Efficient and Generalizable Archetypal Analysis for Discrete Data

2026-10-08

AI Summary

This study addresses the limitation of classical prototype analysis, where least-squares objectives are ill-suited for discrete data, thereby constraining both model fit and interpretability. We propose a likelihood-based prototype analysis framework tailored for discrete data, supporting Bernoulli, Poisson, and multinomial distributions. Methodologically, we design a local quadratic approximation scheme combined with sequential minimal optimization and active-set strategies to achieve efficient computation. Furthermore, predictive likelihood cross-validation replaces heuristic model selection criteria. Experimental results demonstrate that the proposed framework accurately recovers model complexity in tasks such as single-cell sequencing, while exhibiting computational efficiency, structural interpretability, and stable fitting performance.

Abstract

Archetypal Analysis (AA) represents observations as convex combinations of extremal data-driven profiles, yielding interpretable low-dimensional descriptions of complex datasets. Classical AA relies on a least-squares objective, which is poorly suited to discrete observations such as binary, count, and categorical data. We introduce an efficient likelihood-based framework for AA supporting Bernoulli, Poisson, and multinomial observation models. Our optimization scheme employs local quadratic approximations of the negative log-likelihood, enabling constrained updates through sequential minimal optimization (SMO) and an active-set method. Scalability is improved by bounding the active set while preserving simplex feasibility. We further introduce a cross-validated predictive likelihood criterion for selecting the number of archetypes, providing a principled alternative to reconstruction-error heuristics and stability-based diagnostics. Synthetic experiments demonstrate computational efficiency and accurate recovery of model complexity. Applications to single-cell RNA sequencing, microbiome composition, and somatic mutation data show that the learned archetypes capture interpretable domain-specific structures while achieving competitive likelihood fits and stable solutions. Overall, the proposed framework enables efficient likelihood-based archetypal analysis of discrete data, complemented by predictive likelihood-based model selection.

Problem

• Archetypal Analysis
• Discrete Data
• Likelihood-based Framework
• Model Selection

Innovation

• Archetypal Analysis
• Discrete Data
• Likelihood-based Optimization
• Sequential Minimal Optimization
• Predictive Likelihood