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.
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