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<p><b>Abstract</b>—Principal curves and surfaces are nonlinear generalizations of principal components and subspaces, respectively. They can provide insightful summary of high-dimensional data not typically attainable by classical linear methods. Solutions to several problems, such as proof of existence and convergence, faced by the original principal curve formulation have been proposed in the past few years. Nevertheless, these solutions are not generally extensible to principal surfaces, the mere computation of which presents a formidable obstacle. Consequently, relatively few studies of principal surfaces are available. Recently, we proposed the probabilistic principal surface (PPS) to address a number of issues associated with current principal surface algorithms. PPS uses a manifold oriented covariance noise model, based on the generative topographical mapping (GTM), which can be viewed as a parametric formulation of Kohonen's self-organizing map. Building on the PPS, we introduce a unified covariance model that implements PPS <tmath>$\left( 0<\alpha<1\right) $</tmath>, GTM <tmath>$\left( \alpha=1\right) $</tmath>, and the manifold-aligned GTM <tmath>$\left( \alpha>1\right) $</tmath> by varying the clamping parameter <tmath>$\alpha$</tmath>. Then, we comprehensively evaluate the empirical performance (reconstruction error) of PPS, GTM, and the manifold-aligned GTM on three popular benchmark data sets. It is shown in two different comparisons that the PPS outperforms the GTM under identical parameter settings. Convergence of the PPS is found to be identical to that of the GTM and the computational overhead incurred by the PPS decreases to <tmath>$40$</tmath> percent or less for more complex manifolds. These results show that the generalized PPS provides a flexible and effective way of obtaining principal surfaces.</p>
Principal curve, principal surface, probabilistic, dimensionality reduction, nonlinear manifold, generative topographic mapping.

J. Ghosh and K. Chang, "A Unified Model for Probabilistic Principal Surfaces," in IEEE Transactions on Pattern Analysis & Machine Intelligence, vol. 23, no. , pp. 22-41, 2001.
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