# Hierarchical Convex NMF for Clustering Massive Data

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Kristian Kersting,
Mirwaes Wahabzada,
Christian Thurau,
Christian Bauckhage
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Proceedings of 2nd Asian Conference on Machine Learning, PMLR 13:253-268, 2010.

#### Abstract

We present an extension of convex-hull non-negative matrix factorization (CH-NMF) which was recently proposed as a large scale variant of convex non-negative matrix factorization or Archetypal Analysis. CHNMF factorizes a non-negative data matrix V into two non-negative matrix factors V = WH such that the columns of W are convex combinations of certain data points so that they are readily interpretable to data analysts. There is, however, no free lunch: imposing convexity constraints on W typically prevents adaptation to intrinsic, low dimensional structures in the data. Alas, in cases where the data is distributed in a non-convex manner or consists of mixtures of lower dimensional convex distributions, the cluster representatives obtained from CH-NMF will be less meaningful. In this paper, we present a hierarchical CH-NMF that automatically adapts to internal structures of a dataset, hence it yields meaningful and interpretable clusters for non-convex datasets. This is also confirmed by our extensive evaluation on DBLP publication records of 760,000 authors, 4,000,000 images harvested from the web, and 150,000,000 votes on World of Warcraft guilds.

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