  We apply a novel spectral graph technique, that of locally-biased
semi-supervised eigenvectors, to study the diversity of galaxies. This
technique permits us to characterize empirically the natural variations in
observed spectra data, and we illustrate how this approach can be used in an
exploratory manner to highlight both large-scale global as well as small-scale
local structure in Sloan Digital Sky Survey (SDSS) data. We use this method in
a way that simultaneously takes into account the measurements of spectral lines
as well as the continuum shape. Unlike Principal Component Analysis, this
method does not assume that the Euclidean distance between galaxy spectra is a
good global measure of similarity between all spectra, but instead it only
assumes that local difference information between similar spectra is reliable.
Moreover, unlike other nonlinear dimensionality methods, this method can be
used to characterize very finely both small-scale local as well as large-scale
global properties of realistic noisy data. The power of the method is
demonstrated on the SDSS Main Galaxy Sample by illustrating that the derived
embeddings of spectra carry an unprecedented amount of information. By using a
straightforward global or unsupervised variant, we observe that the main
features correlate strongly with star formation rate and that they clearly
separate active galactic nuclei. Computed parameters of the method can be used
to describe line strengths and their interdependencies. By using a
locally-biased or semi-supervised variant, we are able to focus on typical
variations around specific objects of astronomical interest. We present several
examples illustrating that this approach can enable new discoveries in the data
as well as a detailed understanding of very fine local structure that would
otherwise be overwhelmed by large-scale noise and global trends in the data.
