Matrix Variate Distributions

By Andrew L. Rukhin. 1999

PRELIMINARIES Matrix Algebra Jacobians of Transformations Integration Zonal Polynomials Hypergeometric Functions of Matrix Argument LaGuerre Polynomials Generalized Hermite Polynomials Notion of Random Matrix Problems MATRIX VARIATE NORMAL DISTRIBUTION Density Function Properties Singular Matrix Variate Normal distribution Symmetic Matrix Variate Normal Distribution Restricted Matrix Variate Normal Distribution Matrix Variate Q-Generalized Normal Distribution WISHART DISTRIBUTION Introduction Density Function Properties Inverted Wishart Distribution Noncentral Wishart Distribution Matrix Variate Gamma Distribution Approximations MATRIX VARIATE t-DISTRIBUTION Density Function Properties Inverted Matrix Variate t-Distribution Disguised Matrix Variate t-Distribution Restricted Matrix Variate t-Distribution Noncentral Matrix Variate t-Distribution Distribution of Quadratic Forms MATRIX VARIATE BETA DISTRIBUTIONS Density Functions Properties Related Distributions Noncentral Matrix Variate Beta Distribution MATRIX VARIATE DIRICHLET DISTRIBUTIONS Density Functions Properties Related Distributions Noncentral Matrix Variate Dirichlet Distributions DISTRIBUTION OF MATRIX QUADRATIC FORMS Density Function Properties Functions of Quadratic Forms Series Representation of the Density Noncentral Density Function Expected Values Wishartness and Independence of Quadratic Forms of the Type XAX' Wishartness and Independence of Quadratic Forms of the Type XAX'+1/2(LX'+XL')+C Wishartness and Independence of Quadratic Forms of the Type XAX'+L1X'+XL'2+C MISCELLANEOUS DISTRIBUTIONS Uniform Distribution on Stiefel Manifold Von Mises-Fisher Distribution Bingham Matrix Distribution Generalized Bingham-Von Mises Matrix Distribution Manifold Normal Distribution Matrix Angular Central Gaussian Distribution Bimatix Wishart Distribution Beta-Wishart Distribution Confluent Hypergeometric Function Kind 1 Distribution Confluent Hypergeometric Function Kind 2 Distribution Hypergeometric Function Distributions Generalized Hypergeometric Function Distributions Complex Matrix Variate Distributions GENERAL FAMILIES OF MATRIX VARIATE DISTRIBUTIONS Matrix Variate Liouville Distributions Matrix Variate Spherical Distributions Matrix Variate Elliptically Contoured Distributions Orthogonally Invariant and Residual Independent Matrix Distributions GLOSSARY REFERENCES SUBJECT INDEX Each chapter also includes an Introduction and Problems @

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Citations in the corpus

Online Structured Laplace Approximations For Overcoming Catastrophic Forgetting (2018, background): “The block-diagonal Kronecker factored approximation corresponds to assuming independence between the layers and factorizing the covariance between the weights of a layer into the covariance of the columns and rows, resulting in a matrix normal distribution [11].”
A Convex Formulation for Learning Task Relationships in Multi-Task Learning (2010, methodology): “Since W is a matrix variable, it is natural to use a matrix-variate distribution [14] to model it.”
Task-Agnostic Continual Learning Using Online Variational Bayes With Fixed-Point Updates (2020, background): “d We now focus on the case in which the parametric distribution q n ( θ | φ ) and the prior distribution q n − 1 ( θ ) are multivariate gaussian whose covariance matrix is a Kronecker product of two PD matrices (Gupta & Nagar, 2018).”
Bayesian mean–variance analysis: optimal portfolio selection under parameter uncertainty (2018, methodology): “Finally, it has to be noted that the application of the stochastic representation describing the distribution of random quantities has been used both in the frequentist statistics (see, e.g. Givens and Hoeting 2012, Gupta et al. 2013) and in the Bayesian statistics (cf. Bodnar et al. 2017).”
A Survey on Bayesian Deep Learning (2016, background): “Note that N K , J ( 0 , I K ⊗ ( λ l L a ) − 1 ) in Equation (10) is a matrix-variate normal distribution defined as in [33]: where the operator ⊗ denotes the Kronecker product of two matrices [33], tr (·) denotes the trace of a matrix, and L a is the Laplacian matrix incorporating the relational…”
Learning Multiple Tasks with a Sparse Matrix-Normal Penalty (2010, background): “Matrix normal distributions have been studied in probability and statistics for several decades [13, 16, 18] and applied to predictive modeling in the Bayesian literature.”
A Regularization Approach to Learning Task Relationships in Multitask Learning (2014, methodology): “Since W is a matrix variable, it is natural to use a matrix-variate distribution [Gupta and Nagar 2000] to model it.”