  The fields of signal processing and information theory have evolved with the
goal of developing formulations to extract intrinsic information from limited
amount of data. When one considers the modeling of unpredictably varying
processes and complex dynamical signals with a large number of unknowns (such
as those encountered in the fields of finance, NLP, communications, etc.),
there is a need for algorithms to have increased sensitivity with short spans
of data, while maintaining stochastic generalization ability. This naturally
calls for an increased focus on localized stochastic representation. So far,
most metrics developed for characterizing signals envision data from entropic
and probabilistic points of view that lack sensitivity towards quick changes in
signal dynamics. We hypothesize that models that work with the intrinsic
uncertainties associated with local data induced metric spaces would be
significantly more sensitive towards signal characterization. To this end, we
develop a new framework for stochastic signal processing that is based on
decomposing the local metric space of the signal in a Gaussian Reproducing
Kernel Hilbert Space (RKHS). A major advantage of our framework is that we are
able to implement this decomposition on a sample-by-sample basis. The key
aspects of our framework are the following: (1) We use a data defined metric
related to Parzen density estimation for quantifying the local structure of
data in the Gaussian RKHS. (2) We use a quantum description of this metric
which consequently introduces uncertainty in the structure of the local kernel
space. Since the RKHS has been well established and known for providing
universal data fitting capabilities, we submit that local quantifiers of the
kernel space data projection could significantly improve acquisition of signal
information.
