Local Tangent Space Alignment¶

Local tangent space alignment (LTSA) is a method for manifold learning, which can efficiently learn a nonlinear embedding into low-dimensional coordinates from high-dimensional data, and can also reconstruct high-dimensional coordinates from embedding coordinates [1].

This package defines a LTSA type to represent a LTSA results, and provides a set of methods to access its properties.

Properties¶

Let M be an instance of LTSA, 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 LTSA over a given dataset.

transform(LSTA, X; ...)¶

Perform LTSA over the data given in a matrix X. Each column of X is an observation.

This method returns an instance of LTSA.

Keyword arguments:

name	description	default
k	The number of nearest neighbors for determining local coordinate structure.	`12`
d	Output dimension.	`2`

Example:

using ManifoldLearning

# suppose X is a data matrix, with each observation in a column
# apply LTSA transformation to the dataset
Y = transform(LTSA, X; k = 12, d = 2)

References

[1]	Zhang, Zhenyue; Hongyuan Zha. “Principal Manifolds and Nonlinear Dimension Reduction via Local Tangent Space Alignment”. SIAM Journal on Scientific Computing 26 (1): 313–338, 2004. DOI:10.1137/s1064827502419154