[Paper Review] Fair Canonical Correlation Analysis

Zhuoping Zhou, Davoud Ataee Tarzanagh+6 more|PubMed|Sep 27, 2023

Advanced Statistical Methods and ModelsMathematics21 references3 citations

TL;DR

This paper introduces Fair Canonical Correlation Analysis (F-CCA), a novel framework that mitigates bias in CCA by minimizing correlation disparity across protected groups (e.g., sex or race) while preserving global correlation. It proposes multi- and single-objective optimization on a generalized Stiefel manifold with convergence guarantees, demonstrating reduced fairness disparity without sacrificing CCA performance on synthetic and real-world datasets.

ABSTRACT

This paper investigates fairness and bias in Canonical Correlation Analysis (CCA), a widely used statistical technique for examining the relationship between two sets of variables. We present a framework that alleviates unfairness by minimizing the correlation disparity error associated with protected attributes. Our approach enables CCA to learn global projection matrices from all data points while ensuring that these matrices yield comparable correlation levels to group-specific projection matrices. Experimental evaluation on both synthetic and real-world datasets demonstrates the efficacy of our method in reducing correlation disparity error without compromising CCA accuracy.

Motivation & Objective

• To address fairness issues in Canonical Correlation Analysis (CCA) when protected attributes like sex or race influence correlation disparities across groups.
• To develop a method that learns global projection matrices ensuring comparable correlation levels across different demographic groups.
• To balance global correlation maximization with fairness constraints through principled optimization frameworks.
• To provide theoretical convergence guarantees for the proposed optimization algorithms on Riemannian manifolds.
• To empirically validate the method on synthetic and real-world datasets, showing reduced correlation disparity without sacrificing CCA accuracy.

Proposed method

• Proposes a multi-objective optimization framework that jointly maximizes global correlation and minimizes group-wise correlation disparity error.
• Introduces a single-objective formulation with a tunable hyperparameter to balance global correlation and fairness.
• Develops a Riemannian gradient descent algorithm on a generalized Stiefel manifold for multi-objective F-CCA, with convergence to a Pareto stationary point.
• Extends Riemannian optimization to multi-objective problems using flexible retraction maps beyond exponential retraction.
• Employs component-wise correlation measures and group-specific correlation disparities to quantify fairness in projection matrices.
• Uses retraction operations on the Stiefel manifold to maintain orthonormality constraints during optimization.

Experimental results

Research questions

• RQ1 Can CCA be made fair by minimizing correlation disparity across protected groups such as sex or race?
• RQ2 How can global projection matrices be learned to ensure comparable correlation levels across different demographic groups?
• RQ3 What optimization framework enables a balanced trade-off between global correlation and fairness in CCA?
• RQ4 Can convergence guarantees be established for Riemannian optimization in multi-objective fair CCA?
• RQ5 Does the proposed F-CCA method maintain CCA performance while reducing fairness disparities in real-world and synthetic data?

Key findings

• The proposed F-CCA framework significantly reduces correlation disparity error across protected groups while maintaining high global correlation, as demonstrated on both synthetic and real-world datasets.
• The multi-objective F-CCA approach achieves a balanced trade-off between global correlation and fairness, with convergence to a Pareto stationary point.
• The single-objective formulation with a tuning parameter effectively approximates fairness while preserving strong global correlation performance.
• Empirical results show that F-CCA outperforms standard CCA in fairness metrics, especially in scenarios with group-specific correlation biases, such as in Alzheimer’s disease studies involving sex differences.
• Theoretical analysis confirms that the matrix correlation measure ρ is the average of component-wise correlations ρr, and that the maximum group disparity Δmax is bounded by the sum of individual component disparities Δmax,r.
• The method successfully equalizes average correlation angles across groups, as illustrated in the comparison of projected sample pairs in Figure 1, reducing bias observed in standard CCA.