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.

UR - http://www.scopus.com/inward/record.url?scp=84988649794&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84988649794&partnerID=8YFLogxK

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 -