FairKernelAA: Fair Kernelized Archetypal Analysis

• FairKernelAA is a kernelized extension of Fair Archetypal Analysis that produces interpretable, low-dimensional representations while reducing sensitive group influence.
• It integrates fairness regularization with kernel methods to balance data utility and bias mitigation in both linear and nonlinear contexts.
• Empirical evaluations show FairKernelAA minimizes group predictability metrics and maintains high reconstruction accuracy for applications in sensitive domains.

FairKernelAA is a kernelized extension of Fair Archetypal Analysis (FairAA), an unsupervised learning technique designed to produce interpretable and low-dimensional representations that explicitly reduce the influence of sensitive group information. By combining fairness regularization with the expressive power of kernel methods, FairKernelAA yields latent projections that suppress group membership signals in both linear and nonlinear data, achieving a favorable balance between data utility and fairness. This technique is particularly relevant in applications where representations are used for downstream tasks in sensitive domains such as hiring, finance, or healthcare.

1. Foundations and Motivation

Archetypal Analysis (AA) decomposes data into convex combinations of extreme observations, termed archetypes, yielding interpretable representations. However, standard AA and its straightforward kernelized version (KernelAA) may inadvertently encode sensitive attributes—such as race or gender—posing fairness risks. FairAA addresses this by introducing an explicit fairness regularization that discourages projections from reflecting sensitive group information. FairKernelAA extends this fairness constraint to the nonlinear setting by operating in a reproducing kernel Hilbert space (RKHS), where data relationships are captured implicitly through kernel functions rather than explicit feature representations. This extension enables fairness-aware representation learning even in datasets exhibiting complex, nonlinear group structures (Alcacer et al., 16 Jul 2025).

2. Mathematical Formulation and Methodology

The standard AA optimization seeks matrices $S$ and $C$ (with simplex constraints) that minimize the reconstruction error:

$\min_{S, C} \| X - S C X \|^2_F$

where $X \in \mathbb{R}^{n\times d}$ (data), each data point is reconstructed as a convex combination of archetypes, themselves convex combinations of data.

FairAA introduces a fairness term by penalizing the correlation between the latent representation and sensitive group membership. For group attribute vector $z \in \mathbb{R}^n$, after centering $z$, the regularization is incorporated as:

$$\min_{S, C} \| X - S C X \|^2_F + \lambda \| z S \|^2_F$$

with $\lambda \ge 0$ controlling the fairness-utility trade-off. The convexity (simplex) constraints on $S$ and $C$ ensure interpretations and stability.

For FairKernelAA, the methodology is kernelized by replacing $X X^\top$ with a general kernel matrix $K$, allowing the model to capture nonlinear relationships. The gradients of the objective with respect to $S$ and $C$ become:

• For $S$:

$$\nabla_S E = 2(S C K C^\top - K C^\top + \lambda z^\top z S)$$

• For $C$:

$\nabla_C E = 2(S^\top S C K - S^\top K)$

The optimization proceeds over $S$ and $C$ under simplex constraints, iteratively updating parameters using the kernel-based gradients. The regularization parameter $\lambda$ tunes the strength of fairness constraint, with higher $\lambda$ reducing group information but risking loss of explained variance.

3. Empirical Evaluation and Fairness–Utility Trade-off

Extensive experiments on synthetic and real-world datasets demonstrate the effectiveness of FairKernelAA:

• Synthetic Datasets: On linear and multi-class blob datasets, FairAA and FairKernelAA significantly reduced the linear separability and mean maximum discrepancy (MMD) between sensitive groups in the latent space compared to AA and KernelAA. For complex (nonlinear) distributions such as "make_moons" with an RBF kernel, FairKernelAA suppressed group predictive signals far more effectively than the linear version could.
• ANSUR I Dataset: Using anthropometric data with pronounced group (gender) differences, FairKernelAA yielded archetypal latent representations in which group information was drastically diminished—demonstrated by lower MMD and logistic regression accuracy on group membership—while retaining high explained variance.

The following table summarizes core evaluation metrics (as defined in the source):

Method	Explained Variance (EV)	Mean Max Discrepancy (MMD)	Linear Separability (LS)
AA/KernelAA	Highest	High	High
FairAA/FairKernelAA	Slightly reduced	Low	Low

A consistent finding was that FairKernelAA maintained high reconstruction ability while achieving considerable reductions in group-separability metrics, confirming a strong fairness–utility trade-off.

4. Applications and Practical Implications

FairKernelAA is applicable to any scenario where latent representations may be reused for tasks sensitive to bias or disparate impact:

• Preprocessing and Dimensionality Reduction: FairKernelAA can be deployed as a fairness-preserving dimensionality reduction step, yielding lower-dimensional embeddings where sensitive group membership is obfuscated.
• Visualization: When used for visualization of high-dimensional data, FairKernelAA enables plots in which clustering by sensitive attribute is reduced or eliminated, supporting audits for unintentional bias.
• Downstream Supervised Tasks: Representations learned via FairKernelAA can serve as inputs to classifiers or regressors, mitigating the risk that downstream models recover bias-inducing group information.
• Interpretability: By preserving the interpretable structure of archetypal analysis, FairKernelAA supports applications requiring both fairness and transparency, such as human-in-the-loop analytics and policy studies.

These applications are especially important in sectors with fairness scrutiny, including finance, healthcare, personnel selection, and public policy.

5. Relationship to Broader Fair Representation Learning and Kernel Methods

FairKernelAA is closely related to recent advances in kernelized fair representation learning, such as fair kernel regression via fair feature embedding (Okray et al., 2019). Both techniques leverage the implicit feature mapping of kernels to modulate fairness properties of machine-learned representations without requiring explicit understanding of the high-dimensional space.

While FairKernelAA targets unsupervised learning and archetypal interpretability, FKR-F$^2$E operates in supervised kernel regression and constructs explicit fair embeddings by optimizing for distributional similarity in latent space. The kernelization principle in both cases enables flexible, data-adaptive representations where fairness constraints are enforceable via explicit regularization or objective functions.

A plausible implication is that techniques from kernel-based fairness methods—such as those for constructing fair embeddings or adapting kernel matrices—could be further integrated or adapted to enhance FairKernelAA in future research, for example, allowing for richer fairness definitions or multi-attribute fairness.

6. Limitations and Future Directions

Several open questions and directions arise from the current methodology:

• Fairness Beyond Mean Independence: The present regularization penalizes mean correlation ($z S$), but statistical parity or equalized odds may require aligning covariances or higher moments. Extensions to penalize more comprehensive group statistics are suggested in the source.
• Multiple Sensitive Attributes: The extension to multiple or intersecting group attributes, as seen in intersectional fairness research, is indicated as an avenue for further development.
• Efficiency and Scalability: Although kernelization enables nonlinearity, kernel methods can be computationally intensive on large data. Techniques such as fast kernel matrix computations (Ryan et al., 2021) or kernel approximation strategies ("fast kernels" (Yang et al., 2014), kernel exchange algorithms (Wenzel et al., 30 Apr 2024)) represent promising areas to address these challenges.
• Broader Fairness Notions: Future work is suggested to explore balancing reconstruction errors across groups, variable selection, and alternative kernel functions tailored to specific fairness definitions or domains.

7. Summary

FairKernelAA represents a substantive advance in fairness-aware unsupervised learning. By embedding fairness constraints in a kernelized archetypal analysis framework, it achieves interpretable and robust representations where group membership signals are minimized, as demonstrated by substantial reductions in MMD and linear separability with minimal compromise in explained variance. Its methodological rigor and empirical performance across linear, nonlinear, and real-world scenarios position it as a robust tool for responsible representation learning, meeting the demands of modern ethical machine learning practice (Alcacer et al., 16 Jul 2025).