Laplacian Eigenmaps¶

Laplacian Eigenmaps (LEM) method uses spectral techniques to perform dimensionality reduction. This technique relies on the basic assumption that the data lies in a low-dimensional manifold in a high-dimensional space. The algorithm provides a computationally efficient approach to non-linear dimnsionality reduction that has locally preserving properties [1].

This package defines a LEM type to represent a Laplacian Eigenmaps results, and provides a set of methods to access its properties.

Properties¶

Let M be an instance of LEM, 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.

neighbors(M)¶: The number of nearest neighbors used for approximating local coordinate structure.

eigvals(M)¶: The eigenvalues of alignment matrix.

Data Transformation¶

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

transform(LEM, X; ...)¶

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

This method returns an instance of LEM.

Keyword arguments:

name	description	default
k	The number of nearest neighbors for determining local coordinate structure.	`12`
d	Output dimension.	`2`
t	The temperature parameters of the heat kernel.	`1.0`

Example:

using ManifoldLearning

# suppose X is a data matrix, with each observation in a column
# apply Laplacian Eigenmaps transformation to the dataset
Y = transform(LEM, X; k = 12, d = 2, t = 1.0)

References

[1]	Belkin, M. and Niyogi, P. “Laplacian Eigenmaps for Dimensionality Reduction and Data Representation”. Neural Computation, June 2003; 15 (6):1373-1396. DOI:10.1162/089976603321780317