  The Karhunen-Loeve (KL) transform can compactly represent the information
contained in large, complex datasets, cleanly eliminating noise from the data
and identifying elements of the dataset with extreme or inconsistent
characteristics. We develop techniques to apply the KL transform to the
4000-5700A region of 9,800 QSO spectra with z < 0.619 from the SDSS archive. Up
to 200 eigenspectra are needed to fully reconstruct the spectra in this sample
to the limit of their signal/noise. We propose a simple formula for selecting
the optimum number of eigenspectra to use to reconstruct any given spectrum,
based on the signal/noise of the spectrum, but validated by formal
cross-validation tests. We show that such reconstructions can boost the
effective signal/noise of the observations by a factor of 6 as well as fill in
gaps in the data. The improved signal/noise of the resulting set will allow for
better measurement and analysis of these spectra. The distribution of the QSO
spectra within the eigenspace identifies regions of enhanced density of
interesting subclasses, such as Narrow Line Seyfert 1s (NLS1s). The weightings,
as well as the inability of the eigenspectra to fit some of the objects, also
identifies "outliers," which may be objects that are not valid members of the
sample or objects with rare or unique properties. We identify 48 spectra from
the sample that show no broad emission lines, 21 objects with unusual [O III]
emission line properties, and 9 objects with peculiar H-beta emission line
profiles. We also use this technique to identify a binary supermassive black
hole candidate. We provide the eigenspectra and the reconstructed spectra of
the QSO sample.
