  Knowing the redshift of galaxies is one of the first requirements of many
cosmological experiments, and as it's impossible to perform spectroscopy for
every galaxy being observed, photometric redshift (photo-z) estimations are
still of particular interest. Here, we investigate different deep learning
methods for obtaining photo-z estimates directly from images, comparing these
with traditional machine learning algorithms which make use of magnitudes
retrieved through photometry. As well as testing a convolutional neural network
(CNN) and inception-module CNN, we introduce a novel mixed-input model which
allows for both images and magnitude data to be used in the same model as a way
of further improving the estimated redshifts. We also perform benchmarking as a
way of demonstrating the performance and scalability of the different
algorithms. The data used in the study comes entirely from the Sloan Digital
Sky Survey (SDSS) from which 1 million galaxies were used, each having 5-filter
(ugriz) images with complete photometry and a spectroscopic redshift which was
taken as the ground truth. The mixed-input inception CNN achieved a mean
squared error (MSE)=0.009, which was a significant improvement (30%) over the
traditional Random Forest (RF), and the model performed even better at lower
redshifts achieving a MSE=0.0007 (a 50% improvement over the RF) in the range
of z<0.3. This method could be hugely beneficial to upcoming surveys such as
the Vera C. Rubin Observatory's Legacy Survey of Space and Time (LSST) which
will require vast numbers of photo-z estimates produced as quickly and
accurately as possible.
