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(<math>k</math> is positivity preserving).
The kernel constitutes the prior definition of the ''local'' geometry of the data-set. Since a given kernel will capture a specific feature of the data set, its choice should be guided by the application that one has in mind. This is a major difference with methods such as [[principal component analysis]], where correlations between all data points are taken into account at once.
Given <math>(X, k)</math>, we can then construct a reversible Markov chain on <math>X</math> (a process known as the normalized graph Laplacian construction):
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