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 ofX
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 |