Diffusion map: Difference between revisions

Revision as of 09:39, 17 March 2018

Revision as of 03:41, 6 April 2018

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Diffusion maps exploit the relationship between [[heat diffusion]] and [[random walk]] [[Markov chain]]. The basic observation is that if we take a random walk on the data, walking to a nearby data-point is more likely than walking to another that is far away. Let <math>(X, \mathcal{A}, \mu)</math> be a [[measure space]], where <math>X</math> is the data set and <math>\mu</math> represents the distribution on the points on <math>X</math>.

Based on this, the connectivity <math>k</math> between two data points, <math>x</math> and <math>y</math>, can be defined as the probability of walking from <math>x</math> to <math>y</math> in one step of the random walk. Usually, this probability is specified in terms of a kernel function onof the two points: <math>k: X \times X \rightarrow \mathbb{R}</math>. For example, the popular Gaussian kernel:

: <math>