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Non-negative matrix factorization via archetypal analysis

Non-negative matrix factorization via archetypal analysis

An approach to non-negative matrix factorization that does not require data to be separable and provides a generally unique decomposition.

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.



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Andrea Montanari

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Hamid Javadi

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

Anil Damle, Yuekai Sun

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Archetypal analysis for machine learning

Morten Morup, Lars Kai Hansen

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Fast and Robust Archetypal Analysis for Representation Learning

Yuansi Chen, Julien Mairal, Zaid Harchaoui

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Non-negative Matrix Factorization via Archetypal Analysis

Hamid Javadi, Andrea Montanari

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