SOMz: photometric redshift PDFs with self organizing maps and random atlas

In this paper we explore the applicability of the unsupervised machine learning technique of Self Organizing Maps (SOM) to estimate galaxy photometric redshift probability density functions (PDFs). This technique takes a spectroscopic training set, and maps the photometric attributes, but not the redshifts, to a two dimensional surface by using a process of competitive learning where neurons compete to more closely resemble the training data multidimensional space. The key feature of a SOM is that it retains the topology of the input set, revealing correlations between the attributes that are not easily identified. We test three different 2D topological mapping: rectangular, hexagonal, and spherical, by using data from the DEEP2 survey. We also explore different implementations and boundary conditions on the map and also introduce the idea of a random atlas where a large number of different maps are created and their individual predictions are aggregated to produce a more robust photometric redshift PDF. We also introduced a new metric, the $I$-score, which efficiently incorporates different metrics, making it easier to compare different results (from different parameters or different photometric redshift codes). We find that by using a spherical topology mapping we obtain a better representation of the underlying multidimensional topology, which provides more accurate results that are comparable to other, state-of-the-art machine learning algorithms. Our results illustrate that unsupervised approaches have great potential for many astronomical problems, and in particular for the computation of photometric redshifts.