# 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 dimensionality 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.

ManifoldLearning.LEMType
LEM{NN <: AbstractNearestNeighbors, T <: Real} <: AbstractDimensionalityReduction

The LEM type represents a Laplacian eigenmaps model constructed for T type data with a help of the NN nearest neighbor algorithm.

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StatsBase.fitMethod
fit(LEM, data; k=12, maxoutdim=2, ɛ=1.0, nntype=BruteForce)

Fit a Laplacian eigenmaps model to data.

Arguments

• data: a matrix of observations. Each column of data is an observation.

Keyword arguments

• k: a number of nearest neighbors for construction of local subspace representation
• maxoutdim: a dimension of the reduced space.
• nntype: a nearest neighbor construction class (derived from AbstractNearestNeighbors)
• ɛ: a Gaussian kernel variance (the scale parameter)

Examples

M = fit(LEM, rand(3,100)) # construct Laplacian eigenmaps model
R = transform(M)          # perform dimensionality reduction
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## References

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