Multivariate Gaussianization for Data Processing

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Intro Iterative Gaussianization Experiments ConclusionsPropertiesProperty 1: The iterative Gaussianization transform is invertibleGiven the Gaussianization transform:G : x (k+1) = R (k) Ψ (k) (x (k) )by simple manipulation, the inversion transform is:G −1 : x (k) = Ψ −1(k) (R⊤ (k)x (k+1) ).Remark 1: valid for any rotation transformInvertibility is possible for any rotation matrix R (k) .Remark 2: valid for PDF connected supportInvertibility of Ψ (k) is trivially ensured when the PDF support is connected, i.e.no disjoint subspaces in the PDF support.
Intro Iterative Gaussianization Experiments ConclusionsPropertiesProperty 2: The iterative Gaussianization transform is differentiableThe Jacobian of the series of K iterations is the product of the Jacobians:∇ xG = ∏ Kk=1 R (k)∇ x (k)Ψ (k)Marg. Gauss. Ψ (k) is a dimension-wise transform with diagonal Jacobian:⎛⎞∂Ψ 1 (k)· · · 0∂x (k)1∇ x (k)Ψ (k) =⎜.⎝. ... ..⎟⎠0 · · ·Each element in ∇ x (k)Ψ (k) is:∂Ψ d (k)∂x (k)d∂Ψ i (k)∂x (k)i= ∂G∂u∂u∂x (k)i( ∂G−1=∂x i) −1p i (x (k)i) = g(Ψ i (k)(x (k)i)) −1 p i (x (k)i)
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Multivariate Gaussianization for Data Processing

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