Diffusion map: Difference between revisions

Revision as of 01:44, 11 February 2019

Revision as of 04:23, 20 April 2019

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[[File:Diffusion_map_of_a_torodial_helix.jpg|thumb|right|Given non-uniformly sampled data points on a toroidal helix (top), the first two Diffusion Map coordinates with Laplace-Beltrami normalization are plotted (bottom). The Diffusion Map unravels the toroidal helix recovering the underlying intrinsic circular geometry of the data.]]

'''Diffusion maps''' is a [[dimensionality reduction]] or [[feature extraction]] algorithm introduced by [[Ronald Coifman| Coifman]] and Lafon<ref name="PNAS1" /><ref name="PNAS2" /><ref name="DifussionMap" /><ref name="Diffusion" /> which computes a family of [[Embedding|embeddings]] of a data set into Euclidean space (often low-dimensional) whose coordinates can be computed from the eigenvectors and eigenvalues of a diffusion operator on the data. The Euclidean distance between points in the embedded space is equal to the "diffusion distance" between probability distributions centered at those points. Different from linear dimensionality reduction methods such as [[principal component analysis]] (PCA) and [[multi-dimensional scaling]] (MDS), diffusion maps is part of the family of [[nonlinear dimensionality reduction]] methods which focus on discovering the underlying [[manifold]] that the data has been sampled from. By integrating local similarities at different scales, diffusion maps givesgive a global description of the data-set. Compared with other methods, the diffusion maps algorithm is robust to noise perturbation and is computationally inexpensive.

==Definition of diffusion maps==