The document discusses manifold learning and its implementation in MATLAB, detailing the definitions, parameters, and code examples for generating embeddings. It describes how to load input data from various sources and specifies important parameters like target dimension, nearest neighbors, and kernel width for different methods. The output configuration for specific parameters is also provided, illustrating how to generate results using the described methods.

1. Manifold Learning MATLAB
demo

2. 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.

3. 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

4. General Code
 X=rand(10,50);
 k=20;
 dmax=2;
 [Y] = lle(X,k,dmax);

5. Functioning of code in MATLAB

6. 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.

7. 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, where
N 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.

8. 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.

9.  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 Laplacian
Alpha = 1/2 is the Fokker-Plank propagator
Alpha = 1 is the Laplace-Beltrami operator

10. Output
The output for:
d = 2;
k = 8;
Sigma = 10.0;
Alpha = 1.0;