# Fast and Efficient MMD-based Fair PCA via Optimization over Stiefel Manifold

arXiv: https://arxiv.org/abs/2109.11196 · HTML: https://mlsanity.com/paper/2109.11196/

## Key facts

- Cited by 18 papers in the map (data through 14 Oct 2026).
- Builds on 19 of its 52 references that are in the map.
- First posted on arXiv 23 Sep 2021.

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

- [Equality of Opportunity in Supervised Learning](https://mlsanity.com/paper/1610.02413/) (2016)
- [A Survey on Bias and Fairness in Machine Learning](https://mlsanity.com/paper/1908.09635/) (2019)
- [Certifying and removing disparate impact](https://mlsanity.com/paper/1412.3756/) (2014)
- [Fairness Beyond Disparate Treatment & Disparate Impact: Learning Classification without Disparate Mistreatment](https://mlsanity.com/paper/1610.08452/) (2016)
- [Learning Adversarially Fair and Transferable Representations](https://mlsanity.com/paper/1802.06309/) (2018)
- [Fair Clustering Through Fairlets](https://mlsanity.com/paper/1802.05733/) (2018)
- [Data Decisions and Theoretical Implications when Adversarially Learning Fair Representations](https://mlsanity.com/paper/1707.00075/) (2017)
- [Fair Algorithms for Clustering](https://mlsanity.com/paper/1901.02393/) (2019)
- [The Price of Fair PCA: One Extra Dimension](https://mlsanity.com/paper/1811.00103/) (2018)
- [Fair Feature Distillation for Visual Recognition](https://mlsanity.com/paper/2106.04411/) (2021)
- [Toward a better trade-off between performance and fairness with kernel-based distribution matching](https://mlsanity.com/paper/1910.11779/) (2019)
- [A Brief Introduction to Manifold Optimization](https://mlsanity.com/paper/1906.05450/) (2019)
- [Convex Formulations for Fair Principal Component Analysis](https://mlsanity.com/paper/1802.03765/) (2018)
- [A Stochastic PCA and SVD Algorithm with an Exponential Convergence Rate](https://mlsanity.com/paper/1409.2848/) (2014)
- [Simple algorithms for optimization on Riemannian manifolds with constraints](https://mlsanity.com/paper/1901.10000/) (2019)
- [Stochastic Optimization of PCA with Capped MSG](https://mlsanity.com/paper/1307.1674/) (2013)
- [Orthogonal Sparse PCA and Covariance Estimation via Procrustes Reformulation](https://mlsanity.com/paper/1602.03992/) (2016)
- [Covariance-Robust Dynamic Watermarking](https://mlsanity.com/paper/2003.13908/) (2020)
- [Manifold Optimization for High Accuracy Spatial Location Estimation Using Ultrasound Waves](https://mlsanity.com/paper/2103.15050/) (2021)

## Built on by (most cited first, 18 of 18)

- [Efficient fair PCA for fair representation learning](https://mlsanity.com/paper/2302.13319/) (2023)
- [MMD-B-Fair: Learning Fair Representations with Statistical Testing](https://mlsanity.com/paper/2211.07907/) (2022)
- [First-Order Algorithms for Min-Max Optimization in Geodesic Metric Spaces](https://mlsanity.com/paper/2206.02041/) (2022)
- [Fair Representation Learning for Continuous Sensitive Attributes using Expectation of Integral Probability Metrics](https://mlsanity.com/paper/2505.06435/) (2025)
- [A novel approach for Fair Principal Component Analysis based on eigendecomposition](https://mlsanity.com/paper/2208.11362/) (2022)
- [Fair CoVariance Neural Networks](https://mlsanity.com/paper/2409.08558/) (2024)
- [Fair Streaming Principal Component Analysis: Statistical and Algorithmic Viewpoint](https://mlsanity.com/paper/2310.18593/) (2023)
- [Last-Iterate Convergence of Adaptive Riemannian Gradient Descent for Equilibrium Computation](https://mlsanity.com/paper/2306.16617/) (2023)
- [Specification Overfitting in Artificial Intelligence](https://mlsanity.com/paper/2403.08425/) (2024)
- [On Socially Fair Low-Rank Approximation and Column Subset Selection](https://mlsanity.com/paper/2412.06063/) (2024)
- [When Collaborative Filtering is not Collaborative: Unfairness of PCA for Recommendations](https://mlsanity.com/paper/2310.09687/) (2023)
- [Learning Fair Representations with High-Confidence Guarantees](https://mlsanity.com/paper/2310.15358/) (2023)
- [Hidden Convexity of Fair PCA and Fast Solver via Eigenvalue Optimization](https://mlsanity.com/paper/2503.00299/) (2025)
- [PLS-based approach for fair representation learning](https://mlsanity.com/paper/2502.16263/) (2025)
- [Achieving Fair PCA Using Joint Eigenvalue Decomposition](https://mlsanity.com/paper/2502.16933/) (2025)
- [Fair PCA, One Component at a Time](https://mlsanity.com/paper/2503.21563/) (2025)
- [Incorporating Fairness Constraints into Archetypal Analysis](https://mlsanity.com/paper/2507.12021/) (2025)
- [Fair Representation Learning with Controllable High Confidence Guarantees via Adversarial Inference](https://mlsanity.com/paper/2510.21017/) (2025)

## Abstract (arXiv)

This paper defines fair principal component analysis (PCA) as minimizing the maximum mean discrepancy (MMD) between dimensionality-reduced conditional distributions of different protected classes. The incorporation of MMD naturally leads to an exact and tractable mathematical formulation of fairness with good statistical properties. We formulate the problem of fair PCA subject to MMD constraints as a non-convex optimization over the Stiefel manifold and solve it using the Riemannian Exact Penalty Method with Smoothing (REPMS; Liu and Boumal, 2019). Importantly, we provide local optimality guarantees and explicitly show the theoretical effect of each hyperparameter in practical settings, extending previous results. Experimental comparisons based on synthetic and UCI datasets show that our approach outperforms prior work in explained variance, fairness, and runtime.