  Dimension-reduction techniques can greatly improve statistical inference in
astronomy. A standard approach is to use Principal Components Analysis (PCA).
In this work we apply a recently-developed technique, diffusion maps, to
astronomical spectra for data parameterization and dimensionality reduction,
and develop a robust, eigenmode-based framework for regression. We show how our
framework provides a computationally efficient means by which to predict
redshifts of galaxies, and thus could inform more expensive redshift estimators
such as template cross-correlation. It also provides a natural means by which
to identify outliers (e.g., misclassified spectra, spectra with anomalous
features). We analyze 3835 SDSS spectra and show how our framework yields a
more than 95% reduction in dimensionality. Finally, we show that the prediction
error of the diffusion map-based regression approach is markedly smaller than
that of a similar approach based on PCA, clearly demonstrating the superiority
of diffusion maps over PCA for this regression task.
