Isomap.docx - Isomap Isomap stands for isometric mapping Isomap is a non-linear dimensionality reduction method based on the spectral theory which tries

Isomap.docx - Isomap Isomap stands for isometric mapping...

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Isomap Isomap stands for isometric mapping. Isomap is a non-linear dimensionality reduction method based on the spectral theory which tries to preserve the geodesic distances in the lower dimension. Isomap starts by creating a neighborhood network. After that, it uses graph distance to the approximate geodesic distance between all pairs of points. And then, through eigenvalue decomposition of the geodesic distance matrix, it finds the low dimensional embedding of the dataset. In non-linear manifolds, the Euclidean metric for distance holds good if and only if neighborhood structure can be approximated as linear. If neighborhood contains holes, then Euclidean distances can be highly misleading. In contrast to this, if we measure the distance between two points by following the manifold, we will have a better approximation of how far or near two points are. Let's understand this with an extremely simple 2-D example. Suppose our data lies on a circular manifold in a 2-D structure like in the image below. Why geodesic distances are better than the Euclidean distances in nonlinear manifolds? We will reduce the data to 1-D using the Euclidean distances and approximate geodesic distances. Now, if we look at the 1-D mapping based on the Euclidean metric, we see that for points which are far apart(a & b) have been mapped poorly. Only the points which can be
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approximated to lie on a linear manifold(c & d) give satisfactory results. On the other hand, see
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