LSTM-MDNz: Estimating Quasar Photometric Redshifts with an LSTM-Augmented Mixture Density Network

TL;DR

This paper introduces LSTM-MDNz, a deep learning model that combines LSTM and mixture density networks to accurately estimate quasar redshifts from multi-band photometric data, improving over traditional methods.

Contribution

The paper presents a novel end-to-end deep learning approach that directly models photometric fluxes for quasar redshift estimation, eliminating manual feature engineering and providing probabilistic outputs.

Findings

Optimal performance with all 14 bands, $\sigma_{NMAD}$ of 0.037 and 3.5 ext{ extperthousand} outlier rate.

Reductions of 29 ext{ extperthousand} in $\sigma_{NMAD}$ and 56 ext{ extperthousand} in outlier rate compared to SDSS+WISE.

Predicted PDFs closely match true redshift distributions, validated by PIT and CRPS analyses.

Abstract

Quasar photometric redshifts are essential for studying cosmology and large-scale structures. However, their complex spectral energy distributions cause significant redshift-color degeneracy, limiting the accuracy of traditional methods. To overcome this, we introduce LSTM-MDNz, a novel end-to-end deep learning model combining long short-term memory networks (LSTM) with mixture density networks (MDN). The model directly uses multi-band photometric fluxes and associated errors as wavelength-ordered sequential inputs, eliminating the need for manual feature engineering while enabling simultaneous point estimation and probability distribution function (PDF) prediction of quasar redshifts. We integrate data from four major sky surveys-SDSS, DESI-LS, WISE, and GALEX-to assemble a sample of over 550,000 spectroscopically confirmed quasars ($0 \leq z_{\mathrm{spec}} \leq 5$) across 14 ultraviolet to infrared bands for model training and testing. Experimental results show that using all 14 bands yields optimal performance, with a normalized median absolute deviation ($\sigma_{\mathrm{NMAD}}$) of 0.037 and an outlier rate ($f_{\mathrm{out}}$) of 3.5\% on the test set. These values represent reductions of 29\% and 56\%, respectively, compared to the commonly adopted SDSS+WISE band set. Probability integral transform ($\mathrm{PIT}$) and continuous ranked probability score ($\mathrm{CRPS}$) analyses confirm that the predicted PDFs align closely with the true redshift distribution. Band-ablation experiments further highlight the essential role of ultraviolet and infrared data in alleviating color degeneracy and reducing systematic bias. This study demonstrates the effectiveness of multi-band fusion in improving quasar photo-z accuracy and offers a ready-to-use estimation framework for future surveys like LSST, CSST, and Euclid.