A manifold learning approach to dimensionality reduction for modeling data

Manifold learning has gained in recent years a great attention in facing the problem of dimensionality reduction of high-dimensional data. This technique is based on the assumption that data are embedded in a nonlinear manifold of lower dimension. In this context the dimension of the embedding is a key parameter, therefore is of paramount importance for dimensionality reduction to discover the appropriate dimensionality of the reduced feature space, that is the intrinsic dimension (ID) of data. The purpose of this paper is to derive a manifold learning approach to dimensionality reduction for modeling data coming from either causal or noncausal signals. The approach is based on some theoretical results that aim first at giving a practical method for the estimation of the intrinsic dimension and then at deriving a local parametrization of data. Besides, an explicit nonlinear mapping relationship from data to the reduced space can be obtained as the regression of a nonlinear function. Several experiments on both synthetic and real data for the two classes of causal and noncausal signals have been conducted to validate the proposed approach.