Tag Archives: Robust

Dimensionality reduction: comparison of different methods

I’ve already given some answers in one of my first tickets on manifold learning. Here I will give some more complete results on the quality of the dimensionality reduction performed by the most well-known techniques.

First of all, my test is about respecting the geodesic distances in the reduced space. This is not possible for some manifolds like a Gaussian 2D plot. I used the SCurve to create the test, as the speed on the curve is unitary and thus the distances in the coordinate space (the one I used to create the SCurve) are the same as the geodesic ones on the manifold. My test measures the matrix (Frobenius) norm between the original coordinates and the computed one up to an affine transform of the latter.
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More on manifold learning

I hope to present here some result in February, but I’ll expose what I’ve implemented so far :

  • Isomap
  • LLE
  • Laplacian Eigenmaps
  • Hessian Eigenmaps
  • Diffusion Maps (in fact a variation of Laplacian Eigenmaps)
  • Curvilinear Component Analysis (the reduction part)
  • NonLinear Mapping (Sammon)
  • My own technique (reduction, regression and projection)
  • PCA (usual reduction, but robust projection with an a priori term)

The results I will show here are mainly reduction comparison between the techniques, knowing that each technique has a specific field of application : LLE is not made to respect the geodesic distances, Isomap, NLM and my technique are.

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Manifold learning toolbox for Python

As I approach the end of my PhD, I will propose my manifold learning code in a scikit (see this page) in a few weeks. For the moment, I don’t know which scikit will be used, but stay put…

The content of the scikit will be :

  • Isomap
  • LLE
  • Laplacian eigenmaps
  • Diffusion maps
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