Sparse Representation of Photometric Redshift PDFs: Preparing for Petascale Astronomy

One of the consequences of entering the era of precision cosmology is the widespread adoption of photometric redshift probability density functions (PDFs). Both current and future photometric surveys are expected to obtain images of billions of distinct galaxies. As a result, storing and analyzing all of these PDFs will be non-trivial and even more severe if a survey plans to compute and store multiple different PDFs. In this paper we propose the use of a sparse basis representation to fully represent individual photo-$z$ PDFs. By using an Orthogonal Matching Pursuit algorithm and a combination of Gaussian and Voigt basis functions, we demonstrate how our approach is superior to a multi-Gaussian fitting, as we require approximately half of the parameters for the same fitting accuracy with the additional advantage that an entire PDF can be stored by using a 4-byte integer per basis function, and we can achieve better accuracy by increasing the number of bases. By using data from the CFHTLenS, we demonstrate that only ten to twenty points per galaxy are sufficient to reconstruct both the individual PDFs and the ensemble redshift distribution, $N(z)$, to an accuracy of 99.9% when compared to the one built using the original PDFs computed with a resolution of $\delta z = 0.01$, reducing the required storage of two hundred original values by a factor of ten to twenty. Finally, we demonstrate how this basis representation can be directly extended to a cosmological analysis, thereby increasing computational performance without losing resolution nor accuracy.