Parameter selection for principal curves

Principal curves are nonlinear generalizations of the notion of first principal component. Roughly, a principal curve is a parameterized curve in Rd which passes through the "middle" of a data cloud drawn from some unknown probability distribution. Depending on the definition, a principal curve relies on some unknown parameters (number of segments, length, turn. . . ) which have to be properly chosen to recover the shape of the data without interpolating. In the present paper, we consider the principal curve problem from an empirical risk minimization perspective and address the parameter selection issue using the point of view of model selection via penalization. We offer oracle inequalities and implement the proposed approaches to recover the hidden structures in both simulated and real-life data