  To study the effect of supermassive black holes (SMBHs) on their host
galaxies it is important to study the hosts when the SMBH is near its peak
activity. A method to investigate the host galaxies of high luminosity quasars
is to obtain optical spectra at positions offset from the nucleus where the
relative contribution of the quasar and host are comparable. However, at these
extended radii the galaxy surface brightness is often low (20-22 mag per
arcsec$^{2}$) and the resulting spectrum might have such low S/N that it
hinders analysis with standard stellar population modeling techniques. To
address this problem we have developed a method that can recover galaxy star
formation histories (SFHs) from rest frame optical spectra with S/N $\sim$
5~\AA$^{-1}$. This method uses the statistical technique diffusion k-means to
tailor the stellar population modeling basis set. Our diffusion k-means minimal
basis set, composed of 4 broad age bins, is successful in recovering a range of
galaxy SFHs. Additionally, using an analytic prescription for seeing
conditions, we are able to simultaneously model scattered quasar light and the
SFH of quasar host galaxies (QHGs). We use synthetic data to compare results of
our novel method with previous techniques. We also present the modeling results
on a previously published QHG and show that galaxy properties recovered from a
diffusion k-means basis set are less sensitive to noise added to this quasar
host galaxy spectrum. Our new method has a clear advantage in recovering
information from QHGs and could also be applied to the analysis of other low
S/N galaxy spectra such as those typically obtained for high redshift objects
or integral field spectroscopic surveys.
