Dimension-reduction techniques can greatly improve statistical inference in astronomy. A standard approach is to use Principal Components Analysis (PCA). In this letter we apply a recently-developed technique, diffusion maps, to astronomical spectra, and develop a robust, eigenmode-based framework for regression and data parameterization. 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). We analyze 3846 SDSS spectra and show how our framework yields an approximately 99% percent 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 and traditional linear data reduction techniques.