Deriving Photometric Redshifts using Fuzzy Archetypes and
  Self-Organizing Maps. I. Methodology
  We propose a method to substantially increase the flexibility and power of
template fitting-based photometric redshifts by transforming a large numbers of
galaxy spectral templates into a corrresponding collection of "fuzzy
archetypes" using a suitable set of perturbative priors designed to account for
empirical variation in dust attenuation and emission line strengths. To bypass
widely seperated degeneracies in parameter space (e.g., the redshift-reddening
degeneracy), we train Self-Organizing Maps (SOMs) on a large "model catalogs"
generated from appropriate Monte Carlo sampling of our fuzzy archetypes to
cluster the predicted observables in a topologically smooth fashion. Subsequent
sampling over the SOM then allows full reconstruction of the relevant
probability distribution functions (PDFs) using the associated set of inverse
mappings from the SOM to the underlying model parameters. This combined
approach enables the multi-modal exploration of known variation among galaxy
spectral energy distributions (SEDs) using large numbers of archetypes with
minimal modeling assumptions. We demonstrate the power of this approach to
recover full redshift PDFs using discrete Markov Chain Monte Carlo (MCMC)
sampling methods combined with SOMs constructed from model catalogs based on
LSST $ugrizY$ and Euclid $YJH$ mock photometry.
