  The development of fast and accurate methods of photometric redshift
estimation is a vital step towards being able to fully utilize the data of
next-generation surveys within precision cosmology. In this paper we apply a
specific approach to spectral connectivity analysis (SCA; Lee & Wasserman 2009)
called diffusion map. SCA is a class of non-linear techniques for transforming
observed data (e.g., photometric colours for each galaxy, where the data lie on
a complex subset of p-dimensional space) to a simpler, more natural coordinate
system wherein we apply regression to make redshift predictions. As SCA relies
upon eigen-decomposition, our training set size is limited to ~ 10,000
galaxies; we use the Nystrom extension to quickly estimate diffusion
coordinates for objects not in the training set. We apply our method to 350,738
SDSS main sample galaxies, 29,816 SDSS luminous red galaxies, and 5,223
galaxies from DEEP2 with CFHTLS ugriz photometry. For all three datasets, we
achieve prediction accuracies on par with previous analyses, and find that use
of the Nystrom extension leads to a negligible loss of prediction accuracy
relative to that achieved with the training sets. As in some previous analyses
(e.g., Collister & Lahav 2004, Ball et al. 2008), we observe that our
predictions are generally too high (low) in the low (high) redshift regimes. We
demonstrate that this is a manifestation of attenuation bias, wherein
measurement error (i.e., uncertainty in diffusion coordinates due to
uncertainty in the measured fluxes/magnitudes) reduces the slope of the
best-fit regression line. Mitigation of this bias is necessary if we are to use
photometric redshift estimates produced by computationally efficient empirical
methods in precision cosmology.
