Multivariate Gaussianization for Data Processing

Intro Iterative Gaussianization Experiments ConclusionsPreliminariesDefinition 1: PDF estimation under arbitrary transform [Stark86]Let x ∈ R d be a r.v. with PDF, p x(x). Given some bijective, differentiabletransform of x into y using G : R d → R d , y = G(x), the PDFs are related:p x(x) = p y(G(x))∣ dG(x)dx ∣ = py(G(x)) |∇xG(x)|where |∇ xG| is the determinant of the transform’s Jacobian matrix.Remark 1The p x(x) can be obtained if the Jacobian is known, since(1p y(y) = p y(G(x)) =( √ 2π|Σ|) exp − 1 )d 2 (G(x) − µ y )⊤ Σ −1 (G(x) − µ y )