Diffusion map: Difference between revisions

Revision as of 12:24, 6 October 2021

Revision as of 08:50, 22 April 2022

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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|embeddingsembedding]]s 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), 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 give a global description of the data-set. Compared with other methods, the diffusion map algorithm is robust to noise perturbation and computationally inexpensive.

==Definition of diffusion maps==

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==Application==

In the paper <ref name="Nadler05diffusionmaps" /> Nadler et. al. showed how to design a kernel that reproduces the diffusion induced by a [[Fokker–Planck equation]]. They also explained that, when the data approximate a manifold, one can recover the geometry of this manifold by computing an approximation of the [[Laplace–Beltrami operator]]. This computation is completely insensitive

to the distribution of the points and therefore provides a separation of the statistics and the geometry of the

data. Since diffusion maps give a global description of the data-set, they can measure the distances between pairs of sample points in the manifold in which the data is embedded. Applications based on diffusion maps include [[facial recognition system|face recognition]],<ref name="vmrs">{{cite journal