Aleix Alcacer

Aleix Alcacer, Irene Epifanio

Archetypal analysis serves as an exploratory tool that interprets a collection of observations as convex combinations of pure (extreme) patterns. When these patterns correspond to actual observations within the sample, they are termed archetypoids. For the first time, we propose applying archetypoid analysis to nominal observations, specifically for identifying archetypal cases from questionnaires featuring nominal multiple-choice questions with a single possible answer. This approach can enhance our understanding of a nominal data set, similar to its application in multivariate contexts. We compare this methodology with the use of archetype analysis and probabilistic archetypal analysis and demonstrate the benefits of this methodology using a real-world example: the German credit dataset.

Aleix Alcacer, Irene Epifanio, Sebastian Mair et al.

Archetypal analysis (AA) was originally proposed in 1994 by Adele Cutler and Leo Breiman as a computational procedure to extract the distinct aspects called archetypes in observations with each observational record approximated as a mixture (i.e., convex combination) of these archetypes. AA thereby provides straightforward, interpretable, and explainable representations for feature extraction and dimensionality reduction, facilitating the understanding of the structure of high-dimensional data with wide applications throughout the sciences. However, AA also faces challenges, particularly as the associated optimization problem is non-convex. This survey provides researchers and data mining practitioners an overview of methodologies and opportunities that AA has to offer surveying the many applications of AA across disparate fields of science, as well as best practices for modeling data using AA and limitations. The survey concludes by explaining important future research directions concerning AA.

Aleix Alcacer, Irene Epifanio

Archetypal Analysis (AA) is an unsupervised learning method that represents data as convex combinations of extreme patterns called archetypes. While AA provides interpretable and low-dimensional representations, it can inadvertently encode sensitive attributes, leading to fairness concerns. In this work, we propose Fair Archetypal Analysis (FairAA), a modified formulation that explicitly reduces the influence of sensitive group information in the learned projections. We also introduce FairKernelAA, a nonlinear extension that addresses fairness in more complex data distributions. Our approach incorporates a fairness regularization term while preserving the structure and interpretability of the archetypes. We evaluate FairAA and FairKernelAA on synthetic datasets, including linear, nonlinear, and multi-group scenarios, demonstrating their ability to reduce group separability -- as measured by mean maximum discrepancy and linear separability -- without substantially compromising explained variance. We further validate our methods on the real-world ANSUR I dataset, confirming their robustness and practical utility. The results show that FairAA achieves a favorable trade-off between utility and fairness, making it a promising tool for responsible representation learning in sensitive applications.