Principal curves: learning and convergence

Principal curves have been defined as “self consistent” smooth curves which pass through the “middle” of a d-dimensional probability distribution or data cloud. We take a new approach by defining principal curves as continuous curves of a given length which minimize the expected squared distance between the curve and points of the space randomly chosen according to a given distribution. The new definition makes it possible to carry out a theoretical analysis of learning principal curves from training data and it also leads to a new practical construction