6.7.2. Graph-Based Methods
Laplacian eigenmaps
The starting point of this method is the assumption that the data points lie on a smooth manifold (hypersurface) M ⊃ X, whose intrinsic dimension is equal to m < N and it is embedded in RN, that is, M ⊂ RN. The dimension m is given as a parameter by the user. In contrast, this is not required in the kernel PCA, where m is the number of dominant components, which, in practice, is determined so that the gap between λm and λm+1 has a “large” value.
The main philosophy behind the method is to compute the low-dimensional representation of the data so that local neighborhood information in X ⊂ M is optimally preserved. In this way, one attempts to get a solution that reflects the geometric structure ...
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