Title: Fair Canonical Correlation Analysis
Venue: NeurIPS 2023
Accepted: yes
Keywords: Fairness

Canonical Correlation Analysis

Riemannian Optimization

Pareto Optimization
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.

Fair Canonical Correlation Analysis

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.

1 Introduction

Canonical Correlation Analysis (CCA) is a multivariate statistical technique that explores the relationship between two sets of variables [30]. Given two datasets X ∈ R N × D x and Y ∈ R N × D y on the same set of N observations, 1 CCA seeks the R -dimensional subspaces where the projections of X and Y are maximally correlated, i.e. finds U ∈ R D x × R and V ∈ R D y × R such that

CCA finds applications in various fields, including biology [51], neuroscience [2], medicine [79], and engineering [14], for unsupervised or semi-supervised learning. It improves tasks like clustering, classification, and manifold learning by creating meaningful dimensionality-reduced representations [70]. However, CCA can exhibit unfair behavior when analyzing data with protected attributes, like sex or race. For instance, in Alzheimer's disease (AD) analysis, CCA can establish correlations between brain imaging and cognitive decline. Yet, if it does not consider the influence of sex, it may result in disparate correlations among different groups because AD affects males and females differently, particularly in cognitive decline [36, 81].

The influence of machine learning on individuals and society has sparked a growing interest in the topic of fairness [42]. While fairness techniques are well-studied in supervised learning [5, 18, 20], attention is shifting to equitable methods in unsupervised learning [11, 12, 15, 34, 49, 55, 64]. Despite extensive work on fairness in machine learning, fair CCA (F-CCA) remains unexplored. This paper investigates F-CCA and introduces new approaches to mitigate bias in (CCA).

For further discussion, we compare CCA with our proposed F-CCA in sample projection, as illustrated in Figure 1. In Figure 1(a), we have samples x 1 and x 2 from matrix X , and in Figure 1(b), their corresponding samples y 1 and y 2 are from matrix Y . CCA learns U and V to maximize correlation,

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inversely related to the angle between the sample vectors. Figure 1(c) demonstrates the proximity within the projected sample pairs ( U ⊤ x 1 , V ⊤ y 1 ) and ( U ⊤ x 2 , V ⊤ y 2 ) . In Figure 1(d)-(i), we compare the results of different learning strategies. There are five pairs of samples, with female pairs highlighted in red and male pairs shown in blue. Random projection (Figure 1(e)) leads to randomly large angles between corresponding sample vectors. CCA reduces angles compared to random projection (Figure 1(f)), but significant angle differences between male and female pairs indicate bias. Using sex-based projection matrices heavily biases the final projection, favoring one sex over the other (Figures 1(g) and 1(h)). To address this bias, our F-CCA maximizes correlation within pairs and ensures equal correlations across different groups, such as males and females (Figure 1(i)). Note that while this illustration represents individual fairness, the desired outcome in practice is achieving similar average angles for different groups.

Contributions. This paper makes the following key contributions:

We introduce fair CCA (F-CCA), a model that addresses fairness issues in (CCA) by considering multiple groups and minimizing the correlation disparity error of protected attributes. F-CCA aims to learn global projection matrices from all data points while ensuring that these projection matrices produce a similar amount of correlation as group-specific projection matrices.

We propose two optimization frameworks for F-CCA: multi-objective and single-objective. The multi-objective framework provides an automatic trade-off between global correlation and equality in group-specific correlation disparity errors. The single-objective framework offers a simple approach to approximate fairness in CCA while maintaining a strong global correlation, requiring a tuning parameter to balance these objectives.

We develop a gradient descent algorithm on the generalized Stiefel manifold to solve the multiobjective problem, with convergence guarantees to a Pareto stationary point. This approach extends Riemannian gradient descent [8, 9] to multi-objective optimization, accommodating a broader range of retraction maps than exponential retraction [23, 6]. Furthermore, we provide a similar algorithm for single-objective problems, also with convergence guarantees to a stationary point.

We provide extensive empirical results showcasing the efficacy of the proposed algorithms. Comparison against the CCA method on synthetic and real datasets highlights the benefits of the F-CCA approach, validating the theoretical findings 2 .

Organization: Section 2 covers related work. Our proposed approach is detailed in Section 3, along with its theoretical guarantees. Section 4 showcases numerical experiments, while Section 5 discusses implications and future research directions.

2 Related work

Canonical Correlation Analysis (CCA). CCA was first introduced by [28, 29]. Since then, it has been utilized to explore relations between variables in various fields of science, including economics [72], psychology [19, 27], geography [45], medicine [39], physics [76], chemistry [69], biology [62], time-series modeling [26], and signal processing [57]. Recently, CCA has demonstrated its applicability in modern fields of science such as neuroscience, machine learning, and bioinformatics [59, 60]. CCA has been used to explore relations for developing brain-computer interfaces [10, 46] and in the field of imaging genetics [22]. CCA has also been applied for feature selection [47], feature extraction and fusion [61], and dimension reduction [71]. Additionally, numerous studies have applied CCA in bioinformatics and computational biology, such as [54, 56, 58]. The broad range of application domains highlights the versatility of CCA in extracting relations between variables, making it a valuable tool in scientific research.

Fairness. Fairness in machine learning has been a growing area of research, with much of the work focusing on fair supervised methods [5, 16, 18, 20, 67, 78]. However, there has also been increasing attention on fair methods for unsupervised learning tasks [11, 12, 15, 34, 33, 49, 55, 64, 50, 66]. In particular, Samadi et al. [55] proposed a semi-definite programming approach to ensure fairness in PCA. Kleindessner et al. [33, 34] focused on fair PCA formulation for multiple groups and proposed a kernel-based fair PCA. Kamani et al. [32] introduced an efficient gradient method for fair PCA, addressing multi-objective optimization. In this paper, we propose a novel multi-objective framework for F-CCA, converting constrained F-CCA problems to unconstrained ones on a generalized Riemannian manifold. This framework enables the adaptation of efficient gradient techniques for numerical optimization on Riemannian manifolds.

Riemannian Optimization. Riemannian optimization extends Euclidean optimization to smooth manifolds, enabling the minimization of f ( x ) on a Riemannian manifold M and converting constrained problems into unconstrained ones [1, 8]. It finds applications in various domains such as matrix/tensor factorization [31, 63], PCA [21], and CCA [77]. Specifically, CCA can be formulated as Riemannian optimization on the Stiefel manifold [13, 43]. In our work, we utilize Riemannian optimization to develop a multi-objective framework for F-CCAs on generalized Stiefel manifolds.

3 Fair Canonical Correlation Analysis

This section introduces the formulation and optimization algorithms for F-CCA.

3.1 Preliminary

Real numbers are represented as R , with R + for nonnegative values and R ++ for positives. Vectors and matrices use bold lowercase and uppercase letters (e.g., a , A ) with elements a i and a ij . For x , y ∈ R m , x ≺ y and x ⪯ y mean y -x ∈ R m ++ and y -x ∈ R m + , respectively. For a symmetric matrix A ∈ R N × N , A ≻ 0 and A ⪰ 0 denote positive definiteness and positive semidefiniteness (PSD), respectively. I D , J D , and 0 D are D × D identity, all-ones, and all-zeros matrices. Λ i ( A ) stands for the i -th singular values of A . Matrix norms are defined as ∥ A ∥ 1 = ∑ ij | a ij | , ∥ A ∥ = max i Λ i ( A ) , and ∥ A ∥ F := ( ∑ ij | a ij | 2 ) 1 / 2 . We introduce some preliminaries on manifold optimization [1, 6, 8]. Given a PSD matrix B ∈ R D × D , the generalized Stiefel manifold is defined as

The tangent space of the manifold M = St ( D,R, B ) at Z ∈ M is given by

The tangent bundle of a smooth manifold M , which consists of T Z M at all Z ∈ M , is defined as

Definition 1. A retraction on a differentiable manifold M is a smooth mapping from its tangent bundle T M to M that satisfies the following conditions, with R z being the retraction of R to T Z M :

1. R z ( 0 ) = Z , for all Z ∈ M , where 0 denotes the zero element of T Z M .

In the numerical experiments, this work employs a generalized polar decomposition-based retraction. Given a PSD matrix B ∈ R D × D , for any ξ ∈ T Z M with M = St ( D,R, B ) , it is defined as:

where ¯ U Σ ¯ V ⊤ = ξ is the singular value decomposition of ξ , and Q , Λ are obtained from the eigenvalue decomposition QΛ Q ⊤ = ¯ U ⊤ B ¯ U . Further details on retraction choices are in Appendix A.1.

3.2 Correlation Disparity Error

As previously mentioned, applying CCA to the entire dataset could lead to a biased result, as some groups might dominate the analysis while others are overlooked. To avoid this, we can perform CCA separately on each group and compare the results. Indeed, we can compare the performance of CCA on each group's data with the performance of CCA on the whole dataset, which includes all groups' data. The goal is to find a balance between the benefits and sacrifices of different groups so that each group's contribution to the CCA analysis is treated fairly. In particular, suppose the datasets X ∈ R N × D x and Y ∈ R N × D y on the same set of N observations, belong to K different groups { ( X k , Y k ) } K k =1 with X k ∈ R N k × D x and Y k ∈ R N k × D y , based on demographics or some other semantically meaningful clustering. These groups need not be mutually exclusive; each group can be defined as a different weighting of the data.

To determine how each group is affected by F-CCA, we can compare the structure learned from each group's data ( X k , Y k ) with the structure learned from all groups' data combined ( X , Y ) . A fair CCA approach seeks to balance the benefits and drawbacks of each group's contribution to the analysis. Specifically, if we train global subspaces U ∈ R D x × R and V ∈ R D y × R on k -th group dataset ( X k , Y k ) , we can identify the group-specific (local) weights represented by ( U k , V k ) that has the best performance on that dataset. Thus, F-CCA algorithm should be able to learn global weights ( U , V ) on all data points while ensuring that each group's correlation on the CCA learned by the whole dataset is equivalent to the group-specific subspaces learned only by its own data.

To define these fairness criteria, we introduce correlation disparity error as follows:

Definition 2 ( Correlation Disparity Error ) . Consider a pair of datasets ( X , Y ) with K sensitive groups with data matrix { ( X k , Y k ) } K k =1 representing each sensitive group's data samples. Then, for any ( U , V ) , the correlation disparity error for each sensitive group k ∈ [ K ] is defined as

Here, ( U k,⋆ , V k,⋆ ) is the maximizer of the following group-specific CCA problem:

This measure shows how much correlation we are suffering for any global ( U , V ) , with respect to the loss of optimal local ( U k,⋆ , V k,⋆ ) that we can learn based on data points ( X k , Y k ) .

Using Definition 2, we can define F-CCA as follows:

Definition 3 ( Fair CCA ) . A CCA pair ( U ⋆ , V ⋆ ) is called fair if the correlation disparity error among K different groups is equal, i.e.,

̸

A CCA pair ( U ⋆ , V ⋆ ) that achieves the same disparity error for all groups is called a fair CCA.

Next, we introduce the concept of pairwise correlation disparity error for CCA, which measures the variation in correlation disparity among different groups.

Definition 4 ( Pairwise Correlation Disparity Error ) . The pairwise correlation disparity error for any global ( U , V ) and group-specific subspaces { ( U k,⋆ , V k,⋆ ) } K k =1 , is defined as

̸

Here, ϕ : R → R + is a penalty function such as ϕ ( x ) = exp( x ) , ϕ ( x ) = x 2 , or ϕ ( x ) = | x | .

The motivation for incorporating pairwise correlation disparity error in our approach can be attributed to the work by [40, 55] in the context of PCA. To facilitate convergence analysis, we will primarily consider smooth penalization functions, such as squared or exponential penalties.

3.3 A Multi-Objective Framework for Fair CCA

In this section, we introduce an optimization framework for balancing correlation and disparity errors. Let f 1 ( U , V ) := -trace ( U ⊤ X ⊤ YV ) , f 2 ( U , V ) := ∆ 1 , 2 ( U , V ) , . . . , f M ( U , V ) := ∆ K -1 ,K ( U , V ) . The optimization problem of finding an optimal Pareto point of F is denoted by

where U := { U ∈ R D x × R ∣ ∣ U ⊤ X ⊤ XU = I R } and V := { V ∈ R D y × R ∣ ∣ V ⊤ Y ⊤ YV = I R } .

A point ( U , V ) ∈ U × V satisfying Im ( ∇ F ( U , V )) ∩ ( -R M ++ ) = ∅ is called critical Pareto . Here, Im denotes the image of Jacobian of F . An optimum Pareto point of F is a point ( U ⋆ , V ⋆ ) ∈ U × V such that there exists no other ( U , V ) ∈ U × V with F ( U , V ) ≺ F ( U ⋆ , V ⋆ ) . Moreover, a point U ⋆ , V ⋆ ∈ U × V is a weak optimal Pareto of F if there is no ( U , V ) ∈ U × V with F ( U , V ) ⪯ F ( U ⋆ , V ⋆ ) . The multi-objective framework (9) addresses the challenge of handling conflicting objectives and achieving optimal trade-offs between them.

To effectively solve Problem (9), we propose utilizing a gradient descent method on the manifold U ×V that ensures convergence to a Pareto stationary point . The proposed gradient descent algorithm for solving (9) is provided in Algorithm 1 . For each ( U , V ) ∈ U × V , let P := ( P u , P v ) with P u ∈ T U U and P v ∈ T V V . The iterates ( P u t , P v t ) in Step 4 are obtained by solving the following subproblem in the joint tangent plane T U U × T V V :

If ( U t , V t ) / ∈ U × V is not a Pareto stationary point, Problem (10) has a unique nonzero solution P t (see Lemma 7), known as the steepest descent direction for F at ( U t , V t ) . In Steps 5 and 6, R u and R v denote the retractions onto the tangent spaces T U and T V

U V , respectively; refer to Definition 1. Assumption A. For a given subset S of the tangent bundle T U × T V , there exists a constant L F such that, for all ( Z , P ) ∈ S , we have F ( R z ( P )) ⪯ F ( Z ) + ∇ + ( L F / 2) ∥ P ∥ 2 F 1 M , where ∇ i := ⟨∇ f i ( Z ) , P ⟩ , ∇ := [ ∇ 1 , · · · , ∇ M ] ⊤ ∈ R M , and R z is the retraction.

The above assumption extends [8, A 4.3] to multi-objective optimization, and it always holds for the exponential map (exponential retraction) if the gradient of F is L F -Lipschitz continuous [23, 6].

Theorem 5. Suppose Assumption A holds. Let ( U t , V t ) be the sequence generated by MF-CCA . Let f ∗ i := inf { f i ( U , V ) : ( U , V ) ∈ U × V} , for all i ∈ [ M ] and define f i ∗ ( U 0 , V 0 ) -f ∗ i ∗ := min { f i ( U 0 , V 0 ) -f ∗ i : i ∈ [ M ] } . If η u t = η v t = η ≤ 1 /L F for all t ∈ { 0 , . . . , T -1 } , then

Proof Sketch. We employ Lemma 7 to establish the unique solution P t for subproblem (10). Lemmas 9 and 10 provide estimates for the decrease of function F along P t : For any η t ≥ 0 , we have F ( U t +1 , V t +1 ) ⪯ F ( U t , V t ) -( η t -L F η 2 t / 2) ∥ P t ∥ 2 F 1 M . Summing this inequality over t = 0 , 1 , . . . , T -1 and applying our step size condition yields the desired result.

Theorem 5 provides a generalization of [8, Corollary 4.9] to the multi-objective optimization, showing that the norm of Pareto descent directions converges to zero. Consequently, the solutions produced by the algorithm converge to a stationary fair subspace. It is worth mentioning that multi-objective optimization in [23, 6] relies on the Riemannian exponential map, whereas the above theorem covers broader (and practical) retraction maps.

3.4 A Single-Objective Framework for Fair CCA

In this section, we introduce a straightforward and effective single-objective framework. This approach simplifies F-CCA optimization, lowers computational requirements, and allows for fine-tuning fairness-accuracy trade-offs using the hyperparameter λ . Specifically, by employing a regularization parameter λ > 0 , our proposed fairness model for F-CCA is expressed as follows:

̸

where ∆( U , V ) = ∑ i,j ∈ [ K ] ,i = j ∆ i,j ( U , V ) ; see Definiton 4.

The choice of λ in the model determines the emphasis placed on different objectives. When λ is large, the model prioritizes fairness over minimizing subgroup errors. Conversely, if λ is small, the focus shifts towards minimizing subgroup correlation errors rather than achieving perfect fairness. In other words, it is possible to obtain perfectly F-CCA subspaces; however, this may come at the expense of larger errors within the subgroups. The constant λ in the model allows for a flexible trade-off between fairness and minimizing subgroup correlation errors, enabling us to find a balance based on the specific requirements and priorities of the problem at hand.

The proposed gradient descent algorithm for solving (11) is provided as Algorithm 2 . For each ( U , V ) ∈ U × V , let G := ( G u , G v ) with G u ∈ T U U and G v ∈ T V V . The iterates ( G u t , G v t ) are obtained by solving the following problem in the joint tangent plane T U U × T V V :

The solutions ( G u t , G v t ) are maintained on the manifolds using the retraction operations R u and R v . Assumption B. For a subset S ⊆ T U ×T V , there exists a constant L f such that for all ( Z , G ) ∈ S , f ( R z ( G )) ≤ F ( Z ) + ⟨∇ f ( Z ) , G ⟩ +( L f / 2) ∥ G ∥ 2 F , with R z as the retraction.

Theorem 6. Suppose Assumption B holds. Let ( U t , V t ) be the sequence generated by SF-CCA . Let f ∗ := inf { f ( U , V ) : ( U , V ) ∈ U × V} . If η u t = η v t = η ≤ 1 /L f for all t ∈ [ T ] , then

Comparison between MF-CCA and SF-CCA: MF-CCA addresses conflicting objectives and achieves optimal trade-offs automatically, but it necessitates the inclusion of ( K 2 ) additional objectives. SF-CCA, on the other hand, provides a simpler approach but requires tuning an extra hyperparameter λ . When choosing between the two methods, it is crucial to consider the trade-off between complexity and simplicity, as well as the number of objectives and the need for hyperparameter tuning.

4 Experiments

In this section, we provide empirical results showcasing the efficacy of the proposed algorithms.

4.1 Evaluation Criteria and Selection of Tuning Parameter

F-CCA's performance is evaluated on correlation and fairness for each dimension of subspaces. Let U = [ u 1 , · · · , u R ] ∈ R D x × R and V = [ v 1 , · · · , v R ] ∈ R D y × R . The r -th canonical correlation is defined as follows:

Next, in terms of fairness, we establish the following two key measures:

Here, ∆ max ,r measures maximum disparity error, while ∆ sum ,r represents aggregate disparity error. The aim is to reach ∆ max ,r and ∆ sum ,r of 0 without sacrificing correlation ( ρ r ) compared to CCA. We conduct a detailed analysis using component-wise measurements (13) instead of matrix versions; for more discussions, see Appendix C.2.

The canoncorr function from MATLAB and [35] is used to solve (CCA). For MF-CCA and SFCCA, the learning rate is searched on a grid in { 1 e -1 , 5 e -2 , 1 e -2 , . . . , 1 e -5 } , and for SF-CCA, λ is searched on a grid in { 1 e -2 , 1 e -1 , 0 . 5 , 1 , 2 , . . . , 10 } . Sensitivity analysis of λ is provided in Appendix B.2. The learning rate decreases with the square root of the iteration number. Termination of algorithms occurs when the descent direction norm is below 1 e -4 .

4.2 Dataset

4.2.1 Synthetic Data

Following [4, 44], our synthetic data are generated using the Gaussian distribution

Here, µ X ∈ R D x × 1 and µ Y ∈ R D y × 1 are the means of data matrices X and Y , respectively; covariance matrices Σ X , Σ Y and the cross-covariance matrix Σ XY are constructed as follows. Given ground truth projection matrices U ∈ R D x × R , V ∈ R D y × R and canonical correlations ρ = ( ρ 1 , ρ 2 , . . . , ρ R ) defined in (13a). Let U = Q X R X and V = Q Y R Y be the QR decomposition of U and V , then we have

Here, T X ∈ R D x × D x and T Y ∈ R D y × D y are randomly generated by normal distributions, and τ x = 1 and τ y = 0 . 001 are scaling hyperparameters. For subgroup distinction, we added noise to canonical vectors and adjusted sample sizes: 300, 350, 400, 450, and 500 observations each. In the numerical experiment, different canonical correlations are assigned to each subgroup alongside two global canonical vectors U and V to generate five distinct subgroups.

4.2.2 Real Data

National Health and Nutrition Examination Survey (NHANES). We utilized the 2005-2006 subset of the NHANES database https://www.cdc.gov/nchs/nhanes , including physical measurements and self-reported questionnaires from participants. We partitioned the data into two distinct subsets: one with 96 phenotypic measures and the other with 55 environmental measures. Our objective was to apply F-CCA to explore the interplay between phenotypic and environmental factors in contributing to health outcomes, considering the impact of education. Thus, we segmented the dataset into three subgroups based on educational attainment (i.e., lower than high school, high school, higher than high school), with 2,495, 2,203, and 4,145 observations in each subgroup.

Mental Health and Academic Performance Survey (MHAAPS). This dataset is available at https://github.com/marks/convert_to_csv/tree/master/sample_data . It consists of three psychological variables, four academic variables, as well as sex information for a cohort of 600 college freshmen (327 females and 273 males). The primary objective of this investigation revolves around examining the interrelationship between the psychological variables and academic indicators, with careful consideration given to the potential influence exerted by sex.

Alzheimer's Disease Neuroimaging Initiative (ADNI). We utilized AV45 (amyloid) and AV1451 (tau) positron emission tomography (PET) data from the ADNI database ( http://adni.loni.usc. edu ) [73, 74]. ADNI data are analyzed for fairness in medical imaging classification [41, 53, 81], and sex disparities in ADNI's CCA study can harm generalizability, validity, and intervention tailoring. We utilized F-CCA to account for sex differences. Our experiment links 52 AV45 and 52 AV1451 features in 496 subjects (255 females, 241 males).

4.3 Results and Discussion

In the simulation experiment, we follow the methodology described in Section 4.2.1 to generate two sets of variables, each containing two subgroups of equal size. Canonical weights are trained and used to project the two sets of variables into a 2-dimensional space using CCA, SF-CCA, and MF-CCA. From Figure 2, it is clear that the angle between the distributions of the two subgroups, as projected by SF-CCA and MF-CCA, is smaller in comparison. This result indicates that F-CCA has the ability to reduce the disparity between distinct subgroups.

Table 1 shows the quantitative performance of the three models: CCA, MF-CCA, and SF-CCA. They are evaluated based on ρ r , ∆ max ,r , and ∆ sum ,r defined in (13) across five experimental sets. Table 2 displays the mean runtime of each model. Several key observations emerge from the analysis. Firstly, MF-CCA and SF-CCA demonstrate substantial improvements in fairness compared to CCA. However, it is important to note that F-CCA, employed in both MF-CCA and SF-CCA, compromises some degree of correlation due to its focus on fairness considerations during computations. Secondly,

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SF-CCA outperforms MF-CCA in terms of fairness improvement, although it sacrifices correlation. This highlights the effectiveness of the single-objective optimization approach in SF-CCA. Moreover, the datasets consist of varying subgroup quantities (5, 3, 2, and 2) and an imbalanced number of samples in distinct subgroups. F-CCA consistently performs well across these datasets, confirming its inherent scalability. Lastly, although SF-CCA requires more effort to tune hyperparameters, SF-CCA still exhibits a notable advantage in terms of time complexity compared to MF-CCA, demonstrating computational efficiency. Disparities among various CCA methods are visually represented in Figure 3. Notably, the conventional CCA consistently demonstrates the highest disparity error. Conversely, SF-CCA and MF-CCA consistently outperform CCA across all datasets, underscoring their efficacy in promoting fairness within analytical frameworks.

In Table 1, we define the percentage change of correlation ( ρ r ), maximum disparity gap ( ∆ max ,r ), and aggregate disparity ( ∆ sum ,r ), respectively, as follows: Pρ r := ( ρ r of F-CCA -ρ r of CCA ) / ( ρ r of CCA ) × 100 , P ∆ max ,r := -(∆ max ,r of F-CCA -∆ max ,r of CCA ) / (∆ max ,r of CCA ) × 100 , and P ∆ sum ,r := -(∆ sum ,r of F-CCA -∆ sum ,r of CCA ) / (∆ sum ,r of CCA ) × 100 . Here, F-CCA is replaced with either MF-CCA or

(a) Synthetic Data

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SF-CCA to obtain the percentage change for MF-CCA or SF-CCA. Figure 4 illustrates the percentage changes of each dataset. Pρ r is slight, while P ∆ max ,r and P ∆ sum ,r changes are substantial, signifying fairness improvement without significant accuracy sacrifice.

5 Conclusion, Limitations, and Future Directions

We propose F-CCA, a novel framework to mitigate unfairness in CCA. F-CCA aims to rectify the bias of CCA by learning global projection matrices from the entire dataset, concurrently guaranteeing that these matrices generate correlation levels akin to group-specific projection matrices. Experiments show that F-CCA is effective in reducing correlation disparity error without sacrificing much correlation. We discuss potential extensions and future problems stemming from our work.

While F-CCA effectively reduces unfairness while maintaining CCA model accuracy, its potential to achieve a minimum achievable disparity correlation remains unexplored. A theoretical exploration of this aspect could provide valuable insights.

F-CCA holds promise for extensions to diverse domains, including multiple modalities [80], deep CCA [3], tensor CCA [44], and sparse CCA [25]. However, these extensions necessitate novel formulations and in-depth analysis.

Our approach of multi-objective optimization on smooth manifolds may find relevance in other problems, such as fair PCA [55]. Further, bilevel optimization approaches [37, 68, 65] can be designed on a smooth manifold to learn a single Pareto-efficient solution and provide an automatic trade-off between accuracy and fairness.

With applications encompassing clustering, classification, and manifold learning, F-CCA ensures fairness when employing CCA techniques for these downstream tasks. It can also be jointly analyzed with fair clustering [15, 66, 34] and fair classification [78, 18].

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Roadmap. The appendix is organized as follows: In Section A, we present the subproblem solvers for MF-CCA and SF-CCA, along with the proofs for Theorems 5 and 6. In Section B, we provide comprehensive insights into the three real datasets utilized and present supplementary experimental results. Furthermore, Section C outlines the experimental specifics and the procedure for selecting hyperparameters.

Contents

A Addendum to Section 3

A.1 Preliminaries on Retractions

In reference to Definition 1, the following options together with generalized polar decomposition defined in (4) represent commonly employed retractions[1, 6, 7, 21, 23, 38, 75] of a matrix ξ ∈ R D × R :

Exponential mapping. It takes 8 DR 2 + O ( R 3 ) flops and has the closed-form expression:

where QR = -( I r -ZZ ⊤ ) ξ is the unique QR factorization.

Polar decomposition. It takes 3 DR 2 + O ( R 3 ) flops and has the closed-form expression:

QR decomposition. It takes 2 DR 2 + O ( R 3 ) flops and has the closed-form expression:

where qr ( A ) is the Q factor of the QR factorization of A .

Cayley transformation. It takes 7 DR 2 + O ( R 3 ) flops and has the closed-form expression:

where W ( ξ ) = ( I R -ZZ ⊤ / 2) ξ Z ⊤ -Z ξ ⊤ ( I R -ZZ ⊤ / 2) .

This work specifically focuses on generalized polar decomposition defined in (4) for retraction.

A.2 Subproblem Solver for MF-CCA

To solve the optimization problem (9), we introduce a method for finding the descent direction on the joint manifold, as described in Equation (10). In this section, we will provide a detailed explanation of the relevant calculations. We expand the optimization problem into two sub-optimization problems, one for updating the canonical weight U and the other for updating the canonical weight V . Here, we focus on outlining the process of updating U , which is described below.

In the t -th iteration, the gradient of the multi-objective function (9) with respect to U t is computed as follows:

̸

for all M ≥ j ≥ 2 , k, s ∈ [ K ] , k = s , where

Similarly, we can compute the gradient of f w.r.t V t .

Lemma 7, inspired by [23, 6], states that the direction P t can be expressed as a linear combination of gradients {∇ f 1 ( U t , V t ) , ∇ f 2 ( U t , V t ) , . . . , ∇ f M ( U t , V t ) } .

Lemma 7. The unconstrained optimization problem in (10) has a unique solution. Moreover, P t is the solution of the problem in (10) if only if there exist µ i ≥ 0 , i ∈ [ M ] , such that

Proof. Recall that for each ( U , V ) ∈ U × V , we have P = ( P u , P v ) with P u ∈ T U U and P v ∈ T V V . It can be easily seen that the first term of Q ( P ) (defined in (10)) is a summation of the maximum of linear functions in the linear space T U U × T V V . Hence, it is convex, which implies that Q ( P ) is strongly convex. Thus, the solution is unique.

From the convexity of the function, it is well known that P t is the solution of (10) if only if

Therefore, the second statement follows from the formula for the subdifferential of the maximum of convex functions.

The (sub)-gradient formula (15) can be substituted into Equation (10), and subsequently, the fminimax function from the MATLAB optimization toolbox can be applied to solve for the descent direction P t . We found that the Goal Attainment Method [24] gives a more stable and accurate solution for our subproblem 3 . Given P u t , U t can be updated by:

where η t is the adaptive stepsize and R u is the retraction operator onto U = { U ∈ R D x × R | U ⊤ X ⊤ XU = I R } .

Various retraction options are discussed in Sections 3.1 and A.1. Specifically, the generalized polar decomposition method defined in (4) is employed in our numerical experiments described in Section 4. The symmetric nature of the update process for canonical weights U and V in CCA allows us to employ identical procedures for updating V . By substituting X with Y and U with V in the aforementioned steps, we obtain the process for updating V . In each iteration, U and V are updated once in sequence.

A.3 Subproblem Solver for SF-CCA

To solve the optimization problem (11), the method of finding the descent direction is introduced in Equation (12). Detailed calculations will be explained below. We expand it into two sub-optimization problems to update the canonical weight U and the canonical weight V , respectively. First, to update the canonical weight U t in iteration t , we can take gradient of f ( U t , V t ) with respect U t as follows:

̸

where

or equivalently,

Given the subgradients (18) and the single-objective subproblem (12), the direction G t = ( G u t , G v t ) can be effectively computed using subgradient-based methods such as the fmincon function from the MATLAB optimization toolbox. Given G u t , the update for U t can be obtained as follows:

Here, η t is the adaptive stepsize and R u is the retraction operator onto U = { U ∈ R D x × R | U ⊤ X ⊤ XU = I R } .

In our numerical experiments, we utilize polar decomposition as the retraction operator; see (4). Similarly, the canonical weight V can be updated using the same procedure.

A.4 Auxiliary Lemmas

Lemma 8. Suppose Assumption A holds. For any ( U t , V t ) ∈ U × V , let P t be the solution of (10) . Then, for any η t ≥ 0 , we have the following:

where ∇ it := ⟨∇ f i ( U t , V t ) , P t ⟩ and ∇ t := [ ∇ 1 t , · · · , ∇ Mt ] ⊤ ∈ R M .

Proof. The proof is a straightforward consequence of Assumption A.

Lemma 9. For any ( U t , V t ) ∈ U × V , let P t be the solution of (10) . Then,

Hence, trace ( P ⊤ t ∇ f i ( U t , V t ) ) ⪯ -∥ P t ∥ 2 F . Further, P t is critical Pareto point of F if, and only if, ∥ P t ∥ F = 0 .

Proof. It follows from (10) that

Hence, by the second equality in (16), it is easy to verify that (20) holds. The second statement follows by using the definitions of trace ( P ⊤ t ∇ F (( U t , V t )) . We proceed with the proof of the third statement of the lemma. Assuming that P t is a critical Pareto, it follows from the definition that there exists i ∈ [ M ] such that trace( P ⊤ t ∇ f i ( U t , V t ) ≥ 0 . Then, by the first part of the lemma, we have ∥ P t ∥ = 0 . The converse (only if) follows from the application of [6, Lemma 4.2].

The following result provides an estimate for the decrease of a function F , along P . This result is crucial in determining the iteration-complexity bounds for the gradient method on a general Riemannian manifold.

Lemma10 (Descent Property of MF-CCA) . Suppose Assumption A holds. For any ( U t , V t ) ∈ U×V , let P t be the solution of (10) . Then, for any η t ≥ 0 , we have

Proof. The proof is a straight combination of Lemmas 8 and 9.

A.5 Proof of Theorem 5

Proof. It follows from Lemma 10 that

for all t = 0 , 1 , . . . , T -1 .

Setting η t = η , and summing both sides of this inequality for t = 0 , 1 , . . . , T -1 , we get

Thus, by the definition of i ∗ , we obtain

which together with η ≤ 1 L F implies

Remark 11. As a consequence, if we consider a tolerance level of ϵ , we can bound the number of iterations required by the gradient method to obtain P T such that ∥ P T ∥ F ≤ ϵ . This bound can be expressed as O ( ( f i ∗ ( U 0 , V 0 ) -f ∗ i ∗ ) /ϵ 2 ) , where f i ∗ ( U 0 , V 0 ) -f ∗ i ∗ = min { f i ( U 0 , V 0 ) -f ∗ i : i ∈ [ M ] } . In other words, the number of iterations needed by the gradient method to achieve a solution ( U T , V T ) with ∥ P T ∥ F ≤ ϵ is proportional to the difference between the initial objective function value and the optimal objective function value, divided by the square of the tolerance level ϵ . The "big O" notation, O , indicates that the bound is an upper bound, providing an estimate of the worst-case behavior of the algorithm. This result showcases the convergence rate of the gradient method and provides a useful guideline for choosing the tolerance level and estimating the computational resources required to achieve the desired accuracy.

A.6 Proof of Theorem 6

Proof. This proof is a slightly modified version of the proof of Theorem 5. Using Assumption B and a single-objetive variant of Lemma 10, we obtain

Summing both sides of this inequality for t = 0 , 1 , . . . , T -1 , we get

Hence, using the fixed stepsize η t = η ≤ 1 L f , we get

Remark 12. As a result, given a tolerance ϵ , the number of iterations required by the gradient method to obtain ( U T , V T ) such that ∥ G T ∥ F ≤ ϵ is bounded by O ( ( f ( U 0 , V 0 ) -f ∗ ) /ϵ 2 ) .

B Addendum to Section 4

B.1 Detailed Description of Datasets

Table 3 provides a comprehensive overview of the relevant details associated with each real dataset. These details include the number of samples, the number of features, sensitive attributes, and group distribution. Notably, within the group distribution, it is important to acknowledge that in certain datasets, the total number of samples within different sensitive attributes may not be equivalent. This discrepancy arises due to the presence of missing information, as well as the insufficient sample sizes in certain subgroups. Figure 5 visualizes the distribution of all the groups in each dataset. We can see that the groups are imbalanced in synthetic data, NHANES (Education) and NHANES (Race), while balanced in MHAAPS (Sex), ADNI AV45/AV1451 (Sex), and ADNI AV45/Cognition (Sex). Imbalanced groups can make the problem more complicated, which is a good challenge to validate the effectiveness of our methods.

In the following, we provide comprehensive descriptions of all the datasets we used.

B.1.1 National Health and Nutrition Examination Survey (NHANES)

The NHANES dataset is extensively used for CCA analysis [48] and fairness studies [17, 52]. The dataset's diverse attributes give rise to fairness considerations, underscoring the importance of conducting equitable CCA experiments to tackle discrepancies in sample representation stemming from these inequalities.

In our study, we narrow our focus to the 2005-2006 subset of the NHANES database available at https://www.cdc.gov/nchs/nhanes . This subset comprises physical measurements and selfreported questionnaires from participants. To address missing data concerns, we employ the Multiple Imputation by Chained Equations (MICE) Forest methodology. Afterward, we divide the data into two distinct subsets: the "Phenotypic-Measure" dataset and the "Environmental-Measure" dataset, based on the inherent nature of the features. This stratified approach enables us to leverage F-CCA to gain a nuanced understanding of how phenotypic and environmental factors interact in influencing health outcomes while also accounting for the potential influence of education and race.

In our numerical experiments, we utilize education and race as sensitive attributes to partition the two datasets. In the initial experiment, we split the dataset into three subgroups based on participants' educational attainment. These subgroups comprise 2495, 2203, and 4145 observations, respectively. For the subsequent experiment, we work with a total of 8136 observations, dividing them into three

(a) Synthetic Data

(b) NHANES (Education)

(c) NHANES (Race)

(d) MHAAPS (Sex)

(e) ADNI AV45/AV1451 (Sex)

(f) ADNI AV45/Cognition (Sex)

Number of Samples Included

fSamplesIncluded

4000

400-

3000

2000

200-

1000

0

Group1

Group2

Group3

Group4

Group5

<High School

=High School

>High School

GroupName

GroupName

f Samples Included

Number of Samples Included

4000

300

0000

200

2000

100

1000

0

White

Black

Mexican

Male

Female

Group Name

Group Name

Number of Samples Included

pansu sales  aqn m

400

200

300

200

100

100-

Male

Female

Male

Female

Group Name

Group Name

subgroups based on racial categories. Specifically, there are 4576 white subjects, 1793 black subjects, and 1767 Mexican subjects. The "Phenotypic-Measure" dataset encompasses 96 distinct features, while the "Environmental-Measure" dataset includes 55 features.

B.1.2 Mental Health and Academic Performance Survey (MHAAPS)

The dataset employed in this study is obtained from the online repository available at https: //github.com/marks/convert_to_csv/tree/master/sample_data . This particular dataset includes three distinct psychological variables and four academic variables in the form of standardized test scores, as well as gender information for a cohort of 600 individuals classified as college freshmen. The primary objective of this investigation revolves around examining the interrelationship between the aforementioned psychological variables and academic indicators, with careful consideration given to the potential influence exerted by gender. Specifically, the dataset consists of 327 female samples and 273 male samples.

B.1.3 Alzheimer's Disease Neuroimaging Initiative (ADNI)

We utilize AV45 (amyloid), AV1451 (tau) positron emission tomography (PET), and Cognitive Score data from the Alzheimer's Disease Neuroimaging Initiative (ADNI) database ( http://adni.loni. usc.edu ) as three of our experiment datasets. Over- or under-representation of age, sex, or biometric groups might affect ADNI data fairness. Disease distribution among sensitivity features, such as higher AD occurrence in women, can also impact fairness [41]. The unfairness in sex representation within the CCA study of the ADNI dataset arises from an imbalance in the number of male and female participants. This disparity undermines the generalizability and validity of the study findings, as it fails to account for potential sex-related differences in Alzheimer's disease progression and response to treatments. The skewed representation limits the ability to draw accurate conclusions

CCA

MF-CCA

: Disparity

SF-CCA

Aggregate [

2

0

2

3

4

5

6

7

8

Number of Subgroups (K)

and develop tailored interventions for both sexes, perpetuating sex bias in research and healthcare. Consequently, it becomes imperative to utilize F-CCA as a means to acknowledge and integrate sex disparities within research initiatives.

In the conducted numerical study, two distinct experiments are carried out utilizing the ADNI dataset, where the sensitive attribute considered is the sex of the participants. The first experiment aims to explore the correlation between AV45 and AV1451, whereas the second experiment focuses on investigating the correlation between A V45 and Cognitive Scores. In the first experiment, both A V45 and AV1451 include 52 regional features and comprise a total of 496 observations. Among these observations, 255 belong to the female subgroup, and 241 belong to the male subgroup. In the analysis involving the correlation between AV45 and Cognitive Scores, a subset of 785 common samples is extracted, consisting of 431 male samples and 354 female samples. This analysis employs 68 regional features associated with A V45 and 46 cognitive score features.

B.2 Trade-off Analysis of Correlation and Disparity

B.2.1 Sensitivity of Correlation and Disparity Error to K

We conducted an analysis to examine the sensitivity of fairness in relation to the number of subgroups K . To investigate this, we utilized synthetic data consisting of varying numbers of subgroups and compared the resulting aggregate disparity on the first projection dimension.

Figure 6 provides a visual representation of our findings. The results clearly demonstrate that as the number of subgroups increases, there is a substantial amplification in the aggregate disparity when employing CCA. However, both MF-CCA and SF-CCA effectively address this issue and mitigate the observed phenomenon. Specifically, they successfully reduce the level of disparity even as the number of subgroups grows.

Moreover, it is worth noting that the aggregate discrepancy of SF-CCA exhibits a certain degree of sensitivity beyond a threshold of six subgroups. Beyond this point, the effectiveness of SF-CCA in reducing disparity begins to diminish. Conversely, MF-CCA remains consistently unaffected by variations in the number of subgroups, consistently maintaining its fairness performance. This analysis underscores the robustness and scalability of MF-CCA in handling an increasing number of subgroups, as it consistently maintains fairness irrespective of this parameter. On the other hand, SF-CCA demonstrates effective fairness mitigation but exhibits limitations when faced with a large number of subgroups beyond a certain threshold.

B.2.2 Sensitivity of Correlation and Disparity Error to r

Table 4 presents the numerical results of three measures described in Section 4.1, namely correlation ( ρ r ), maximum disparity ( ∆ max ,r ), and aggregate disparity ( ∆ sum ,r ). The results are with respect to

(a) Correlation of CCA

(b) Correlation of MF-CCA

(c) Correlation of SF-CCA

1.0

1.0

1.0

Overall

Overall

Overall

ation

0.9

Group 1

ation

0.9

Group 1

Correlation

0.9

Group 1

Correlat

0.8

Group 2

Correlat

0.8

Group2

0.8

Group 2

Group3

Group3

Group3

0.7

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on NHANES (Education)

(b) Correlation of MF-CCA

on NHANES (Education)

(c) Correlation of SF-CCA

on NHANES (Education)

(d) Correlation of CCA

on NHANES (Race)

(e) Correlation of MF-CCA

on NHANES (Race)

(f) Correlation of SF-CCA

on NHANES (Race)

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the first r projection dimensions. It is an extension of Table 1. Specifically, we present the results of the first seven projection dimensions for synthetic data; the first five projection dimensions for NHANES and ADNI; and the first two projection dimensions for MHAAPS, respectively. Overall, we have consistent results and conclusions in Table 1. Firstly, MF-CCA and SF-CCA show substantial improvements in fairness compared to CCA with mild compromises of correlation. Secondly, SFCCA outperforms MF-CCA in terms of fairness improvement, although it sacrifices correlation. This highlights the effectiveness of the single-objective optimization approach in SF-CCA. F-CCA consistently performs well across these datasets, confirming its inherent scalability.

(a) Correlation of CCA on MHAAPS (Sex)

(b) Correlation of MF-CCA on MHAAPS (Sex)

(c) Correlation of SF-CCA on MHAAPS (Sex)

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Table 5 quantifies the improvement in fairness and the minor compromises in accuracy presented in Table 4 by depicting the percentage performance shift from baseline CCA to MF-CCA and SF-CCA. The table is a reference and an extension of Figure 4 in Section 4.1. Figure 4 only visualizes the percentage shifts of the second and fifth projection dimension for Synthetic Data, NHANES, and ADNI and the first two projection dimensions for MHAAPS regarding the results presented in Table 1 for coherence. In Table 5, the metrics are formulated as: Pρ r = ρ r of F-CCA -ρ r of CCA ρ r of CCA × 100 , P ∆ max ,r = -∆ max ,r of F-CCA -∆ max ,r of CCA ∆ max ,r of CCA × 100 , P ∆ sum ,r = -∆ sum ,r of F-CCA -∆ sum ,r of CCA ∆ sum ,r of CCA × 100 , where F-CCA is switched to MF-CCA and SF-CCA to obtain the corresponding shift results. The formulations are designed regarding the properties of the metrics where ρ r is the larger the better while ∆ max ,r and ∆ sum ,r are the smaller the better. According to the table, compared to CCA, MFCCA, and SF-CCA sacrifice slightly the performance in correlation ( ρ r ) in exchange for significant fairness improvements in terms of maximum disparity ( ∆ max ,r ) and aggregate disparity ( ∆ sum ,r ) across all datasets in general, which demonstrate the good performance of F-CCA.

We also provide visualizations demonstrating how the correlation changes as the dimensionality changes. Figures 7, 8, 9, and Figure 10 display these correlation curves. Our expectation is that the correlation curves of our methods will show a stronger tendency to concentrate around the overall correlation curve compared to the baseline method. The findings indicate that the utilization of both MF-CCA and SF-CCA approaches yields a convergence of subgroup performance towards the overall canonical correlation. Furthermore, the overall canonical correlations derived from these two models exhibit a comparable level of performance to that of the CCA model.

Upon examining the synthetic data in Figure 7, we observe a clear concentration tendency. However, in the case of real data, this tendency is less pronounced. This is due to the fact that the disparities between different groups in the real data are already quite small. As a result, all the dashed curves in the CCA are already close to the solid red curve.

B.2.3 Sensitivity of Correlation and Disparity of SF-CCA to λ

The SF-CCA experiment incorporates the hyperparameter λ to strike a balance between correlation and fairness considerations. This hyperparameter allows us to modulate the emphasis placed on fairness within the SF-CCA model. Figure 11 provides a visual representation of the effects of different λ values on both correlation and fairness metrics.

The illustration clearly demonstrates that increasing the magnitude of λ enhances the emphasis on fairness within the SF-CCA model. Consequently, this leads to a reduction in aggregate disparity, indicating improved fairness. Conversely, lower values of λ prioritize correlation, emphasizing the preservation of the underlying relationship between the datasets.

This experimental finding aligns with the discussion presented in Section 3 of our work, where we discussed the trade-off between correlation and fairness. By adjusting the value of λ , we can effectively control the balance between these two objectives in the SF-CCA model.

(a) Correlation of CCA on ADNI AV45/AV1451 (sex)

(b) Correlation of MF-CCA on ADNI AV45/AV1451 (sex)

(d) Correlation of CCA on ADNI AV45/Cognition (Sex)

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B.3 Runtime Sensitivity of MF-CCA and SF-CCA

When it comes to the issue of running time, it is important to consider the trade-off between computational efficiency and hyperparameter tuning effort. While SF-CCA may require more effort in tuning its hyperparameters, it still demonstrates a notable advantage in terms of time complexity compared to MF-CCA. To conduct a sensitivity analysis concerning the sample count ( N ), the feature count ( d ), and the subgroup count( K ), we conduct a series of experiments on synthetic datasets, ensuring the hyperparameters remained constant. These experiments are performed in three distinct settings, each replicated ten times, employing techniques including CCA, MF-CCA, and SF-CCA. The specifics of outcomes are depicted in Figures 12, 13, and 14.

B.3.1 Runtime Sensitivity of MF-CCA and SF-CCA to d

In the first experimental setup, with the sample size held constant at N = 2000 and the number of groups held constant at K = 5 , the dimensionality of features is varied across the set [50, 100, 150, 200, 250, 300, 350, 400]. According to Figure 12, it is discerned that as feature dimensionality increases, the runtime of MF-CCA exhibits a concomitant augmentation. Conversely, the runtime associated with SF-CCA remains comparatively stable throughout the varying feature sizes.

B.3.2 Runtime Sensitivity of MF-CCA and SF-CCA to N

In the subsequent experimental configuration, the feature size is held constant at d = 100 , while the sample size varies across the set [600, 800, 1000, 1200, 1400, 1600, 1800, 2000]. Notably, according to Figure 13, the increment in the sample size exerts a minimal impact on the runtime duration of SF-CCA. This observation can be rationalized by considering that an increase in the number of features would lead to a covariance matrix of greater dimensionality. Consequently, the

corresponding numerical computations, such as the resolution of eigenvalues, become increasingly time-intensive.

B.3.3 Runtime Sensitivity of MF-CCA and SF-CCA to K

Finally, we examined the sensitivity of both SF-CCA and MF-CCA to the number of subgroups. Synthetic data was generated with varying numbers of subgroups, allowing us to assess the corresponding running time. The results are presented in Figure 14. From the figure, it is evident that conventional CCA is not significantly sensitive to the number of groups. However, MF-CCA exhibits a higher level of sensitivity to the number of groups compared to SF-CCA.

C Experimental Details and Hyperparameter Selection Procedure

In this section, we delve into the particulars of our experiments, which encompass the choice of the penalty function and hyperparameters. We initiate our discussion with the penalty function ϕ outlined in Equation (8). This function takes on various forms, such as absolute, square, and exponential functions. For our experimental purposes, we concentrate specifically on the absolute function.

The distinctive characteristic of the absolute function lies in its robustness against minor discrepancies, as opposed to the square function, which tends to converge rapidly towards zero. It's worth noting

that due to the non-differentiability of the absolute function at the origin, its identification within the MATLAB environment requires the utilization of the sign function. When the input to the penalty function reaches zero, the sign function indicates that the discrepancy error has already reached zero, leading to the termination of the training process.

C.1 Hyperparameters Selection

When it comes to selecting hyperparameters, the key decisions revolve around providing the appropriate learning rate and the parameter λ within the SF-CCA framework. Our approach consists of a two-step process. Firstly, we ascertain the optimal λ for each dataset through an extensive sensitivity analysis, as detailed in Section B.2.3. With these derived values in hand, we then conduct a comprehensive grid search to determine the best-fitting learning rate. The unique parameter combinations utilized for each experimental set are elucidated as follows:

Synthetic Data : λ = 10 , learning rate of 2e-2 in SF-CCA, and 4e-1 in MF-CCA.

NHANES (Education) : λ = 5 , learning rate of 2e-3 in SF-CCA, and 5e-2 in MF-CCA.

(a) Synthetic Data

(b) NHANES (Education)

(c) NHANES (Race)

(d) MHAAPS (sex)

(e) ADNI AV45/AV1451 (sex)

(f) ADNI AV45/Cognition (sex))

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Aggregate Disparity (x10-2)

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Aggregate Disparity(x10-2)

NHANES (Race) : λ = 5 , learning rate of 1e-3 in SF-CCA, and 2e-2 in MF-CCA.

MHAAPS : λ = 10 , learning rate of 2e-2 in SF-CCA, and 4e-1 in MF-CCA.

ADNI (AV45 and AV1451) : λ = 5 , learning rate of 1e-3 in SF-CCA, and 1e-2 in MF-CCA.

ADNI (AV45 and Cognition) : λ = 5 , learning rate of 1e-3 in SF-CCA, and 2e-2 in MF-CCA.

C.2 Fairness and Correlation Measures

The correlation and fairness between X and Y under projections of U and V within R -dimensional spaces can be quantitatively measured by

In Section 4.1, instead of employing matrix-based measurements (21), we adopted component-wise measurements (13) to facilitate detailed observations on each individual projection dimension r . In the following discussion, we demonstrate that the component-wise measurements, across all r dimensions, provide a more aggressive measurement approach compared to the matrix variants (21). In other words, it is evident that having large component-wise correlation values ρ r and small

700

CCA

600

MF-CCA

SF-CCA

500

e

400

itim

n

Ru

300

200

100

0

50

100

150

200

250

300

350

400

Number of Features (d)

CCA

140

....

MF-CCA

120

SF-CCA

100

e

80

60

R

40

20

0

600

800

1000

1200

1400

1600

1800

2000

Number of Samples (n)

disparity errors ∆ max ,r and ∆ sum ,r for all r ∈ [ R ] can guarantee large values of ρ and small values of ∆ max and ∆ sum.

The following lemma rigorously supports this observation.

Lemma 13. Let ρ r , ∆ max ,r , ∆ sum ,r be defined as in (13) , and let ρ, ∆ max , ∆ sum be defined as in (21) . Then,

600

CCA

MF-CCA

SF-CCA

1

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400

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R

200

2

3

4

5

6

7

8

Number of Subgroups (K)

Note that U = { U ∣ ∣ U ⊤ X ⊤ XU = I R } and V = { V ∣ ∣ V ⊤ Y ⊤ YV = I R } . Using the implementation of the retraction operation on U and V (see, (19) and (17)), we obtain

Thus, for the component-wise measure ρ r defined in (13a) and the matrix variant ρ defined in (21a), we have

For component-wise measure ∆ max ,r and matrix measure ∆ max , we have

Similarly, for ∆ sum ,r and ∆ sum, we have ∆ sum ≤ ∑ R r =1 ∆ sum ,r .

Peer reviews:
{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper proposes a fair CCA algorithm that aims to find fair CCA projection matrices. The authors claim that it is necessary to develop an appropriate algorithm for CCA with fairness guarantee. In the presence of sensitive attribute and unfairness in observed data, the proposed algorithms successfully reduces certain unfairness of the learned projection matrices. Specifically, the two proposed algorithms, MF-CCA and SF-CCA, are optimized using gradient descent algorithms on Stiefel manifolds. The convergences of these algorithms are also theoretically guaranteed, and several experiments conducted provide empirical support for the proposed algorithms.\n\nSTRENGTHS:\n- The paper is generally well-written and easy to follow in overall.\n- To the best of my knowledge, this work is the first to address the issue of unfairness issue in CCA, which could be a milestone.\n- The fairness metric targeted (i.e., Correlation Disparity Error in Definition 1) is well-defined and fits well with other general fairness notions used in fair prediction tasks (e.g., demographic parity).\n- The proposed algorithms (MF-CCA and SF-CCA) align well with the theoretical studies presented (Theorems 4 and 5).\n- The roles of the two algorithms well-specified. MF-CCA finds the optimal solution by minimizing CCA error and unfairness losses simultaneously, while SF-CCA provides an advantage in controlling the trade-off between CCA error and unfairness losses.\n- Empirical results validate the efficacy of MF-CCA and SF-CCA, not only on synthetic data but also on real datasets.\n\nWEAKNESSES:\n- There is no theoretical guarantee for fairness provided. If the authors could theoretically demonstrate that the solutions to equations 7 and 9 have low $\\mathcal{E}^{k} (U, V),$ as has already been empirically shown, the contribution would be more novel.\n- To illustrate the ability of SF-CCA in controlling the trade-off between error and fairness, a visualization such as Pareto-front lines (commonly used in fair classification problems [1, 2]) would be beneficial. Table 1 only presents results using a single $\\lambda$ selected from the set [1e-2, 1e-1, 1, 10, 100].\n\n[1] https://arxiv.org/abs/1802.06309\n\n[2] https://arxiv.org/abs/2103.06503\n\nQUESTIONS:\n- Which penalty function, $\\phi$, was used in the experiments?\n- Is the computation time more significantly impacted by the size of the training data or the dimension of the input feature?\n- Could the authors provide the definition of a 'componentwise Lipschitz continuous function' as stated in Assumption A?\n\nLIMITATIONS:\n- Unlike the vanilla CCA which solves the objective by optimizing two matrices, $U$ and $V$, the proposed algorithms require more matrices to be optimized, the number of which increases with the number of sensitive attributes, $K.$\n- Naturally, the computational cost of these proposed algorithms is higher than that of the vanilla CCA due to the fairness objectives they must minimize. Any future work aimed at reducing this computational cost would certainly be novel.", "TITLE": "Official Review", "RECOMMENDATION": 6, "REVIEWER_CONFIDENCE": 4, "SOUNDNESS_CORRECTNESS": 3, "DATE": "2023-06-28"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper addresses fairness and bias in Canonical Correlation Analysis (CCA). The authors propose a framework that minimizes correlation disparities associated with protected attributes, reducing unfairness without compromising accuracy. Experimental evaluation validates the effectiveness of the approach. The findings emphasize the importance of fairness in CCA applications.\n\nSTRENGTHS:\n- Novel Contribution: The paper introduces a framework to address fairness and bias concerns in Canonical Correlation Analysis (CCA), making a valuable contribution to the field.\n\n- Practical Relevance: By focusing on CCA, a widely used statistical technique, the paper addresses a real-world problem and emphasizes the importance of considering fairness in data analysis.\n\n- Experimental Validation: The authors conduct experiments on both synthetic and real-world datasets, providing empirical evidence of the effectiveness of their proposed framework in reducing unfairness without compromising the accuracy of the CCA model.\n\n- Clear Presentation: The abstract provides a concise overview of the paper's objectives, approach, and findings, making it easy to understand the key contributions of the research.\n\nWEAKNESSES:\n- Lack of Detailed Metrics: The paper could benefit from providing more specific details about the metrics used to measure unfairness and bias in CCA. This would enhance the transparency and reproducibility of the experimental evaluation. \n\n- Limited Comparison: The paper does not explicitly compare the proposed framework with existing fairness-aware CCA methods. Including such comparisons would provide insights into the relative performance and advantages of the proposed approach. \n\n- Scope and Generalizability: While the paper addresses fairness concerns in CCA, the focus is limited to this specific technique. It would be beneficial to discuss the potential implications of the findings for other statistical methods or broader machine learning applications.\n\nQUESTIONS:\n- In the experimental setup, what considerations were made in selecting the synthetic and real-world datasets? Were there any specific characteristics of these datasets that influenced the results or generalizability of the findings? \n\n- While the paper focuses on fairness and bias in CCA, could you elaborate on the potential implications of the findings for other statistical methods or broader machine learning applications? How transferable do you believe the proposed framework is beyond the scope of CCA?\n\n- Are there any additional factors or considerations that should be taken into account when applying the proposed framework in practical settings? For instance, how would the framework handle missing data, outliers, or high-dimensional datasets?\n\n- Given the goal of reducing unfairness, how does the proposed framework balance the trade-off between fairness and overall predictive accuracy? Were there any cases in the experiments where the framework significantly sacrificed accuracy to achieve fairness?\n\nLIMITATIONS:\n- Fairness Metrics: The paper could provide a more detailed discussion on the fairness metrics used to evaluate the proposed framework. Further elaboration on the choice and justification of these metrics would enhance the clarity and interpretability of the experimental results.\n\n- Generalizability: The generalizability of the findings may be limited by the specific characteristics and distribution of the datasets used in the experiments. The authors could discuss the potential challenges or variations that may arise when applying the framework to other datasets or domains.\n\n- Trade-off between Fairness and Accuracy: The paper briefly mentions that the proposed framework minimizes unfairness without compromising the accuracy of the CCA model. However, a more in-depth analysis of the potential trade-off between fairness and accuracy would provide a clearer understanding of the framework's limitations and its impact on prediction performance.\n\n- Real-world Application Challenges: While the framework demonstrates effectiveness in reducing unfairness, the paper does not extensively discuss the practical challenges that may arise when applying the proposed approach to real-world scenarios, such as handling missing data, complex feature distributions, or scalability issues.", "TITLE": "Official Review", "RECOMMENDATION": 7, "REVIEWER_CONFIDENCE": 2, "SOUNDNESS_CORRECTNESS": 4, "DATE": "2023-07-06"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper investigates the concept of Fair CCA, focusing on addressing the potential bias that arises when analyzing the relationship between two sets of variables using CCA, a widely utilized statistical technique. The conventional application of CCA fails to account for the impact of sensitive attributes like gender or race, leading to potential biases. In response, this study aims to bridge this gap by integrating fairness principles into CCA. The authors introduce the fairness issue within the context of CCA and propose two distinct methods to tackle it: a multi-objective approach and a single-objective approach, each offering unique strengths. The effectiveness of the proposed methods is substantiated through empirical and theoretical analyses, confirming their value in addressing the fairness concerns in CCA.\n\nSTRENGTHS:\n1. The problem addressed in this paper holds significant importance. Given the increasing influence of machine learning algorithms and methods on individuals and society, it becomes crucial to delve into the study of fairness within this domain. By mitigating bias issues in machine learning, we can contribute to a more equitable outcome and benefit vulnerable groups. While numerous works have explored fairness in machine learning, the majority of them focus on the supervised learning scenario. In contrast, this paper ventures into uncharted territory by examining the fairness issue in CCA, an unsupervised learning approach. This unique perspective underscores the urgency and significance of studying fairness within the context of CCA.\n2. The concepts and methods presented in this paper exhibit a high degree of novelty. To the best of my knowledge, this is the first study to explore fairness within the context of CCA. Fairness, being a multifaceted concept, encompasses various definitions. In the realm of supervised learning, researchers have proposed different definitions such as demographic parity, equalized odds, and group sufficiency. Therefore, establishing a practical and reasonable definition becomes crucial. This paper introduces the notion of fairness criteria through correlation disparity error, which takes into account both global and group-wise correlations. The resulting fairness definition is intuitive and reasonable. The incorporation of fairness as additional objectives (in the multi-objective framework) or constraints (in the single-objective framework) is accomplished seamlessly, aligning with the natural progression of the problem. Furthermore, the authors introduce the Riemannian manifold in their solutions, which promotes convergence and facilitates computation, thereby introducing a novel aspect to the research.\n3. The paper exhibits good writing quality, characterized by clarity and soundness. It effectively guides readers through its content, ensuring easy comprehension from the motivation and definition to the methods and solutions. Notably, Figure 1 provides a clear and intuitive visualization that enables immediate understanding of the proposed method's functionality. The effectiveness of the methods is supported by robust experimental results on synthetic and real data. Furthermore, Figure 3 serves as a compelling validation of fairness, as it visually demonstrates the improved proximity between the two groups after the projection using the proposed methods. Overall, the paper is meticulously crafted, maintaining a high level of clarity and rigor throughout.\n\nWEAKNESSES:\n1. Limited discussion on multiple modalities: While CCA is not restricted to two modalities, the paper primarily focuses on this scenario. It would be beneficial to discuss a more general setting involving multiple modalities and computing correlations under the fairness setting.\n2. Figure 4 lacks an obvious trend: The authors could consider including Figure 9 from the supplementary file in the main body, as it provides a more intuitive demonstration of the method's effectiveness.\n\nQUESTIONS:\n1. Clarification on \"critical Pareto\" (line 164): The definition of \"critical Pareto\" is unclear. It would be helpful to provide a more intuitive explanation for better understanding.\n2. Elaboration on optimization problem (8) and steepest descent direction (lines 174-175): The statement regarding the optimization problem and steepest descent direction requires further elaboration to enhance clarity.\n\nLIMITATIONS:\nThe method is currently limited to two modalities. Even though this is the most common scenario in CCA, it would be interesting to see how the method can be extended to more than two modalities.\n\nOverall, this is a well-written and informative paper that makes a significant contribution to the field of machine learning. The proposed methods are novel and effective, and the experimental results are convincing. However, the method is currently limited to two modalities, and it would be interesting to see how it can be extended to more than two modalities.", "TITLE": "Official Review", "RECOMMENDATION": 7, "REVIEWER_CONFIDENCE": 4, "SOUNDNESS_CORRECTNESS": 3, "DATE": "2023-07-06"}

comments: SUMMARY:
This paper addresses a fairness issue that arises in CCA, proposing a fair CCA that well trade-offs correlation disparity errors w.r.t. sensitive attributes against correlation w.r.t. global projection subspaces. It introduces two optimization frameworks (multi-objective and single-objective), then developing corresponding efficient algorithms based on the generalized Stiefel manifold together with convergence analysis. Experimental results both on synthetic and real datasets are provided to validate the theoretical findings.

STRENGTHS:
S1. The paper is very well written. Many illustrations (e.g., Fig. 1) are greatly helpful in grasping the ideas. 

S2. The proposed optimization frameworks are convincing, and the corresponding translation techniques enable the use of efficient algorithms in the manifold literature. In addition, the theoretical analysis (Theorems 3 and 4) provides convergence guarantees of the algorithms. 

S3. Experimental results emphasize the efficacy of the proposed approach, and the discussions therein are insightful.

WEAKNESSES:
W1. The translations for efficient algorithms, (8) and (9), can further be detailed for those who are not familiar with the manifold literature. 

W2. For Theorems 3 and 4, the proof sketches (or technical contributions if any) are preferred to be included in the main body.

QUESTIONS:
See Weakness in the above.

LIMITATIONS:
See Weakness in the above.
TITLE: Official Review
SOUNDNESS_CORRECTNESS: 3
DATE: 2023-07-06