  Many estimation problems in astrophysics are highly complex, with
high-dimensional, non-standard data objects (e.g., images, spectra, entire
distributions, etc.) that are not amenable to formal statistical analysis. To
utilize such data and make accurate inferences, it is crucial to transform the
data into a simpler, reduced form. Spectral kernel methods are non-linear data
transformation methods that efficiently reveal the underlying geometry of
observable data. Here we focus on one particular technique: diffusion maps or
more generally spectral connectivity analysis (SCA). We give examples of
applications in astronomy; e.g., photometric redshift estimation, prototype
selection for estimation of star formation history, and supernova light curve
classification. We outline some computational and statistical challenges that
remain, and we discuss some promising future directions for astronomy and data
mining.
