Diffusion maps¶

Diffusion maps leverages the relationship between heat diffusion and a random walk; an analogy is drawn between the diffusion operator on a manifold and a Markov transition matrix operating on functions defined on the graph whose nodes were sampled from the manifold [1].

This package defines a DiffMap type to represent a Hessian LLE results, and provides a set of methods to access its properties.

Properties¶

Let M be an instance of DiffMap, n be the number of observations, and d be the output dimension.

outdim(M)¶: Get the output dimension d, i.e the dimension of the subspace.

projection(M)¶: Get the projection matrix (of size (d, n)). Each column of the projection matrix corresponds to an observation in projected subspace.

kernel(M)¶: The kernel matrix.

Data Transformation¶

One can use the transform method to perform DiffMap over a given dataset.

transform(DiffMap, X; ...)¶

Perform DiffMap over the data given in a matrix X. Each column of X is an observation.

This method returns an instance of DiffMap.

Keyword arguments:

name	description	default
d	Output dimension.	`2`
t	Number of time steps.	`1`
ɛ	The scale parameter.	`1.0`

Example:

using ManifoldLearning

# suppose X is a data matrix, with each observation in a column
# apply DiffMap transformation to the dataset
Y = transform(DiffMap, X; d=2, t=1, ɛ=1.0)

References

[1]	Coifman, R. & Lafon, S. “Diffusion maps”. Applied and Computational Harmonic Analysis, Elsevier, 2006, 21, 5-30. DOI:10.1073/pnas.0500334102