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Abstract:
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Dimensionality reduction techniques are used for representing higher-dimensional data by a more parsimonious and meaningful lower-dimensional structure. Such methods have potential application to visualizing and interpreting high-dimensional data. In this paper, we will study Carroll's Parametric Mapping (PARAMAP). The PARAMAP algorithm relies on iterative minimization of a cost function measuring "smoothness" of the mapping from the low- to the high-dimensional space. We will develop a measure of congruence based on preservation of local structure between the input data and the mapped low-dimensional embedding and demonstrate the application of PARAMAP to various sets of nonlinear manifolds, including points located on the surface of a sphere, "Swiss Roll Data," and truncated spheres.
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