smile.manifold

Class LaplacianEigenmap

java.lang.Object

smile.manifold.LaplacianEigenmap

All Implemented Interfaces:

java.io.Serializable

public class LaplacianEigenmap
extends java.lang.Object
implements java.io.Serializable

Laplacian Eigenmap. Using the notion of the Laplacian of the nearest neighbor adjacency graph, Laplacian Eigenmap computes a low dimensional representation of the dataset that optimally preserves local neighborhood information in a certain sense. The representation map generated by the algorithm may be viewed as a discrete approximation to a continuous map that naturally arises from the geometry of the manifold.

The locality preserving character of the Laplacian Eigenmap algorithm makes it relatively insensitive to outliers and noise. It is also not prone to "short circuiting" as only the local distances are used.

See Also:

IsoMap, LLE,

References

1. Mikhail Belkin and Partha Niyogi. Laplacian Eigenmaps and Spectral Techniques for Embedding and Clustering. NIPS, 2001.

, Serialized Form

Field Summary

Fields

Modifier and Type	Field and Description
double[][]	coordinates The coordinate matrix in embedding space.
smile.graph.AdjacencyList	graph Nearest neighbor graph.
int[]	index The original sample index.
double	width The width of heat kernel.

Constructor Summary

Constructors

Constructor and Description
LaplacianEigenmap(double width, int[] index, double[][] coordinates, smile.graph.AdjacencyList graph) Constructor with Gaussian kernel.
LaplacianEigenmap(int[] index, double[][] coordinates, smile.graph.AdjacencyList graph) Constructor with discrete weights.

Method Summary

Modifier and Type	Method and Description
static LaplacianEigenmap	of(double[][] data, int k) Laplacian Eigenmaps with discrete weights.
static LaplacianEigenmap	of(double[][] data, int k, int d, double t) Laplacian Eigenmap with Gaussian kernel.
static <T> LaplacianEigenmap	of(T[] data, smile.math.distance.Distance<T> distance, int k) Laplacian Eigenmaps with discrete weights.
static <T> LaplacianEigenmap	of(T[] data, smile.math.distance.Distance<T> distance, int k, int d, double t) Laplacian Eigenmap with Gaussian kernel.

Methods inherited from class java.lang.Object

clone, equals, finalize, getClass, hashCode, notify, notifyAll, toString, wait, wait, wait

Field Detail

width

public final double width

The width of heat kernel.

index

public final int[] index

The original sample index.

coordinates

public final double[][] coordinates

The coordinate matrix in embedding space.

graph

public final smile.graph.AdjacencyList graph

Nearest neighbor graph.

Constructor Detail

LaplacianEigenmap

public LaplacianEigenmap(int[] index,
                         double[][] coordinates,
                         smile.graph.AdjacencyList graph)

Constructor with discrete weights.

Parameters:

index - the original sample index.

coordinates - the coordinates.

graph - the nearest neighbor graph.

LaplacianEigenmap

public LaplacianEigenmap(double width,
                         int[] index,
                         double[][] coordinates,
                         smile.graph.AdjacencyList graph)

Constructor with Gaussian kernel.

Parameters:

width - the width of heat kernel.

index - the original sample index.

coordinates - the coordinates.

graph - the nearest neighbor graph.

Method Detail

of

public static LaplacianEigenmap of(double[][] data,
                                   int k)

Laplacian Eigenmaps with discrete weights.

Parameters:

data - the input data.

k - k-nearest neighbor.

of

public static LaplacianEigenmap of(double[][] data,
                                   int k,
                                   int d,
                                   double t)

Laplacian Eigenmap with Gaussian kernel.

Parameters:

data - the input data.

d - the dimension of the manifold.

k - k-nearest neighbor.

t - the smooth/width parameter of heat kernel e^{-||x-y||2 / t}. Non-positive value means discrete weights.

of

public static <T> LaplacianEigenmap of(T[] data,
                                       smile.math.distance.Distance<T> distance,
                                       int k)

Laplacian Eigenmaps with discrete weights.

Parameters:

data - the input data.

distance - the distance measure.

k - k-nearest neighbor.

of

public static <T> LaplacianEigenmap of(T[] data,
                                       smile.math.distance.Distance<T> distance,
                                       int k,
                                       int d,
                                       double t)

Laplacian Eigenmap with Gaussian kernel.

Parameters:

data - the input data.

distance - the distance measure.

k - k-nearest neighbor.

d - the dimension of the manifold.

t - the smooth/width parameter of heat kernel e^{-||x-y||2 / t}. Non-positive value means discrete weights.