## Regularization-Free Principal Curve Estimation

*Samuel Gerber, Ross Whitaker*; 14(May):1285−1302, 2013.

### Abstract

Principal curves and manifolds provide a framework to formulate
manifold learning within a statistical context. Principal curves
define the notion of a curve passing through the middle of a
distribution. While the intuition is clear, the formal
definition leads to some technical and practical difficulties.
In particular, principal curves are saddle points of the mean-
squared projection distance, which poses severe challenges for
estimation and model selection. This paper demonstrates that the
difficulties in model selection associated with the saddle point
property of principal curves are intrinsically tied to the
minimization of the mean-squared projection distance. We
introduce a new objective function, facilitated through a
modification of the principal curve estimation approach, for
which all critical points are principal curves and minima. Thus,
the new formulation removes the fundamental issue for model
selection in principal curve estimation. A gradient-descent-
based estimator demonstrates the effectiveness of the new
formulation for controlling model complexity on numerical
experiments with synthetic and real data.

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