scholarly journals Orthogonal polynomials and analytic functions associated to positive definite matrices

2006 ◽  
Vol 315 (1) ◽  
pp. 54-67 ◽  
Author(s):  
Christian Berg ◽  
Antonio J. Durán
Keyword(s):  
Orthogonal Polynomials ◽  
Analytic Functions ◽  
Positive Definite ◽  
Positive Definite Matrices

scholarly journals Multi-variable weighted geometric means of positive definite matrices

2011 ◽  
Vol 435 (2) ◽  
pp. 307-322 ◽  
Author(s):  
Hosoo Lee ◽  
Yongdo Lim ◽  
Takeaki Yamazaki
Keyword(s):  
Positive Definite ◽  
Positive Definite Matrices ◽  
Geometric Means

Identifying graph clusters using variational inference and links to covariance parametrization

2009 ◽  
Vol 367 (1906) ◽  
pp. 4407-4426
Author(s):  
David Barber
Keyword(s):  
Mean Field ◽  
Matrix Decomposition ◽  
Difficult Problem ◽  
Positive Definite ◽  
Variational Inference ◽  
Field Theories ◽  
Variational Approximation ◽  
Positive Definite Matrices ◽  
The Matrix ◽  
Non Gaussian

Finding clusters of well-connected nodes in a graph is a problem common to many domains, including social networks, the Internet and bioinformatics. From a computational viewpoint, finding these clusters or graph communities is a difficult problem. We use a clique matrix decomposition based on a statistical description that encourages clusters to be well connected and few in number. The formal intractability of inferring the clusters is addressed using a variational approximation inspired by mean-field theories in statistical mechanics. Clique matrices also play a natural role in parametrizing positive definite matrices under zero constraints on elements of the matrix. We show that clique matrices can parametrize all positive definite matrices restricted according to a decomposable graph and form a structured factor analysis approximation in the non-decomposable case. Extensions to conjugate Bayesian covariance priors and more general non-Gaussian independence models are briefly discussed.

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