The Matrix Generalized Inverse Gaussian Distribution: Properties and Applications
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- 刊名：Lecture Notes in Computer Science
- 出版年：2016
- 出版时间：2016
- 年：2016
- 卷：9851
- 期：1
- 页码：648-664
- 全文大小：646 KB
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Arindam Banerjee (17)
17. Department of Computer Science and Engineering, University of Minnesota, Twin Cities, USA - 丛书名：Machine Learning and Knowledge Discovery in Databases
- ISBN：978-3-319-46128-1
- 刊物类别：Computer Science
- 刊物主题：Artificial Intelligence and Robotics
Computer Communication Networks
Software Engineering
Data Encryption
Database Management
Computation by Abstract Devices
Algorithm Analysis and Problem Complexity - 出版者：Springer Berlin / Heidelberg
- ISSN：1611-3349
- 卷排序：9851
文摘
While the Matrix Generalized Inverse Gaussian (\(\mathcal {MGIG}\)) 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 \(\mathcal {MGIG}\) 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 \(\mathcal {MGIG}\) 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 \(\mathcal {MGIG}\)’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 \(\mathcal {MGIG}\) 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.