Continuum Isomap for manifold learnings

Recently, the Isomap algorithm has been proposed for learning a parameterized manifold from a set of unorganized samples from the manifold. It is based on extending the classical multidimensional scaling method for dimension reduction, replacing pairwise Euclidean distances by the geodesic distances on the manifold. A continuous version of Isomap called continuum Isomap is proposed. Manifold learning in the continuous framework is then reduced to an eigenvalue problem of an integral operator. It is shown that the continuum Isomap can perfectly recover the underlying parameterization if the mapping associated with the parameterized manifold is an isometry and its domain is convex. The continuum Isomap also provides a natural way to compute low-dimensional embeddings for out-of-sample data points. Some error bounds are given for the case when the isometry condition is violated. Several illustrative numerical examples are also provided.