## Abstract

While the Matrix Generalized Inverse Gaussian (MGIG) distribution arises naturally in some settings as a distribution over symmetric positive semi-definite matrices, certain key properties of the distribution and effective ways of sampling from the distribution have not been carefully studied. In this paper, we show that the MGIG is unimodal, and the mode can be obtained by solving an Algebraic Riccati Equation (ARE) equation [7]. Based on the property, we propose an importance sampling method for the MGIG where the mode of the proposal distribution matches that of the target. The proposed sampling method is more efficient than existing approaches [32,33], which use proposal distributions that may have the mode far from the MGIG’s mode. Further, we illustrate that the the posterior distribution in latent factor models, such as probabilistic matrix factorization (PMF) [24], when marginalized over one latent factor has the MGIG distribution. The characterization leads to a novel Collapsed Monte Carlo (CMC) inference algorithm for such latent factor models. We illustrate that CMC has a lower log loss or perplexity than MCMC, and needs fewer samples.

Original language | English (US) |
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Title of host publication | Machine Learning and Knowledge Discovery in Databases - European Conference, ECML PKDD 2016, Proceedings |

Editors | Jilles Giuseppe, Niels Landwehr, Giuseppe Manco, Paolo Frasconi |

Publisher | Springer Verlag |

Pages | 648-664 |

Number of pages | 17 |

ISBN (Print) | 9783319461274 |

DOIs | |

State | Published - 2016 |

Event | 15th European Conference on Machine Learning and Principles and Practice of Knowledge Discovery in Databases, ECML PKDD 2016 - Riva del Garda, Italy Duration: Sep 19 2016 → Sep 23 2016 |

### Publication series

Name | Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics) |
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Volume | 9851 LNAI |

ISSN (Print) | 0302-9743 |

ISSN (Electronic) | 1611-3349 |

### Other

Other | 15th European Conference on Machine Learning and Principles and Practice of Knowledge Discovery in Databases, ECML PKDD 2016 |
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Country | Italy |

City | Riva del Garda |

Period | 9/19/16 → 9/23/16 |

### Bibliographical note

Funding Information:The research was supported by NSF grants IIS-1447566, IIS-1447574, IIS-1422557, CCF-1451986, CNS-1314560, IIS-0953274, IIS-1029711, NASA grant NNX12AQ39A, and gifts from Adobe, IBM, and Yahoo. F.F. acknowledges the support of DDF (2015?2016) from the University of Minnesota.