Preliminaries Wishart Processes Relationship between GP and WP o For any kernel function A:'x -R,there exists a function B:Fs.t.A(xi,xj)=B(xi)'B(xj). where A'is the input space and FCR9 is some latent(feature) space(in general the feature space may also be infinite-dimensional). Our previous result:A(xi,xj)is a Wishart process iff [Bk(x)1 are q mutually independent Gaussian processes. Let A=[A(xi,xj)]2j=1 and B=[B(x1),...,B(xn)]'=[b1,...,bn]'. Then b;are the latent vectors,and A BB'is a linear kernel in the latent space but is a nonlinear kernel w.r.t.the input space. Theorem Let E be an nxn positive definite matrix.Then A is distributed according to the Wishart distribution Wn(q,if and only if B is distributed according to the (matrix-variate)Gaussian distribution Nn.g(0,) Li,Zhang and Yeung (CSE,HKUST) LWP AISTATS 2009 7/23Preliminaries Wishart Processes Relationship between GP and WP For any kernel function A : X × X → R, there exists a function B : X → F s.t. A(xi , xj) = B(xi) 0B(xj), where X is the input space and F ⊂ R q is some latent (feature) space (in general the feature space may also be infinite-dimensional). Our previous result: A(xi , xj) is a Wishart process iff {Bk (x)} q k=1 are q mutually independent Gaussian processes. Let A = [A(xi , xj)]n i,j=1 and B = [B(x1), . . . ,B(xn)]0 = [b1, . . . , bn] 0 . Then bi are the latent vectors, and A = BB0 is a linear kernel in the latent space but is a nonlinear kernel w.r.t. the input space. Theorem Let Σ be an n×n positive definite matrix. Then A is distributed according to the Wishart distribution Wn(q, Σ) if and only if B is distributed according to the (matrix-variate) Gaussian distribution Nn,q(0, Σ⊗Iq). Li, Zhang and Yeung (CSE, HKUST) LWP AISTATS 2009 7 / 23