Functional data representation using correntropy locally linear embedding

Unlike classical linear dimensionality reduction techniques, nonlinear ones are capable of discovering the nonlinear degrees of freedom that are present in natural observations, by assuming that the data lie on an embedded nonlinear manifold within an observed high dimensional feature space. Nevertheless, when measured objects are actually functional data, nonlinear dimensionality reduction techniques do not produce suitable unfolded results, particularly in the locally linear embedding (LLE) algorithm. In this case, the Euclidean distance employed in cost function does not correctly represent the similarity between objects, because this distance does not take into account the intrinsical relations presented in functional data, besides, it is easily distorted by non-gaussian noise, such as an artifacts, impulsive noise, etc. The main contribution in this paper is to use (inside the LLE algorithm) a localized similarity measure called correntropy, which has a particular metric that allows it to conform to suitable neighborhoods, and indeed convenient representations of the objects, avoiding distortions.