Hidden Convexity of Fair PCA and Fast Solver via Eigenvalue Optimization

2025-03-01

AI Summary

Existing semidefinite relaxation (SDR)-based methods for Fair PCA (FPCA) suffer from high computational cost and often violate strict subgroup-wise reconstruction error equality—a core fairness requirement. Method: We first establish the intrinsic convex structure of the FPCA model, enabling an exact reformulation of the original NP-hard problem as an efficient eigenvalue optimization problem—bypassing SDR entirely. Contribution/Results: Based on this insight, we propose a novel convex optimization algorithm that provably enforces identical reconstruction errors across all subgroups while achieving substantial speedups: up to 8× faster than SDR-based approaches on real-world datasets, and at most 85% slower than standard PCA. To our knowledge, this is the first method to simultaneously guarantee strict fairness and computational scalability for FPCA.

Technology Category

• Computer Vision: Bias, Fairness & Privacy
• Search and Optimization: Non-convex Optimization
• Machine Learning: Learning with Manifolds

Application Category

• User Modeling, Personalization and Recommendation: Fairness-aware retrieval and ranking
• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Economics, Online Markets and Human Computation: Fairness, privacy, and diversity in economic environments

Abstract

Principal Component Analysis (PCA) is a foundational technique in machine learning for dimensionality reduction of high-dimensional datasets. However, PCA could lead to biased outcomes that disadvantage certain subgroups of the underlying datasets. To address the bias issue, a Fair PCA (FPCA) model was introduced by Samadi et al. (2018) for equalizing the reconstruction loss between subgroups. The semidefinite relaxation (SDR) based approach proposed by Samadi et al. (2018) is computationally expensive even for suboptimal solutions. To improve efficiency, several alternative variants of the FPCA model have been developed. These variants often shift the focus away from equalizing the reconstruction loss. In this paper, we identify a hidden convexity in the FPCA model and introduce an algorithm for convex optimization via eigenvalue optimization. Our approach achieves the desired fairness in reconstruction loss without sacrificing performance. As demonstrated in real-world datasets, the proposed FPCA algorithm runs $8 imes$ faster than the SDR-based algorithm, and only at most 85% slower than the standard PCA.

Problem

• Address bias in PCA by equalizing reconstruction loss between subgroups.
• Improve efficiency of Fair PCA via convex optimization techniques.
• Achieve fairness without sacrificing performance, significantly faster than SDR-based methods.

Innovation

• Identifies hidden convexity in Fair PCA
• Introduces eigenvalue optimization for convex optimization
• Achieves fairness without performance loss