Kernel embedding of distributions: Difference between revisions

Revision as of 19:28, 27 March 2020

Revision as of 06:34, 6 October 2021

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===Joint distribution embedding===

If <math>Y</math> denotes another random variable (for simplicity, assume the co-___domain of <math>Y</math> is also <math>\Omega</math> with the same kernel <math>k</math> which satisfies <math> \langle \varphi(x) \otimes \varphi(y), \varphi(x') \otimes \varphi(y') \rangle = k(x,x') \otimes k(y,y')</math>), then the [[Joint probability distribution|joint distribution]] <math> P(x,y)) </math> can be mapped into a [[tensor product]] feature space <math>\mathcal{H} \otimes \mathcal{H} </math> via <ref name = "Song2013"/>

:<math> \mathcal{C}_{XY} = \mathbb{E} [\varphi(X) \otimes \varphi(Y)] = \int_{\Omega \times \Omega} \varphi(x) \otimes \varphi(y) \ \mathrm{d} P(x,y) </math>