  Diffusion Maps framework is a kernel based method for manifold learning and
data analysis that defines diffusion similarities by imposing a Markovian
process on the given dataset. Analysis by this process uncovers the intrinsic
geometric structures in the data. Recently, it was suggested to replace the
standard kernel by a measure-based kernel that incorporates information about
the density of the data. Thus, the manifold assumption is replaced by a more
general measure-based assumption.
  The measure-based diffusion kernel incorporates two separate independent
representations. The first determines a measure that correlates with a density
that represents normal behaviors and patterns in the data. The second consists
of the analyzed multidimensional data points.
  In this paper, we present a representation framework for data analysis of
datasets that is based on a closed-form decomposition of the measure-based
kernel. The proposed representation preserves pairwise diffusion distances that
does not depend on the data size while being invariant to scale. For a
stationary data, no out-of-sample extension is needed for embedding newly
arrived data points in the representation space. Several aspects of the
presented methodology are demonstrated on analytically generated data.
