Modern multi- and hyper-dimensional processing problems, such as those encountered in many applications
involving image processing, adaptive beamforming, hyperspectral IR detection, medical imaging, STAP,
Volterra calibration, etc., are numerically very demanding due to the vast amounts of data involved. Further
compounding the situation is the fact that many such applications require estimating a set of parameters of
interest that may be so large that the data available, despite its massiveness, may not be enough to properly
calculate the pertinent statistics.
The approach presented here addresses such problems by projecting the available data - both, modeled and
measured - into a reduced-dimensionality domain where the estimation process is then performed. This
strategy is extremely useful when the parameter set is not the final objective per se, but rather just a means to
an end (e.g., a classification decision, detecting a signal of interest, etc.). In particular, we will concentrate on
the case of finding the optimal projector for a given problem of interest where a priori information may be
available. This means that the reduced-dimensionality domain must be selected as one incorporating and
preserving that knowledge. We explore the use of Krylov Subspaces to achieve this end, as they inherently
allow the inclusion of such data.
In order to maintain a visage of practicality, we have chosen to present our developments from the
perspective of the adaptive processing (filtering) problem, as this enables our presentation to be applicable to
the endless expanse of optimization problems that can be addressed via a Least Squares formulation.
Regularization issues, as well as extensions to non-linear filters (Taylor/Volterra/polynomial), will also be
presented so as to provide additional ideas regarding the usefulness and malleability of our methods.