  In this paper, we consider the problem of non-linear dimensionality reduction
under uncertainty, both from a theoretical and algorithmic perspectives. Since
real-world data usually contain measurements with uncertainties and artifacts,
the input space in the proposed framework consists of probability distributions
to model the uncertainties associated with each sample. We propose a new
dimensionality reduction framework, called NGEU, which leverages uncertainty
information and directly extends several traditional approaches, e.g., KPCA,
MDA/KMFA, to receive as inputs the probability distributions instead of the
original data. We show that the proposed NGEU formulation exhibits a global
closed-form solution, and we analyze, based on the Rademacher complexity, how
the underlying uncertainties theoretically affect the generalization ability of
the framework. Empirical results on different datasets show the effectiveness
of the proposed framework.
