Optimizing Photometric Redshift Training Sets I: Efficient Compression of the Galaxy Color-Redshift Relation with UMAP

Abstract

Spectroscopic datasets are essential for training and calibrating photometric redshift (photo-$z$) methods. However, spectroscopic redshifts (spec-$z$'s) constitute a biased and sparse sampling of the photometric galaxy population, which creates difficulties for the common grid-based approach for mapping color to redshift using self-organizing maps (SOMs). Instead, we utilized the uniform manifold approximation and projection (UMAP) algorithm to compress a Rubin--Roman-like $ugrizyJH$ color space into a thin and densely-sampled manifold. Crucially, the manifold varies continuously and monotonically in redshift and specific star formation rate in roughly orthogonal directions. Using COSMOS2020 many-band photo-$z$'s and compiled spec-$z$'s as representative and non-representative samples, respectively, we trained and tested redshift prediction from a SOM, from nearest neighbors in UMAP coordinates (UMAP-$k$NN-$z$), and directly from nearest neighbors in the color space to assess how well location in each space maps to redshift. For the representative training set, UMAP-$k$NN-$z$ exhibited smaller photo-$z$ scatter and fraction of outliers than SOM-based methods. When training with the highly-biased spec-$z$ sample, UMAP-$k$NN-$z$ maintained similar performance while SOM- and color-based methods predictions were significantly degraded, especially at $z>1.5$ where training sets are the most sparse. The physically-meaningful trends across the UMAP manifold allow for accurate redshift prediction even in such poorly-sampled regions of color space. This suggests that representative, spectroscopically-anchored training sets can be produced by interpolating between spectroscopic sources at the UMAP coordinates of photometric objects, maximizing the performance of photo-$z$ algorithms.