  Expanding upon the work of Way and Srivastava 2006 we demonstrate how the use
of training sets of comparable size continue to make Gaussian process
regression (GPR) a competitive approach to that of neural networks and other
least-squares fitting methods. This is possible via new large size matrix
inversion techniques developed for Gaussian processes (GPs) that do not require
that the kernel matrix be sparse. This development, combined with a
neural-network kernel function appears to give superior results for this
problem. Our best fit results for the Sloan Digital Sky Survey (SDSS) Main
Galaxy Sample using u,g,r,i,z filters gives an rms error of 0.0201 while our
results for the same filters in the luminous red galaxy sample yield 0.0220. We
also demonstrate that there appears to be a minimum number of training-set
galaxies needed to obtain the optimal fit when using our GPR rank-reduction
methods. We find that morphological information included with many photometric
surveys appears, for the most part, to make the photometric redshift evaluation
slightly worse rather than better. This would indicate that most morphological
information simply adds noise from the GP point of view in the data used
herein. In addition, we show that cross-match catalog results involving
combinations of the Two Micron All Sky Survey, SDSS, and Galaxy Evolution
Explorer have to be evaluated in the context of the resulting cross-match
magnitude and redshift distribution. Otherwise one may be misled into overly
optimistic conclusions.
