# Probabilistic Contrastive Principal Component Analysis

arXiv: https://arxiv.org/abs/2012.07977 · HTML: https://mlsanity.com/paper/2012.07977/

## Key facts

- Cited by 10 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 14 Dec 2020.

## Built on by (most cited first, 10 of 10)

- [Moment Matching Deep Contrastive Latent Variable Models](https://mlsanity.com/paper/2202.10560/) (2022)
- [Separating common from salient patterns with Contrastive Representation Learning](https://mlsanity.com/paper/2402.11928/) (2024)
- [Toward the Identifiability of Comparative Deep Generative Models](https://mlsanity.com/paper/2401.15903/) (2024)
- [PCA++: How Uniformity Induces Robustness to Background Noise in Contrastive Learning](https://mlsanity.com/paper/2511.12278/) (2025)
- [Feature Selection in the Contrastive Analysis Setting](https://mlsanity.com/paper/2310.18531/) (2023)
- [Contrastive inverse regression for dimension reduction](https://mlsanity.com/paper/2305.12287/) (2023)
- [Contrastive independent component analysis](https://mlsanity.com/paper/2407.02357/) (2024)
- [Contrastive Dimension Reduction: A Systematic Review](https://mlsanity.com/paper/2510.11847/) (2025)
- [Multi-context principal component analysis](https://mlsanity.com/paper/2601.15239/) (2026)
- [Disentangling Shared and Target-Enriched Topics via Background-Contrastive Non-negative Matrix Factorization](https://mlsanity.com/paper/2602.22387/) (2026)

## Abstract (arXiv)

Dimension reduction is useful for exploratory data analysis. In many applications, it is of interest to discover variation that is enriched in a "foreground" dataset relative to a "background" dataset. Recently, contrastive principal component analysis (CPCA) was proposed for this setting. However, the lack of a formal probabilistic model makes it difficult to reason about CPCA and to tune its hyperparameter. In this work, we propose probabilistic contrastive principal component analysis (PCPCA), a model-based alternative to CPCA. We discuss how to set the hyperparameter in theory and in practice, and we show several of PCPCA's advantages over CPCA, including greater interpretability, uncertainty quantification and principled inference, robustness to noise and missing data, and the ability to generate data from the model. We demonstrate PCPCA's performance through a series of simulations and case-control experiments with datasets of gene expression, protein expression, and images.