287,872 Supermassive Black Holes Masses: Deep Learning Approaching Reverberation Mapping Accuracy

TL;DR

This work develops QuasarSpecNet, a physically informed encoder–decoder autoencoder trained on RM-based black hole masses from the SDSS-RM sample to infer SMBH masses from single-epoch SDSS spectra. The model jointly reconstructs spectra and predicts RM-calibrated masses, enforcing a latent space that captures global virial features while preserving spectral detail via skip connections. Applied to 287,872 quasars across z ≈ 0–4, it delivers masses with RMSE ≈ 0.058 dex (≈14% relative) and R^2 ≈ 0.91 relative to RM masses, outperforming traditional single-line virial estimators and providing mass estimates where those methods fail. The resulting catalog enables robust population analyses, including a differential mass function with a break near 3×10^8 M⊙ and insights into broad vs. narrow line quasars, redshift distribution, and high-redshift limitations, while highlighting the need for infrared data to extend RM-calibrated accuracy to z > 4.

Abstract

We present a population-scale catalogue of 287,872 supermassive black hole masses with high accuracy. Using a deep encoder-decoder network trained on optical spectra with reverberation-mapping (RM) based labels of 849 quasars and applied to all SDSS quasars up to $z=4$, our method achieves a root-mean-square error of $0.058$\,dex, a relative uncertainty of $\approx 14\%$, and coefficient of determination $R^{2}\approx0.91$ with respect to RM-based masses, far surpassing traditional single-line virial estimators. Notably, the high accuracy is maintained for both low ($<10^{7.5}\,M_\odot$) and high ($>10^{9}\,M_\odot$) mass quasars, where empirical relations are unreliable.

Figures (9)

• Figure 1: Architecture of the proposed deep autoencoder-based neural network.
• Figure 2: UMAP projections of the learned latent representation for latent dimensionality $d=1,2,4,5,10,$ and $50$. Points are colored by $\log M_{\rm BH}$. A smooth mass gradient is visible in all panels. For $d=1$, the embedding collapses to a single arc with a monotonic gradient; for $d=2$, a principal ridge with transverse structure appears; for $d=4$-$5$, the manifold thickens while the gradient remains continuous; for $d=10$, the global pattern and gradient are preserved with a slightly more compact cloud; for $d=50$, the central manifold persists and a small fraction of points is displaced to the periphery. UMAP preserves local neighborhoods rather than global geometry, so axis orientation is not physically meaningful.
• Figure 3: Comparison between black hole mass estimates from our autoencoder-based model and those obtained using traditional single-epoch virial methods for the test dataset. The blue points show the predictions from the neural-network model, while the purple, yellow, and green points correspond to single-line estimates based on H$\beta$, Mg ii, and C iv, respectively. The dashed line marks the one-to-one relation expected for perfect agreement with the reverberation-mapping (RM) masses. The scatter of the blue points mainly reflects variations among individual spectra from the multi-epoch SDSS-RM observations, as each spectrum is treated independently in the estimation. Traditional estimators exhibit larger scatter and systematic deviations, particularly at the low- and high-mass ends, and many sources lack reliable single-line measurements. In contrast, our model maintains a tight correlation with the RM masses ($R^{2}=0.909$) and achieves a low overall RMSE of 0.058 dex with respect to the RM-based mass, and also providing predictions for objects where the conventional methods fail.
• Figure 4: Heatmap of predicted versus RM-based black hole masses across different redshifts, with two overlaid layers: The blue colormap shows RM-based masses; the semi-transparent red colormap shows network-predicted masses.
• Figure 5: SDSS DR16 quasar sample: hexbin map of predicted black-hole mass versus redshift, with a logarithmic color scale for counts. Two LOWESS curves are overplotted: solid for $\log M$ as a function of $z$, dashed for $z$ as a function of $\log M$, tracing the central trend. Top and right panels show marginal histograms. The redshift is limited to $0 \le z \le 4$.