PARAMAP vs. Isomap: A Comparison of Two Nonlinear Mapping Algorithms |
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Authors: | Ulas Akkucuk J Douglas Carroll |
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Institution: | (1) Department of Management, Bogazici University, Bebek, 34342, Istanbul, Turkey;(2) Rutgers Business School, Rutgers University, Management Education Center, MEC #125, 111 Washington Street, Newark, NJ 07102-3207, USA |
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Abstract: | Dimensionality reduction techniques are used for representing higher dimensional data by a more parsimonious and meaningful
lower dimensional structure. In this paper we will study two such approaches, namely Carroll’s Parametric Mapping (abbreviated
PARAMAP) (Shepard and Carroll, 1966) and Tenenbaum’s Isometric Mapping (abbreviated Isomap) (Tenenbaum, de Silva, and Langford,
2000). The former relies on iterative minimization of a cost function while the latter applies classical MDS after a preprocessing
step involving the use of a shortest path algorithm to define approximate geodesic distances. We will develop a measure of
congruence based on preservation of local structure between the input data and the mapped low dimensional embedding, and compare
the different approaches on various sets of data, including points located on the surface of a sphere, some data called the
"Swiss Roll data", and truncated spheres. |
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Keywords: | |
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