Laplacian Eigenmaps for dimensionality reduction and data representation
Neural Computation
A kernel view of the dimensionality reduction of manifolds
ICML '04 Proceedings of the twenty-first international conference on Machine learning
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The problem addressed in nonlinear dimensionality reduction, is to find lower dimensional configurations of high dimensional data, thereby revealing underlying structure. One popular method in this regard is the Isomap algorithm, where local information is used to find approximate geodesic distances. From such distance estimations, lower dimensional representations, accurate on a global scale, are obtained by multidimensional scaling. The property of global approximation sets Isomap in contrast to many competing methods, which approximate only locally. A serious drawback of Isomap is that it is topologically instable, i.e., that incorrectly chosen algorithm parameters or perturbations of data may abruptly alter the resulting configurations. To handle this problem, we propose new methods for more robust approximation of the geodesic distances. This is done using a viewpoint of electric circuits. The robustness is validated by experiments. By such an approach we achieve both the stability of local methods and the global approximation property of global methods.