TY - GEN
T1 - The matrix generalized inverse Gaussian distribution
T2 - 15th European Conference on Machine Learning and Principles and Practice of Knowledge Discovery in Databases, ECML PKDD 2016
AU - Fazayeli, Farideh
AU - Banerjee, Arindam
N1 - 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.
Publisher Copyright:
© Springer International Publishing AG 2016.
PY - 2016
Y1 - 2016
N2 - 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.
AB - 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.
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U2 - 10.1007/978-3-319-46128-1_41
DO - 10.1007/978-3-319-46128-1_41
M3 - Conference contribution
AN - SCOPUS:84988649794
SN - 9783319461274
T3 - Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)
SP - 648
EP - 664
BT - Machine Learning and Knowledge Discovery in Databases - European Conference, ECML PKDD 2016, Proceedings
A2 - Giuseppe, Jilles
A2 - Landwehr, Niels
A2 - Manco, Giuseppe
A2 - Frasconi, Paolo
PB - Springer
Y2 - 19 September 2016 through 23 September 2016
ER -