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With PCA, it is possible to reconstruct high-dimensional data from its low-dimensional point by $$ x_i' = Pb + \bar X $$ Where $\bar X$ is the mean of training set $X$, $P$ is the eigenvectors and $b$ is a principal component.

Can this be achieved with kernel PCA (KPCA)? If so, how?

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    $\begingroup$ +1. See my answer here stats.stackexchange.com/questions/131142, in particular around the reference to the Mika et al. 1998 paper. They call it "pre-image". You can also try googling pre-image kernel PCA (try this google scholar query) and browse through the literature. I am not particularly familiar with it. $\endgroup$
    – amoeba
    Commented Jun 22, 2016 at 13:09

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