Non-negative matrix factorization via archetypal analysis

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.