# When Collaborative Filtering is not Collaborative: Unfairness of PCA for Recommendations

arXiv: https://arxiv.org/abs/2310.09687 · HTML: https://mlsanity.com/paper/2310.09687/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 19 of its 53 references that are in the map.
- First posted on arXiv 15 Oct 2023.

## Builds on (most cited first, 19 of 19)

- [Equality of Opportunity in Supervised Learning](https://mlsanity.com/paper/1610.02413/) (2016)
- [LightGCN: Simplifying and Powering Graph Convolution Network for Recommendation](https://mlsanity.com/paper/2002.02126/) (2020)
- [Recommendations as Treatments: Debiasing Learning and Evaluation](https://mlsanity.com/paper/1602.05352/) (2016)
- [Why Is My Classifier Discriminatory?](https://mlsanity.com/paper/1805.12002/) (2018)
- [Managing Popularity Bias in Recommender Systems with Personalized Re-ranking](https://mlsanity.com/paper/1901.07555/) (2019)
- [Ranking with Fairness Constraints](https://mlsanity.com/paper/1704.06840/) (2017)
- [FairRec: Two-Sided Fairness for Personalized Recommendations in Two-Sided Platforms](https://mlsanity.com/paper/2002.10764/) (2020)
- [The Price of Fair PCA: One Extra Dimension](https://mlsanity.com/paper/1811.00103/) (2018)
- [A Faster Interior Point Method for Semidefinite Programming](https://mlsanity.com/paper/2009.10217/) (2020)
- [A Survey on Popularity Bias in Recommender Systems](https://mlsanity.com/paper/2308.01118/) (2023)
- [User-centered Evaluation of Popularity Bias in Recommender Systems](https://mlsanity.com/paper/2103.06364/) (2021)
- [Alleviating Matthew Effect of Offline Reinforcement Learning in Interactive Recommendation](https://mlsanity.com/paper/2307.04571/) (2023)
- [Two-sided fairness in rankings via Lorenz dominance](https://mlsanity.com/paper/2110.15781/) (2021)
- [Online certification of preference-based fairness for personalized recommender systems](https://mlsanity.com/paper/2104.14527/) (2021)
- [Convex Formulations for Fair Principal Component Analysis](https://mlsanity.com/paper/1802.03765/) (2018)
- [Efficient Fair Principal Component Analysis](https://mlsanity.com/paper/1911.04931/) (2019)
- [Fast and Efficient MMD-based Fair PCA via Optimization over Stiefel Manifold](https://mlsanity.com/paper/2109.11196/) (2021)
- [Stochastic Optimization of PCA with Capped MSG](https://mlsanity.com/paper/1307.1674/) (2013)
- [A novel approach for Fair Principal Component Analysis based on eigendecomposition](https://mlsanity.com/paper/2208.11362/) (2022)

## Built on by (most cited first, 1 of 1)

- [humancompatible.detect: a Python Toolkit for Detecting Bias in AI Models](https://mlsanity.com/paper/2509.24340/) (2025)

## Abstract (arXiv)

We study the fairness of dimensionality reduction methods for recommendations. We focus on the fundamental method of principal component analysis (PCA), which identifies latent components and produces a low-rank approximation via the leading components while discarding the trailing components. Prior works have defined notions of "fair PCA"; however, these definitions do not answer the following question: why is PCA unfair? We identify two underlying popularity mechanisms that induce item unfairness in PCA. The first negatively impacts less popular items because less popular items rely on trailing latent components to recover their values. The second negatively impacts highly popular items, since the leading PCA components specialize in individual popular items instead of capturing similarities between items. To address these issues, we develop a polynomial-time algorithm, Item-Weighted PCA, that flexibly up-weights less popular items when optimizing for leading principal components. We theoretically show that PCA, in all cases, and Normalized PCA, in cases of block-diagonal matrices, are instances of Item-Weighted PCA. We empirically show that there exist datasets for which Item-Weighted PCA yields the optimal solution while the baselines do not. In contrast to past dimensionality reduction re-weighting techniques, Item-Weighted PCA solves a convex optimization problem and enforces a hard rank constraint. Our evaluations on real-world datasets show that Item-Weighted PCA not only mitigates both unfairness mechanisms, but also produces recommendations that outperform those of PCA baselines.