This method relies on results from Spectral Graph Theory to find a low-dimensional representation of high-dimensional data points embedded in a non-linear manifold.
Similarly to Isomaps_Old Kiwi, the Laplacian Eigenmaps work by constructing a connected graph connecting neighboring data points. Instead of the adjacency matrix, the algorithm deals with the Laplacian matrix of the graph defined as:
The reasons for this choice are results from the Spectral Graph Theory and are far beyond the scope of this article.
Also from the Spectral Graph Theory, we use the results that the eigenvectors corresponding to the lowest eigenvalues that are different than zero produce a good low-dimensional map of the points.
Bibliography
- Spectral Graph Theory and its Applications, Lecture notes from Prof. Dan Spielman at Yale, 2004 [[1]].
- Laplacian Eigenmaps and Spectral Techniques for Embedding and Clustering, M. Belkin and, P. Niyogi, 2001[[2]].
