Principal Curve Algorithms for Data Sets
Curves
Manifolds (surfaces)
Most of the extensions of principal curves use
multidimensional base functions to represent
principal manifolds. The algorithms differ in the selection of the
set of base functions and in the
approach of optimizing the parameters. Another
popular approach is to use variants of the
self-organizing map algorithm.
- Michael LeBlanc and Robert Tibshirani
optimizes linear base functions by using
multivariate adaptive regression splines.
- D. Dong and T. J. McAvoy use autoassociative neural networks
and optimize the parameters by using the conjugate gradient method.
- Alex Smola,
Sebastian Mika, and
Bernhard Schölkopf
optimize Gaussian kernels by using an
expectation-minimization-like iteration. They also point
out the similarities between their method and the
generative topographic mapping.
- Kui-yu Chang and
Joydeep Ghosh use a
parametric model based on the
generative topographic mapping, and use the
expectation-minimization algorithm to optimize the parameters.
- Ralf Der, Ulrich Steinmetz,
Gerd Balzuweit, and Gerrit Schüürmann apply the
self-organizing map algorithm to extract principal
manifolds and extended principal curves ("principal graphs")
from data.