Non-negative matrix factorization via archetypal analysis is named after two well-known techniques of statistics and unsupervised learning, non-negative matrix factorization (NMF) and archetypal analysis (AA).

Unlike the original archetypal analysis method developed by Cutler and Breiman, NMF via archetypal analysis does not require the data in a given data set to be separable. The method aims to optimize the trade-off between two objectives:

- Minimizing the distance of the data points from the convex envelope of archetypes (which can be interpreted as an empirical risk); and
- Minimizing the distance of the the archetypes from the convex envelope of data (which can be interpreted as a data-dependent regularization).

NMF via archetypal analysis introduces a 'uniqueness condition' on the data which is necessary for exactly recovering the archetypes from noiseless data. The approach requires solving a non-convex optimization problem, but early experiments showed that the standard optimization methods succeeded in finding good solutions.

### Timeline

### People

Andrea Montanari

Creator

Hamid Javadi

Creator

### Further reading

A geometric approach to archetypal analysis and non-negative matrix factorization

Anil Damle, Yuekai Sun

Archetypal analysis for machine learning

Morten Morup, Lars Kai Hansen

Fast and Robust Archetypal Analysis for Representation Learning

Yuansi Chen, Julien Mairal, Zaid Harchaoui

Non-negative Matrix Factorization via Archetypal Analysis

Hamid Javadi, Andrea Montanari