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 ofX
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 |