## Abstract

Cluster analysis by nonnegative low-rank approximations has experienced a remarkable progress in the past decade. However, the majority of such approximation approaches are still restricted to nonnegative matrix factorization (NMF) and su er from the following two drawbacks: 1) they are unable to produce balanced partitions for large-scale manifold data which are common in real-world clustering tasks; 2) most existing NMF-type clustering methods cannot automatically determine the number of clusters. We propose a new low-rank learning method to address these two problems, which is beyond matrix factorization. Our method approximately decomposes a sparse input similarity in a normalized way and its objective can be used to learn both cluster assignments and the number of clusters. For efficient optimization, we use a relaxed formulation based on Data-Cluster-Data random walk, which is also shown to be equivalent to low-rank factorization of the doublystochastically normalized cluster incidence matrix. The probabilistic cluster assignments can thus be learned with a multiplicative majorization-minimization algorithm. Experimental results show that the new method is more accurate both in terms of clustering large-scale manifold data sets and of selecting the number of clusters.

Original language | English |
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Number of pages | 25 |

Journal | Journal of Machine Learning Research |

Volume | 17 |

Publication status | Published - 1 Oct 2016 |

MoE publication type | A1 Journal article-refereed |

## Keywords

- Cluster analysis
- Doubly stochastic matrix
- Manifold
- Multiplicative updates
- Probabilistic relaxation