 
Summary: Localized Diffusion, Part II: CoarseGrained Process
Guy Wolfa
, Aviv Rotbarta
, Gil Davidb
, Amir Averbucha,
aSchool of Computer Sciene, Tel Aviv University, Tel Aviv 69978, Israel
bDepartment of Mathematics, Program in Applied Mathematics, Yale University, New
Haven, CT 06510, USA
Abstract
Dataanalysis methods nowadays are expected to deal with increasingly large
amounts of data. Such massive datasets often contain many redundancies. One
effect from these redundancies is the highdimensionality of datasets, which is
handled by dimensionality reduction techniques. Another effect is the duplic
ity of very similar observations (or datapoints) that can be analyzed together
as a cluster. We propose an approach for dealing with both effects by coarse
graining the popular Diffusion Maps (DM) dimensionality reduction framework
from the datapoint level to the cluster level. This way, the size of the an
alyzed dataset is decreased by only referring to clusters instead of individual
datapoints. Then, the dimensionality of the dataset can be decreased by the
DM embedding. We show that the essential properties (e.g., ergodicity) of the
