Understand Manifolds using MATLAB

Manifold Learning MATLAB demo

Manifolds

“According to mathematics, manifold is a collection of points forming a certain kind of set, such as those of topologically closed surface.”

Example: Surface, Curve & point.

Terms to be used in code

X = data as D x N matrix (D = dimensionality, N = #points)

K = number of neighbors

dmax = max embedding dimensionality

Y = embedding as dmax x N matrix

General Code

X=rand(10,50);

k=20;

dmax=2;

[Y] = lle(X,k,dmax);

Functioning of code in MATLAB

Animated Version using MATLAB

The input data can be read from a matrix in the workspace, from a text file, or selected from one of 8 built-in examples.

Description to every option

Load Matrix: Enter the name of a matrix in the MATLAB workspace and press this button. The matrix should be NxD, whereN is the number of data items;D is the dimension of the manifold.

Load File: Enter the name of a text file and press this button. The text file should contain numbers separated by spaces with each data element as a row.

Parameters used in simulation

Target Dimension d: The desired dimension of the embedding. In general, the target dimension d should be less than the input manifold dimension D.

Nearest Neighbors K: Specifies the number of nearest neighbors (KNN) used to build the graph for the following methods: ISOMAP, LLE, Hessian LLE, Laplacian, and LTSA.

Sigma: This specifies the width of the Gaussian kernel in the Diffusion Map method.

Alpha: This parameter controls the normalization used by Diffusion Map.Alpha = 0 is the Graph LaplacianAlpha = 1/2 is the Fokker-Plank propagatorAlpha = 1 is the Laplace-Beltrami operator

Output

The output for:

d = 2; k = 8;Sigma = 10.0;Alpha = 1.0;