Statisticians use various methods in order to sum up information and represent the

Abstract – Principal curves are nonlinear generalizations of the notion of first principal component. Roughly, a principal curve is a parame-terized 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 (num-ber 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 us-ing 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. Index terms – Principal curves, parameter selection, model selection, oracle in-equality, penalty calibration, slope heuristics