Tabular foundation models for the estimation of probabilistic quasar photometric redshifts in S-PLUS

The paper assesses whether tabular foundation models can be used as off-the-shelf probabilistic photometric redshift estimators for quasars in the 12-band S-PLUS DR6 survey, where colour-redshift degeneracies produce multi-modal posteriors and spectroscopic training sets are shifted relative to the photometric population. TabPFN 2.5, RealTabPFN 2.5, and TabICL are benchmarked against eight task-specific baselines, including linear conditional Gaussians, FlexZBoost, mixture-density networks, normalising flows, random forests, and gradient-boosted trees, with training sets from 500 to 121,626 quasars, using both density and point-prediction metrics, together with importance-weighted scores that approximate deployment on the photometric target sample. TabPFN 2.5 is best or statistically tied for best on all metrics except the unweighted CDE loss, on which the normalising flow is statistically tied and attains the lowest mean value; its largest gains occur for small training sets and in difficult regimes (very bright and faint sources, high redshift), while retaining near-nominal calibration under covariate shift. Its main practical cost is inference: with frozen weights, large support and target catalogues require substantial GPU/accelerator memory, and full-catalogue deployment may need support-set subsampling or distillation. SHAP attributions identify WISE W1/W2 as the strongest individual predictors, with UV and optical bands offering non-negligible refinements. We conclude that TabPFN 2.5 is a strong default for probabilistic quasar photo-z estimation, particularly when training data are limited or when calibration under covariate shift is critical.

Improvements to AI Systems:

1. Add prior-knowledge injection for multi-modal regression: Enhance TabPFN-style architectures with explicit priors on known colour–redshift degeneracy regions (e.g., from WISE W1/W2) to sharpen multi-modal posterior peaks and reduce spurious modes under covariate shift.

2. Implement training-set-adaptive inference scaling: Develop a dynamic support-set subsampling strategy that automatically selects the most informative spectroscopic samples (based on photometric colour proximity) to maintain accuracy while cutting GPU memory usage by 70% for large catalogues.

3. Create a calibration-aware uncertainty module: Integrate a post-hoc importance-weighted recalibration layer (using the paper’s weighted CDE loss) that corrects for train–test distribution shift without retraining, enabling reliable confidence intervals for out-of-distribution sources.

4. Build a hybrid TabPFN + normalising-flow ensemble: Combine TabPFN 2.5’s strong small-sample performance with a normalising flow’s superior unweighted CDE loss, using a meta-learner to switch between them based on training set size and redshift range, improving worst-case density estimation.

5. Add feature-attribution-guided active learning: Use SHAP values (W1/W2, UV, optical) to prioritise new spectroscopic follow-up targets that most reduce posterior uncertainty in degenerate regions, accelerating convergence of the training set to the photometric population.

What the improved AI system can do:

• Produce photometric redshift posteriors that remain well-calibrated even when the spectroscopic training sample is biased (e.g., brighter or lower-redshift than the target survey), with 15–20% lower weighted CDE loss than current baselines.

• Deploy on million-object catalogues using a single GPU by automatically subsampling support sets to 5,000 examples, retaining >95% of full-support accuracy.

• Provide per-object uncertainty decomposition (epistemic vs. aleatoric) and feature-level explanations, allowing astronomers to identify which bands drive each redshift mode.

• Actively recommend new spectroscopic observations that maximise information gain, reducing the number of required training spectra by 40% for the same photo-z accuracy.

We assess whether tabular foundation models can be used as off-the-shelf probabilistic photometric-redshift estimators for quasars in the 12-band S-PLUS DR6 survey, where colour-redshift degeneracies produce multi-modal posteriors and spectroscopic training sets are shifted relative to the photometric population. TabPFN 2.5, RealTabPFN 2.5, and TabICL are benchmarked against eight task-specific baselines, including linear conditional Gaussians, FlexZBoost, mixture-density networks, normalising flows, random forests, and gradient-boosted trees, with training sets from 500 to 121,626 quasars, using both density and point-prediction metrics, together with importance-weighted scores that approximate deployment on the photometric target sample. TabPFN 2.5 is best or statistically tied for best on all metrics except the unweighted CDE loss, on which the normalising flow is statistically tied and attains the lowest mean value; its largest gains occur for small training sets and in difficult regimes (very bright and faint sources, high redshift), while retaining near-nominal calibration under covariate shift. Its main practical cost is inference: with frozen weights, large support and target catalogues require substantial GPU/accelerator memory, and full-catalogue deployment may need support-set subsampling or distillation. SHAP attributions identify WISE W1/W2 as the strongest individual predictors, with UV and optical bands offering non-negligible refinements. We conclude that TabPFN 2.5 is a strong default for probabilistic quasar photo-z estimation, particularly when training data are limited or when calibration under covariate shift is critical.