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

TL;DR

This work tackles the challenge of quasar photometric redshift estimation under redshift–color degeneracy by introducing LSTM-MDNz, an end-to-end model that fuses 14-band photometric fluxes and their errors in wavelength order. A bidirectional LSTM feature extractor captures spectral dependencies, while a Gaussian Mixture MDN outputs a full conditional redshift distribution p(z|x), enabling accurate point estimates and well-calibrated PDFs. Band-ablation analyses demonstrate that ultraviolet and infrared data substantially reduce degeneracy and bias, with all bands providing a synergistic gain over optical data alone. The approach yields state-of-the-art performance on a large, multi-survey quasar sample and offers a scalable framework for upcoming surveys like LSST, CSST, and Euclid, delivering both precise redshifts and quantified uncertainties for cosmological 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 ($σ_{\mathrm{NMAD}}$) 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.

Figures (7)

• Figure 1: Redshift distribution of the quasar sample with complete 14-band photometry. The vertical axis shows the number of sources, and the horizontal axis the redshift.
• Figure 2: Architecture and data flow of the LSTM-MDNz model. The input consists of multi-band photometric fluxes and their corresponding errors. These sequences are processed by the LSTM encoding module (red dashed box), which includes bidirectional LSTM layers, dropout, and batch normalization for feature extraction and stabilization. The resulting features are transformed via a fully connected layer before being passed to the MDN module (blue dashed box), where a Gaussian mixture model is used to compute the conditional probability density function (PDF) of the photometric redshift.
• Figure 3: Photometric vs. spectroscopic redshifts for test set samples under full-band (14 bands) input. The solid line is the diagonal; dashed lines show $|z_{\mathrm{spec}} - z_{\mathrm{phot}}|/(1 + z_{\mathrm{spec}})=0.15$. Color indicates point density.
• Figure 4: Distribution of CRPS values for the test set under full-band (14 bands) input. The red dashed line marks the mean CRPS.
• Figure 5: Examples of photometric redshift predictions for four randomly selected quasars under full-band (14 bands) input. The solid blue line in the upper panel represents the predicted PDF, while the solid blue line in the lower panel shows the corresponding CDF. The red dashed line indicates the position of the true spectroscopic redshift ($z_{\mathrm{spec}}$), and the blue dashed line represents the photometric redshift ($z_{\mathrm{phot}}$).