Matthieu Brucher's blog

Some of the widely used method are based on a similarity graph made with the local structure. For instance LLE uses the relative distances, which is related to similarities. Using similarities allows the use of sparse techniques. Indeed, a lot of points are not similar, and then the similarities matrix is sparse. This also means that a lot of manifold can be reduced with these techniques, but not with Isomap or the other geodesic-based techniques.

It is worth mentioning that I only implemented Laplacian Eigenmaps with a sparse matrix, due to the lack of generalized eigensolver for sparse matrix, but it will be available in a short time, I hope.

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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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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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