2301.08852/) (2023)
- [T-Rex: Fitting a Robust Factor Model via Expectation-Maximization](https://mlsanity.com/paper/2505.12117/) (2025)
- [Defining Energy Indicators for Impact Identification on Aerospace Composites: A Structured Feature Selection Approach Guided by Domain Knowledge](https://mlsanity.com/paper/2511.01592/) (2025)
- [Heteroskedastic Canonical Polyadic Tensor Decomposition](https://mlsanity.com/paper/2610.00498/) (2026)

## Abstract (arXiv)

Principal component analysis (PCA) is a classical and ubiquitous method for reducing data dimensionality, but it is suboptimal for heterogeneous data that are increasingly common in modern applications. PCA treats all samples uniformly so degrades when the noise is heteroscedastic across samples, as occurs, e.g., when samples come from sources of heterogeneous quality. This paper develops a probabilistic PCA variant that estimates and accounts for this heterogeneity by incorporating it in the statistical model. Unlike in the homoscedastic setting, the resulting nonconvex optimization problem is not seemingly solved by singular value decomposition. This paper develops a heteroscedastic probabilistic PCA technique (HePPCAT) that uses efficient alternating maximization algorithms to jointly estimate both the underlying factors and the unknown noise variances. Simulation experiments illustrate the comparative speed of the algorithms, the benefit of accounting for heteroscedasticity, and the seemingly favorable optimization landscape of this problem. Real data experiments on environmental air quality data show that HePPCAT can give a better PCA estimate than techniques that do not account for heteroscedasticity.