isomap

Isometric feature mapping. Isomap is a widely used low-dimensional embedding methods, where geodesic distances on a weighted graph are incorporated with the classical multidimensional scaling. Isomap is used for computing a quasi-isometric, low-dimensional embedding of a set of high-dimensional data points. Isomap is highly efficient and generally applicable to a broad range of data sources and dimensionalities.

To be specific, the classical MDS performs low-dimensional embedding based on the pairwise distance between data points, which is generally measured using straight-line Euclidean distance. Isomap is distinguished by its use of the geodesic distance induced by a neighborhood graph embedded in the classical scaling. This is done to incorporate manifold structure in the resulting embedding. Isomap defines the geodesic distance to be the sum of edge weights along the shortest path between two nodes. The top n eigenvectors of the geodesic distance matrix, represent the coordinates in the new n-dimensional Euclidean space.

The connectivity of each data point in the neighborhood graph is defined as its nearest k Euclidean neighbors in the high-dimensional space. This step is vulnerable to "short-circuit errors" if k is too large with respect to the manifold structure or if noise in the data moves the points slightly off the manifold. Even a single short-circuit error can alter many entries in the geodesic distance matrix, which in turn can lead to a drastically different (and incorrect) low-dimensional embedding. Conversely, if k is too small, the neighborhood graph may become too sparse to approximate geodesic paths accurately.

This class implements C-Isomap that involves magnifying the regions of high density and shrink the regions of low density of data points in the manifold. Edge weights that are maximized in Multi-Dimensional Scaling(MDS) are modified, with everything else remaining unaffected.

====References:====

• J. B. Tenenbaum, V. de Silva and J. C. Langford A Global Geometric Framework for Nonlinear Dimensionality Reduction. Science 290(5500):2319-2323, 2000.

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