Kernel embedding of distributions: Difference between revisions

Revision as of 18:43, 27 March 2020

Revision as of 18:44, 27 March 2020

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By the equivalence between a [[tensor]] and a [[linear map]], this joint embedding may be interpreted as an uncentered [[cross-covariance]] operator <math>\mathcal{C}_{XY}: \mathcal{H} \to \mathcal{H}</math> from which the cross-covariance of mean-zero functions <math>f,g \in \mathcal{H}</math> can be computed as <ref name = "SongCDE">L. Song, J. Huang, A. J. Smola, K. Fukumizu. (2009).[http://www.stanford.edu/~jhuang11/research/pubs/icml09/icml09.pdf Hilbert space embeddings of conditional distributions]. ''Proc. Int. Conf. Machine Learning''. Montreal, Canada: 961–968.</ref>

:<math>\operatorname{Cov}_{XY} (f(X), g(Y)) := \mathbb{E}_{XY} [f(X) g(Y)] = \langle f , \mathcal{C}_{XY} g \rangle_{\mathcal{H}} = \langle f \otimes g , \mathcal{C}_{XY} \rangle_{\mathcal{H} \otimes \mathcal{H}}</math>

Given <math>n</math> pairs of training examples <math>\{(x_1, y_1), \dots, (x_n, y_n)\} </math> drawn i.i.d. from <math>P</math>, we can also empirically estimate the joint distribution kernel embedding via