Multivariate Gaussianization for Data Processing

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Intro Iterative Gaussianization Experiments ConclusionsPropertiesTheorem 2: redundacy reduces independently of the rotationGiven a marginally Gaussianized variable, Ψ(x), any rotation reduces theredundancy:∆I = I (Ψ(x)) − I (RΨ(x)) ≥ 0, ∀ RProof.RememberJ(x) = I (x) + J m(x) → I (x) = J(x) − J m(x)Apply it on I (Ψ(x)) and I (RΨ(x)):∆I = J(Ψ(x)) − J m(Ψ(x)) − J(RΨ(x)) + J m(RΨ(x))= J m(RΨ(x)) ≥ 0, ∀ Rsince (1) negentropy is rotation invariant, and (2) the marginal negentropy of amarginally Gaussianized r.v. is 0.
Intro Iterative Gaussianization Experiments ConclusionsPropertiesCorollary 1: convergence ensuredThe combination of marginal Gaussianization and rotation gives rise toredundancy reduction since marginal Gaussianization does not changeredundancy, I (Ψ(x)) = I (x).
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