Deep Learning Improves Photometric Redshifts in All Regions of Color Space

TL;DR

This work tackles the challenge of accurate photometric redshifts across the full color space relevant to large surveys by comparing image-based deep learning photo-$z$ methods with traditional photometry-based ML on SDSS MGS data. By partitioning color space with a self-organizing map, it reveals that deep learning substantially reduces attenuation bias and scatter in most regions, especially for galaxies with varying star-formation histories, due to exploiting pixel-level color information. The study combines global performance metrics with per-cell analyses and supports its conclusions with Monte Carlo experiments inspired by attenuation-bias theory. The findings have practical significance for upcoming surveys (e.g., Euclid, LSST, Roman) by guiding the design of photo-$z$ pipelines that robustly handle local color-space variations and complex galaxy morphologies.

Abstract

Photometric redshifts (photo-$z$'s) are crucial for the cosmology, galaxy evolution, and transient science drivers of next-generation imaging facilities like the Euclid Mission, the Rubin Observatory, and the Nancy Grace Roman Space Telescope. Previous work has shown that image-based deep learning photo-$z$ methods produce smaller scatter than photometry-based classical machine learning (ML) methods on the Sloan Digital Sky Survey (SDSS) Main Galaxy Sample, a testbed photo-$z$ dataset. However, global assessments can obscure local trends. To explore this possibility, we used a self-organizing map (SOM) to cluster SDSS galaxies based on their $ugriz$ colors. Deep learning methods achieve lower photo-$z$ scatter than classical ML methods for all SOM cells. The fractional reduction in scatter is roughly constant across most of color space with the exception of the most bulge-dominated and reddest cells where it is smaller in magnitude. Interestingly, classical ML photo-$z$'s suffer from a significant color-dependent attenuation bias, where photo-$z$'s for galaxies within a SOM cell are systematically biased towards the cell's mean spectroscopic redshift and away from extreme values, which is not readily apparent when all objects are considered. In contrast, deep learning photo-$z$'s suffer from very little color-dependent attenuation bias. The increased attenuation bias for classical ML photo-$z$ methods is the primary reason why they exhibit larger scatter than deep learning methods. This difference can be explained by the deep learning methods weighting redshift information from the individual pixels of a galaxy image more optimally than integrated photometry.

Figures (7)

• Figure 1: Global $z_\mathrm{phot}$ vs. $z_\mathrm{spec}$ for SDSS MGS test set galaxies using the classical ML method of beck2016 (left) and the deep learning model (encapzulate) of dey2022. The gray solid line denotes the one-to-one relation. Points outside of the gray dashed lines ($| \Delta z \, / \,(1+z_\mathrm{spec}) | > 0.05$) are considered outliers. The encapzulate photo-$z$'s have smaller global scatter ($\sigma_\mathrm{NMAD}$), lower fraction of catastrophic outliers ($f_\mathrm{outlier}$), and a smaller bias ($\langle \Delta z / (1 + z_\mathrm{spec} \rangle$) than the beck2016 photo-$z$'s. We also note that the attenuation bias (see Section \ref{['sec:attenuation_bias']}) for the classical ML photo-$z$'s is subtle when the entire dataset is considered, but it manifests as a systematic overestimation of $z_\mathrm{phot}$ at low $z_\mathrm{spec}$ and vice versa.
• Figure 2: A SOM derived from $ugriz$ colors for $\sim$500,000 SDSS MGS galaxies. Cells are color-coded by the mean spec-$z$ (left panel) and mean bulge-to-total ratio of the objects in each cell (right panel). Generally, mean spec-$z$ increases left-to-right, while $\langle$B/T$\rangle$ increases bottom-to-top. We highlight four cells: Cells A, B, and C represent typical cells with high, low, and intermediate $\langle$B/T$\rangle$, respectively, while Cell D contains very low-$z$ galaxies with a large fraction of catastrophic photo-$z$ outliers in the beck2016 catalog.
• Figure 3: The ratio of a robust measure of scatter ($\sigma_\mathrm{NMAD}$) of beck2016 photo-$z$'s to the $\sigma_\mathrm{NMAD}$ for encapzulate photo-$z$'s for each SOM cell (gray points) plotted as a function of the cell's mean bulge-to-total ratio ($\langle$B/T$\rangle$). The solid blue line shows the median $\sigma_\mathrm{NMAD}$ ratio computed in bins of $\langle$B/T$\rangle$ with its uncertainty shown as the shaded region. The binned median $\sigma_\mathrm{NMAD}$ ratio is always greater than $1.0$ (dotted black line). The $\sigma_\mathrm{NMAD}$ ratio for Cell D is a major outlier from the general trend due to its extremely high fraction of catastrophic outliers for beck2016 photo-$z$'s (see Figure \ref{['fig:zphot-zspec-cells']}). The scatter for the beck2016 photo-$z$'s is higher than that of the encapzulate photo-$z$'s for all cells but approaches the performance of the latter for cells with high $\langle$B/T$\rangle$ values (e.g., Cell A). This suggests that deep learning models achieve as good or better photo-$z$'s compared to classical ML methods across all regions of color space, and they are especially adept at extracting additional redshift information from images of intermediate- and low-$\langle$B/T$\rangle$ galaxies.
• Figure 4: For the four SOM cells highlighted in Figure \ref{['fig:som']}, we show a 0.42$^\prime$$\times$ 0.42$^\prime$ DESI-LS $grz$ image of a representative galaxy from the cell (first column), as well as $z_\mathrm{phot}$--$z_\mathrm{spec}$ plots for each of three photo-$z$ methods: beck2016 (second column), encapzulate (third column), and random forest (fourth column). From top-to-bottom, the rows correspond to Cell A (high $\langle$B/T$\rangle$ galaxies), Cell B (low $\langle$B/T$\rangle$ galaxies), Cell C (intermediate $\langle$B/T$\rangle$ galaxies), and Cell D (very low-$z$ galaxies). In each $z_\mathrm{phot}$--$z_\mathrm{spec}$ panel, the solid black line corresponds to the one-to-one relation, the dashed black lines ($\Delta z / (1+z_\mathrm{spec}) = \pm 0.05$) demarcate outliers from typical data, and the red line shows the best-fit line. Both classical ML learning methods (beck2016 and random forest) yield flatter best-fit lines than the deep learning algorithm (encapzulate). The beck2016 photo-$z$'s for Cell D suffer from a very high catastrophic outlier rate, which is not an issue for the other two methods.
• Figure 5: Left panel: the angle of the best-fit $z_\mathrm{phot}$--$z_\mathrm{spec}$ line from the horizontal axis, $\theta$, as a function of $\langle$B/T$\rangle$ for each SOM cell. Right panel: a histogram of the number of SOM cells with each $\theta$ value. A perfect correlation would correspond to $\theta$ = 45$^{\circ}$, while a flat best-fit line would have $\theta$ = 0$^{\circ}$. The encapzulate$\theta$ values are closer to the ideal angle (with a average of 41$^{\circ}$) than beck2016's $\theta$ values (average of 29$^{\circ}$) due to the latter suffering from a more severe local attenuation bias.