Photometric Redshift PDFs via Neural Network Classification for DESI Legacy Imaging Surveys and Pan-STARRS

TL;DR

This work introduces a neural network classification (NNC) framework that produces well-calibrated photometric redshift PDFs by discretizing redshift into bins and optimizing the Continuous Ranked Probability Score (CRPS). Applied to LSDR10 and PS1DR2 with an expansive spectroscopic training set from DESI DR1 and SDSS DR19, the method yields precise photo-$z$ estimates and robust uncertainty quantification, outperforming standard regression and other ML baselines. Infrared data from unWISE substantially improves PS1DR2 photo-$z$ (NMAD down from ~0.028 to ~0.022 and outliers from ~0.62% to ~0.34%), while LSDR10 benefits from deeper photometry and W1/W2 coverage, achieving NMAD ~0.015 and outliers ~0.50%. A five-level hierarchical photometric redshift catalog combining LSDR10 and PS1DR2 enables broad sky coverage with optimized accuracy, and the calibrated PDFs—validated via PIT and stacked $N(z)$ analyses—are suitable for cosmology and scalable to future surveys like CSST, Euclid, and LSST.

Abstract

We present a neural network classification (NNC) method for photometric redshift estimation that produces well-calibrated redshift probability density functions (PDFs). The method discretizes the redshift space into ordered bins and optimizes the Continuous Ranked Probability Score (CRPS), which respects the ordinal nature of redshift and naturally provides uncertainty quantification. Unlike traditional regression approaches that output single point estimates, our method can capture multi-modal posterior distributions arising from color-redshift degeneracies. We apply this method to the DESI Legacy Imaging Surveys Data Release 10 (LSDR10) and Pan-STARRS Data Release 2 (PS1DR2), using an unprecedented spectroscopic training sample from DESI DR1 and SDSS DR19. Our method achieves $σ_{\mathrm{NMAD}} = 0.0153$ and $η= 0.50\%$ on LSDR10, and $σ_{\mathrm{NMAD}} = 0.0222$ and $η= 0.34\%$ on PS1DR2 combined with unWISE infrared photometry. The NNC method outperforms Random Forest, XGBoost, and standard neural network regression. We demonstrate that DESI DR1 significantly improves photo-$z$ performance at $z > 1$, while the combination of deep optical photometry and mid-infrared coverage is essential for achieving high precision across the full redshift range. We provide a unified photometric redshift catalog combining LSDR10 and PS1DR2 with a hierarchical model selection strategy based on available photometry. The well-calibrated PDFs produced by our method are valuable for cosmological studies and can be extended to next-generation surveys such as CSST, Euclid, and LSST.

Figures (8)

• Figure 1: Redshift distributions of the spectroscopic training samples for LSDR10 (red solid) and PS1DR2 (blue dashed).
• Figure 2: Comparison of photometric redshifts ($z_{\mathrm{phot}}$) and spectroscopic redshifts ($z_{\mathrm{spec}}$) for the test sets. Left: LSDR10; Middle: PS1DR2 (optical only); Right: PS1DR2+unWISE. The red dashed lines indicate the one-to-one relation, and the red dotted lines show the outlier boundaries ($|\Delta z_{\mathrm{norm}}| = 0.15$). Performance metrics are shown in each panel.
• Figure 3: $\sigma_{\mathrm{NMAD}}$ as a function of spectroscopic redshift. Left: LSDR10 with different training configurations (combined, SDSS-only, DESI-only). Right: PS1DR2 configurations (optical only, +unWISE, with LSDR10 reference). The comparison demonstrates the impact of training sample composition and infrared photometry.
• Figure 4: Representative examples of redshift PDFs from the LSDR10 test set. Upper row (a--c): high-confidence predictions where the spectroscopic redshift (blue vertical dashed line) agrees well with the PDF expectation (red vertical solid line). Lower row (d--f): outlier cases showing various failure modes including peaked predictions at incorrect redshifts and broader distributions. The shaded regions indicate the $1\sigma$ confidence intervals.
• Figure 5: PIT histograms for the LSDR10 test set before (left) and after (right) temperature scaling calibration. The red dashed line indicates the ideal uniform distribution expected for perfectly calibrated PDFs. After calibration, the PIT distribution closely matches the uniform expectation.