Fair Archetypal Analysis (FairAA)

• Fair Archetypal Analysis (FairAA) is an extension of archetypal analysis that integrates fairness constraints to minimize sensitive attribute influence in data representations.
• It employs regularization and alternating minimization techniques to balance reconstruction quality with reduced group leakage in the latent space.
• FairAA is applied in diverse fields such as healthcare and personnel selection, ensuring more equitable outcomes by mitigating bias in unsupervised learning.

Fair Archetypal Analysis (FairAA) is a principled extension of classical archetypal analysis (AA) that integrates fairness constraints into the process of learning interpretable, low-dimensional representations of high-dimensional data. By introducing regularization terms or explicit constraints aimed at reducing the influence of sensitive attributes in the latent space, FairAA aims to provide representations that are both useful for downstream tasks and less likely to encode or perpetuate bias. This approach responds to increasing concern that unsupervised representation learning can inadvertently capture and reinforce undesirable group information, especially in applications with significant ethical or societal implications.

1. Foundations of Archetypal Analysis and the Motivation for FairAA

Archetypal Analysis (AA) seeks to model each data point as a convex combination of "archetypes," which are themselves required to be convex combinations of actual data points. This yields highly interpretable representations, as both the archetypes and the decompositions can be traced directly to observed data or pure extremes in the dataset. The standard AA formulation is given by:

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

subject to the constraints that each row of $S$ and each row of $C$ belong to the probability simplex (i.e., are non-negative and sum to one) (Alcacer et al., 16 Apr 2025).

However, in practical settings, the archetypal representations learned by standard AA may encode sensitive group information (such as gender, race, or other protected characteristics). If these representations are used in downstream applications (such as classification, clustering, or recommendation), they risk introducing or perpetuating unfair treatment of individuals from marginalized or minority groups. The FairAA framework is thus explicitly motivated by the need to mitigate these risks by reducing or eliminating the presence of sensitive attribute information in the learned latent space (Alcacer et al., 16 Jul 2025).

2. Formal FairAA Methodology: Objectives and Formulations

The central methodological innovation in FairAA lies in augmenting the standard AA objective with a fairness regularization term. In the canonical FairAA approach, the objective is modified as follows:

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

where $X \in \mathbb{R}^{n \times d}$ is the data matrix, $z \in \mathbb{R}^n$ is the (centered) sensitive attribute indicator vector (e.g., for group membership), $S$ is the matrix of latent representations (rows of S correspond to observations), $C$ is the coefficient matrix constructing archetypes as convex mixtures of data, and $\lambda \ge 0$ controls the tradeoff between reconstruction quality and fairness (Alcacer et al., 16 Jul 2025).

The fairness constraint in this formulation enforces that the sensitive attribute $z$ is uncorrelated with the learned representation $S$ (i.e., $z S = 0$). This is a tractable relaxation of the stricter requirement of total statistical independence, and in many practical scenarios, removing linear dependence suffices to significantly reduce group leakage in the archetypal coefficients.

Optimization proceeds via alternating updates of $S$ and $C$, where gradient steps for $S$ are corrected by the fairness penalty term ($\lambda z^\top z S$), and projections ensure that the updated $S$ and $C$ remain in their respective simplices. The structure and interpretability of the archetypes are preserved by the convexity constraints on the variables.

In addition to this linear setting, the FairKernelAA variant generalizes to data with nonlinear relationships by "kernelizing" the algorithm: replacing all inner products $X X^\top$ with a kernel matrix $K$ in a reproducing kernel Hilbert space. Gradients and updates are then carried out analogously using the kernel trick, thus extending FairAA to more complex data distributions while enforcing the same fairness constraint (Alcacer et al., 16 Jul 2025).

3. Theoretical and Empirical Properties: Trade-offs and Guarantees

FairAA is designed to achieve a balance between preservation of utility (i.e., explained variance or data fidelity) and reduction of group-specific information in the latent space. The trade-off is controlled via the regularization parameter $\lambda$—higher values enforce stronger fairness at the potential cost of reconstruction error, while lower values prioritize fidelity with possible group leakage.

Three primary metrics are employed to evaluate the efficacy of FairAA and FairKernelAA:

• Explained Variance (EV): Measures how much of the original data variability is captured by the archetypal component. High EV indicates effective representation, while a modest decrease is generally acceptable when offset by strong fairness gains.
• Mean Maximum Discrepancy (MMD): Quantifies the difference between the distributions of latent codes for different sensitive groups. Lower MMD implies that group information is less present in the latent space.
• Linear Separability (LS): Assessed by the accuracy of a linear classifier (e.g., logistic regression) predicting $z$ from $S$. Lower LS suggests less group information can be extracted from the representations.

Empirical results demonstrate that FairAA and FairKernelAA can substantially reduce MMD and linear separability (i.e., group leakage) while maintaining only a minor reduction in explained variance across a suite of synthetic and real-world datasets (including the ANSUR I anthropometric survey, where features are highly group-specific by design) (Alcacer et al., 16 Jul 2025).

4. Extensions and Generalizations Within the AA Framework

The core FairAA methodology can be extended in several ways:

• Multiple Sensitive Groups: Although the baseline objective is presented for binary group membership, extensions to multiple (possibly overlapping) groups can be achieved by stacking group indicator vectors and enforcing orthogonality across all such attributes simultaneously (Alcacer et al., 16 Jul 2025). This suggests a scalable approach to multi-attribute fairness in archetypal analysis.
• Nonlinear and Kernelized FairAA: FairKernelAA employs positive-definite kernels to embed data into high-dimensional feature spaces, enabling fairness-constrained archetypal analysis of data with complex or nonlinear relationships.
• Soft Constraint Variants: While the canonical FairAA penalizes correlation between sensitive attributes and latent codes, stricter or alternative fairness criteria—such as covariate balancing, demographic parity penalties, or distribution-matching terms—may be considered depending on application requirements. A plausible implication is that incorporating higher-order statistics (beyond the mean, such as groupwise covariances) might further reduce subtle forms of group leakage.

5. Practical Implementation and Real-World Use Cases

Implementing FairAA requires the following key steps:

1. Data Preparation: Assemble the data matrix $X$ and the sensitive attribute vector $z$. Center $z$ if necessary for the fairness constraint.
2. Objective Specification: Choose the regularization strength $\lambda$, reflecting the desired trade-off between utility and fairness.
3. Optimization: Employ an alternating minimization strategy, projecting $S$ and $C$ back to the simplex after each gradient step. In kernel settings, replace inner products with kernel evaluations.
4. Evaluation: After training, quantify group leakage (via LS and MMD) and representation quality (via EV).
5. Tuning/Diagnostics: Optionally adjust $\lambda$ as informed by results on validation data; visualize archetypes and representation distributions for interpretability and post-hoc fairness audit.

Applications include scenarios where archetypal structure is valuable for interpretation yet fairness is a concern—for instance, personnel selection, healthcare outcome modeling, marketing segmentation, or sensitive customer analytics—ensuring that latent representations or derived decisions do not amplify protected group signals (Alcacer et al., 16 Jul 2025).

6. Relationship to Broader Fairness and Representation Learning Literature

FairAA stands out compared to earlier approaches by embedding fairness constraints directly within the core AA optimization, as opposed to post-processing, pre-processing, or data reweighting alone. This approach parallels trends in fair principal component analysis (PCA), fair clustering, and fair matrix factorization, yet is notable for preserving the distinctive interpretability property of archetypal representations.

Previous generalizations in AA (such as probabilistic AA, geometric or kernel AA, robust and sparse variants) are compatible with the FairAA paradigm, as fairness regularization can be integrated into the respective optimization objectives (Seth et al., 2013, Alcacer et al., 16 Apr 2025). This suggests that future work could leverage regularization or priors at multiple levels (e.g., stochastic, Bayesian, or hierarchical extensions) for nuanced fairness control.

7. Limitations, Open Directions, and Outlook

While FairAA provides a compelling foundation for fair and interpretable unsupervised representation learning, several challenges and directions remain:

• Utility-Fairness Trade-off: Achieving perfect fairness may come at the expense of meaningful clustering or representation of genuine group-distinguishing structure vital for some applications. The optimal balance is context-specific.
• Scalability and Computational Cost: As with classical AA, computational complexity can be significant; approximate or scalable variants (e.g., via randomized projection or active set optimization) are essential for large-scale settings (Han et al., 2021, Damle et al., 2014).
• General Fairness Notions: Current formulations focus on group uncorrelatedness in representation space. Extending to richer notions—such as individual fairness, equalized odds, or rich group-specific parity metrics—requires development of new penalties or constraints.
• Theoretical Guarantees: Consistency and convergence of archetypal analysis with fairness constraints remain active areas of research. Recent studies provide statistical consistency under standard AA; analogous guarantees for FairAA require further formalization (Osting et al., 2020).

The emergence of FairAA marks a significant development in the responsible deployment of interpretable unsupervised learning methods. By explicitly addressing fairness in the representation layer, FairAA facilitates applications in high-stakes domains where equitable treatment, transparency, and interpretability must be balanced. Research in this area is poised to continue integrating fairness at deeper levels of the representation learning stack and in increasingly complex, heterogeneous, or nonlinear data regimes (Alcacer et al., 16 Jul 2025, Alcacer et al., 16 Apr 2025).