Some Topics Deserved Concerns

Some Topics deserved Concerns Songcan Chen 2013.3.6

Some Topics Deserved Concerns Songcan Chen 2013.3.6

Outlines Copula its applications Kronecker decomposition for matrix Covariance Descriptors Metric on manifold

Outlines • Copula & its applications • Kronecker Decomposition for Matrix • Covariance Descriptors & Metric on manifold

Copula its applications [1 Fabrizio durante and Carlo Sempi, Copula Theory: An Introduction(Chapt. 1), P. Jaworski et al. (eds ) Copula Theory and Its Applications, Lecture Notes in Statistics 198.2010 [2]Jean-David Fermanian, An overview of the goodness-of-fit test problem for copulas(Chapt 1), arXiv: 19 NoV 2012 Applications [Al] David Lopez-Paz, Jose Miguel Hernandez-Lobato, Bernhard Scholkopf, Semi- Supervised Domain Adaptation with Non-Parametric Copulas NIPS2012/arXiv: 1 Jan, 2013 [A2]David Lopez-Paz, et al, Gaussian Process vine Copulas for Multivariate Dependence, ICML2013/arXiV: 16 Feb 2013 [A3 Carlos Almeida, et al, Modeling high dimensional time-varying dependence using D-vine SCAR models, arXiv: 9 Feb 2012 [A4] Alexander Baue, et al, Pair-copula Bayesian networks, arXiv: 23 NoV. 2012

[1] Fabrizio Durante and Carlo Sempi, Copula Theory: An Introduction (Chapt. 1), P. Jaworski et al. (eds.), Copula Theory and Its Applications, Lecture Notes in Statistics 198,2010. [2] Jean-David Fermanian, An overview of the goodness-of-fit test problem for copulas (Chapt 1), arXiv: 19 Nov. 2012. Applications [A1] David Lopez-Paz, Jose Miguel Hernandez-Lobato, Bernhard Scholkopf, Semi￾Supervised Domain Adaptation with Non-Parametric Copulas, NIPS2012/arXiv:1 Jan,2013. [A2] David Lopez-Paz, et al, Gaussian Process Vine Copulas for Multivariate Dependence, ICML2013/arXiv: 16 Feb. 2013. [A3] Carlos Almeida, et al, Modeling high dimensional time-varying dependence using D-vine SCAR models, arXiv: 9 Feb. 2012. [A4] Alexander Baue, et al, Pair-copula Bayesian networks, arXiv:23 Nov. 2012. … … Copula & its applications

Kronecker Decomposition for matrix []C.V. Loan and N. Pitsianis, Approximation with kronecker products, in Linear Algebra for Large Scale and Real Time Applications. Kluwer Publications, 1993, pp. 293-314 [2] T. Tsiligkaridis, A Hero, and s Zhou, On Convergence of Kronecker Graphical Lasso Algorithms, to appear in IEEE TSP, 2013 [3-, Convergence Properties of Kronecker Graphical Lasso Algorithms, aXv:12040585,July2012 [4]-, Low Separation Rank Covariance Estimation using Kronecker Product Expansions, google 2013 [5---Covariance Estimation in High Dimensions via Kronecker Product Expansions, arXiv: 12 Feb 2013 [6] SPARSE COVARIANCE ESTIMATION UNDER KRONECKER PRODUCT STRUCTURE, ICCASP2012, pp: 3633-3636 7 Marco F Duarte, Richard G Baraniuk, Kronecker Compressive Sensing IEEE TIE,21(24945042012 8]MARTIN SINGULL, et al, More on the Kronecker Structured Covariance Matrix, Communications in Statistics-Theory and Methods, 41: 2512-2523 2012

Kronecker Decomposition for Matrix [1] C. V. Loan and N. Pitsianis, Approximation with kronecker products, in Linear Algebra for Large Scale and Real Time Applications. Kluwer Publications, 1993, pp. 293–314. [2] T. Tsiligkaridis, A. Hero, and S. Zhou, On Convergence of Kronecker Graphical Lasso Algorithms, to appear in IEEE TSP, 2013. [3] ---, Convergence Properties of Kronecker Graphical Lasso Algorithms, arXiv:1204.0585, July 2012. [4] ---, Low Separation Rank Covariance Estimation using Kronecker Product Expansions, google 2013. [5] --- Covariance Estimation in High Dimensions via Kronecker Product Expansions, arXiv:12 Feb. 2013. [6] --- SPARSE COVARIANCE ESTIMATION UNDER KRONECKER PRODUCT STRUCTURE, ICCASP2012,pp:3633-3636. [7] Marco F. Duarte, Richard G. Baraniuk, Kronecker Compressive Sensing, IEEE TIP, 21(2)494-504 2012 [8] MARTIN SINGULL, et al, More on the Kronecker Structured Covariance Matrix, Communications in Statistics—Theory and Methods, 41: 2512–2523, 2012

Covariance Descriptor [1]Oncel Tuzel, Fatih Porikli, and Peter Meer, Region Covariance-A Fast Descriptor for Detection and Classification, Tech Report 2005 2 Yanwei Pang, Yuan Yuan, Xuelong Li, Gabor-Based Region Covariance Matrices for Face Recognition, IEEE T CIRCUITS AND SYSTEMS FOR VIDEO TECHNOLOGY 18(7)9899932008 [3]Anoop Cherian, et al, Jensen-Bregman LogDet Divergence with Application to Efficient Similarity Search for Covariance Matrices, IEEE TPAMI, in press, 2012 [4] Pedro Cortez Cargill,et al, Object Tracking based on Covariance Descriptors and On-Line Naive Bayes Nearest Neighbor Classifier, 2010 4th Pacific-Rim Symp Image and video Technology, pp 139-144 5] Ravishankar Sivalingam, et al, Positive Definite Dictionary Learning for Region Covariances, ICCV 2011 [6] Mehrtash T Harandi, et al, Kernel Analysis over Riemannian Manifolds for Visual Recognition of Actions Pedestrians and Textures, cVPR2012

Covariance Descriptor [1] Oncel Tuzel, Fatih Porikli, and Peter Meer,Region Covariance-A Fast Descriptor for Detection and Classification, Tech. Report 2005. [2] Yanwei Pang, Yuan Yuan, Xuelong Li, Gabor-Based Region Covariance Matrices for Face Recognition, IEEE T CIRCUITS AND SYSTEMS FOR VIDEO TECHNOLOGY, 18(7):989-993,2008 [3] Anoop Cherian, et al, Jensen-Bregman LogDet Divergence with Application to Efficient Similarity Search for Covariance Matrices, IEEE TPAMI, in press, 2012. [4] Pedro Cortez Cargill,et al, Object Tracking based on Covariance Descriptors and On-Line Naive Bayes Nearest Neighbor Classifier, 2010 4th Pacific-Rim Symp. Image and Video Technology,pp.139-144. [5] Ravishankar Sivalingam, et al, Positive Definite Dictionary Learning for Region Covariances, ICCV 2011. [6] Mehrtash T. Harandi, et al, Kernel Analysis over Riemannian Manifolds for Visual Recognition of Actions, Pedestrians and Textures, CVPR2012

Copula its applications

Copula & its applications

What is Copula? Definition Copulas are statistical tools that factorize multivariate distributions into the product of its marginals and a function that captures any possible form of dependence among them(marginals). This function is referred to as the copula, and it links the marginals together into the joint multivariate model

What is Copula? • Definition Copulas are statistical tools that factorize multivariate distributions into the product of its marginals and a function that captures any possible form of dependence among them (marginals). This function is referred to as the copula, and it links the marginals together into the joint multivariate model

What is Copula? Mathematical formulation p(x)=I(x;)c(Px),…,P(za).( 1 copula P(Xi is the marginal cdf of the random variable Xi Interestingly, this density has uniform marginals, since P(z)U[0; 1] for any random variable Z When P(X);.; P(Xa are continuous, the copula c( )is unique

What is Copula? • Mathematical formulation: P(xi ) is the marginal cdf of the random variable xi . Interestingly, this density has uniform marginals, since P(z)~ U[0; 1] for any random variable z. When P(x1 ); … ; P(xd ) are continuous, the copula c(.) is unique (2)

Especially, When factorizing multivariate densities into a product of marginal distributions and bivariate copula functions (called as vines) Each of these factors corresponds to one of the building blocks that are assumed either constant or varying across different learning domains applicable to DA, TL and mtL

Especially, when factorizing multivariate densities into a product of marginal distributions and bivariate copula functions (called as vines). Each of these factors corresponds to one of the building blocks that are assumed either constant or varying across different learning domains. → applicable to DA, TL and MTL!

Characteristics Infinitely many multivariate models share the same underlying copula function Figure 1: Left, sample from a Gaussian copula with correlation p=0.8. Middle and right, two samples drawn from multivariate models with this same copula but different marginal distributions, depicted as rug plots

Characteristics Infinitely many multivariate models share the same underlying copula function!