Locally Linear Embedding¶

Locally Linear Embedding (LLE) technique builds a single global coordinate system of lower dimensionality. By exploiting the local symmetries of linear reconstructions, LLE is able to learn the global structure of nonlinear manifolds [1].

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

Properties¶

Let M be an instance of LLE, 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 HLLE over a given dataset.

transform(LLE, X; ...)¶

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

This method returns an instance of LLE.

Keyword arguments:

name	description	default
k	The number of nearest neighbors for determining local coordinate structure.	`12`
d	Output dimension.	`2`

Example:

using ManifoldLearning

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

References

[1]	Roweis, S. & Saul, L. “Nonlinear dimensionality reduction by locally linear embedding”, Science 290:2323 (2000). DOI:10.1126/science.290.5500.2323